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International Conference

Asymptotic Problems:Elliptic and Parabolic Issues

June 1–5, 2015, Vilnius, Lithuania

Abstracts

Vilnius, 2015

Conference Program

Plenary Talks 112 Parabolic equations on non-compact Riemannian manifolds

Herbert Amann

3 Long-time Behaviour of Nonlinear Free Boundary ProblemsYihong Du

4 Positive solutions of an elliptic equation with a dynamical boundary conditionMarek Fila

5 Self-Oscillations of a Navier-Stokes Liquid past a CylinderGiovanni P. Galdi

6 On a non-blow up criterion involving vorticity direction under the non-slip boundary conditionfor the three-dimensional Navier-Stokes flowYoshikazu Giga

7 Nonlinear Diffusions Inspired by Image ProcessingPatrick Guidotti

8 The Existence Theorems and the Liouville Theorem for the Steady-State Navier-Stokes Prob-lemsMikhail Korobkov

9 Asymptotic analysis of the non-steady Navier–Stokes equations in thin structuresGrigory Panasenko, Konstantin Pileckas

10 Liouville theorems for superlinear parabolic problemsPavol Quittner

11 Finite difference discretizations of nonlinear elliptic boundary value problemsWolfgang Reichel

12 Free boundaries arising from semi- and quasilinear problemsHenrik Shahgholian

13 Boundary layers in the presence of characteristic pointsRoger Temam

1 To be announcedVsevolod Solonnikov

Special Session

Propagation phenomena and free boundary problems 1414 Organizer Yihong Du

15 Existence, uniqueness and numerical simulation of the long range spatially segregation ellipticsystemFarid Bozorgnia

16 Semilinear elliptic equations in convex domains and convex ringFrançois Hamel

17 Spreading speeds and profiles of solutions in nonlinear free boundary problemsHiroshi Matsuzawa

18 Weakly interacting front dynamics in the FitzHugh-Nagumo systemYoshihisa Morita

19 The Hele-Shaw asymptotics for mechanical models of tumor growthFernando Quirós

20 On a free boundary problem with two fluidsJürgen Socolowsky

21 On a free boundary problem for a competition system: two invasive species caseChang-Hong Wu

22 Multiple spreading phenomena for a free boundary problem in ecologyYoshio Yamada

1 To be announcedXing Liang

Special Session

Navier-Stokes Equations and Other Models of Fluid Mechanics 2323 Organizer Reinhard Farwig

24 Regular solutions to the compressible, viscous heat-conducting Navier-Stokes equationsJan Burczak, Wojciech Zajaczkowski, Yoshihiro Shibata

25 Solvability of the Initial-Boundary value problem of the Navier-Stokes equations with nonho-mogeneous dataTongkeun Chang, Bum Ja Jin

26 A class of large solutions for the compressible Navier-Stokes systemRaphael Danchin

27 Exponential decay of the vorticity of steady incompressible viscous flow around a rotating rigidbodyPaul Deuring, Paolo G. Galdi

28 Quasi-optimal initial value conditions for the Navier-Stokes equationsReinhard Farwig

29 Regular Lagrangian flows - on the application to problems of polymeric fluidsPiotr Gwiazda

30 Asymptotic structure of steady Stokes flow around a rotating obstacle in 2DToshiaki Hishida

31 Fundamental solution to the time-periodic Stokes equationsMads Kyed

32 On Ukai-type solution formula for the Stokes system in a domain with graph boundaryHideyuki Miura

33 Regularizing effects of a thin elastic interface in fluid-structure interaction problemsBoris Muha, Suncica Canic

34 Inviscid incompressible limits on expanding domainsŠárka Necasová, Eduard Feireisl, Yongzhong Sun

35 Navier-Stokes flow in the weighted Hardy spaceTakahiro Okabe

36 Inflow-outflow nonstationary flowJoanna Rencławowicz

37 The Stokes Operator in Periodical Domains with BoundaryJonas Sauer

38 Time periodic Stokes problems in a layer: the spatial asymptoticsMaria Specovius-Neugebauer, Konstantin Pileckas

39 Polymeric fluidsAgnieszka Swierczewska-Gwiazda

40 Decay of Non-stationary Navier-Stokes Flow with Nonzero Dirichlet Boundary DataDavid Wegmann

41 Some stability problems for the Navier-Stokes and magnetohydrodynamics equationsWojciech Zajaczkowski

1 To be announcedWerner Varnhorn

Special Session

Blow-up and large time behaviour of solutions of parabolic problems 4242 Organizers Marek Fila and Pavol Quittner

43 Critical mass in a volume filling modelTomasz Cieslak

44 Symmetry and convergence properties for asymptotically symmetric parabolic problemsJuraj Földes, Peter Polácik

45 Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradientnonlinearityJohannes Lankeit

46 Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes systemLi Yuxiang, Zhang Qingshan

47 Evolution of nonlocal planar curvature flowXiaoliu Wang

48 Mathematical challenges arising in the analysis of chemotaxis-fluid interactionMichael Winkler

Special Session

Surface Tension and Capillarity 4949 Organizer Robert Finn

50 Bounded and Unbounded Capillary Surfaces in a Cusp DomainYasunori Aoki, Hans De Sterck, David Siegel

51 Viscous capillary flows inside a rotating cylinderEugene Benilov

52 Isovolumetric and isoperimetric inequalities for a class of capillarity functionalsPaolo Caldiroli

53 The Floating Cylinder ReconsideredHanzhe Chen, David Siegel

54 Mutual Attractions of Floating Objects: an Idealized ExampleRobert Finn, Rajat Bhatnagar

55 Stable capillary hypersurfaces in a wedge and uniqueness of the minimizerMiyuki Koiso

56 Capillary Surfaces: Necessary and Sufficient Conditions for Continuity at CornersKirk Lancaster

57 Symmetries on capillary surfaces in Euclidean spaceRafael López

58 Numerical limits of stability for cylindrical pendent dropsJohn McCuan, Manxi Wu

59 Capillary and free boundary surfaces obtained by deformation of minimal surfacesFilippo Morabito

60 On the monotonicity of energy in the blow-up limit at the triple junction of three fluids inequilibriumRay Treinen

61 Liquid Bridges between Contacting SpheresThomas I Vogel

Special Session

Asymptotic and Numerical Methods for Viscous and Elastic Media 6262 Organizer Grigory Panasenko

63 On the Existence of Singular Solutions to the Stokes Problem in the Power Cusp DomainsAlicija Eismontaite, Konstantin Pileckas

64 Error estimates in shell theoryCristinel Mardare

65 Homogenization of elliptic systems with periodic coefficients: twoparametric operator errorestimatesYulia Meshkova, Tatiana Suslina

66 Homogenization of the Stokes equation with stress boundary conditions in periodic porousmediaElena Miroshnikova, John Fabricius, Peter Wall

67 Evolutional contact with Coulomb’s friction on a periodic microstructureJulia Orlik

68 Asymptotic analysis of the fluid flow with a pressure-dependent viscosityIgor Pažanin, Eduard Marušic-Paloka

69 On the Dirichlet type problem to degenerate at a line elliptic systemsStasys Rutkauskas

70 ADI method for two-dimensional pseudo-parabolic equation with integral boundary conditionsArturas Štikonas, Mifodijus Sapagovas, Olga Štikoniene

Contributed talks 717172 Solvability of a surface reaction model taking into account diffusion of both adsorbates

Algirdas Ambrazevicius, Vladas Skakauskas

73 On Poincaré’s and Lions’ lemmas and on De Rham’s theoremChérif Amrouche, Philippe Ciarlet, Cristinel Mardare

74 Analysis of asynchronous multi-time-step methods for parabolic problemsMichal Beneš

75 Free boundary problem of magnetohydrodynamicsElena Frolova

76 A simplified approach to the regularising effect of nonlinear semigroupsDaniel Hauer

77 Asymptotic behavior of solutions of systems described nonlinear viscoelastic flowsNatalia Karazeeva

78 On Nonhomogeneous Boundary Value Problems for the Stationary Navier–Stokes Equationsin 2D Symmetric Semi-Infinite Outlets IKristina Kaulakyte, Michel Chipot, Konstantin Pileckas, Wei Xue

79 Linearized Navier-Stokes problems in the periodic domainsNeringa Kloviene

80 Local in time solutions of MHD system for bounded and exterior domainAdam Kubica, Giovanni P. Galdi

81 On a very singular parabolic equationMichał Łasica

82 The Brezis–Nirenberg effect for Navier and Dirichlet fractional LaplaciansAlexander I. Nazarov

83 Parabolic problems in oscillating thin domainsMarcone Pereira

84 Interaction of particles through a singular potentialJan Peszek

85 A remark on the orthogonal decomposition of the Hilbert space W 1,2

Reimund Rautmann86 The Stokes and Navier-Stokes equations with pressure-velocity boundary conditions in Lp

spacesNour Seloula, Hind Al Baba

87 Poincaré Constants in 2D-annuliGudrun Thaeter, Bernd Rummler, Michael Ružicka

88 Asymptotics for Ventssel’s problems in fractal domainsPaola Vernole

89 On Nonhomogeneous Boundary Value Problems for the Stationary Navier–Stokes Equationsin 2D Symmetric Semi-Infinite Outlets IIWei Xue, Michel Chipot, Kristina Kaulakyte, Konstantin Pileckas

90 Linearized Non-Stationary Navier-Stokes Problem in an Infinite Periodic PipeStephanie Zube, Michel Chipot, Neringa Kloviene, Konstantin Pileckas

Participants 91

919293Index of Authors 94

Int. Conference: Asymptotic Problems, Elliptic and Parabolic IssuesVilnius, Lithuania, June 1–5, 2015

Plenary Talks

1


Parabolic equationson non-compact Riemannian manifolds

Herbert AmannInstitut fur Mathematik, Universitat Zurich, Switzerland,

[email protected]

We discuss optimal well-posedness results for parabolic boundary value problems on non-compactRiemannian manifolds. We explain, in particular, the intimate connection between manifolds withsingular ends and degenerate equations. The results are illustrated by some model problems.

2


Long-time Behaviour of Nonlinear Free BoundaryProblems

Yihong DuUniversity of New England, Australia, [email protected]

In this talk I will report some recent results on the long-time behavior of a class of nonlinearfree boundary problems, based on work in [1, 2, 3, 4, 5]. I will in particular compare the resultswith several well-known theorems for the corresponding Cauchy problems.

References

[1] Du, Y., Lou, B. Spreading and vanishing in nonlinear diffusion problems with free boundaries,J. Eur. Math. Soc., to appear. (arXiv1301.5373).

[2] Du, Y., Lou, B., Zhou, M. Nonlinear diffusion problems with free boundaries: Convergence,transition speed and zero number arguments, preprint (arXiv1501.06258).

[3] Du, Y., Matano, H., Wang, K. Regularity and asymptotic behavior of nonlinear Stefan prob-lems, Arch. Rational Mech. Anal., 212, 957-1010 (2014).

[4] Du, Y., Matsuzawa, H., Zhou, M. Spreading speed and profile for nonlinear Stefan problemsin high space dimensions, J. Math. Pures Appl., 103, 741-787 (2015).

[5] Du, Y., Matsuzawa, H., Zhou, M. Sharp estimate of the spreading speed determined by non-linear free boundary problems, SIAM J. Math. Anal., 46, 375-396 (2014).

3


Positive solutions of an elliptic equation with a dynamicalboundary condition

Marek FilaComenius University, Slovakia, [email protected]

We consider a semilinear elliptic equation in the half-space. On the boundary, a linear dynam-ical boundary condition is imposed. We present sharp results on existence and non-existence ofpositive solutions. We also study the large-time behavior of small solutions. This is a joint workwith Kazuhiro Ishige and Tatsuki Kawakami.

References

[1] Fila M., Ishige K., Kawakami T. Large-time behavior of solutions of a semilinear elliptic equa-tion with a dynamical boundary condition, Adv. Differential Equations, 18, 69–100 (2013).

[2] Fila M., Ishige K., Kawakami T. Large-time behavior of small solutions of a two-dimensionalsemilinear elliptic equation with a dynamical boundary condition, Asymptotic Analysis, 85,107–123 (2013).

[3] Fila M., Ishige K., Kawakami T. Existence of positive solutions of a semilinear elliptic equa-tion with a dynamical boundary condition, Calc. Var. Partial Differential Equations, DOI:10.1007/s00526-015-0856-8.

[4] Fila M., Ishige K., Kawakami T. Minimal solutions of a semilinear elliptic equation with adynamical boundary condition, preprint.

4


Self-Oscillations of a Navier-Stokes Liquid past aCylinder

Giovanni P. GaldiUniversity of Pittsburgh, U.S.A., [email protected]

Objective of this talk is to provide sufficient conditions for the branching out of a time pe-riodic solution from a steady-state solution to the two-dimensional Navier-Stokes equations pasta body. We shall first show that the problem can be reduced to the study of a coupled elliptic-parabolic nonlinear system. We then prove that time-periodic bifurcation may occur provided therelevant linear, time-independent operator of the parabolic problem possess a simple eigenvaluethat crosses the imaginary axis when the Reynolds number passes through a (suitably defined)critical value. [1]

References

[1] Galdi G. P., On bifurcating time-periodic flow of a Navier-Stokes liquid past a cylinder, sub-mitted.

5


On a non-blow up criterion involving vorticity directionunder the non-slip boundary condition for the

three-dimensional Navier-Stokes flowYoshikazu Giga

University of Tokyo, Japan, [email protected]

We give a geometric non-blow up criterion on the direction of the vorticity for the three-dimensional half-space Navier-Stokes flow under the non-slip boundary condition whose initialdata is just bounded and may have infinite energy. We prove that under a restriction on behavior intime (type I condition) the solution does not blow up if the vorticity direction is uniformly contin-uous at the place where the vorticity is large even if we impose the Dirichlet boundary conditions.A similar geometric regularity criterion for non-blow up has been proved by P. Constantin and C.Fefferman (1993) under Lipschitz regularity condition for the whole space has been establishedby H. Miura and the author (2011), and for the half space with the slip boundary condition. Theirargument does not directly apply to the non-slip boundary condition since a key Liouville resultfor the two dimensional flow does not directly extend to the case of non-slip boundary. We apply arepresentation formula for the vorticity (Y. Maekawa (2012)) and establish a Liouville type resultunder the non-slip boundary condition for type I blow-up. This enables us to prove that a contin-uous alignment condition for the vorticity prevents the blow-up even under the non-slip boundarycondition which may produce a lot of vorticity near the boundary.

This is a joint work with P.-Y. Hsu and Y. Maekawa.

6


Nonlinear Diffusions Inspired by Image ProcessingPatrick Guidotti, UC Irvine, USA, [email protected]

The talk focusses on two mild regularizations of the well-know Perona-Malik equation (PME)ut = ∇ ·

(1

1+|∇u|2∇u)

in Ω for t > 0,

u(0, ·) = u0 in Ω,

on a bounded domain Ω ⊂ Rn and complemented with either homogeneous Neumann or periodicboundary conditions, where in applications one mostly has n = 2. The first regularization

ut = ∇ ·([

1

1 + |∇u|2 + δ]∇u)

is considered for δ > 0 and preserves the forward-backward nature of Perona-Malik while allow-ing for the construction of global solutions with properties which help explain the experimentallyobserved phenomenon of staircasing for solutions of PME. The second reads

ut = ∇ ·( 1

1 + |∇1−εu|2∇u),

and can be shown to be locally well-posed, to preserve the main features of the dynamical behaviorof PME solutions while avoiding (the practical artifact) of staircasing. As the parameter ε ∈(0, 1] varies, a transition from non-trivial to triviall dynamical behavior is observed, which will bedescribed in the talk. Results and open questions will be discussed.

This is partly joint work with James Lambers (University of Southern Mississippi, USA) andYunho Kim (Ulsan National Institute of Science and Technology, South Korea).

References

[1] Guidotti Patrick. A class of weakly degenerate elliptic equations, in preparation.[2] Guidotti Patrick. Anisotropic Diffusions of Image Processing From Perona-Malik on, Ad-

vanced Studies in Pure Mathematics, 67:131–156, 2015.[3] Guidotti Patrick, Kim Yunho, Lambers James. Image restoration with a new class of forward-

backward-forward diffusion equations of Perona-Malik type with Applications to SatelliteImage Enhancement, SIAM J. Imag. Sci., 6(3):1416–1444, 2013.

[4] Guidotti Patrick.A backward-forward regularization of the Perona-Malik equation, JDE252(4):3226-3244, 2012.

[5] Guidotti Patrick.A Family of Nonlinear Diffusions Connecting Perona-Malik to LinearDiffusionDCDS-S, 5(3):581–590, 2012.

[6] Guidotti Patrick.A New Well-posed Nonlinear Nonlocal Diffusion for Noise Reduction, Non-linear Analysis TMA, 72(12):4625–4637, 2010.

