286 9th AIMS CONFERENCE – ABSTRACTS

Special Session 77: The Navier-Stokes Equations and Related Problems

Sarka Necasova, Mathematical Institute, Academy of Sciences, Prague, Czech RepublicReimund Rautmann, Paderborn University, Germany

Werner Varnhorn, Institute of Mathematics - Kassel University, Germany

The Navier-Stokes equations represent the fundamental equations in mathematical fluid dynamics. Recentlyresearchers from many countries around the world have developed important new results to the theory ofthese important equations and their applications: Existence proofs for weak or even strong solutions for time-dependent as well as for stationary boundary value problems of the fully nonlinear Navier-Stokes equationsor for the linearized Stokes equations, considered with a wide variety of boundary conditions, uniquenessclasses in the frame of suitably adapted abstract spaces, questions of asymptotic behavior and stability ofsolutions, and of maximum regularity, and convergence results with vanishing viscosity. A further importantpart of research is the interaction of fluid flow with moving bodies or particles or with additional heat flow,leading to enlarged dynamical systems which combine the Navier-Stokes equations with other equationsof evolution. The strong progress in theory opens the way to e�cient numerical schemes for problems intechnology and medicine. The aim of our special session will be to bring together leading researchers fromall parts of the world and from di↵erent working directions as mentioned above, and to initiate exchange ofideas as well as future cooperations.

The L1-Stokes semigroup in exterior domains

Ken AbeUniversity of Tokyo, Japankabe@ms.u-tokyo.ac.jpYoshikazu Giga

Analyticity of semigroups is one of fundamental prop-erties for semigroups associated with parabolic equa-tions. In this talk we study analyticity of the Stokessemigroup in spaces of bounded functions both inbounded and unbounded domains. In particular, weconsider the Stokes system in an exterior domain asa typical example of an unbounded domain. Eventhe existence of solution is nontrivial in that case.For the proof of analyticity of the Stokes semigroup,we appeal to a priori L1-estimates which is derivedfrom a blow-up argument.

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Lp-theory for Stokes and Navier-Stokes equa-tions with non-standard boundary conditions

Cherif AmroucheUniversity of Pau, Francecherif.amrouche@univ-pau.frNour El Houda Seloula

We consider here elliptical systems as Stokes andNavier-Stokes problems in a bounded domain, even-tually multiply connected, whose boundary consistsof multi-connected components. We investigate thesolvability in Lp theory, with 1 < p < 1, under thenon standard boundary conditions

u · n = g, curlu⇥ n = h or

u⇥ n = g , ⇡ = ⇡⇤ on�.

The main ingredients for this solvability are given bythe Inf-Sup conditions, some Sobolev’s inequalities

for vector fields and the theory of vector potentialssatisfying

· n = 0, or ⇥ n = 0 on�.

Those inequalities play a fundamental key and areobtained thanks to Calderon-Zygmund inequalitiesand integral representations. In the study of ellpiti-cal problems, we consider both generalized solutionsand strong solutions that very weak solutions.

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Boundary regularity for the steady stokestype flow with shear thickening viscosity

Hyeong-ohk BaeAjou University, Koreahobae@ajou.ac.krJorg Wolf

We work on the boundary regularity for weak so-lutions to the stationary Stokes type equations withshear dependent viscosity. Using a weighted estimatenear the boundary, we obtain the Holder continuityof the solution for the shear thickening fluid withoutthe convection term.

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An obstacle problem for capillary surfaces

Josef BemelmansRWTH Aachen University, Germanybemelmans@instmath.rwth-aachen.deG.P. Galdi, M. Kyed

Let G := ⌦ ⇥ R+, ⌦ ✓ R2 a bounded domain,be a cylinder that is partly filled with liquid; B isa rigid body that is floating on it, and the interfacebetween the fluid and the air above is described as agraph of a real function u.

SPECIAL SESSION 77 287

If we assume that the shape of the interface isgoverned by surface tension then the unknowns ofthe problems, which are the function u and the po-sition of B, are determined by a variational problemfor the energy E of the configuration.

