Volume of Abstracts

7th Chinese-German Workshop

1

Sub-Riemannian structures on compact nilmanifolds andspectral theory of related operators

Wolfram Bauer (Göttingen, Germany)

Coauthors: K. Furutani ,C. Iwasaki

Abstract. In a natural way one can dene a sub-Riemannian structure on compactnilmanifolds M = L\G where G is a nilpotent Lie group and L denotes a uniform lattice inG. We assume that the distribution in the tangent bundle is bracket generating and spannedby a collection of globally dened vector elds X1, · · · , Xn. In this geometric frameworkwe study the spectral-zeta function of the corresponding sub-Laplacian ∆sub (a negativesum of squares of the vector elds Xj) which is an hypo-elliptic operator. We compareit with the spectral zeta-function of the Laplacian on M with respect to a closely relatedRiemannian structure. In particular, these considerations apply to Heisenberg manifoldswhere the spectral zeta-function of ∆sub has a more explicit form.These results have recently been obtained as a joint work with Prof. K. Furutani (ScienceUniversity of Tokyo) and Prof. C. Iwasaki (Hyogo University).

2

Lax Pairs and Inverse Scattering

Swanhild Bernstein (Freiberg, Germany)

Abstract. We demonstrate that it is possible to obtain Lax pairs for linear dierentialoperators with a polynomial of Dirac operators and use the AKNS method for the nonlinearKdV equation. The Ablowitz, Kaup, Newel, Segur method works as follows. Given alinear operator X associated with the system vx = X v, we are interested in nding anotheroperator T , and the operators X and T are said to form an AKNS pair, when:

(i) The spectral parameter λ does not change in time.

(ii) The quantity vt − T v must remain a solution to vx = X v.

(iii) The quantity Xt − Tx + XT − T X must be a (matrix) multiplication operator.

For compatibility, we are forced to have

Xt − Tx + XT − T X = 0,

which is again can be interpreted as an integrable PDE and is in general nonlinear.

The KdV equation can also be obtained by the AKNS method. We will use this method toobtain a nonlinear system which possesses an AKNS pair build up with Dirac operators.

3

Energy estimates for Klein-Gordon type models withtime-dependent potential

Christiane Böhme (Freiberg, Germany)

Coauthors: Fumihiko Hirosawa, Yamaguchi University, Japan;

Key words: Klein-Gordon equation, energy estimates, Lp-Lq decay estimates

Abstract. In this talk we investigate Klein-Gordon type models whose mass term isessentially described by a decreasing function and an oscillating perturbation. In dependenceon the interplay between the decay and the oscillations of the coecient we are interestedin statements about generalized energy conservation and Lp-Lq decay estimates. Therefore,we take advantage of a rened diagonalization procedure.

4

On the energy estimates for second order hyperbolic equationswith time dependent coecients

Bui Tang Bao Ngoc (Freiberg, Germany (Hanoi, Vietnam))

Coauthors: Fumihiko Hirosawa

Key words: Generalized energy conservation, stabilization, Ck-type Levi conditions

Abstract. We consider the energy estimates for the second order homogeneous hyperbolicequations with time dependent coecients. The property of energy conservation, whichholds in the case of constant coecients, does not hold in general for variable coecients;in fact, the energy can be unbounded as t → ∞. Conditions to the coecients for thegeneralized energy conservation = (GEC), which means, that the energy can be estimatedboth-sided by the initial energy, have been studied precisely for wave type equations. In thiscase, only the propagation speed is variable. However, it is not true that the same conditionsto the coecients conclude (GEC) for general homogeneous hyperbolic equations of secondorder. The main purpose of this paper is to give additional conditions to the coecientswhich provide (GEC); they will be called as Ck-type Levi conditions due to the essentiallysame meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.

5

On Summability of Formal Solutions for a Class of NonlinearSingular Partial Dierential Equations

Chen Hua (Wuhan, China)

Abstract. In this talk, we study a class of rst order nonlinear degenerated partialdierential equations with singularity at origin in complex plane. By means of exponentialtype Nagumo norm approach, Gevrey asymptotic analysis can be used to get the exactcondition to deduce the so-called k-summability for the formal solutions. Furthermore, theanalytic solutions in conical domains are found for each type of those nonlinear singularPDEs.