[7] Guidotti PatrickA New Nonlocal Nonlinear Diffusion of Image Processing, JDE,246(12):4731–4742, 2009.

[8] Guidotti Patrick, Lambers James.Two New Nonlinear Nonlocal Diffusions for Noise Reduc-tion, JMIV, 33(1):25–37, 2009.

7


The Existence Theorems and the Liouville Theorem forthe Steady-State Navier-Stokes Problems

Mikhail KorobkovSobolev Institute of Mathematics, Novosibirsk, Russia, [email protected]

In the talk we present a survey of recent results (see [4]–[6]) on the existence theorems for thesteady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D casesfor bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]).One of the main tools is the Morse–Sard Theorem for the Sobolev functions f ∈W 2

1 (R2) [1] (seealso [2]–[3] for the multidimensional case). This theorem guaranties that almost all level lines ofsuch functions are C1–curves besides the function f itself could be not C1-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equationsfor axially symmetric 3D solutions in the absence of swirl (see [1]).

References

[1] Bourgain J., Korobkov M. V., Kristensen J. On the Morse– Sard property andlevel sets of Sobolev and BV functions, Rev. Mat. Iberoam., 29(1):1–23, 2013.http://dx.doi.org/10.4171/RMI/710

[2] Bourgain J., Korobkov M. V., Kristensen J. On the Morse–Sard property and level sets ofWn,1 Sobolev functions on Rn, Journal fur die reine und angewandte Mathematik (CrellesJournal) (Online first 2013). http://dx.doi.org/10.1515/crelle-2013-0002

[3] Korobkov M. V., Kristensen J. On the Morse-Sard Theorem for the sharpcase of Sobolev mappings, Indiana Univ. Math. J., 63(6):1703–1724, 2014.http://dx.doi.org/10.1512/iumj.2014.63.5431

[4] Korobkov M. V., Pileckas K., Russo R. The existence theorem for steady Navier-Stokes equa-tions in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 14(1):233–262,2015. http://dx.doi.org/10.2422/2036-2145.201204 003

[5] Korobkov M. V., Pileckas K., Russo R. Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains, Ann. of Math., 181(2):769–807, 2015. http://dx.doi.org/10.4007/annals.2015.181.2.7

[6] Korobkov M. V., Pileckas K., Russo R. The existence theorem for the steadyNavier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp.,http://arXiv.org/abs/1403.6921.

[7] Korobkov M. V., Pileckas K., Russo R. The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. FluidMech. (Online first 2015). http://dx.doi.org/10.1007/s00021-015-0202-0

[8] Leray J. Etude de diverses equations integrals non lineaires et de quelques problemes que posel’hydrodynamique, J. Math. Pures Appl. Ser. 9, 12:1–82, 1933.

8


Asymptotic analysis of the non-steady Navier–Stokesequations in thin structuresGrigory Panasenko1,∗, Konstantin Pileckas2

1 Institute Camille Jordan UMR CNRS 5208, PRES University of Lyon/University of SaintEtienne, France and UMI 2615 J.-V.Poncelet,

∗[email protected] Vilnius University, Lithuania

Thin structures are some finite unions of thin rectangles (in 2D settings) or cylinders (in 3Dsettings) depending on small parameter ε 1 that is, the ratio of the thickness of the rectangle(cylinder) to its length. We consider the non-steady Navier–Stokes equations in thin structures withthe no-slip boundary condition at the lateral boundary and with the inflow and outflow conditionswith the given velocity of order one. The steady state Navier–Stokes equations in thin structureswere considered in [1-3]. The asymptotic expansion of the solution is constructed. The errorestimates for high order asymptotic approximations are proved. Asymptotic analysis is appliedfor an asymptotically exact condition of junction of 1D and 2D (or 3D) models. These results arepresented in [4-8]. The present work is supported by the grant number 14-11-00306 of RussianScientific Foundation, by the Research Federative Structures MODMAD FED 4169 and FR CNRS3490, and Lithuanian-Swiss cooperation programme to reduce economic and social disparitieswithin the enlarged European Union under project agreement No. CH-3-SMM-01/01.

References

[1] Panasenko G.P.Asymptotic expansion of the solution of Navier-Stokes equation in a tubestructure, C.R.Acad.Sci.Paris, 326 Serie IIb, 867-872, 1998.

[2] Panasenko G.P.Partial asymptotic decomposition of domain: Navier-Stokes equation in tubestructure, C.R.Acad.Sci.Paris, 326, Serie IIb, 893-898,1998.

[3] Panasenko G.P. Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht,2005, 398 pp.

[4] Panasenko G.,Pileckas K.Asymptotic analysis of the nonsteady viscous flow with a given flowrate in a thin pipe, Applicable Analysis, 91(3):559-574, 2012.

[5] Panasenko G., Pileckas K.Divergence equation in thin-tube structure, Applicable Analysis,2014, http://dx.doi.org/10.1080/00036811.2014.933476.

[6] Panasenko G., Pileckas K.Flows in a tube structure: equation on the graph, Journal of Math-ematical Physics, 55, 081505, 2014. http://dx.doi.org/10.1063/1.4891249.

[7] Panasenko G., Pileckas K.Asymptotic analysis of the non-steady Navier-Stokes equa-tions in a tube structure. I. The case without boundary layer-in-time, Nonlin-ear Analysis, Series A, Theory, Methods and Applications, 122:125-168, 2015.http://dx.doi.org/10.1016/j.na.2015.03.008.

[8] Panasenko G., Pileckas K.Asymptotic analysis of the non-steady Navier-Stokes equations ina tube structure. II. General case, Nonlinear Analysis, Series A, Theory, Methods and Appli-cations, 2015, in print.

9


Liouville theoremsfor superlinear parabolic problems

Pavol QuittnerComenius University, Slovakia, [email protected]

It is known that Liouville theorems for entire solutions of scaling invariant parabolic prob-lems guarantee universal estimates for solutions of more general problems. We prove such Li-ouville theorems for two classes of problems by either reducing parabolic problems to the cor-responding elliptic ones or by reducing parabolic systems to scalar parabolic equations. We firstconsider several parabolic problems with gradient structure and show that each positive boundedentire solution of such problems has to be time-independent. Then we consider a class of two-component parabolic systems without gradient structure and show that the components of anypositive bounded entire solution of such systems have to be proportional. In particular, the uni-versal estimates guaranteed by our Liouville theorems imply global existence and boundedness ofthreshold solutions lying on the boundary between global existence and blow-up and also yieldoptimal blow-up rate estimates for solutions which blow-up in finite time. Finally, we use theuniversal estimates to prove the existence of positive periodic solutions of strongly cooperativeparabolic Lotka-Volterra systems with equal diffusion.

References

[1] Quittner P. Liouville theorems for scaling invariant superlinear parabolic problems with gra-dient structure, Math. Ann. http://dx.doi.org/10.1007/s00208-015-1219-7

[2] Quittner P. Liouville theorems, universal estimates and periodic solutions for cooperativeparabolic Lotka-Volterra systems, Preprint 2015.

10


Finite difference discretizations of nonlinear ellipticboundary value problems

Wolfgang ReichelDepartment of Mathematics, Karlsruhe Institute of Technology (KIT), Germany,

[email protected]

We consider the nonlinear boundary value problem

−∆u = f(x, u) in Ω, u = 0 on ∂Ω, (1)

and its discretization by a finite-difference mesh of uniform meshsize h > 0

−∆hu = f(x, u) in Ωh, u = 0 on ∂Ωh. (2)

Here −∆h is the finite difference Laplacian given by

∆hu(x) =

n∑

i=1

u(x+ hei)− 2u(x) + u(x− hei)h2

where e1, . . . , en are the unit normal vectors of the standard basis of Rn. The main questions, which willbe addressed in this talk both for bounded and unbounded domains Ω, are the following:

(A) A-priori bounds: Under which conditions on f(x, s) does there exist K > 0 (independent of themeshisize h) such that for every solution u of (2) one has ‖u‖∞ ≤ K?

(B) Convergence: in which sense do solutions of (2) converge to solutions of (1) when h→ 0?(C) Symmetry: in which sense do positive solutions of (2) inherit symmetries of the domain?

A typical answer for (A) reads as follows: if f(x, s) ≈ sp as s → ∞ then there exists a criticalexponent p∗ > 1 such that for 1 < p < p∗ all positive solutions of (1) or (2) are uniformly boundedwhereas for p > p∗ a-priori bounds fail and sequences of solutions forming singularities exist. For classicalC2-solutions and weakW 1,2

0 -solutions of (1) the critical exponent has the value p∗ = n+2n−2 and is essentially

independent of the domain geometry. For solutions of (2) however, p∗ is typically smaller than n+2n−2 and it

does depend on Ω, cf. [4]. I will show for some examplary cases that the finite-difference critical exponentfor (2) is the same as the critical exponent of so-called very-weak solutions of the continuous problem (1),which solve the problem in a distributional sense, cf. [1], [2], [3].

Once a-priori bounds for (2) have been obtained, there is still the question (B) of how solutions of (2)converge to solutions of (1) when the meshsize tends to 0. In order to prove the expected result, additionalcompactness properties of the families of discrete solutions are necessary. We will show how the uniformL∞-bounds on solutions transfer into discrete analogues of W 1,q-bounds.

Finally, I will explain that one cannot expect solutions of(2) to exactly inherit the symmetries of the un-derlying domain. The best one can expect is the notion of asymptotic symmetry with quantitative estimates,cf.[5]. A prerequisite for such symmetry theorems are the a-priori bounds addressed in question (A).

References[1] Mandel R., Reichel W. Distributional solutions of the stationary nonlinear Schrodinger equation: sin-

gularities, regularity and exponential decay, Z. Anal. Anwend., 32: –82, 2013.[2] Horak J., McKenna P.J., Reichel W. Very weak solutions with boundary singularities for semilinear

elliptic Dirichlet problems in domains with conical corners, J. Math. Anal. Appl., 352: 496 – 514,2009.

[3] McKenna P.J., Reichel W. A priori bounds for semilinear equations and a new class of critical expo-nents for Lipschitz domains, J. Funct. Anal., 244: 220–246, 2007.

[4] McKenna P.J., Reichel W., Verbitsky A. Mesh-independent a priori bounds for nonlinear elliptic finitedifference boundary value problems, J. Math. Anal. Appl., 419: 496–524, 2014.

[5] McKenna P.J., Reichel W. Gidas-Ni-Nirenberg results for finite difference equations: estimates of ap-proximate symmetry, J. Math. Anal. Appl., 334: 206–222, 2007.

11


Free boundaries arising from semi- and quasilinearproblems

Henrik ShahgholianThe Royal Institute of Technology, Stockholm, [email protected]

In this talk I shall present recent developments on the regularity theory of free boundary problems,that arrive from semi- and quasi-linear problems.

I shall present a few methodology and approach as well as some sketchy ideas of proofs. Thetalk is at survey character.

12


Boundary layers in the presence of characteristic pointsRoger Temam

Indiana University Bloomington, USA, [email protected]

We study the boundary layers for a singularly perturbed convection - diffusion equation ina circle. We consider both the time independent and the time dependent cases. In the time in-dependent case two characteristic points appear at the boundary. We determine the correctors atthese points, solutions of the analogue of the Prandtl equation of the problem, and we use them tobuild efficient numerical methods which do not necessitate mesh refinements, thus reducing con-siderably the discretization costs. Similar considerations are also developed in the time dependentcase.

13


Special Session

Propagation phenomena and free boundary problemsOrganizer

Yihong DuUniversity of New England, Australia

[email protected]

14


Existence, uniqueness and numerical simulation of thelong range spatially segregation elliptic system

Farid BozorgniaInstituto Superior Tecnico, Lisbon, Portugal, [email protected]

One of the important problems in population ecology is modeling of the interactions betweenbiological components. To achieve this aim, different models based on reaction−diffusion equa-tions are studied.

For spatial segregation two models have been studied:

• A) Adjacent segregation: In this model particles annihilate on contact, and there is a com-mon surface of separation.The adjacent segregation model has been studied extensively. In [2, 3] the steady state ofm competing species coexisting in the same area Ω has been considered.

• B) Segregation at distance: Species interact at a distance from each other. Recently in [1],the modeling of species that keep a positive distance is considered.

We study a class of elliptic competition-diffusion systems of long range segregation modelsfor two and more competing species. The existence and uniqueness of the solution are shown.We prove that as the competition rate goes to infinity the solution converges, along with suitablesequences, to a spatially long range segregated state satisfying some free boundary problems.Moreover, we use properties of limiting problem to construct efficient numerical simulations forelliptic and parabolic systems.

References

[1] L. Caffarelli, S. Patrizi and V. Quitala. A nonlocal segregation model, preprint.

[2] M. Conti, S. Terracini, and G. Verzini. Asymptotic estimate for spatial segregation of compet-itive systems, Advances in Mathematics, 195:524–560, 2005.

[3] E.N. Dancer, D. Hilhorst, M. Mimura, L.A. Peletier. Spatial segregation limit of a competitiondiffusion system, European J. Appl. Math., 10:97–115, 1999.

15


Semilinear elliptic equations in convex domains andconvex ring

Francois HamelAix Marseille Universite, France, [email protected]

In this talk, I will discuss some geometrical properties of positive solutions of some semilin-ear elliptic equations in bounded convex domains or convex rings, with Dirichlet-type boundaryconditions. A solution is called quasiconcave if its superlevel sets are convex. I will present twocounterexamples, that is, two cases of semilinear elliptic equations for which the solutions are notquasiconcave. This talk is based on a joint work with N. Nadirashvili and Y. Sire.

16


Spreading speeds and profiles of solutions in nonlinearfree boundary problems

Hiroshi MatsuzawaNational Institute of Technology, Numazu College, Japan, [email protected]

This talk is based on joint works [2, 4] with Professor Yihong Du, Dr. Yuki Kaneko and Dr.Maolin Zhou. In this talk, we concern with the following free boundary problem:

ut − uxx + βux = f(u), t > 0, g(t) < x < h(t),u(t, g(t)) = u(t, h(t)) = 0, t > 0,g′(t) = −µux(t, g(t)), h′(t) = −µux(t, h(t)), t > 0,−g(0) = h(0) = h0, u(0, x) = u0(x), −h0 ≤ x ≤ h0,

where x = h(t) and x = g(t) are the moving boundaries to be determined together with u. β, µand h0 are given positive constants. We assume that f is monostable, bistable or combustion type.Such problems may be used to describe the spreading of a biological or chemical species underan advective environment. Du and Lou [1] considered the problem for the case where β = 0 andstudied the long-time dynamical behavior of solutions (spreading and vanishing) and determinedthe asymptotic spreading speed limt→∞(−g(t)/t) = limt→∞(h(t)/t) of the free boundaries whenspreading happens. For the problem with the advection term βux, Gu, Lin and Lou [3] showed thatthe rightward and leftward asymptotic spreading speeds are different due to the advection term.The aim of this talk is to give much sharper estimates for the spreading speeds of the fronts thanthat in [1] and [3] and obtain how the solution approaches the traveling semi-wave when spreadinghappens.

References

[1] Du Y., Lou B. Spreading and vanishing dichotomy in the diffusive logistic model with a freeboundary, J. Eur. Math. Soc., to appear.

[2] Du Y., Matsuzawa H., Zhou M. Sharp estimate of the spreading speed determined by nonlinearfree boundary problems, SIAM J. Math Anal., 46(1):499–531, 2014.

[3] Gu H, Lin Z., Lou B. Different asymptotic spreading speeds induced by advection in a diffu-sion problem with free boundaries, Proc. Amer. Math. Soc., 143(3):1109–1117, 2015.

[4] Kaneko Y., Matsuzawa H. Spreading speed and sharp asymptotic profiles of solutions infree boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl.,428(1):43–76.

17


Weakly interacting front dynamics in theFitzHugh-Nagumo system

Yoshihisa MoritaRyukoku Univeristy, Japan, [email protected]

We are concerned with the following FitzHugh-Nagumo equations, a reaction-diffusion modelof activator-inhibitor type:

ut − d∆u = f(u)− v,τvt −∆v = u− γv,

where d > 0, γ > 0, τ > 0, and f(u) = k(u − u3). This system allows a standing wavefrontsolution (u, v) = (U(x), V (x)) satisfying

limx→±∞

(U(x), V (x)) = (a±, a±/γ), a± := ±√

1− 1/kγ,

for kγ > 3√

6/2 and the solution is asymptotically stable up to phase-shift if γ is sufficiently large([2]).

Using the following ansatz with the front solution

(u, v) = (U(x− x1), V (x− x1)) + (U(−x+ x2), V (−x+ x2)) + (w1, w2),

x1 = x1(t), x2 = x2(t),

and applying the result of [3], we derive a dynamical law for (x1(t), x2(t)) when x2(t)−x1(t) >>1. Investigating the equation for h(t) := x2(t) − x1(t), we show that the dynamics for the twofronts can be classified as (i) attractive (ii) repulsive (iii) equilibrium states. Moreover, we exhibitparameter regimes for those three cases ([1]).