E consists of the interfacial energy which is pro-portional to the area of the graph of u, the adhesionenergy, which is proportional to the wetted part ofthe boundary of B and of the cylindrical boundary@⌦ ⇥ R+ as well as the gravitational energies of thefluid and of the floating body.

Because of the presence of B the capillary sur-face ⌃ is bounded by some curve on @⌦⇥R+ as wellas a contact line � on @B; therefore in our formula-tion of the variational problem the body B acts as anobstacle for u.

We show the existence of a minimizer and in-vestigate some of its properties, in particular theregularity. This is based on the first variation ofthe energy which also gives a variant of Archimedes’principle that includes the forces exerted on B by ⌃.

References

[1] J. Bemelmans, G. P. Galdi, M. Kyed: Fluid Flowsaround Floating Bodies, I: The Hydrostatic Case,preprint, RWTH Aachen University, to appear inJournal of Mathematical Fluid Mechanics, 2012

[2] J. Bemelmans, G. P. Galdi, M. Kyed: Capil-lary Surfaces and Floating Bodies, preprint, RWTHAachen University, 2012

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Maximum modulus estimate in nonstationaryStokes system

Hi Jun ChoeYonsei University, Koreachoe@yonsei.ac.krTongKeun Chang

A maximum modulus estimate for the nonstation-ary Stokes system in smooth domain is found. Thesingular part and regular part of Poisson kernel areanalyzed. The singular part is a gradient potentialrelated to only normal component of the bound-ary data. Furthermore, the normal velocity nearthe boundary is bounded if the boundary data isbounded. If the normal component of the boundarydata is Dini-continuous and the tangential compo-nent of the boundary data is bounded, then themaximum modulus of velocity is bounded in wholedomain

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Some properties of solutions to liquid crystalsystems

Mimi DaiUniversity of California, Santa Cruz, USAmdai@ucsc.eduJie Qiang, Maria Schonbek

In this talk, I will present some work on LiquidCrystal systems. The existence of global weak so-lutions and regularity of solutions to the nematicliquid crystals with non-constant fluid density wereestablished. We also established a long time behaviorresult for the regular solutions to the liquid crystalsystem with constant density, providing small initialcondition.

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Pointwise decay of incompressible flowsaround rigid bodies

Paul DeuringUniversite du Littoral, FrancePaul.Deuring@lmpa.univ-littoral.fr

We consider pointwise spatial decay of flows aroundrigid bodies moving steadily in an incompressibleviscous fluid.

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Well-posedness of the Euler equations in pla-nar convex domains

Francesco Di PlinioIndiana University, USAfradipli@indiana.eduMadalina Petcu, Roger Temam

Let ⌦ ⇢ R2 be a bounded convex domain. Weconsider the well-posedness of the Euler equations

8<

:

@tu(x, t) + (u ·r)u(x, t) +rp(x, t) = f(x, t),x 2 ⌦, t 2 (0, T )

r · u(x, t) = 0, x 2 ⌦, t 2 (0, T )

with impermeability boundary conditions. Fordivergence-free initial data u0 = u(t = 0) 2 H1(⌦),we show the existence of a weak solution

u = u(x, t) 2 L1((0, T ), H1(⌦)2)

\W 1,2((0, T );Lp(⌦)2)

to the Euler equations. Under the additional assump-tions that ⌦ is a polygonal-type domain, and thatcurlu0 2 L1(⌦), the obtained solution u is unique,and possesses the additional regularity

Du 2 L1((0, T ),ExpL1(⌦)).

Even for smooth domains, these results improve onthe classical theorem of Kato; they also extend to

288 9th AIMS CONFERENCE – ABSTRACTS

polygonal-type domains the uniqueness theorem ofYudovich, thus partly improving results by Taylor,and Gerard-Varet and Lacave.

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On the steady equations for compressible ra-diative gas

Ondrej KremlInstitute of Mathematics, Academy of Sciences ofthe Czech Rep., Czech Repkreml@math.cas.czSarka Necasova, Milan Pokorny

We study the equations describing the steady flow ofa compressible radiative gas with newtonian rheol-ogy. Under suitable assumptions on the data whichinclude the physically relevant situations (i.e. thepressure law for monoatomic gas, the heat conduc-tivity growing with square root of the temperature)we show the existence of a variational entropy so-lution to the corresponding system of partial di↵er-ential equations. Under additional restrictions wealso show the existence of a weak solution to thisproblem.