6

Rate of Convergence in Nonlinear Hartree Dynamics withfactorized initial data

Chen Li (Tsinghua University Beijing, China)

Coauthors: Ji Oon Lee

Key words: Many body Schrödinger, mean eld limit, quantum uctuation.

Abstract. The mean eld dynamics of N weekly interacting Boson system can be describedby the nonlinear Hartree equation. In this talk, I will present estimates on the 1/N rate ofconvergence of many-body Schrodinger dynamics to the one-body nonlinear Hartree dynam-ics with factorized initial data with two-body interaction potential V in L3(R3) + L∞(R3).

7

ON THE COMPARISON FUNCTION IN SPECTRALTHEORY

Michael Demuth (Clausthal- Zellerfeld, Germany)

Abstract. Let A and B be selfadjoint lower-semibounded operators in L2(E,m). Assumingthat the semigroups e−tA, t ≥ 0 and e−tB , t ≥ 0 are L2−L∞ smoothing, the comparisonfunction is given by

Dt(x) =

∫E

|e−tB(x, y)− e−tA(x, y)|dm(y).

We show that, provided the spectrum of A is known, integral conditions for Dt determine theessential spectrum, the absolutely continuous spectrum and the moments of the eigenvaluesof B.

8

Interface Layers for Quantum Semiconductor Models

Michael Dreher (Konstanz, Germany)

Coauthors: Li Chen (Tsinghua University Beijing) and Johannes Schnur (University of Kon-stanz)

Key words: Elliptic system, singular perturbation, variational methods

Abstract. We discuss a stationary state of a quantum drift-diusion model which de-scribes the densities of electrons and holes in a MOSFET device. Near the gate contact ofthat device, quantum eects generate an inversion layer, for which we derive rigorously itsasymptotic expansion, after proving the existence of the solution by variational methods.In a second part of the talk, we present some results on stationary states of a quantumhydrodynamical system that includes a barrier potential.

9

The inuence of oscillations on global existence for a class ofsemi-linear wave equations

Marcelo R. Ebert (University of Sao Paulo, Brazil)

Abstract. In this talk we describe how the interplay between the increasing and oscillatingbehavior of the time variable propagation speed can inuence, on the global existence ofsmall data solutions, of the Cauchy problem for the semi-linear wave equations

utt − λ2(t)b2(t)∆u = u2t − λ2(t)b2(t)|∇u|2,

where λ(t) describes the asymptotic behavior for large t (shape function) and b(t) allows usto include certain oscillations. In particular we are interested in statements for the 1D case.We will present a way how to explain optimality of our results.This is a joint work with Michael Reissig.

10

Global Stability of E-H type Regular Refraction of Shocks onthe Interface between Two Media

Fang Beixiang (Shanghai Jiao Tong University, China)

Coauthors: HU Dian

Key words: Stability, E-H Type Regular Refraction, Nonlinear Mixed Type Equations,Generalized Tricomi Problems.

Abstract.In this paper, we are concerned with the global stability of the 2-D steady nonlinear wavestructure of the E-H type regular refraction of shocks on the interface between two dierentmedia. In general, when a shock front attacks an interface between two media with theincident angle less than some critical value, a reected wave and a refracted shock will appearwith the interface being deected. Such a reection-refraction wave structure is called regularrefraction. According to dierent parameters of the incident shock and the media, both thereected wave and the refracted shock can be a supersonic-supersonic shock(supersonic shockfor short) or a supersonic-subsonic shock(transonic shock for short). Correspondingly, thesecases of regular refraction can be classied as H-H type, E-E type and E-H type. In thispaper, we concentrate on the E-H type regular refraction. In particular, without loss ofgenerality, we are going to study the case that the reected wave is a supersonic shock andthe refracted shock is a transonic shock. Thus the ow between the reected shock frontand the deected interface is relatively supersonic, while it is relatively subsonic betweenthe refracted shock front and the deected interface. Our result indicates that the steadyat E-H type regular refraction is globally stable.We use the 2-D steady potential equations as the mathematical model to describe the motionof the uid. The system of potential equations is hyperbolic for supersonic ows, while itis elliptic for subsonic ones. Hence, the stability problem of the at E-H type regularrefraction can be reduced to a free boundary problem of nonlinear mixed type equationsin an unbounded domain for the unique existence of the solution near an unperturbedpiecewise constant solution. The corresponding linearized problem has similarities to thegeneralized Tricomi problem of the linear Lavrentiev-Bitsadze mixed type equation, and itcan be reduced to a nonlocal boundary value problem of an elliptic system in the ellipticregion. Solving the nonlocal problem can be further reduced to verifying the bijectivenessof a nonlocal operator in a weighted Hölder space, which is established via a careful analysisfor the operator. Then nonlinear iteration schemes are employed to approach the solution tothe nonlinear free boundary problem. The unique existence of the solution to the nonlinearfree boundary problem implies the global stability of the at E-H type regular refraction.