References

[1] Chen C.-N., Ei S., Morita Y. Weakly interacting wavefront dynamics in FitzHugh-Nagumosystems, in preparation.

[2] Chen C.-N., Kung S.-Y., Morita Y. Planar standing wavefronts in FitzHugh-Nagumo equa-tions, SIAM J.Math. Anal., 46:657–690, 2014.

[3] Ei S. The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Dif-ferential Equations, 14:85–137 2002.

18


The Hele-Shaw asymptotics for mechanical models oftumor growth

Fernando QuirosU. Autonoma de Madrid, Spain, [email protected]

Models of tumor growth, now commonly used, present several levels of complexity, both interms of the biomedical ingredients and the mathematical description. The models can describethe tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case bymeans of a free boundary problem.

The goal of this talk is to report on recent work on a connection between cell density modelsand free boundary models of Hele-Shaw type which can be established by passing to a certain‘incompressible” limit; see [1], [2], and [3].

References

[1] Perthame B., Quiros F., Vazquez J. L. The Hele-Shaw asymptotics for mechanical models oftumor growth, Arch. Ration. Mech. Anal., 212(1):33–127, 2014.

[2] Perthame B., Quiros F., Tang M., Vauchelet N. Derivation of a Hele-Shaw type system froma cell model with active motion, Interfaces Free Bound., 16(4):489–508, 2014.

[3] Mellet A., Perthame B., Quiros F. In preparation.

19


On a free boundary problem with two fluidsJurgen Socolowsky

Brandenburg University of Applied Sciences,Engineering Department - Mathematics Group, Germany,

[email protected]

In this lecture a particular plane steady-state flow problem for viscous fluids is studied in-cluding evaporation effects. The corresponding free boundary value problem describes the mo-tion of two heavy immiscible fluids flowing in a horizontal channel. The flow is generated by alower moving wall and by pressure gradient. The governing equations create a coupled system ofNavier-Stokes and Stephan equations. The flow domain is unbounded in two directions and thefree interface separating both fluids is infinite. Furthermore, the upper wall of the channel has asmall geometrical perturbation. A corresponding isothermal problem was solved in full detail inthe paper [1]. Later on the methods developed and applied in [1] for the proof of existence anduniqueness of solutions have been applied to a similar nonisothermal problem describing Boussi-nesq approximation and Marangoni convection effects in [2]. In the present talk the same methodsare applied to a particular problem including evaporation effects. Existence and uniqueness resultsof suitable weak solutions in weighted Sobolev spaces can be proved for small data (i.e. for smallfluxes and for a small distortion of the channel) of the problem.

References

[1] Pileckas K., Socolowsky J. Viscous two-fluid flows in perturbed unbounded domains, Mathe-matische Nachrichten, 278(5):589–623, 2005.

[2] Socolowsky J. On nonisothermal two-fluid channel flows, Mathematical Modelling and Anal-ysis – The Baltic Journal on Mathematical Applications, Numerical Analysis and DifferentialEquations, 12(1):143–156, 2007.

20


On a free boundary problem for a competition system:two invasive species case

Chang-Hong WuDepartment of Applied Mathematics, National University of Tainan, Taiwan,

[email protected]

We focus on a two-species competition-diffusion model with two free boundaries in one-dimensional homogeneous environments. Here, two free boundaries which may intersect eachother are used to describe the spreading fronts of two competing species, respectively. The spread-ing mechanism for species is assumed to satisfy a Stefan condition, which is proposed by Du andLin [1]. We mainly study the dynamics and offer some biological insight.

References

[1] Y. Du and Z.G. Lin. Spreading-vanishing dichotomy in the diffsive logistic model with a freeboundary, SIAM J. Math. Anal., 42:377–405, 2010.

21


Multiple spreading phenomena for a free boundaryproblem in ecology

Yoshio Yamada, Waseda University, Japan, [email protected]

We consider the following free boundary problem:

(P)

ut = duxx + f(u), t > 0, 0 < x < h(t),ux(0, t) = u(t, h(t)) = 0, t > 0,h′(t) = −µux(t, h(t)), t > 0,h(0) = h0, u(0, x) = u0(x), 0 ≤ x ≤ h0,

where d, µ and h0 are positive constants and u0 is a nonnegative function. In 2010, Du and Lin[1] proposed (P) with f(u) = u(a − bu) to describe invasion of a species and established thespreading-vanishing dichotomy theorem. Here f is assumed to satisfy the following conditions(A) f(u) = 0 has solutions 0, u1, u2, u3 (0 < u1 < u2 < u3) such that

f ′(0) > 0, f ′(u1) < 0, f ′(u2) > 0, f ′(u3) < 0 and∫ u3

u1

f(u)du > 0.

A typical example satisfying (A) is given by a nonlinear function with the Holling-type III func-tional response, f(u) = u(a− bu)− u2/(1 + u2).

By the result of [1], (P) has a unique global solution. So our main purpose is to study theasymptotic behaviors of global solutions as t → ∞. It will be shown that any solution (u, h) of(P) satisfies one of the following properties:(i) lim

t→∞h(t) ≤ (π/2)

√d/f ′(0) and lim

t→∞‖u(t)‖C[0,h(t)] = 0,

(ii) limt→∞

h(t) =∞ and limt→∞

u(t, x) = u1 locally uniformly in [0,∞),

(ii) limt→∞


u(t, x) = u3 locally uniformly in [0,∞),

(iv) limt→∞


u(t, x) = v∗(x) locally uniformly in [0,∞),

where v∗ is a unique positive decreasing solution of dvxx + f(v) = 0 withvx(0) = 0.Furthermore, studying the related semi-wave problem as in [2] we will give precise informa-tion on an asymptotic speed of h(t) and a transient profile of u(t, x) for large t when h satisfieslimt→∞

h(t) =∞.

References

[1] Yihong Du and Zhigui Lin. Spreading-vanishing dichotomy in the diffusive logistic modelwith a free boundary, SIAM J. Math. Anal., 42:377–405, 2010.

[2] Yihong Du and Bendong Lou. Spreading and vanishing in nonlinear diffusion problems withfree boundaries, to appear in J. Eur. Math. Soc.

22


Special Session

Navier-Stokes Equations and Other Models of FluidMechanics

Organizer

Reinhard FarwigTechnische Universitat Darmstadt, Germany

[email protected]

23


Regular solutions to the compressible, viscousheat-conducting Navier-Stokes equations

Jan Burczak1,∗, Wojciech Zajaczkowski1, Yoshihiro Shibata21 Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland, ∗[email protected]

2 Department of Mathematics, Waseda University, Tokyo, Japan

I will present a proof of short-time or small-data existence of regular solutions to the equationsof compressible, heat-conducting fluids in a bounded domain, with the Dirichlet boundary condi-tions for velocity and temperature. We allow for very general constitutive relations for pressurep = p(%, θ) and for the specific heat cv = cv(%, θ).

24


Solvability of the Initial-Boundary value problem of theNavier-Stokes equations with nonhomogeneous data

Tongkeun Chang1,∗, Bum Ja Jin21 Mokpo National Maritime University

South Korea, ∗[email protected] Mokpo National University, South Korea

In this talk, we introduce our recent works on the initial-boundary value problem of the Navier-Stokes equations in the half space with nonhomogeneous data. An initial data h in some Besovspace and a boundary data g in some anisotropic Besov space have been considered. Indepen-dently, a Holder continuous initial h and a anisotropic Holder continuous boundary data g havealso been considered. Our study is motivated by [1, 2, 3] and references therein.

The followings are the main results to be presented.

1) Let 0 ≤ α < 2 and∞ > q > n+2α+1 .

We prove the unique existence of weak solution u ∈ Bα,α2

q (R+ × (0, T )) (u ∈ Lq(R+ ×(0, T )) if α = 0) with ∇u ∈ Lrloc(R+ × (0, T )) (for some r > 1) for some time interval

(0, T ) when the initial data h ∈ Bα− 2q

q (R+) and the boundary data g ∈ Bα− 1q,α2− 1

2qq (Rn−1×

R+) with gn ∈ B12α

q (R+; B− 1q

q (Rn−1))∩Lq(R+; Bα− 1

qq (Rn−1)) (If α ≥ 3

q , then some com-patibility condition should be satisfied. In particular, if α > 3

q , then the usual compatibilitycondition g|t=0 = h|xn=0 is required).

2) Let q =∞ and 0 < α < 1.

We prove the unique solvability of the weak solution u ∈ Cα,α2 (R+ × (0, T )) for some

short time interval (0, T ) with ∇u ∈ L∞loc(R+ × (0, T ]) when the given initial data h ∈Cα(R+) (R′hn ∈ L∞(R+)) and the given boundary data g ∈ Cα,

α2 (Rn−1 × (0, T ))

(R′gn ∈ L∞(Rn−1;Cα2 (0, T ))) with compatibility condition g|t=0 = h|xn=0.

References

[1] H. Amann. Navier-Stokes equations with nonhomogeneous Dirichlet data, J. Nonlinear Math.Phys. 10(1): 1–11, 2003.

[2] R. Farwig, H. Kozono, and H. Sohr. Very weak solutions of the Navier-Stokes equations inexterior domains with nonhomogeneous data, J. Math. Soc. Japan 59(1):127–150, 2007.

[3] G. Grubb. Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity Lp Sobolevspaces, J. Math. Fluid Mech. 3(1):57–81, 2001.

25


A class of large solutions for the compressibleNavier-Stokes system

Raphael DanchinUniversite Paris-Est Creteil, France, [email protected]

We are concerned with the construction of global unique solutions for the barotropic compressibleNavier-Stokes system. We show that if the the volume viscosity is large then initial data having(possibly) large potential viscosity give rise to global solutions, even though the shear viscosityremains of typical size. We expect this result to be a step toward a new approach for the derivationof the incompressible Navier-Stokes system with variable density.

26


Exponential decay of the vorticity of steadyincompressible viscous flow around a rotating rigid body

Paul Deuring1,∗, Paolo G. Galdi21 Univ Lille Nord de France, France, ∗[email protected]

2 University of Pittsburgh, USA

We consider the steady flow of a viscous incompressible fluid around a body which performs arotation with constant angular velocity and a translation with constant velocity. The axis of rotationand the direction of translation are supposed to coincide. This type of flow is usually describedby the stationary incompressible Navier-Stokes system with additional terms corresponding to thetranslation (Oseen term) and the rotation of the body. Our main result states that outside the wakeregion, the vorticity of the flow decays with an exponential rate.

27


Quasi-optimal initial value conditionsfor the Navier-Stokes equations

Reinhard FarwigTechnische Universitat Darmstadt, Germany

[email protected]

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional do-main Ω. It is well-known that an initial value u0 ∈ H1

0,σ(Ω) or even u0 ∈ D(A1/4) ⊂ L3σ(Ω),

where A = −P∆ denotes the Stokes operator, admits a unique regular solution in Serrin’s classLsq(0, T ;Lq(Ω)), 2

sq+ 3

q = 1, 2 < sq < ∞, for some T = T (u0). The optimal class of initialvalues u0 ∈ L2

σ(Ω) with this property was determined by H. Sohr, W. Varnhorn and myself [1] in2009 and is given by the Besov space B−2/sqq,sq of solenoidal vector fields u0 satisfying the condition

∫ ∞

0

(‖e−τAu0‖q

)sq dτ <∞.

This condition is used at (almost) all t0 > 0 along a given weak solution u to find various newregularity conditions [2].

The above optimal condition can be weakened to a quasi-optimal condition on u0 ∈ L2σ(Ω)

with finite integral ∫ ∞

0

(τα‖e−τAu0‖q

)sdτ <∞

where s > sq satisfies 2s + 3

q = 1 − 2α, 0 < α < 12 , cf. [3]. This condition can be described by

the Besov space B−1+3/qq,s , q > 3. A weak solution with such an initial value is contained in an

Ls(Lq)-space with time weight τα, still satisfies the energy equality on some interval [0, T ) andSerrin’s condition u ∈ Ls(ε, T ;Lq(Ω)), ε > 0, but the classical Serrin weak-strong uniquenesstheorem holds only under additional assumptions to be described in this talk.

References

[1] Farwig, R., Sohr, H., Varnhorn, W. On optimal initial value conditions for local strong solu-tions of the Navier-Stokes equations, Ann. Univ. Ferrara, 55:89–110, 2009.

[2] Farwig, R. On regularity of weak solutions to the instationary Navier-Stokes system – a reviewon recent results, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 60:91–122, 2014.

[3] Farwig, R., Giga, Y., Hsu, P.-Y. Initial values for the Navier-Stokes equations in spaces withweights in time. FB Mathematik, TU Darmstadt, Preprint No. 2691, 2014.

28


Regular Lagrangian flows - on the application toproblems of polymeric fluids

Piotr GwiazdaInstitute of Applied Mathematics and Mechanics, University of Warsaw,

[email protected]

Our interest is directed to the studies on transport equation with integral operators. The moti-vations for such considerations come from the description of polymeric flows. The problem is tofind a macroscopic velocity v(t, x) : (0, T )×Ω→ R3, a pressure q(t, x) : (0, T )×Ω→ R+ anda density of the polymer molecules of the length z namely µ(t, x, z) : (0, T ) × Ω × R+ → R+

satisfying the following system of equations

vt + v∇xv +∇xq − divxT = f (1)

divxv = 0, (2)

where the stress tensor T is given by the formula

T (µ,Dxv(t, x)) = ν(µ, |Dxv|)Dxv(t, x). (3)

The quantity µ satisfies the following transport equation

∂tµ+ divx(vµ) = ∂z(τµ)−Bµ+ 2

∫ ∞

zB(y)b(z, y)µ(t, x, y) dy. (4)

where B(z) : R+ → R+ is the rate of fragmentation of polymer molecules, b(z, y) : R+×R+ →[0, 1] is the probability that a given particle of length y brakes into a particle of length z and y− z,and τ(z) : R+ → R+ is the rate of coagulation of monomers.

We will show how the approach of regular Lagrangian flows can be used to equations of type(4). The talk is based on [1].

References

[1] De Lellis, C., Gwiazda, P., Swierczewska-Gwiazda, A. Transport equation with integralterms, submitted.

29


Asymptotic structure of steady Stokes flow around arotating obstacle in 2D

Toshiaki HishidaNagoya University, Japan, [email protected]

Let Ω be an exterior domain in the plane R2 with smooth boundary, and consider the motionof a viscous incompressible fluid around an obstacle (rigid body) R2 \ Ω. As compared with 3Dproblem, we have less knowledge about exterior steady flows in 2D. The difficulty is to analyzethe asymptotic behavior of the flow at infinity. This is related to the hydrodynamical paradoxfound by Stokes. In this presentation it is shown that the rotation of the obstacle leads to theresolution of the Stokes paradox in the sense that: (i) The flow can be bounded (and even goes to aconstant vector at the rate |x|−1) at infinity even if the net force does not vanish; (ii) Given externalforce f(x) decaying sufficiently fast, there exists a linear flow which enjoys u(x) = O(|x|−1) as|x| → ∞. We also provide a remarkable asymptotic representation of the flow at infinity, in whichthe leading term involves the rotating profile x⊥/|x|2 whose coefficient is given by the torque,where x⊥ = (−x2, x1)T . Here, the linear system arising from the flow around a rotating obstaclewith constant angular velocity a ∈ R \ 0 is described as

−∆u− a(x⊥ · ∇u− u⊥

)+∇p = f, div u = 0 in Ω

in the reference frame attached to the obstacle. The essential reason why there is no longer Stokes’paradox is better decay structure of the fundamental solution of the system above in the wholeplane R2.

References

[1] Hishida. T. Asymptotic structure of steady Stokes flow around a rotating obstacle in two di-mensions, ArXiv 1503.02321v1.

30


Fundamental solution to the time-periodic Stokesequations

Mads KyedTechnische Universitat Darmstadt, Germany,[email protected]

The concept of a fundamental solution to the time-periodic Stokes equations in dimensionn ≥ 2 will be introduced. It will then be shown how to identify and establish integrability andpointwise estimates of such a fundamental solution. Applications to different problems in fluiddynamics will be discussed. In particular, how to analyze the asymptotic structure of time-periodicfluid flows.

31


On Ukai-type solution formula for the Stokes system in adomain with graph boundary

Hideyuki MiuraTokyo Institute of Technology, Japan, [email protected]

We consider the space of solenoidal vector fields in an unbounded domain Ω ⊂ Rn whoseboundary is given as a Lipschitz graph. It is shown that, under suitable functional setting, thespace of solenoidal vector fields is isomorphic to the n − 1 product space of the space of scalarfunctions. This leads to a natural and systematic reduction of the equations describing the motionof incompressible flows. Our approach provides a formula for the Stokes semigroup, which canbe considered as a generalization of Ukai’s formula in Rn

+[1]. This is a joint work with YasunoriMaekawa.