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Weak solutions for the motion of a self-propelled deformable structure in a viscousincompressible fluid

Sarka NecasovaMathematical Institute, Czech Repmatus@math.cas.czM. Tucsnak, T. Takahashi

We deal with a model of swimming where we con-sider that the muscles of the creature are enoughto perform a given deformation. Using this modelwe investigate the self-propelled motion in a viscousfluid. We study the existence of weak solution. Theproof is based on a penalization method.

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On a stationary compressible flow with slip-inflow boundary conditions

Tomasz PiaseckiUniversity of Warsaw, Polandtpiasecki@mimuw.edu.pl

In my talk I am going to discuss the issue of ex-istence of stationary solutions to the Navier-Stokessystem describing the flow of a compressible fluid ina cylindrical domain. On the boundary we prescribeinhomogeneous slip conditions on the velocity. I willconsider both the barotropic flow and the completesystem with thermal e↵ects. In both cases we showthe existence of a solution in a vicinity of a given

special solution such as a constant flow with nonzerovelocity or a Poiseuille-like profile. The main prob-lem to face in the proof is the lack of compactess inthe continuity equation, I will discuss di↵erent waysto overcome this problem, such as elliptic regular-ization, successive approximations or reformulationof the problem in a kind of Lagrangian coordinates.The results have been obtained together with PiotrB. Mucha and Milan Pokorny.

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Hopf-Galerkin approach to vorticity transport& di↵usion.

Reimund RautmannUniversity of Paderborn, Germanyrautmann@uni-paderborn.de

To the initial boundary value problem of vortic-ity transport & di↵usion in a smoothly bounded3-dimensional domain we present a convergent Hopf-Galerkin scheme including error estimates locally intime.

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Maximal regularity on cross-sections impliesmaximal regularity on a cylinder

Juergen SaalCenter of Smart Interfaces, TU Darmstadt, Germanysaal@csi.tu-darmstadt.deTobias Nau

The proof of maximal regularity for general bound-ary value problems on domains with non-compactboundaries relies typically on intricate localizationprocedures. In the talk presented, I will show thatfor the Lp-approach such procedures can be avoided,if the domain (and the corresponding di↵erential op-erator) is cylinder like, i.e., of the form ⌦ = ⌦1 ⇥⌦2

and maximal regularity is known on ⌦1 ⇢ Rk and⌦2 ⇢ Rm. For domains of this type there is a muchmore elegant approach based on operator-valuedfunctional-calculus techniques. The approach hasalso some advantages in comparison to localizationprocedures, as e.g. it also applies to Lipschitz cross-sections. The methods apply to Stokes and Navier-Stokes equations as well, as will be demonstrated inthe talk, too. This is a joint project with Tobias Nauat the University of Konstanz.

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On questions of decay for solutions to liquidcrystals

Maria SchonbekUCSC, USAschonbek@ucsc.eduM. Dai, J. Qing

I will discuss the asymptotic behavior for solutions toa nematic liquid crystals system in the whole spaceof three dimensions. The fluid under considerationhas constant density and small initial data.

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The fractional Laplacian in an exterior domain

Tomas SchonbekFlorida Atlantic University, USAschonbek@fau.eduLeonardo Koslo↵

We study the fractional Laplacian in the comple-ment of an obstacle and use our result to obtainresults about solutions of some fluid dynamics equa-tions in exterior domains. The work relies heavily ona generalization of the Fourier transform due to Ikebeand Ramm, and the characterization of the fractionalLaplacian as a local operator due to Ca↵arelli andSilvestre.