11

Lp-approximation of the integrated density of states forSchrödinger operators with nite local complexity

Michael J. Gruber (Clausthal- Zellerfeld, Germany)

Coauthors: Daniel H. Lenz, Ivan Veseli¢

Key words: integrated density of states, random Schrödinger operators, nite local complex-ity

Abstract. We study spectral properties of Schrödinger operators on Rd. The electromag-netic potential is assumed to be determined locally by a colouring of the lattice points inZd, with the property that frequencies of nite patterns are well dened. We prove thatthe integrated density of states (spectral distribution function) is approximated by its nitevolume analogues, i.e. the normalised eigenvalue counting functions. The convergence holdsin the space Lp(I) where I is any nite energy interval and 1 ≤ p <∞ is arbitrary.

12

Uniqueness and non-uniqueness in the Cauchy problem forelliptic and backward parabolic operators

Christian Jäh (Freiberg, Germany)

Key words:

Abstract. We consider the two model operators

Eu = D2t u+ akl(t)Dxk

Dxlu+ bm(t, x)Dxm

u+ β(t, x)Dtu+ c(t, x)u (0.1)

andPu = Dtu+ akl(t)Dxk

Dxlu+ bm(t, x)Dxm

u+ c(t, x)u. (0.2)

For them we investigate the inuence of the interplay between global and local conditions onthe coecients of the principal part on the uniqueness property of the solution of the Cauchy

problem for (0.1) and (0.2). We show that the local condition∣∣∣ ddtakl(t) ξkξl|ξ|2

∣∣∣ ≤ Ct−1 ensuresuniqueness for both operators with the global regularity akl ∈ C0 if the constant C is smallenough. Furthermore we give a precise bound for C and construct Pli²-type counterexamplesto show that our result is sharp. We also show that one can choose a bigger constant Cif one shrikes the space of admissible solutions to certain suitable Gevrey-space γ(s). At

the end we construct counterexamples with the local condition∣∣∣ ddtakl(t) ξkξk|ξ|2 ∣∣∣ ≤ CM(η−1(t))

η−1(t)

and the global regularity assumption akl ∈ CM with a modulus of continuityM satisfying∫ 1

0dsM(s) < +∞.

13

On transonic shocks in a divergent nozzle

Li Jun (Nanjing, China)

Abstract. The talk focous on transonic shocks in a divergent nozzle. In the book "Su-personic ow and shock waves", Courant and Friedrichs described the following transonicshock phenomena in a de Laval nozzle: Given the appropriately large receiver pressure pr, ifthe upstream ow is still supersonic behind the throat of the nozzle, then at a certain placein the diverging part of the nozzle a shock front intervenes and the gas is compressed andslowed down to subsonic speed. The position and the strength of the shock front are auto-matically adjusted so that the end pressure at the exit becomes pr. When the end pressurepr varies and lies in an appropriate scope, in general, it is expected that a curved transonicshock is still formed in a nozzle. I want to talk about some progress on this problem. Theuniquely existence, stability of the transonic shock are contained. Furthermore, dependenceof the existence and regularity of the shock on the geometry of the nozzle is also introduced.