References

[1] Ukai S. A solution formula for the Stokes equation in Rn+, Comm. Pure Appl. Math., 40:611–

621, 1987.

32


Regularizing effects of a thin elastic interface influid-structure interaction problems

Boris Muha1,∗, Suncica Canic21 Department of Mathematics, University of Zagreb, Croatia, ∗[email protected]

2 Department of Mathematics, University of Houston, United States

Motivated by modeling blood flow in human arteries, we study a fluid-structure interactionproblem in which the structure is composed of multiple layers, each with possibly different me-chanical characteristics and thickness. In the problem presented in this talk the structure is com-posed of two layers: a thin layer modeled by the 1D wave equation, and a thick layer modeled bythe 2D equations of linear elasticity. The flow of an incompressible, viscous fluid is modeled bythe Navier-Stokes equations. The thin structure is in contact with the fluid thereby serving as afluid-structure interface with mass. The coupling between the fluid and the structure is nonlinear.The resulting problem is a nonlinear, moving-boundary problem of parabolic-hyperbolic type. Wepresent the main ideas behind the constructive proof of the existence of a weak solution for theconsidered problem which reveals that the thin elastic interface has a regularizing effect.

In the second part of the talk we consider a linear heat-wave system which serves as a sim-plified fluid-structure interaction problem. The system is coupled in two different ways: thefirst, when the interface does not have mass and the second, when the interface does have mass.We prove an optimal regularity result in Sobolev spaces for both cases. Furthermore, we showthat point mass coupling regularizes the problem and quantify this regularization in the sense ofSobolev spaces.

33


Inviscid incompressible limits on expanding domainsSarka Necasova1,∗, Eduard Feireisl2, Yongzhong Sun3

1 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Czech Republic,∗[email protected]

2 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Czech Republic3Department of Mathematics, Nanjing University,China

We consider the inviscid incompressible limit of the compressible Navier-Stokes system on alarge domain, the radius of which becomes infinite in the asymptotic limit. We show that the limitsolutions satisfy the incompressible Euler system on the whole physical space R3 as long as theradius of the domain is larger than the speed of acoustic waves inversely proportional to the Machnumber. The rate of convergence is estimated in terms of the Mach and Reynolds numbers and theradius of the family of spatial domains [1].

References

[1] Feireisl E.,Necasova,S.,Sun, Y. Inviscid incompressible limits on expanding domains, Nonlin-earity, 27(10):2465–2478, 2014.

34


Navier-Stokes flow in the weighted Hardy spaceTakahiro Okabe

Hirosaki University, Japan, [email protected]

The asymptotic expansions of the Navier-Stokes flow in Rn and the rates of decay are dis-cussed with aid of weighted Hardy spaces. Fujigaki and Miyakawa [2], Miyakawa [3] proved thenth order asymptotic expansion of the Navier-Stokes flow if initial data decays like (1+ |x|)−n−1

and if nth moment of initial data is finite. In this presentation, it is clarified that the moment con-dition for initial data is essential in order to obtain higher order asymptotic expansion of the flowand to consider the rapid time decay problem. The second author [5] established the weighted es-timates of the strong solutions in the weighted Hardy spaces with small initial data which belongsto Ln(Rn) and a weighed Hardy space. Firstly, as a refinement, we derived the existence theoremin the weighted Hardy spaces assuming smallness of initial data only in Ln(Rn). Moreover, intwo dimensional case the smallness condition on initial data is completely removed. As an appli-cation, the rapid time decay of the flow are investigated with aid of asymptotic expansions and ofthe symmetry conditions introduced by Brandolese [1].

References

[1] L. Brandolese. On the localization of symmetric and asymmetric solutions of the Navier-Stokes equations in Rn, C. R. Acad. Sci. Paris Ser. I Math., 332:125–130, 2001.

[2] Y. Fujigaki and T. Miyakawa. Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space, SIAM J. Math. Anal., 33:523–544, 2001.

[3] T. Miyakawa. On space-time decay properties of nonstationary incompressible Navier-Stokesflows in Rn, Funkcial. Ekvac., 43:541–557, 2000.

[4] T. Okabe and Y. Tsutsui. Navier-Stokes flow in the weighted Hardy space with applications totime decay problem, preprint, submitted.

[5] Y. Tsutsui. An application of weighted Hardy spaces to the Navier-Stokes equations, J. Funct.Anal., 266:1395–1420, 2014.

35


Inflow-outflow nonstationary flowJoanna Rencławowicz

Institute of Mathematics, Polish Academy of Sciences, Poland, [email protected]

The problem examined is a nonstationary, incompressible motion in cylindrical domain. Themotion is modeled with Navier-Stokes system of equations, with slip boundary condition. Wediscuss the existence result for the inflow-outflow problem with arbitrarily large flux, where theinitial velocity does not change to much along the axis of cylinder and the inflow does not changemuch along directions perpendicular to this axis either with respect to time. In our results thereis no restrictions on magnitude of flux, moreover, in the proof of the existence of global regularsolutions we admit arbitrarily large L2 norm of initial velocity.

36


The Stokes Operator in Periodical Domains withBoundary

Jonas SauerTechnische Universitat Darmstadt, Germany, [email protected]

We investigate maximal regularity in Lq-spaces of the abstract Stokes operator Aq in spa-tially periodic domains with boundary, that is, in cylindrical domains. The main step is to showcorresponding resolvent estimates in the periodic whole space G := Rn−1 × T , where T is theone-dimensional torus, and the periodic half space G+ = x ∈ G : x1 > 0. In the wholespace case, these estimates have been obtained in [2] by applying Fourier techniques on the lo-cally compact abelian group G. In the half space case, however, the resolvent estimates followfrom a sophisticated mirroring technique. If one wants to read offR-boundedness of the resolventoperator directly from the obtained resolvent estimates, one has to use an extrapolation theorem inthe style of Garcıa-Cuerva and Rubio de Francia [1]. In particular, the resolvent estimates have tohold in weighted Lq

ω-spaces, where ω is a Muckenhoupt weight. Unfortunately, ω(x1, ·) is not an(n − 1)-dimensional Muckenhoupt weight in general. Therefore, classical mirroring techniquesthat rely on (n− 1)-dimensional estimates for fixed x1 > 0 cannot be applied. The main aspect ofmy talk will be to show how to overcome these difficulties arising in the periodic half space case.Once the periodic half space is understood, results for cylindrical domains follow easily by well-known perturbation and localization methods.

References

[1] Garcıa-Cuerva J., Rubio de Francia J. L. Weighted Norm Inequalities and Related Topics,North-Holland Publishing Co., Amsterdam, Volume 116 (1985).

[2] Sauer J. Weighted Resolvent Estimates for the Spatially Periodic Stokes Equations, Ann. Univ.Ferrara, 2014, http://dx.doi.org/10.1007/s11565-014-0221-4.

37


Time periodic Stokes problems in a layer: the spatialasymptotics

Maria Specovius-Neugebauer1,∗, Konstantin Pileckas21 University of Kassel, Germany, ∗[email protected]

2 Vilnius University, Lithuania

We consider the solutions to the time periodic Stokes problem in a layer Ω = R2 × (0, 1) 3x = (y, z):

ut −∆u+∇p = f, div u = g in Ω

u|z=1 = 0, u|z=0, u|t=0 = u|t=2π,

where the data f, g are also time periodic and smooth with bounded support for simplicity. Resultson spatial asymptotics are typically of the form

u = U + u, p = P + P ,

where U,P are explicitely known and the remainder u, p is in a space with a stronger decay (see[1] for results for the stationary problem). Starting from solutions with u ∈ L2(L2

β), p ∈ L2(L2β),

where L2β(Ω) is a weighted L2-space with polynomial weights we derive the main asymptotic

terms of u, p as |y| tends to infinity, including estimates for the remainder.

References

[1] Nazarov, S.A., Pileckas, K.I. The asymptotic properties of the solution to the Stokes problemin domains that are layer-like at infinity, J. Math. Fluid Mech, 1:131–167, 1999.

38


Polymeric fluidsAgnieszka Swierczewska-Gwiazda

Institute of Applied Mathematics and Mechanics, University of Warsaw,[email protected]

The essence of modelling polymeric fluids is encapsulated in the coupling of equations de-scribing the evolution of macroscopic quantities (like velocity, pressure and eventually also densityand temperature) with an additional equation describing the microscopic structure. The influenceof the processes of polymerization and fragmentation will be accounted through the dependence ofviscosity on the level of polymerization and/or appearance of the extra stress tensor. We describethe regime of concentrated solution of polymer chains suspended in a non-Newtonian solvent, weassumed that polymer chains can be convected by the macroscopic velocity field, and are alsosubject to polymerization and fragmentation processes. We will discuss various approaches tomodeling of polymers and briefly present the analytical results. The talk is based on [1].

References

[1] Bulicek, M., Gwiazda, P., Suli, E., Swierczewska-Gwiazda, A. Analysis of a viscosity modelfor concentrated polymers, submitted, arXiv:1501.05766.

39


Decay of Non-stationary Navier-Stokes Flow withNonzero Dirichlet Boundary Data

David WegmannTechnische Universitat Darmstadt, [email protected]

We consider the Navier-Stokes equations in a domain with compact boundary and nonzeroDirichlet boundary data β. Recently, in the case of an exterior domain the existence of a global intime weak solution which satisfies the strong energy inequality was shown in [1]. This solution isconstructed as a sum of a very weak solution b to the instationary Stokes equation with nonzeroboundary data and a weak solution v to a system of Navier-Stokes type with zero boundary data.Assuming that β(t) → 0 as t → ∞, we prove in [2] that the corresponding solution v fulfills‖v(t)‖2 → 0 as t → ∞. Furthermore, in a bounded domain the solution v tends exponentially to0 if the corresponding data are exponentially decreasing.As a last result, we calculate a lower polynomial bound for the decay rate if Ω is unbounded.Therefor, we use a suitable spectral decomposition of the Stokes operator as introduced in [3] andDuhamel’s formula.

References

[1] Farwig, R., Kozono, H. Weak solutions of the Navier-Stokes equations with non-zero bound-ary values in an exterior domain satisfying the strong energy inequality, J. Differential Equa-tions, 7(10):2633–2658, 2014.

[2] Farwig, R., Kozono, H., Wegmann, D. Decay of non-stationary Navier-Stokes flow withnonzero Dirichlet boundary data, in preparation.

[3] Borchers, W., Miyakawa, T. Algebraic L2 decay for Navier-Stokes flows in exterior domains.II, Hiroshima Math. J., 3:621–640, 1991.

40


Some stability problems for the Navier-Stokes andmagnetohydrodynamics equations

Wojciech Zajaczkowski,Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

[email protected]

First we examine two-dimensional and axially symmetric solutions to the Navier-Stokes (NS) andmagnetohydrodynamics (MHD) equations. We call them special solutions. We consider suchsolutions which correspond to nonvanishing in time external force. Therefore we obtain estimatesfor these solutions step by step in time. To show stability of these solutions we assume that theexternal and the initial data are sufficiently close to the corresponding force and initial data of thespecial solutions. In this way we prove also existence of global regular solutions to NS and MHD,which are close to the special solutions in all time. We use the energy method.

We are going to present the following cases:

1. Stability of two-dimensional solutions to NS equations with periodic, nonslip, slip andNavier boundary conditions.

2. Stability of two-dimensional solutions to NS equations coupled with the heat convectionwith slip and Navier boundary conditions.

3. Stability of axially symmetric solutions to MHD equations.

41


Special Session

Blow-up and large time behaviour of solutions ofparabolic problems

Organizers

Marek FilaComenius University, Slovakia, [email protected]

Pavol QuittnerComenius University, Slovakia, [email protected]

42


Critical mass in a volume filling modelTomasz Cieslak

IMPAN, Poland, [email protected]

In my talk I will review our recent common results with Christian Stinner related to the fullyparabolic volume filling Keller-Segel model with a probability jump function given by

q(u) = (1 + u)−γ , γ ≥ 0.

The most interesting one is a critical mass phenomenon in dimension 2. It states that for γ ≥ 1there exists a value of intial mass m∗ distinguishing between global-in-time bounded solutions forinitial data with mass m < m∗ (in the case of radially symmetric solutions the critical value is8π(1 + γ), for any data m∗ = 4π(1 + γ)) and an existence of solutions which become infinitewhen time goes to ∞, however existing globally in time. The second situation takes place forinitial mass of radially symmetric data exceeding 8π(1 + γ). For 0 < γ < 1 we have a similarresult, the only difference is that it is open whether in the supercritical case solutions blow up infinite or infinite time.

43


Symmetry and convergence properties for asymptoticallysymmetric parabolic problems

Juraj Foldes1,∗, Peter Polacik21 Universite Libre de Bruxelles, Belgium, ∗[email protected]

2 University of Minnesota, USA

We discuss qualitative properties of positive solutions of the equation

ut = ∆u+ f(t, x, u,∇u) + h(x, t) , (1)

on either a bounded domain Ω ⊂ RN complemented with Dirichlet boundary conditions, or onRN with decay assumptions. The domain and the nonlinearity f are assumed to be invariantunder the reflection about the x1-axis, and the function h accounts for a nonsymmetric decayingperturbation: h(·, t) → 0 as t → ∞. With additional decay assumptions on h, we show theasymptotic symmetry of each bounded positive solution u of (1). The novelty of these results isthat the asymptotic symmetry is established even for solutions not assumed uniformly positive,what prevents an application of common techniques based on the method of moving hyperplanes.For asymptotically time autonomous problems, that is, for f independent of t, we also discusssufficient conditions that guarantee convergence of u to a single equilibrium, and we classify allbounded entire solutions.

44


Equilibration of unit mass solutions to a degenerateparabolic equation with a nonlocal gradient nonlinearity

Johannes LankeitUniversitat Paderborn, Germany, [email protected]

The solutions to

ut = u∆u+ u

∫

Ω|∇u|2, u

∣∣∂Ω

= 0, u(·, 0) = u0

in a bounded smooth domain Ω ⊂ Rn, n ≥ 1, blow up at finite time if∫

Ω u0 > 1 and converge to0 as t→∞ if

∫Ω u0 < 1.[1]

In this talk we will consider the case of∫

Ω u0 = 1 and identify the (nontrivial) W 1,20 (Ω)-limit

of u(t) as t→∞.[2]

References

[1] Kavallaris N. I., Lankeit J., Winkler M. On a degenerate non-local parabolic problem describ-ing infinite dimensional replicator dynamics, in preparation.

[2] Lankeit J. Equilibration of unit mass solutions to a degenerate parabolic equation with anonlocal gradient nonlinearity, in preparation.

45


Convergence rates of solutions for a two-dimensionalchemotaxis-Navier-Stokes system

Li Yuxiang∗

joint with Zhang QingshanDepartment of Mathematics, Southeast UniversityNanjing 211189, China, ∗[email protected]

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation

nt + u · ∇n = ∆n− χ∇ · (n∇c), x ∈ Ω, t > 0,

ct + u · ∇c = ∆c− nc, x ∈ Ω, t > 0,

ut + κ(u · ∇)u = ∆u+∇P + n∇φ, x ∈ Ω, t > 0,

∇ · u = 0, x ∈ Ω, t > 0,

in a bounded domain Ω ⊂ R2. It is known that if χ > 0, κ ∈ R and φ ∈ C2(Ω), for suffi-ciently smooth initial data, the model possesses a unique global classical solution which satisfies(n, c, u) → (n0, 0, 0) as t → ∞ uniformly with respect to x ∈ Ω, where n0 := 1

|Ω|∫

Ω n(x, 0)dx.In the present paper, we prove this solution converges to (n0, 0, 0) exponentially in time.

46


Evolution of nonlocal planar curvature flowXiaoliu Wang

Math. Dep., Southeast University, China, [email protected]

In this talk, we will introduce some recent results on nonlocal curvature flows, especially aboutthe global asymptotic behaviour and the finite-time singularity of the non-simple convex closedcurves’ evolution. Part of the results are from [1].

References

[1] Wang Xiaoliu, Kong Linghua. Area-preserving evolution of nonsimple symmetric planecurves, Journal of Evolution Equations, 14(2):387–401, 2014.

47


Mathematical challenges arising in the analysis ofchemotaxis-fluid interaction

Michael WinklerUniversity of Paderborn, Germany

[email protected]

We consider models for the spatio-temporal evolution of populations of microorganisms, mov-ing in an incompressible fluid, which are able to partially orient their motion along gradients of achemical signal. According to modeling approaches accounting for the mutual interaction of theswimming cells and the surrounding fluid, we study study parabolic chemotaxis systems coupledto the (Navier-)Stokes equations through transport and buoyancy-induced forces. The presenta-tion discusses mathematical challenges encountered even in the context of basic issues such asquestions concerning global existence and boundedness, and attempts to illustrate this by review-ing some recent developments. A particular focus will be on strategies toward achieving a prioriestimates which provide information sufficient not only for the construction of solutions, but alsofor some qualitative analysis.