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Fluid-structure interaction problems in hemo-dynamics

Adelia SequeiraTechnical University of Lisbon, Portugaladelia.sequeira@math.ist.utl.pt

Blood flow interacts mechanically with the vesselwall, giving rise to pressure waves propagating inarteries, which deform under the action of bloodpressure. In order to capture these phenomena, com-plex fluid-structure interaction (FSI) problems mustbe considered, coupling physiologically meaningfulmodels for both the blood and the vessel wall. Fromthe theoretical point of view, this is extremely di�-cult because of the high non-linearity of the problemand the low regularity of the displacement of thefluid-structure interface. So far, mathematical re-sults have been obtained only in simplified cases. Inthis talk, simulations of the mechanical interactionbetween blood flow and vessel walls will be shown,based on a partitioned approach. A 3D FSI modelin a compliant vessel is used to describe the pressurewave propagation. The 3D fluid is described throughthe Navier-Stokes equations (or a shear-thinninggeneralized Newtonian model) and the structure bya 3D hyperelastic model. In order to cope withthe spurious reflections due to the truncation of thecomputational domain, several absorbing boundary

conditions are analyzed. This work has been donein collaboration with A. Moura, J. Janela, and A.Gambaruto.

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On the motion of a fluid-rigid ball system atthe zero limit of the rigid ball radius

Ana SilvestreInstituto Superior Tecnico, PortugalAna.Silvestre@math.ist.utl.ptTakeo Takahashi

We investigate the limit of a coupled system “rigidball + Navier-Stokes liquid” when the radius of theball goes to 0. Dashti and Robinson [Arch. Ration.Mech. Anal., 2011] solved this problem in dimensiontwo, in the absence of rotation. Our aim is to im-prove their result, considering the two-dimensionalcase when the disk is also spinning and the generalthree-dimensional case.

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(Almost) everything you always wanted toknow about the Helmholtz decomposition butwere afraid to ask.

Maria Specovius-NeugebauerUniversity of Kassel, Germanyspecovi@mathematik.uni-kassel.de

The Helmholtz decomposition of Lp-spaces (and inits wake the definition of the Stokes operator) is acrucial tool in the theory of Navier-Stokes equations.This contribution collects results on this topic witha particular emphasis on why certain arguments donot work in particular situations.

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Compressible Navier-Stokes equations withslip boundary conditions in time dependentdomains

Jan StebelInstitute of Mathematics of the AS CR, Czech Repstebel@math.cas.czEduard Feireisl, Ondrej Kreml, SarkaNecasova, Jirı Neustupa

We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supple-mented with slip boundary conditions. Our approachis based on penalization of the boundary behaviour,viscosity, and the pressure in the weak formulation.Global-in-time weak solutions are obtained.

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290 9th AIMS CONFERENCE – ABSTRACTS

Extensions of Serrin’s condition for weak so-lutions of the Navier-Stokes equations

Werner VarnhornInstitute of Mathematics, Kassel University, Ger-manyvarnhorn@mathematik.uni-kassel.deReinhard Farwig, Hermann Sohr

We consider weak solutions of the non-stationaryNavier-Stokes equations in a smoothly boundedthree-dimensional domain and develop some exten-sions of Serrin’s uniqueness and regularity conditionfor this case.

�!1 ⇧1 �On a new 3D model for incompressible Eulerand Navier-Stokes equations

Shu WangBeijing University of Technology, Peoples Rep ofChinawangshu@bjut.edu.cn

In this talk, we investigate some new propertiesof the incompressible Euler and Navier-Stokes equa-tions by studying a 3D model for axisymmetric 3Dincompressible Euler and Navier-Stokes equationswith swirl. The 3D model is derived by reformu-lating the axisymmetric 3D incompressible Eulerand Navier-Stokes equations and then neglecting theconvection term of the resulting equations. Someproperties of this 3D model are reviewed. Finally,some potential features of the incompressible Eulerand Navier-Stokes equations such as the stabilizinge↵ect of the convection are presented. (The researchwas supported by National Basic Research Programof China (973 Program, 2011CB808002), the NSFC(11071009) and PHR-IHLB (200906103))

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Chemically reacting mixtures

Ewelina ZatorskaUniversity of Warsaw, Polande.zatorska@mimuw.edu.pl

Development of rigorous mathematical theory forreactive flows is of fundamental need for purposesof many practical applications. In most of them,one has to deal with multicomponent mixtures un-dergoing reactions that are, in general, completlyreversible. In this talk I will summarize the recentresults on the existence of solutions to the full Navier-Stokes system coupled with the species mass balanceequations and present the open problems.

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