14

Multiple solutions for semilinear totally characteristic ellipticequations with subcritical or critical cone Sobolev exponents

Liu Xiaochun (Wuhan, China)

Coauthors: Hua Chen and Yawei Wei

Key words: cone Sobolev exponent, multiplicity, existence, totally characteristic

Abstract. In this talk, we will study a class of semilinear totally characteristic elliptic equa-tions with subcritical or critical cone Sobolev exponents and get the existence of innitelymany solutions in both cases.

15

Distributional solutions of a nonlinear Schrödinger equation

Rainer Mandel (Karlsruhe Institute of Technology, Germany)

Coauthors: Wolfgang Reichel (Karlsruhe Institute of Technology)

Abstract. In this talk the existence of locally unbounded distributional solutions of

−∆u+ V (x)u = Γ(x)|u|p−1u in Rn (∗)

is discussed for functions V,Γ ∈ L∞(Rn) with 0 < minσ(−∆ + V (x)).A recent result is presented saying that in the case n ≥ 3 one can nd such solutions withsingularity at x0 ∈ Rn at least for p ∈ ( n

n−2 ,nn−2 +ε) where ε = ε(V,Γ) > 0 is a small number

depending on the behaviour of V and Γ near x0. In the case 1 < p < n(n−2)+ , however, we

nd that (∗) does not admit positive locally unbounded solutions in the case Γ(x) ≥ Γ0 > 0.The results have been obtained in collaboration with Wolfgang Reichel (Karlsruhe Instituteof Technology)

16

Existence of ground states in semi-relativistic QED at criticalcoupling

Oliver Matte (Clausthal- Zellerfeld, Germany)

Abstract. We consider two dierent models of a hydrogen-like atom interacting withthe quantized electromagnetic eld, namely the semi-relativistic Pauli-Fierz model and ano-pair model of quantum electrodynamics. Our aim is to prove the existence of energyminimizing ground state eigenvectors in both models. Since the electronic Hamiltonians aregiven as the sum of relativistic, pseudodierential kinetic energy operators and a Coulomb-like singular potential, there are critical values for the Coulomb singularity above which thesystem becomes unstable. In this talk, which is based on joint work with Martin Könenbergand Edgardo Stockmeyer, we present in particular recent results dealing with both systemsat critical coupling.

17

Global solutions to the Cauchy problem for a system ofdamped wave equations

Narazaki Takashi (Tokai University, Japan)

Abstract. In this talk we study the global existence of solutions to the Cauchy problemfor a system of nonlinear damped wave equations

∂2t uj −∆uj + ∂tuj = cjuσj,1

1 · · ·uσj,N

N =: Fj(u), t > 0, x ∈ Rn, (0.3)

for 1 ≤ j ≤ N , where u = (u1, · · · , uN )T is unknown, and σj,k (1 ≤ j, k ≤ N) is a constantsuch that σj,k ≥ 1 or σj,k = 0. When N = 1 several authors have studied the above Cauchyproblem, and it is known that σF := 1 + 2/n is the critical exponent to problem (0.3).Recently, when n ∈ 1, 2, 3, Ogawa-Takeda [1] proved that problem (0.3) admits a globalsolution provided that σ1,1, · · · σ1,N

... · · ·...

σN,1, · · · σN,N

ν1

...νN

=

1 + ν1...

1 + νN

(0.4)

admits a solution (ν1, · · · , νN ) such that

0 < min (ν1, · · · , νN ) ≤ max (ν1, · · · , νN ) <n

2. (0.5)

Takeda [2] proved that the solutions of (0.3) with nonlinear term replaced by |uσj,1

1 · · ·uσj,N

N |blow up in nite time when condition (0.5) does not hold. It should be noted that condition(0.4)(0.5) is equivalent to σ1,1 > σF when N = 1. The aim of this talk is to remove therestriction n ≤ 3. We will show that the Cauchy problem (0.3) admits a global solutionwhen n ≥ 4 under certain conditions. Moreover, we show the asymptotic behavior of theabove solution. To prove the result we use weighted Sobolev spaces.

References

[1] T. Ogawa and H. Takeda, Global existence of solutions for a system of nonlinear dampedwave equations, Dierential and Integral Equations, Vol. 23, No. 7-8(2010), 635657.