48


Special Session

Surface Tension and CapillarityOrganizer

Robert FinnStanford University, USA

[email protected]

49


Bounded and Unbounded Capillary Surfacesin a Cusp Domain

Yasunori, Aoki1,∗, Hans De Sterck2, David Siegel21 Department of Mathematics Uppsala University, Sweden ∗[email protected]

2 Department of Applied Mathematics University of Waterloo, Canada

Figure 1: Domain and Boundary conditions,where γ1,2 are physical constants.

A capillary surface, a liquid surface atequilibrium, can be modelled mathemati-cally by a nonlinear elliptic PDE calledLaplace-Young equation:

∇ · ∇u√1 + |∇u|2

= κu

where u is the height of the capillary sur-face and κ is a physical constant. In thistalk, we consider a domain and the bound-ary conditions as depicted in Figure 1, andpresent the following theorems [1]:

Theorem 1: u is bounded if cos γ1 + cos γ2 = 0 and f1,2 have finite curvatures.

Theorem 2: u = cos γ1+cos γ2f1(x)−f2(x) + O

(f ′1(x)−f ′2(x)f1(x)−f2(x)

)in a non-osculatory cusp domain if cos γ1 +

cos γ2 6= 0.Then we show a numerical method developed based on Theorem 2 to accurately approximate

unbounded capillary surfaces [2].

00.2

0.40.6

0.81 −0.2

−0.10

0.10.2

0

1000

2000

3000

4000

5000

6000

7000

8000

Finally using this method, we propose the following conjectures:Conjecture 1: u is bounded if cos γ1 + cos γ2 = 0.Conjecture 2: u = cos γ1+cos γ2

f1(x)−f2(x) +O(f ′1(x)−f ′2(x)f1(x)−f2(x)

)if cos γ1 + cos γ2 6= 0.

References

[1] Aoki Y., Siegel D.Bounded and Unbounded Capillary Surfaces in a cusp domain, PacificJournal of Mathematics, 257(1):143–165, 2012.

[2] Aoki Y., De Sterck H.Numerical Study of Unbounded Capillary Surfaces, Pacific Journal ofMathematics, 267,(1):1–34, 2014.

50


Viscous capillary flows inside a rotating cylinderEugene Benilov

University of Limerick, Ireland, [email protected]

Flows of a thin liquid film on the inside of a cylinder with horizontal axis, rotating about thisaxis, are usually referred to as rimming flows. If the amount of liquid in the cylinder is sufficientlysmall, all of it is entrained by rotation and the film is distributed more or less evenly. For mediumamounts, the liquid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools atthe cylinder’s bottom.

In this talk, two problems associated with rimming flows are presented, illustrating highlyunusual mathematical properties of this physical setting.

1. Firstly, we examine rimming flows with a pool, affected by weak surface tension [1]. Usingthe lubrication approximation and the method of matched asymptotics, we find a solutiondescribing the pool, the ‘outer’ region, and two transitional regions, one of which includes– quite unusually – an infinite number of asymptotic zones. Interestingly, infinitely manyasymptotic zones arise in another two problems involving liquid films [2, 3].

2. Secondly, we examine the stability of evenly-distributed rimming flows [4, 5]. It is shownthat all eigenmodes of the linearized stability problem are neutrally stable, and the corre-sponding eigenfunctions form a complete set. However, even though all solutions of thelinearized problem can be represented by a series in stable eigenfunctions, the problem isstill unstable with respect to ‘exploding’ perturbations, such that develop a singularity in afinite time. It is shown that the singularity is due to a divergence of the above-mentionedseries in the eigenfunctions.

References

[1] Benilov E. S., Benilov M. S., Kopteva N. Steady rimming flows with surface tension, J. FluidMech., 597:91–118, 2008.

[2] Wilson S. D. R., Jones, A. F. The entry of a falling film into a pool and the air-entrainmentproblem, J. Fluid Mech., 128:53–69, 1983.

[3] Benilov E. S., Chapman S. J., Mcleod J. B., Ockendon J. R., Zubkov V. S. “On liquid films onan inclined plate,” J. Fluid Mech., 663, 53–69 (2010).

[4] Benilov E. S., O’Brien S. B. O. G., Sazonov I. A. “A new type of instability: explosive distur-bances in a liquid film inside a rotating horizontal cylinder,” J. Fluid Mech., 497, 201–224(2003).

[5] Benilov E. S. “Explosive instability in a linear system with neutrally stable eigenmodes. Part2: Multi-dimensional disturbances,” J. Fluid Mech., 501, 105–124 (2004).

51


Isovolumetric and isoperimetric inequalitiesfor a class of capillarity functionals

Paolo CaldiroliUniversity of Torino, Italy, [email protected]

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionalparametric surfaces in R3 as the sum of the area integral and an anisotropic term of suitable form(see [2], Sect. 4.13). In the class of parametric surfaces with the topological type of S2 and withfixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribedup to a constant. For a certain class of anisotropies vanishing at infinity, we prove existence andnonexistence of volume-constrained, S2-type, minimal surfaces for the corresponding capillarityfunctionals. Moreover, in some cases, we show existence of extremals for the full isoperimetricinequality. These results are contained in [1].

References

[1] Caldiroli P. Isovolumetric and isoperimetric inequalities for a class of capillarity functionals,Preprint arXiv:1411.7541 (2014).

[2] Dierkes U., Hildebrandt S., Sauvigny F. Minimal Surfaces, GMW 339, Springer, 2010.

52


The Floating Cylinder ReconsideredHanzhe Chen1,∗, David Siegel2

1 University of Waterloo, Canada, ∗[email protected] University of Waterloo, Canada

We reconsider the floating cylinder problem studied in the ground-breaking paper of Bhatnagarand Finn (2006)[1]. We derive the energy relative to the undisturbed state E and the total forceF and show that F = −dE

dh , where h is the height of the center of the cylinder relative to theundisturbed fluid level. Let γ be the contact angle between the fluid interface and the cylinder.The goal is to find the equilibrium configurations and their stability. Bhatnagar and Finn gavethe first example with two equilibrium configurations, one stable and the other unstable. We givea complete discussion in terms of γ and two non-dimensional parameters A and C, C =

√B,

where A and B are those of Bhatnagar and Finn. In all cases there are at most two equilibriumconfigurations and when there are two equilibrium configurations, one is stable and the other isunstable.

References

[1] Bhatnagar Rajat, Finn Robert. Equilibrium Configurations of an Infinite Cylinder in an Un-bounded Fluid, Physics of Fluids, 18, 047103, 2006.

[2] Finn Robert. Equilibrium Capillary Surface, Springer-Verlag, New York, 1986.

53


Mutual Attractions of Floating Objects:an Idealized Example

Rajat Bhatnagar1, Robert Finn2,∗1 UC San Francisco, USA

2 Stanford University, USA, ∗[email protected]

During the 17th Century, Edme Mariotte observed that two rigid balls floating in a commonwater-bath could attract or repel each other, and he attempted (without success!) to develop aformal theory describing the phenomenon.

In 1806 Laplace substituted the formally simpler configuration of two infinite vertical parallelplates of possibly differing materials, partially immersed in an infinite liquid bath and rigidlyconstrained. He displayed particular configurations leading respectively to attraction or repulsion,and he provided quantitative estimates for the forces.

The present work continues our study of that configuration from a more geometrical pointof view, characterizing all modes of behavior that can occur. Our attention focuses on an exoticdiscontinuity in behavior relative to prescribed data, which can occur as the plates approach eachother.

We present here a quantitative description of that behavior, which contains features we couldnot have predicted. In one example, for which a universal upper bound for repelling force mag-nitude can be demonstrated, a range of data is exhibited, for the entirety of which that bound isactually achieved, at a (unique) positive plate separation, as the plates come together. Any furtherapproach of the plates to each other yields a rapid switch to attracting mode, with forces becomingunbounded as the inverse square of plate separation.

We emphasize that this work is based on the original (nonlinear) equation, going back toThomas Young in 1805 and later reaffirmed by Gauss, and entails no linearization or other com-promise with the original physical content.

54


Stable capillary hypersurfaces in a wedge and uniquenessof the minimizer

Miyuki KoisoInstitute of Mathematics for Industry, Kyushu University, Japan

[email protected]

Let Σ be a compact immersed stable capillary hypersurface in a wedge bounded by two hy-perplanes Π1, Π2 in Rn+1. Suppose Σ meets each Πi in constant contact angle not less than π/2.We prove that if ∂Σ is embedded for n = 2, or if ∂Σ is convex for n ≥ 3, then Σ is part of theround sphere.

It is known that the only compact immersed stable constant mean curvature (CMC) hypersur-face in Rn+1 is the round sphere (Barbosa - do Carmo, 1984). In this talk, we generalize this resultto hypersurfaces with free boundary.

Let Π1, Π2 be hyperplanes in Rn+1 with intersection Π0 := Π1 ∩Π2 = xn = 0, xn+1 = 0and making angles α,−α (0 < α < π/2) with xn+1 = 0, respectively. Let Ω ⊂ xn > 0 bethe wedge-shaped domain bounded by Π1 ∪Π2. Let X : (Σ, ∂Σ)→ (Ω, ∂Ω) be an immersion ofan n-dimensional oriented compact connected C∞ manifold Σ with boundary into Ω.

The n-dimensional area Hn(X) of X and the algebraic (n + 1)-dimensional volume V (X)enclosed by X(Σ) ∪ Π1 ∪ Π2 are given by Hn(X) =

∫Σ dΣ, V (X) = (n + 1)−1

∫Σ〈X, ν〉 dΣ,

where ν is the the unit normal vector field along X . The wetting energy W (X) and the totalenergy E(X) of X are defined by

W (X) = ω1Hn(D1) + ω2Hn(D2), E(X) = Hn(X) +W(X),

where ωi is a constant with |ωi| < 1 andHn(Di) is the n-dimensional area of the domainDi ⊂ Πi

bounded by Ci = (X(∂Σ) ∩Πi) ∪Π0.Consider the variational problem of E(X) for volume-preserving variations. Each critical

point (we call it a capillary hypersurface) has CMC and it meets Πi with contact angle θi =− cos−1(ωi). It is said to be stable if the second variation of the energy E is nonnegative for allvolume-preserving variations.

We show the following uniqueness result ([1], [2]). Assume ω1, ω2 are nonnegative. Supposean immersionX : (Σ, ∂Σ)→ (Ω, ∂Ω) is a stable capillary hypersurface with embedded boundary.If n = 2, then Σ is part of the round 2-sphere. If n ≥ 3 and D1, D2 are convex, then Σ is part ofthe round n-sphere.

We also discuss the uniqueness of the energy-minimizer. Our results are generalized to hyper-surfaces in a domain bounded by several hyperplanes.

References

[1] Choe, J., Koiso, M. Stable capillary hypersurfaces in a wedge, preprint,http://arxiv.org/abs/1405.5407

[2] Koiso, M., Mitsuo, Y. Stable capillary hypersurfaces in a wedge II, in preparation.

55


Capillary Surfaces: Necessary and Sufficient Conditionsfor Continuity at Corners

Kirk LancasterWichita State University, USA, [email protected]

A variational solution of the capillary problem (in non-negative gravity) consists of a mini-mizer f ∈ BV (Ω) of the functional

J(u) =

∫

Ω

√1 + |Du|2 dA+

1

2

∫

Ωκu2 dA−

∫

∂Ωcos(γ)u ds, u ∈ BV (Ω),

where Ω is a domain in the plane, γ : ∂Ω → [0, π] and κ ∈ [0,∞). When a solution f exists, thedomain has a corner at a point x0 ∈ ∂Ω and

∣∣γ(x)− π2

∣∣ ≤ π2−δ when x ∈ ∂Ω is near x0, for some

δ > 0, then the radial limits of f at x0 exist. When Ω has a convex corner at x0, necessary andsufficient conditions for the continuity of f at x0 were obtained in [1]. When Ω has a nonconvexcorner at x0, necessary and sufficient conditions for the continuity of f at x0, modulo a solutionof the “central fan question”, were obtained in [2]; the “central fan question” was introduced in[3]. We will discuss these results.

References

[1] Lancaster K. A Proof of the Concus-Finn conjecture, Pacific Journal of Mathematics, 247(1):75–108, 2010.

[2] Lancaster K. Remarks on the behavior of nonparametric capillary surfaces at corners, PacificJournal of Mathematics, 258(2):369–392, 2012.

[3] Athanassenas M., Lancaster K. CMC capillary surfaces at reentrant corners, Pacific Journalof Mathematics, 234(2):201–228, 2008.

56


Symmetries on capillary surfaces in Euclidean spaceRafael Lopez

Departamento de Geometrıa y Topologıa. University of Granada, Spain, [email protected]

Consider T a surface in Euclidean space which separates the ambient space in two componentsand denote by W one of such connected components. A capillary surface S on T is a compactembedded surface of constant mean curvature such that int(S) is included in W , the boundary ofS lies on T and S meets T with constant angle along its boundary. We ask whether the geometryof T imposes restrictions to the possible configurations of S, such as, if the symmetries of T areinherited by S.

We study different settings where T has high symmetries, as for example, when T is a cone,a sphere and a cylinder, obtaining conditions that assure that S inherits the symmetries of T . Themain ingredient in the proofs is the Alexandrov reflection principle [1], but we use the processwith a uniparametric family of planes which are not parallel but that keep the symmetry of T . See[2, 3, 4]. We also consider capillary surfaces in a wedge under the assumption of stability [5].

References

[1] Alexandrov, A.D. Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ.Math., 13:5–8, 1958; English translation: AMS Transl. 21, 412–416, 1962.

[2] Lopez, R., Pyo, J. Constant mean curvature surfaces with boundary on a sphere, Appl.Math.Comput., 220:316 –323, 2013.

[3] Lopez, R., Pyo, J. Capillary surfaces of constant mean curvature in a right solid cylinder,Math. Nachr., 287:1312–1319, 2014.

[4] Lopez, R., Pyo, J. Capillary surfaces in a cone, J. Geom. Phys., 76:256–262, 2014.

[5] Lopez, R. Capillary surfaces with free boundary in a wedge, Adv. Math., 262:476–483, 2014.

57


Numerical limits of stability for cylindrical pendent dropsJohn McCuan1,∗, Manxi Wu 2

1 Georgia Institute of Technology, USA, ∗[email protected] Peking University, China

Cylindrical pendent drops are a hybrid between pendent drops and horizontal liquid bridgesin a downward gravity field [1] and [2]. They are somewhat remarkable for admitting discon-tinuous stable families indexed by increasing volume and stable drops with volume larger thansome unstable drops adhering to the same support geometry in particular. This feature, amongothers, complicates the stability analysis, and while some stability properties of these drops canbe rigorously proved, some assertions presently depend on numerical calculations. In this talk Iwill describe the basic stability behavior distinguishing between rigorous and numerical resultsand giving particular emphasis to some recently discovered fine properties exposed by the numer-ics. Time permitting, I will discuss details of our numerical approach and some technical aspectsrelying on asymptotic analysis.

References

[1] McCuan J. The stability of cylindrical pendent drops, Memoirs of the AMS, to appear, 2015.

[2] McCuan J. Extremities of stability for cylindrical pendant drops, in: Geometric analysis,mathematical relativity, and nonlinear partial differential equations, AMS, Providence, 2013,pp. 157–173.

58


Capillary and free boundary surfaces obtained bydeformation of minimal surfaces

Filippo MorabitoKorea Advanced Institute Science and Technology, South Korea, [email protected]

I will present the construction of new examples of free boundary/capillary surfaces embeddedin convex sets of R3. In [1] I showed the existence of capillary surfaces in a unit ball B of R3

of arbitrary positive genus and which meet the boundary ∂B along three closed curves. Thesesurfaces are a deformation of the Costa-Hoffman-Meeks minimal surfaces. In [2] I constructedfree boundary surfaces in a solid cylinder C of R3 by deforming Scherk / Saddle tower minimalsurfaces. These surfaces meet ∂C along 2l curves of infinite length, l ≥ 2.

References

[1] Filippo Morabito. Higher genus capillary surfaces in a unit ball of R3, Boundary Value Prob-lems, art. n. 130, 2014.

[2] Filippo Morabito. Singly periodic free boundary surfaces in a solid cylinder of R3, DiscreteContinuous Dynamical Systems, 35(10) October 2015, to appear.

59


On the monotonicity of energy in the blow-up limit at thetriple junction of three fluids in equilibrium

Ray TreinenTexas State University, USA, [email protected]

We conduct a local analysis of the triple junction of three fluids. Energy minimizers have beenshown to exist in the class of functions of bounded variations, and the classical theory impliesthat an interface between two fluids is an analytic surface. We prove a monotonicity formula atthe triple junction for the three-fluid configuration, and show that there exist blow-up cones. Wediscuss some of the geometric consequences of our results. This is joint work with Ivan Blank andAlan Elcrat.

60


Liquid Bridges between Contacting SpheresThomas I Vogel

Texas A&M University, College Station, TX, [email protected]

The problem studied is that of a rotationally symmetric liquid bridge between two contactingballs of equal radius, with the same contact angle with both balls, and in the absence of gravity.The bridge surface must be of constant mean curvature, hence a Delaunay surface. If the contactangle is less than π

2 , existence of a rotationally symmetric bridge is shown for a large range ofthe relevant parameter, giving unduloidal, catenoidal, and nodoidal bridges. If the contact angleis greater than or equal to π

2 , it is shown that no stable rotationally symmetric bridge which issymmetric across the perpendicular bisector of the line segment between the two centers of theballs exists. Existence therefore depends discontinuously on contact angle.

61


Special Session

Asymptotic and Numerical Methods for Viscous andElastic Media

Organizer

Grigory PanasenkoInstitute Camille Jordan UMR CNRS 5208, PRES University of Lyon/University of Saint

Etienne, France and Laboratory J.V. Poncelet UMI CNRS 2615 Moscow, [email protected]

62


On the Existence of Singular Solutions to the StokesProblem in the Power Cusp Domains

Alicija Eismontaite∗, Konstantin PileckasFaculty of Mathematics and Informatics, Vilnius University, Lithuania,

∗[email protected]

We consider the boundary value problem for the time-periodic Stokes problem

∂u∂t − ν∆u +∇p = f , x ∈ Ω,

divu = 0, x ∈ Ω,

u|∂Ω = a, u(x, 0) = u(x, 2π),

in a bounded domain Ω = ΩH ∪ Ω0, where ΩH is a power cusp domain

ΩH =x ∈ Rn : |x′| < ϕ(xn), xn ∈ (0, T )

,

ϕ(xn) = γ0xλn, γ0 = const, λ > 1, x′ = (x1, ..., xn−1), n = 2, 3, suppa ∈ ∂Ω0 ∩ ∂Ω and ∂Ω0 is

Lipschitz. We look for the solutions satisfying the additional nonzero flux condition∫

σ(t)

u · n ds = F (t)

whereσ(t) = x ∈ Ω : xn = t = const

is the cross section of ΩH by the plane xn = h.The complete asymptotic expansion near the singularity point is constructed and justified. The

solution to the time-periodic Stokes problem is then constructed as a sum of asymptotic and other,better-behaved, terms.

63


Error estimates in shell theoryCristinel Mardare

University Pierre et Marie Curie, France, [email protected]

The deformation of a thin elastic shell in response to applied forces is described either by theequations of three-dimensional elasticity, or by a (two-dimensional) shell model. While the equa-tions of three-dimensional elasticity are justified by physical laws, a shell model has to be justifiedby an asymptotic analysis of the equations of three-dimensional elasticity when the thickness ofthe shell goes to zero.

The particularity of the asymptotic analysis in shell theory is that the equations of the three-dimensional elasticity depend not only on the small parameter (the thickness of the shell) that goesto zero, but also on the geometry of the middle surface of the shell. For this reason, these equationsmay not possess a limit when the thickness of the shell goes to zero in the general case (however,they do possess a limit if specific assumptions are made on the geometry of the middle surface, onthe boundary conditions, and on the applied forces).

To alleviate this inconvenient, several two-dimensional shell models depending on the thick-ness of the shell have been proposed in the literature. Among them, of particular interest are theshell models of Koiter and of Naghdi, since they are in some sense the simplest ones that are validfor a general shell.

We will show that this is indeed the case, by showing that the difference between the solutionof the shell model of Koiter, or of Naghdi, and the solution to the equations of three-dimensionalelasticity goes to zero in an appropriate norm when the thickness of the shell goes to zero.

References

[1] Lods V., Mardare C. Asymptotic justification of the Kirchhoff-Love assumptions for a linearlyelastic clamped shell, J. Elasticity, 58(2):105–154, 2000.

[2] Lods V., Mardare C. Error estimates between the linearized three-dimensional shell equationsand Naghdi’s model, Asymptot. Anal., 28(1):1–30, 2001.

64


Homogenization of elliptic systemswith periodic coefficients:

twoparametric operator error estimatesYulia Meshkova1,2,∗, Tatiana Suslina2

1 Chebyshev Laboratory, St. Petersburg State University, Russia,∗[email protected]

2 Department of Physics, St. Petersburg State University, Russia

Let Γ be a lattice in Rd and let Ω be the cell of Γ. InL2(Rd;Cn), we consider matrix selfadjointsecond order differential operators Bε, ε > 0. The coefficients of Bε are periodic with respect toΓ and depend on x/ε. So, they oscillate rapidly, as ε → 0. The principal part of Bε is given in afactorized form b(D)∗g(x/ε)b(D). It is assumed that the matrix-valued function g(x) is boundedand uniformly positive definite. Next, b(D) is a first order differential operator with constantcoefficients. The symbol b(ξ) is subject to some condition which ensures strong ellipticity ofthe operator Bε. The coefficients of the lower order terms belong to some Lp(Ω)-spaces. It isassumed that Bε > 0.

Let Q be a Γ-periodic uniformly positive definite and bounded matrix-valued function, andlet Qε be an operator of multiplication by Q(x/ε). Our goal is to study the behaviour of thegeneralized resolvent (Bε − ζQε)−1 in a point ζ ∈ C \ [0,∞), |ζ| > 1, in dependence of thesmall parameter ε and the spectral parameter ζ. It turns out that (Bε − ζQε)−1 converges in theL2-operator norm to the operator (B0 − ζQ)−1, as ε → 0. Here B0 is the so-called effectiveoperator with constant coefficients and Q = |Ω|−1

∫ΩQ(x) dx. We prove that

‖(Bε − ζQε)−1 − (B0 − ζQ)−1‖L2(Rd)→L2(Rd) 6 C(φ)ε|ζ|−1/2.

Here C(φ) depends on φ = arg ζ and on the problem data. We also obtain approximation for(Bε − ζQε)−1 in the L2 → H1-operator norm with the corrector taken into account. For a fixedζ, both L2 → L2 and L2 → H1 error estimates are of sharp order O(ε).

Now, let O ⊂ Rd be a bounded domain of class C1,1. We also study operators BD,ε actingin L2(O;Cn) and given by the same differential expression as before with the Dirichlet boundarycondition. For ζ ∈ C \ [0,∞), |ζ| > 1, and sufficiently small ε we prove that

∥∥(BD,ε − ζQε)−1 − (B0 − ζQ)−1∥∥L2(O)→L2(O)

6 C(φ)(ε|ζ|−1/2 + ε2).

Here C(φ) depends on φ = arg ζ and on the problem data. For a fixed ζ, this estimate is of sharporder O(ε). Approximation for (BD,ε − ζQε)−1 in the L2 → H1-norm was also obtained. For afixed ζ, the L2 → H1-error estimate is of order O(ε1/2). The estimate becomes worse than in Rd

because of the boundary influence.

65


Homogenization of the Stokes equation with stressboundary conditions in periodic porous media

John Fabricius1, Elena Miroshnikova1,∗, Peter Wall11 Lulea University of Technology, Sweden, ∗[email protected]

There exist several mathematical approaches for deriving Darcy’s law for an incompressibleviscous fluid flowing in a porous medium (see e.g. [1, 2, 3, 4, 5] and the references therein).

We study stationary incompressible fluid flow in a bounded periodic domain Ωε perforatedby obstacles which are placed at the distance ε from each other. The boundary ∂Ωε = Γε ∪ Sε

of the domain consists of two disjoint parts: Sε — the boundary of micro–inclusions, and Γε —the global boundary of the domain. The flow in Ωε is governed by the Stokes equation with no-slip boundary condition on the micro-scale (on Sε) and the prescribed stress vector on the globalboundary Γε.

For this situation Darcy’s law is derived in terms of solutions of the corresponding homoge-nized problem for a non-perforated domain Ω(⊃ Ωε). The convergence result is proved by usingtwo-scale convergence technique (see e.g. [1, 4]).

To illustrate our results numerically we calculate the permeability tensorKε which is obtainedby solving the local problem for the unit representative cell. The numerics give a good agreementwith the Gebart formula (see [6]) for both quadratic and hexagonal types of packing obstacles.

References

[1] Lions J.-L. Some methods in the Mathematical Analysis of Systems and their Control, SciencePress and Gordon and Breach, Beijing and New York (1981).

[2] Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory, in: Lecture Notes inPhysics, V. 129, Springer-Verlag, Berlin (1980).

[3] Tartar L. Incompressible fluid flow in a porous medium — convergence of the homogenizationprocess, in: Lecture Notes in Physics, V. 129, Springer-Verlag, Berlin, (1980).

[4] Allaire G. Homogenization of the Stokes flow in a connected porous medium, AsymptoticAnal., 2, 203–222 (1989).

[5] Hornung U. Homogenization and Porous Media, Springer-Verlag, New York (1997).

[6] Gebart B. R. Permeability of Unidirectional Reinforsements for RTM, Journal of CompositeMaterials, 26, No. 8, 1100–1133 (1989).

66


Evolutional contact with Coulomb’s friction on a periodicmicrostructure

Julia OrlikFraunhofer ITWM, Germany, [email protected]

We consider here the elasticity problem in a heterogeneous domain with an ε−periodic micro-structure, ε 1, including a multiple micro-contact in a simply connected matrix domain withinclusions completely surrounded by cracks, which do not connect the boundary or a textile-likematerial, but are locked to a matrix on a piece of the boundary. The contact is described bythe Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, thedissipative functional is state dependent, like in [1]. A time discretization scheme from [1] reducesthe contact problem to the frictional traction known from the previous step on each time-incrementand is then solved by fixed point argument. For a fixed ε, the necessary condition for the keepingthe contact continuous in time (for the contraction mapping) is given in [2] in the form of thebound on the frictional coefficient by lower and upper bounds on the elastic tensor and normsof the direct and inverce trace operators. We further look for the spatial homogenization of thecontact problems on each time-increment and introduce scaling of Sobolev-Slobodetsy norms [4](just modify them for jumps) and Bessel potentials. By shifting argument [2, 4], we obtain thepreliminary eslimates for normal tractions in a better space and proof its strong convergence. Thelimiting energy and the dissipation term in the stability condition obtained for the contact withTresca’s friction law in [3] are then valid also for the Colomb one. Using these results and theconcept of energetic solutions for evolutional quasi-variational problems from [1], for a uniformtime-step partition, the existence can be proved for the solution of the continuous problem and asubsequence of incremental solutions weakly converging to the continuous one uniformly in time.

References

[1] Mielke, A., Rossi, R. Existence and uniqueness results for general rate-independent hysteresisproblems, WIAS (2005).

[2] Eck, C., Jarusek, J., Krbec, M. Unilateral contact problems variational methods and existencetheorems, Springer (2005).

[3] Cioranescu, D., Damlamian, A. & Orlik, J. Homogenization via unfolding in periodic elastic-ity with contact on closed and open cracks, Asymptotic Analysis 82, Issue 3–4 (2013).

[4] Gahn, M., Knabner, P. & Neuss-Radu, M. Homogenization of reaction-diffusion processes ina two-component porous medium with a nonlinear flux condition at the interface, and appli-cation to metabolic processes in cells, Preprint Angew. Math., Uni Erlangen, No. 384 (2014).

67


Asymptotic analysis of the fluid flow with apressure-dependent viscosity

Igor Pazanin1,∗, Eduard Marusic-Paloka21 Department of Mathematics, Faculty of Science, University of Zagreb, Croatia,

∗[email protected] Department of Mathematics, Faculty of Science, University of Zagreb, Croatia

Our goal is to present recent results on the stationary motion of incompressible viscous fluidwith a pressure-dependent viscosity. Under general assumptions on the viscosity-pressure relation(satisfied by the Barus formula and other empiric laws), first we discuss the existence and unique-ness of the solution of the corresponding boundary value problem. The main part of the talk isdevoted to asymptotic analysis of such systems in thin domains naturally appearing in the appli-cations. We address the problems of fluid flow in pipe-like domains and also study the behaviorof a lubricant flowing through a narrow gap. In each setting we rigorously derive a new asymp-totic model describing the effective flow. The key idea is to conveniently transform the governingproblem into the Stokes system with small nonlinear perturbation.

References

[1] Marusic-Paloka E., Pazanin I. A note on the pipe flow with a pressure-dependent viscosity,J. Non-Newtonian Fluid Mech., 197, 5–10 (2013).

[2] Pazanin I. On the helical-pipe flow with a pressure-dependent viscosity,Theor. Appl. Mech. Lett., 4, 062006 (2014), 8 pages.

[3] Marusic-Paloka E., Pazanin I. Asymptotic modeling of the thin-film flow with a pressure-dependent viscosity, J. Appl. Math., 2014, Article ID 217174 (2014), 8 pages.

[4] Marusic-Paloka E. An analysis of the Stokes system with pressure dependent viscosity,Rend. Istit. Mat. Univ. Trieste, 46, 123–136 (2014).

68


On the Dirichlet type problem to degenerate at a lineelliptic systems

Stasys RutkauskasVilnius University, Lithuania, [email protected]

Let D ⊂ Rn+1, n ≥ 1, Γ := ∂D ∈ C2,α, 0 < α < 1, be a domain of points x = (x0, x′),

x′ = (x1, . . . , xn), containing the cylinder CR = (x0, x′) : |x′| < R, 0 < x0 < H, bothbases of which lie on Γ. Thus, the x′ = 0 cross the domain D and intersect with Γ by two pointsO (0, 0) and P (H, 0) ∈ Rn+1.

We consider in D the system of equations

n∑

i,j=0

Aij(x)uxixj +

n∑

i=0

Bi(x)uxi + C (x)u = F (x) , (1)

assuming that matrixes Aij = diag (a(1)ij , . . . , a

(m)ij ), Bi = diag (b

(1)i , . . . , b

(m)i ), C = (ckl) (k, l =

1,m) and right-hand side F = (f1, . . . , fm) are smooth enough in D.Let Ω be the projection of D onto the plane x′ = 0. We assume that there exist continuous in

Ω and positive for |x′| 6= 0 functions a1 and a2 such that a2 (0) = 0 and

a1(x′) |ξ|2 ≤

n∑

i,j=0

a(k)ij (x)ξiξj ≤ a2(x′) |ξ|2 , k = 1,m,

in D for each ξ = (ξ0, . . . , ξn) ∈ Rn+1. Therefore, system (1) is elliptic in D0 = D\x′ = 0 inthe sense of Petrovskii and its order degenerates at the line x′ = 0.

Let Dδ = D\x : |x′| ≤ δ < R. Introduce the class of vector-functions

C2,αloc (D0) := u : u ∈ C2,α

(Dδ

)∀δ > 0.

The following two Dirichlet type problems to system (1) are studied:

u = g on Γ0 = Γ\O ∪ P, |u| <∞ in D0; (2)

u = g on Γ0, limx′→0

(u(x0, x

′)− h(x0, x/∣∣x′∣∣))

= 0. (3)

The sufficient conditions of existence and uniqueness of the solution u ∈ C2,αloc (D0) of both

problems (2) and (3) are given.

69


ADI method for two-dimensional pseudo-parabolicequation with integral boundary conditions†

Arturas Stikonas∗, Mifodijus Sapagovas, Olga StikonieneVilnius University, Lithuania, ∗[email protected]

We consider the third order linear pseudo-parabolic equation in rectangle with nonlocal integralconditions

ut = uxx + uyy + η(uxx + uyy)t + f, (x, y) ∈ (0, Lx)× (0, Ly), η ≥ 0,

u|x=0 = γ0

∫ Lx

0u(x, y, t)dx+ vl, u|x=Lx = γ1

∫ Lx

0u(x, y, t)dx+ vr, (1)

u|y=0 = vb, u|y=Ly = vt, u|t=0 = v0.

Pseudo-parabolic equations with nonlocal boundary conditions arise from various physical phe-nomena, particulary, the dynamics of ground moisture. A very close mathematical model arises inthe study of the incompressible non-newtonian flow problem.

We provide a numerical algorithm for the approximations of problem (1) based on the Peacemen-Rachford alternating direction implicit (ADI) method [1]. According to this algorithm it is nec-essary to solve alternately two systems of difference equations with tridiagonal matrices, one ofthese systems is solved with nonlocal conditions. Each of systems is three-layer difference schemeant it approximates the initial problem with truncation error O(τ + h2).

For investigation of stability of this method we rewrite the three-layer scheme in an equivalentform of a two-layer scheme

Yn+1 = SYn,

where Yn is vector defined into two layers, S is a nonsymmetric matrix.The stability conditions are derived by investigating of the spectrum of S. For this end we

analyse the auxillary nonlinear eigenvalue problem.

References

[1] Peaceman D. W., Rachford H. H. The numerical solution of parabolic and elliptic differentialequations, J. Soc. Ind. Appl. Math., 3(1):28–41, 1955.

[2] Sapagovas M., Kairyte G., Stikoniene O., Stikonas A. Alternating direction method for atwo-dimensional parabolic equation with a nonlocal boundary condition, Math. Model. Anal.,12(1):131–142, 2007.http://dx.doi.org/10.3846/1392-6292.2007.12.131-142

†The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).

70


Contributed talks

71


Solvability of a surface reaction model taking intoaccount diffusion of both adsorbates

Algirdas Ambrazevicius∗, Vladas SkakauskasVilnius University, Lithuania, ∗[email protected]

We prove the existence and uniqueness of a classical solution to a coupled system of parabolicequations, some of them being determined on the boundary of the domain. This system describesthe model of surface reactions between carbon monoxide and nitrous oxide.

72


On Poincare’s and Lions’ lemmasand on De Rham’s theorem

Cherif Amrouche1,∗, Philippe Ciarlet2, Cristinel Mardare31 University Pau, France, ∗[email protected]

2 City University, Hong-Kong3 University Paris 6, France

We prove here the equivalence between many important properties concerning: the divergenceoperator, the Lions lemma, the Necas inequality, the Korn inequality and the weak lemma ofPoincare. Using then the Bogovskii operator and the Calderon-Zygmund theory, we give someisomorphism concerning the divergence operator. We give also a new proof of the original DeRham theorem and we obtain some extension to the irrotational fields (see [1], [2]).

References

[1] Amrouche Cherif, Ciarlet Philippe G., Mardare Cristinel. Remarks on a lemma by Jacques-Louis Lions, C. R. Math. Acad. Sci. Paris, 352(9):691–695, 2014.

[2] Amrouche Cherif, Ciarlet Philippe G., Mardare Cristinel. On a lemma of Jacques-Louis Lionsand its relation to other fundamental results, Journal de Mathematiques Pures et Appliquees,2015.

73


Analysis of asynchronous multi-time-step methods forparabolic problems

Michal BenesDepartment of Mathematics, Faculty of Civil Engineering,

Czech Technical University in Prague, Czech Republic,[email protected]

Evolution of time dependent physical quantities such as current, heat etc., in composite materi-als are modelled by initial boundary value problems for parabolic PDEs. These physical quantitiesfollow different evolution patterns in different parts of the computational domain depending on thematerial properties, size of constituent material subdomains, coupling scheme, etc. Therefore, thestability and accuracy requirements of a numerical integration scheme may necessitate domain de-pendent time discretization. Parabolic problems are usually solved by discretizing spatially usingfinite elements and then integrating over time using discrete solvers. We propose an asynchronousmulti-domain time integration scheme for parabolic problems. For efficient computing of largescale problems, we present the dual decomposition method with local Lagrange multipliers to en-sure the continuity of the primary unknowns at the interface between subdomains. The proposedmethod enables us to use domain dependent Rothe method on different parts of a computationaldomain and thus provide an efficient and robust approach to solving large scale problems.

References

[1] Benes M., Nekvinda A., Yadav M.K. Multi-time-step domain decomposition method withnon-matching grids for parabolic problems, Appl. Math. Comput, 2015, to appear.

74


Free boundary problem of magnetohydrodynamicsElena Frolova

St.Petersburg Electrotechnical University, St. Petersburg State University, Russia,[email protected]

We consider a free boundary problem governing the motion of a finite isolated mass of aviscous incompressible electrically conducting fluid in vacuum. The initial position of the freeboundary is assumed to be a small normal perturbation of a sphere. Media is moving under theaction on magnetic field and volume forces. We prove solvability of this free boundary problem inan infinite time interval under the additional smallness assumptions on initial data and the externalforces [1]. In the case of absence of the external forces this result is proved in the joint paper withV.A. Solonnikov [2]. We discuss a question of stability of the rest state with zero velocity in a ball.

References

[1] Frolova E. V. Free boundary problem of magnetohydrodynamics, Zap.nauchn.sem. POMI,425:149–178, 2014.

[2] Solonnikov V. A., Frolova E. V. Solvability of a free boundary problem of magnetohydrody-namics in an infinite time interval, Zap.nauchn.sem. POMI, 410:131–167, 2013.

75


A simplified approach to the regularising effect ofnonlinear semigroups

Daniel Hauer1,∗

In joint work with Prof. Thierry Coulhon2

1 The University of Sydney, School of Mathematics and Statistics, Australia,∗[email protected]

2 Paris Sciences et Lettres, Departement de Mathematiques et Applications, Ecole NormaleSuperieure, 62 bis rue Gay-Lussac, 75005 Paris, France

Since the beginning of the 21st century, there appeared a huge flow of papers written on theregularising effect of nonlinear semigroups. Most authors of these papers follow the same ap-proach: As a first step, a Log-Sobolev inequality is derived from a known Sobolev inequality.Then by using the Log-Sobolev inequality, one shows that the function t 7→ ‖Tt|r(t) satisfies adifferential inequality which is strong enough to conclude an Lp-Lq-regularisation of the trajecto-ries t 7→ Ttϕ of the given semigroup Tt. In this talk, we present a simplified approach to thisregularity effect.

76


Asymptotic behavior of solutions of systems describednonlinear viscoelastic flows

Natalia KarazeevaSteklov Mathematical Institute, Sankt-Petersburg Department, Russia,

[email protected]

The system of the type

d

dtvε + (vε · ∇)vε − ν∆vε +

1

2vεdiv vε − div KDε − 1

εgrad div vε = 0 (1)

is considered in the cylinder QT = Ω× [0, T ], where Ω is a bounded 2-dimensional domain. Thissystem is an ε-approximation of the system describing motion of viscoelastic fluid

d

dtv + (v · ∇)v − div KD − ν∆v + grad p = f, (2)

div v = 0.

Here v is the velocity vector, p is the pressure, D is the rate of stress tensor. The operator Kmust satisfy certain assumptions, namely it must be Lipschitzian, nonnegatively definite and mustsatisfy an upper bound. For such systems (1) the initial boundary value problem

vε(x, t)|x∈∂Ω = 0, (3)

vε(x, 0) = vε0(x), x ∈ Ω. (4)

is considered. It is shown that problem (1), (3), (4) has a unique solution. This solution forε → 0 asymptotically tends to the solution of corresponding initial boundary value problem fornonperturbed system (2).

The examples of viscoelastic flows satisfying this system are represented.

77


On Nonhomogeneous Boundary Value Problems for theStationary Navier–Stokes Equations in 2D Symmetric

Semi-Infinite Outlets IKristina Kaulakyte1,2,∗, Michel Chipot1, Konstantin Pileckas2, Wei Xue1

1 University of Zurich, Switzerland, ∗[email protected] Vilnius University, Lithuania

We study the stationary nonhomogeneous Navier–Stokes problem in a two dimensional sym-metric domain with a semi-infinite outlet (either paraboloidal or channel-like). We assume that thedomain, boundary value and external force are symmetric with respect to one of the axis. Noticethat we do not impose any restrictions on the size of the fluxes, i.e. the fluxes of the boundaryvalue over the inner and the outer boundaries may be arbitrarily large. Only the necessary com-patibility condition (the total flux is equal to zero) has to be satisfied. Under these assumptionswe construct a solenoidal symmetric extension of the boundary value satisfying the Leray–Hopfinequality. Such an extension reduces the nonhomogeneous boundary value problem to the homo-geneous one.

78


Linearized Navier-Stokes problems in the periodicdomains

Neringa KlovieneDepartment of Mathematics and Informatics, Vilnius University,

Institute of Informatics, Mathematics and E-studies, Siauliai University, [email protected]

We consider the non-stationary linearized Navier-Stokes problem:

∂∂tu− ν∆u + (U · ∇)u +∇p = f,div u = 0,u|SL

= 0, u(x, 0) = a(x),u(x′, 0, t) = u(x′, L, t),

(1)

with the prescribed flux condition∫σun(x, t)dx′ = F (t).

The flow of the incompressible, homogeneous viscosity is analyzed in n-dimensional infinite,periodic with respect to xn, pipe Π = x = (x′, xn) ∈ Rn | (x′, xn) ∈ σ × R, here σ denotesthe cross-section of the tube. The period of the pipe is denoted by L, the aria between the plotsxn = 0 and xn = L by ΠL and the lateral boundary of ΠL is denoted by SL.

In the problem (1) function U is given and satisfies the following conditions:

U ∈ L∞(0, T ;W 12 (ΠL) ∩ L∞(ΠL)), div U = 0, U|SL

= 0,

U(x′, xn, t) = U(x′, xn + L, t), ‖∇U‖L2(ΠL) → 0, t→∞.Let remark that the case U(x, t) = 0 coincides with the non-stationary Stokes problem (see

[1]) and the case U(x, t) = u(x, t) consides with Navier-Stokes.For the problem (1) we prove the existence of the unique weak solution of the Poiseuille type.Instead of the initial condition taking the time periodicity condition, i.e. u(x, t) = u(x, t+2π),

we prove the existence of the weak solution and find the relation between the flux and pressuredrop function.

References

[1] Kloviene N. The Poiseuille type solution for the non-stationary Stokes problem in the infiniteperiodic pipe, Siauliai mathematical seminar, 10(18), accepted (2015).

79


Local in time solutions of MHD system for bounded andexterior domain

Adam Kubica1,∗, Giovanni P. Galdi21 Warsaw University of Technology, Poland, ∗[email protected]

2 University of Pittsburgh, USA

Assume that Ω ⊂ RN , N = 2, 3 is bounded or exterior domain, the boundary of which isuniformly of class C3. We will consider the non-resistive MHD problem

ut −∆u+ u∇u+∇p = b∇b in ΩT

bt + u∇b = b∇u in ΩT

divu = 0 in ΩT (1)divb = 0 in ΩT

u|∂Ω = 0u|t=0 = u0, b|t=0 = b0,

Under the assumption that u0, b0 ∈ H2(Ω), divu0 = 0, divb0 = 0 and u0|∂Ω = 0 we showthat there exists c and T > 0 such that the problem (1) has unique solution (u, b) such thatu ∈ L2(0, T ;H3(Ω)), ut ∈ L2(0, T ;H1(Ω)), b ∈ L∞(0, T ;H2(Ω)) and bt ∈ L∞(0, T ;H1(Ω))and the following estimate

‖u‖L2(0,T ;H3(Ω)) + ‖ut‖L2(0,T ;H1(Ω)) + ‖b‖L∞(0,T ;H2(Ω)) + ‖bt‖L∞(0,T ;H1(Ω)) ≤ c,

holds, where T depends on ‖u0‖H2(Ω), ‖b0‖H2(Ω) and the C3-regularity of ∂Ω and c in additiondepends on T .

The reasoning is based on Schauder fixed point theorem. First, in the case of bounded domainwe examine auxiliary problem

bt + w∇b = b∇w in ΩT

divb = 0 in ΩT

b|t=0 = b0,,

where w ∈ L2(0, T ;H3(Ω)) is given and divw = 0, w · n|∂Ω = 0. Next using the solvability ofStokes problem we define the mapping which fixed point is a solution of problem (1).

In the case of exterior domain we define a sequence of bounded domains Ωn such that Ω =⋃∞n=1 Ωn and show that un solutions of (1) in Ωn converge to u, solution in the exterior domain

Ω.

80


On a very singular parabolic equationMichał Łasica

University of Warsaw, Poland, [email protected]

We consider a model very singular uniformly parabolic equation

ut = uxx +1

2(sgnux)x.

First, we collect known results concerning existence of semigroup solutions and interpretationof formal expression 1

2(sgnux)x. Our goal is however to understand regularity and qualitativebehaviour of solutions. Strong directional diffusion produced by the one-dimensional 1-Laplaceoperator 1

2(sgnux)x leads to appearance of flat facets in the graphs of solutions. This effect isreduced by smoothing due to Laplace equation. This competition leads to non-trivial behaviour ofendpoints of facets. We partially solve the problem of its description by identifying the solutionswith solutions to a certain system of Stefan-like free boundary problems.

References

[1] Łasica M. Analysis of solutions to a model parabolic equation with strongly singular diffusion,preprint, http://arxiv.org/abs/1406.1775

81


The Brezis–Nirenberg effect for Navier and Dirichletfractional Laplacians

Alexander I. NazarovSt.Petersburg Dept of Steklov Institute and St.Petersburg University, Russia,

[email protected]

Let m, s be two given real numbers, with 0 ≤ s < m < n2 . Let Ω ⊂ Rn be a bounded smooth

domain. Denote by 2∗m = 2nn−2m the critical Sobolev exponent for the embedding Wm

2 → Lq.We study equations

(−∆)mDu = λ(−∆)sDu+ |u|2∗m−2u in Ω, (1)(−∆)mNu = λ(−∆)sNu+ |u|2∗m−2u in Ω. (2)

Here fractional Laplacians (−∆)mD and (−∆)mN (Dirichlet and Navier, respectively) are self-adjoint operators defined by their quadratic forms:

QDm[u] ≡

∫

Ω

(−∆)mDu · udx :=

∫

Rn

|ξ|2m|F [u]|2dξ ,

QNm[u] ≡

∫

Ω

(−∆)mNu · udx :=∑

k∈Nλmk (u, ϕk)2,

respectively. Here F stands for the Fourier transform while λk and ϕk are eigenvalues and (nor-malized) eigenfunctions of conventional Dirichlet–Laplacian in Ω. The domains of quadraticforms satisfyDom(QD

m) = Hm(Ω) ⊂ Dom(QNm), where Hm(Ω) = u ∈Wm

2 (Rn) : suppu ⊂Ω.

Theorem. Let s ≥ 2m− n2 . Then each of problems (1) and (2) has a nontrivial weak solution

for arbitrarily small positive λ.

The case s = 0 and m integer or m ∈ (0, 1) was considered earlier in a number of papersbeginning with the celebrated paper [1] (for m = 1).

This talk is based on a joint papers with Roberta Musina, see [2], [3]. Author was supportedby RFBR grant 14-01-00534 and by St.Petersburg University grant 6.38.670.2013.

References

[1] Brezis H., Nirenberg L., Positive solutions of nonlinear elliptic equations involving criticalSobolev exponents, Comm. Pure Appl. Math. 36(4):437–477, 1983.

[2] Musina R., Nazarov A. I., Non-critical dimensions for critical problems involving fractionalLaplacians, To appear in Revista Matematica Iberoamericana.

[3] Musina R., Nazarov A. I., On fractional Laplacians–3. Preprint available athttp://arxiv.org/abs/1503.00271. 13p.

82


Parabolic problems in oscillating thin domainsMarcone Pereira

Universidade de Sao Paulo, Brazil, [email protected]

In this talk we discuss some recent results obtained in [1, 2] about the asymptotic behavior ofthe solutions of semilinear parabolic problems with homogeneous Neumann boundary conditionsposed on two dimensional thin domains with locally periodic structure on the boundary. We firstobtain the limit problem assuming the thin domain degenerates to the unit interval also analyzingits dependence with respect to the geometry of the thin channel. Next we study the convergence ofthe nonlinear semigroup investigating the upper and lower semicontinuity of the family of globalattractors taking dissipative assumptions to the system.

References

[1] Pereira M. C. Parabolic problems in highly oscillating thin domains, Ann. Mat. Pura Appl.,http://dx.doi.org/10.1007/s10231-014-0421-7, 2014.

[2] Arrieta J. M., Carvalho, A. N., Pereira, M. C., Silva R. P. Semilinear parabolic problems in thindomains with a highly oscillatory boundary, Nonlinear Analysis, 74(15):5111–5132, 2011.

83


Interaction of particles through a singular potentialJan Peszek

University of Warsaw, Poland, [email protected]

Mathematical description of the collective dynamics of interacting particles finds many appli-cations ranging from the modeling of flocks of birds, optimal control over sensor networks to someseemingly unrelated subjects as the modeling of a consensus in societies. One of such models isCucker-Smale’s (C-S) flocking model associated with the Vlasov-type equation

∂tf + v · ∇f + divv(F (f)f) = 0,

F (f)(x, v, t) :=

∫

R2d

ψ(|x− y|)(w − v)f(y, w, t)dydw,

where ψ is a given function called the communication weight. In the case of regular ψ, C-S modelwas thoroughly studied and it’s properties are well known (see e.g. [3] or [4]). However, with asingular ψ of the form ψ(s) = s−α for α > 0, it becomes much more difficult. Depending on αthe trajectories of the particles may exhibit tendencies to behave in various ways: stick together,collide or avoid each other altogether (see [4, 1, 2, 5]). Though much effort was put into the studyof dynamics of C-S model with a singular weight, yet the existence of the solutions is still notknown in most cases. In my talk I will present the proof of existence of solutions to the C-S modelprovided that 0 < α < 1

2 (see [6]). I will also show some examples of it’s dynamics.

References

[1] Ahn S. M. and Choi H. and Ha S.-Y. and Lee H. On collision-avoiding initial configurationsto Cucker-Smale type flocking models, Commun. Math. Sci., 10(2): 625–643, 2012.

[2] Carrillo J., A. and Choi Y.-P. and Hauray M. Local well-posedness of the generalized Cucker-Smale model, preprint, arXiv:1406.1792, 2014.

[3] Ha S.-Y. and Liu, J.-G. A simple proof of the Cucker-Smale flocking dynamics and mean-fieldlimit, Commun. Math. Sci., 7(2):297–325, 2009.

[4] Ha S.-Y. and Tadmor, E. From particle to kinetic and hydrodynamic descriptions of flocking,Kinet. Relat. Models, 1(3):415–435, 2008.

[5] Peszek J. Existence of piecewise weak solutions of a discrete Cucker–Smale’s flocking modelwith a singular communication weight, J. Differential Equations, 257(8):2900–2925, 2014.

[6] Peszek J. Discrete Cucker-Smale’s flocking model with a weakly singular weight, submitted,arXiv:1412.6458v1, 2014.

84


A remark on the orthogonal decompositionof the Hilbert space W 1,2

Reimund RautmannUniversity of Paderborn, Germany, ∗[email protected]

The well known results on the generalized Stokes boundary value problem in domains Ω ⊂Rn, n ≥ 2, imply the decomposition of the homogeneous space W 1,2(Ω) with respect to theDirichlet form 〈5u,5v〉. Considering the transition from slip- to no-slip boundary condition inLipschitz-bounded domains Ω, from this decomposition we find lower and upper bounds

(a) to the change of the Dirichlet seminorm ‖ 5u ‖ in Ω ⊂ Rn, n ≥ 2, and

(b) to the change of the seminorm ‖ curlu ‖ in Ω ⊂ R3.

The results apply to the diffusion steps in transport-diffusion splitting schemes.

85


The Stokes and Navier-Stokes equations withpressure-velocity boundary conditions in Lp spaces

Hind Al Baba1, Cherif Amrouche2, Nour Seloula3,∗1,2 Universite de Pau, France

3 Universite de Caen, France, ∗[email protected]

In a three dimensional bounded domain, eventually multiply connected, we consider theStokes and Navier-Stokes problems under boundary conditions involving the normal componentof the velocity and the pressure. In a first part, we consider the stationary problem. We provethe solvability in Lp spaces for 1 < p < ∞ where the main ingredients for this solvability aregiven by the Inf-Sup conditions, some Sobolev’s inequalities for vector fields and the theory ofvector potentials (see [2] and [3]). In a second part, we will consider the nonstationary case for theStokes equations. We first prove that the Stokes operator generates a bounded analytic semigroupand then show the existence of weak and strong solutions (see [1]).

References

[1] Al Baba H., Amrouche C., Seloula N. Time dependent Stokes problem with normal and pres-sure boundary conditions on Lp spaces, to appear in Analysis (2015).

[2] Amroyuche C., Seloula N. Lp-theory for vector potentials and Sobolev’s inequalities for vec-tor fields: application to the Stokes equations with pressure boundary conditions, Math. Mod-els. Methods Appl. Sci.,23(1):37–92, 2013.

[3] Amroyuche C., Seloula N. Lp-theory for the Navier-Stokes equations with pressure boundaryconditions, Discrete Contin. Dyn. Syst. Ser. S, 6(5):1113–1137, 2013.

86


Poincare Constants in 2D-annuliGudrun Thaeter1,∗, Bernd Rummler2, Michael Ruzicka3

1 Karlsruhe Institute of Technology, Germany, ∗[email protected] Otto-von-Guericke-Universitat Magdeburg, Germany

3 Universitat Freiburg, Germany

In the study of such divers objects as thermal energy storage systems, aircraft cabin insulation,cooling of electronic components, electrical power cable, and thin films one has to understand thefluid flow between two horizontal coaxial cylinders. The usual model is Natural convection, i.e. thefluid moves (i) under the influence of gravity and (ii) because of temperature differences causingchanges in density [1]. As is to be expected, the Poincare constant turns out to be essential for allresults obtained referring to the stability behaviour. For that we are interested in optimal analyticalbounds for this constant. With Ri and Ro beeing inner and outer radius of our 2D-configurationthe geometry is characterized by the nondimensional paramter

A :=2Ri

Ro −Ri. (1)

We investigate and improve estimates of the Poincare constants for scalar functions and solenoidalvector functions with vanishing Dirichlet traces on the boundary by the use of the first eigenvaluesof the scalar Laplacian and the Stokes operator and can compare some of it to results in [2].Especially the limiting cases when A → ∞ and A → 0, respectively, are interesting to get thewhole picture.

References

[1] Ferrario C., Passerini A., Ruzicka M., Thater G. Theoretical results on steady convective flowsbetween horizontal coaxial cylinders, SIAM Journal on Applied Mathematics, 71(2):465–486,2010.

[2] Lee D., S., Rummler B. The Eigenfunctions of the Stokes Operator in Special Domains III,ZAMM, 82:399–407, 2002.

87


Asymptotics for Ventssel’s problems in fractal domainsPaola Vernole

Dept. of Mathematics Sapienza University of Rome, Italy, [email protected]

In this talk we study a boundary value problem (P ) for a second order operator in diver-gence form L with Venttsel’s boundary conditions in a three dimensional fractal domain Q. Fromthe point of view of numerical analysis it is also crucial to study the corresponding approximat-ing (prefractal) problems (Ph) in the domains Qh, where Qhh∈N is a sequence of increasing(invading) domains approximating Q. To this aim the asymptotic behavior, as h → ∞, of theapproximating solutions is studied.

Venttsel problems are also known as boundary value problems with dynamical boundary con-ditions, since the time derivative appears in the boundary condition as well as a second orderoperator. The presence of the second order operator in the boundary condition has an importantconsequence. The fractal set has both a static and a dynamical role, that is on one side it is theboundary of an Euclidean domain and on the other side it supports the notion of a Laplacian, fromthe point of view of PDEs this fact has a counterpart, since the associated energy functional is thesum of the bulk energy and of the boundary (fractal) energy. By a semigroup approach we provethe existence and uniqueness of a strict solution of the abstract Cauchy problems (P ) and (Ph)respectively. The asymptotic behaviour of the pre-fractal solution is studied via the Mosco-Kuwae-Shyoia convergence of the energies which in turn implies the convergence of the semigroups [2].We also give the corresponding strong interpretations by proving that the solutions of abstractCauchy problems solve suitable parabolic partial differential problems with dynamical boundaryconditions. As to the asymptotic behavior of the solutions, the presence of the time derivative inthe boundary conditions requires, as a natural functional setting for these problems, suitable vary-ing Hilbert spaces. So we use the Mosco convergence adapted to this setting, studied by Kuwaeand Shioya. When studying the M-K-S convergence in our approach, a crucial role is played bythe existence of a core of smooth functions dense in the domain V (Q,S) of the energy form. Inthe two-dimensional case one can prove a complete characterization of the energy space on thefractal curve in terms of ”fractal” Lipschitz spaces, which in turn are subsets of Holder continuousfunctions on the fractal set [3]. In the three-dimensional case, as far as we know, this character-ization does not hold. Therefore it is of the utmost importance to approximate the functions inthe energy form domains by ”smooth” functions. We have proved in a joint paper with Lanciaand Regis Durante,(one of our PHD students), [1] density results for the energy spaces V (Q,S).These results allow us to prove the Mosco-Kuwae-Shioya convergence of the energy forms E(h),which in turn implies the convergence of the associated semigroups.

References

[1] Lancia, M.R, Regis Durante V., Vernole P. Density results for energy spaces on fractafold, toappear on Z. Anal. Anwendungen, 2015.

[2] Lancia, M.R, Regis Durante V., Vernole P. Asymptotic behaviour of Venttsel problems in ir-regular domains, PhD. Thesis 2015.

[3] Lancia, M.R. and Vernole P. Venttsel’s problems in fractal domains, Jour. of Evol. Eq., 14:681-714, 2014. http://dx.doi.org/10.1007/s00028-014-0233-7

88


On Nonhomogeneous Boundary Value Problems for theStationary Navier–Stokes Equations in 2D Symmetric

Semi-Infinite Outlets IIWei Xue1,∗, Michel Chipot1, Kristina, Kaulakyte1,2, Konstantin Pileckas2

1 University of Zurich, Switzerland, ∗[email protected] Vilnius University, Lithuania

We study the stationary nonhomogeneous Navier–Stokes problem in a two dimensional sym-metric domain with a semi-infinite outlet (either paraboloidal or channel-like). Under the symme-try assumptions on the domain, boundary value and external force we prove the existence of atleast one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxesof the boundary value over the inner and the outer boundaries may be arbitrarily large. After theconstruction of a suitable extension of the boundary value we show the existence of at least oneweak solution. Notice that the Dirichlet integral of the solution can be finite or infinite dependingon the geometry of the domain.

89


Linearized Non-Stationary Navier-StokesProblem in an Infinite Periodic Pipe

Stephanie Zube1,∗, Michel Chipot1, Neringa Kloviene2, Konstantin Pileckas21 University of Zurich, Switzerland, ∗[email protected]

2 Vilnius University, Lithuania

One of the fundamental practical Navier-Stokes motions in infinite domains is the so-calledPoiseuille type flow, when the pressure field is characterized by an axial gradient q(t). ThePoiseuille flow in an infinite cylinder was considered in several papers, e.g. [2].

We study a linearized non-stationary incompressible Navier-Stokes problem

∂tu− ν∆u + (U · ∇)u +∇p = f

with prescribed flux in a two or three dimensional L-periodic, with respect to the xn-axis, pipe.We look for the pressure p(x, t) having the following form

p(x, t) = −q(t)xn + p0(t) + p(x, t),

where p0(t) is an arbitrary function, p(x, t) is a L-periodic function and q(t) is associated to theflux condition. We will also focus on the asymptotic behavior in an infinite periodic pipe.

References

[1] Chipot C., Mardare S. Asymptotic behaviour of the Stokes problem in cylinders becomingunbounded in one direction, J. Math. Pures Appl., 90(2):133–159, 2008.

[2] Galdi G. P., Pileckas K., Silvestre A. On the unsteady Poiseuille flow in a pipe, Zeitschrift furangew. Mathematik und Physik, 58(6):994–1007, 2007.

[3] Ladyzhenskaya O. A. The Mathematical theory of viscous incompressible flow, Gordon andBreach, New York-London-Paris, 1969.

[4] Temam R. Navier-Stokes Equations Theory and Numerical Analysis, North-Holland Publish-ing Company, Amsterdam-New York-Oxford, 1977.

90


Participants

Amann Herbert [email protected] Algirdas [email protected] Cherif [email protected] Yasunori [email protected]

Benes Michal [email protected] Eugene [email protected] Farid [email protected] Jan [email protected]

Caldiroli Paolo [email protected] Tongkeun [email protected] Hanzhe [email protected] Michel [email protected] Tomasz [email protected] Paul [email protected]

Danchin Raphael [email protected] Paul [email protected] Yihong [email protected]

Eismontaite Alicija [email protected]

Farwig Reinhard [email protected] Marek [email protected] Robert [email protected] Juraj [email protected] Elena [email protected]

Galdi Giovanni [email protected] Mi-Ho [email protected] Yoshikazu [email protected] Patrick [email protected] Piotr [email protected]

Hajduk Karol [email protected] Francois [email protected] Daniel [email protected] Toshiaki [email protected]

91

Karazeeva Natalia [email protected] Kristina [email protected] Mads [email protected] Neringa [email protected] Miyuki [email protected] Michai [email protected] Adam [email protected]

Lancaster Kirk [email protected] Johannes [email protected]Łasica Michal [email protected] Yuxiang [email protected] Xing [email protected] Rafael [email protected]

Mardare Cristinel [email protected] Hiroshi [email protected] John [email protected] Yulia [email protected] Elena [email protected] Hideyuki [email protected] Filippo [email protected] Yoshihisa [email protected] Boris [email protected]

Nazarov Alexander [email protected] Sarka [email protected]

Okabe Takahiro [email protected] Julia [email protected]

Panasenko Grigory [email protected] Igor [email protected] Marcone [email protected] Jan [email protected] Konstantin [email protected] Maria Assunta [email protected]

Quiros Fernando [email protected] Pavol [email protected]

Rautmann Reimund [email protected] Wolfgang [email protected]ławowicz Joanna [email protected] Ludmila [email protected] Stasys [email protected]

Sauer Jonas [email protected] Nour [email protected] Henrik [email protected] Vladas [email protected] Mindaugas [email protected] Jurgen [email protected] Vsevolod [email protected] Maria [email protected] Agnieszka [email protected] Artras [email protected] Olga [email protected]

Temam Roger [email protected] Gudrun [email protected] Ray [email protected]

Varnhorn Werner [email protected] Paola [email protected] Thomas [email protected]

Wang Xiaoliu [email protected] David [email protected] Michael [email protected] Chang-Hong [email protected]

Xue Wei [email protected]

Yamada Yoshio [email protected]

Zajczkowski Wojciech [email protected] Stephanie [email protected]

Index of Authors

Al Baba, Hind, 86Amann, Herbert, 3Ambrazevicius, Algirdas, 72Amrouche, Chérif, 73, 86Aoki, Yasunori, 50

Beneš, Michal, 74Benilov, Eugene, 51Bhatnagar, Rajat, 54Bozorgnia, Farid, 15Burczak, Jan, 24

Caldiroli, Paolo, 52Canic, Suncica, 33Chang, Tongkeun, 25Chen, Hanzhe, 53Chipot, Michel, 78, 89, 90Ciarlet, Philippe, 73Cieslak, Tomasz, 43Coulhon, Thierry, 76

Danchin, Raphael, 26De Sterck, Hans, 50Deuring, Paul, 27Du, Yihong, 3

Eismontaite, Alicija, 63

Földes, Juraj, 44Fabricius, John, 66Farwig, Reinhard, 28Feireisl, Eduard, 34Fila, Marek, 4Finn, Robert, 54Frolova, Elena, 75

Galdi, Giovanni P., 5, 80Galdi, Paolo G., 27Giga, Yoshikazu, 6Guidotti, Patrick, 7Gwiazda, Piotr, 29

Hamel, François, 16Hauer, Daniel, 76Hishida, Toshiaki, 30

Jin, Bum Ja, 25

Karazeeva, Natalia, 77Kaulakyte, Kristina, 78, 89Kloviene, Neringa, 79, 90

Koiso, Miyuki, 55Korobkov, Mikhail, 8Kubica, Adam, 80Kyed, Mads, 31

Lancaster, Kirk, 56Łasica, Michał, 81Lankeit, Johannes, 45Li, Yuxiang, 46López, Rafael, 57

Mardare, Cristinel, 64, 73Marušic-Paloka, Eduard, 68Matsuzawa, Hiroshi, 17McCuan, John, 58Meshkova, Yulia, 65Miroshnikova, Elena, 66Miura, Hideyuki, 32Morabito, Filippo, 59Morita, Yoshihisa, 18Muha, Boris, 33

Nazarov, Alexander I., 82Necasová, Šárka, 34

Okabe, Takahiro, 35Orlik, Julia, 67

Pažanin, Igor, 68Panasenko, Grigory, 9Pereira, Marcone, 83Peszek, Jan, 84Pileckas, Konstantin, 9, 38, 63, 78, 89, 90Polácik, Peter, 44

Quirós, Fernando, 19Quittner, Pavol, 10

Ružicka, Michael, 87Rautmann, Reimund, 85Reichel, Wolfgang, 11Rencławowicz, Joanna, 36Rummler, Bernd, 87Rutkauskas, Stasys, 69

Sapagovas, Mifodijus, 70Sauer, Jonas, 37Seloula, Nour, 86Shahgholian, Henrik, 12Shibata, Yoshihiro, 24Siegel, David, 50, 53

Skakauskas, Vladas, 72Socolowsky, Jürgen, 20Specovius-Neugebauer, Maria, 38Štikonas, Arturas, 70Štikoniene, Olga, 70Sun, Yongzhong, 34Suslina, Tatiana, 65Swierczewska-Gwiazda, Agnieszka, 39

Temam, Roger, 13Thaeter, Gudrun, 87Treinen, Ray, 60

Vernole, Paola, 88Vogel, Thomas I, 61

Wall, Peter, 66Wang, Xiaoliu, 47Wegmann, David, 40Winkler, Michael, 48Wu, Chang-Hong, 21Wu, Manxi, 58

Xue, Wei, 78, 89

Yamada, Yoshio, 22

Zajaczkowski, Wojciech, 24, 41Zhang, Qingshan, 46Zube, Stephanie, 90

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