[2] H. Takeda, Global existence and non existence of solutions for a system of nonlineardamped wave equations, J. Math. Anal. Appl., Vol. 360(2009), 631650.

18

On Existence and Stability for Heat ConductingReissner-Mindlin Plates

Michael Pokojovy (Konstanz, Germany)

Key words: Reissner-Mindlin, plate equation, nonlinear existence, exponential stability, sec-ond sound

Abstract. We consider nonlinear thermoelastic plate equations of Reissner-Mindlin typein a bounded domain. The elastic behavior of the plate is described by a hypoelastic lawwhereas the thermal part is modelled through the linearized Cattaneo's law of heat conduc-tion.We prove the existence and exponential stability (under a certain damping) of the associatedC0-semigroup of contractions for the linearized problem. A local existence theorem for aclass of nonlinearities including the geometric one will be discussed. Global solvability andexponential stability in the nonlinear situation will also be addressed.

19

Geometric aspects and wave breaking phenomena ofµ-Camassa-Holm type equations

Qu Changzheng (Northwest University, Xi'an, China)

Coauthors: Yue Liu, Ying Fu

Key words: Invariant geometric ow, µ-Camassa-Holm equation, wave breaking, blow-up

Abstract. Recently, Camassa-Holm equation and its various extensions have draw muchattention in the past twenty years. In this talk, we show that some of these equationsarise from invariant geometrical ows in certain geometries. In particular, µ-Camassa-Holmequation and its two-component extension are investigated. The local well-posedeness andglobal existence to initial value problem are studied. Several blow up criterions for initialvalue problem in the periodic setting are provided.

20

INSTABILITY OF COUPLED SYSTEMS WITH DELAY

Reinhard Racke (Konstanz, Germany)

Coauthors: B. Said-Houari

Key words: Delay problems, ill-posedness

Abstract. We consider linear and nonlinear initial-boundary value problems that area coupling like second-order thermoelasticity, or the thermoelastic plate equation or itsgeneralization (the α-β-system). Now, there is a delay term given in part of the coupledsystem, and we demonstrate that the expected inherent damping will not prevent the systemfrom being not stable; indeed, the systems will shown to be not well-posed, a sequence ofbounded initial data may lead to exploding solutions (at a xed time).

21

The Mellin-edge quantisation of corner operatorsB.-Wolfgang Schulze (Potsdam, Germany)

Operators on manifolds with second order corners, degenerate symbols, weighted

corner spaces

Abstract. The content of the talk is based on a joint work with Yawei Wei from the NankaiUniversity, Tianjin, China. The ideas belong to the general program of generating algebrasof pseudo-dierential operators on a corner manifold, containing the typical dierentialoperators together with the parametrices of elliptic elements. Singular manifolds containmany interesting special cases, e.g. manifolds with boundary; in that case the calculuscorresponds to boundary value problems with or without the transmission property at theboundary. Other examples are manifolds with conical singularities, locally modelled oncones with a smooth base, or manifolds with edges, locally described by Cartesian productsbetween such a cone and a smooth manifold. If we replace the smooth base of the coneby a manifold B with edge in the former sense, then we obtain a manifold M with secondorder corners (or edges). Those are specic stratied spaces (not topological manifolds inthe standard sense), here with three strata s0(M), s1(M), s2(M) that are smooth manifoldsof dierent dimension where M = s0(M)∪ s1(M)∪ s2(M) is a disjoint union. s0(M)) is theinterior part of M of maximal dimension, while s1(M) ∪ s2(M) is a manifold with smoothedge s2(M) and M \ s2(M) is a manifold with smooth edge s1(M). The strata contributethree principal symbolic levels σ0(A), σ1(A), σ2(A) of operators A in the calculus. One of themajor issues of the calculus are quantisations which lead to operators that are continuousin adequate weighted spaces. We adopt here a method of edge quantisation with the help ofthe Mellin transform, originally established by Gil, the speaker, and Seiler for the rst orderedge calculus. The result are certain operator-valued symbols taking values in operators onan innite stretched cone where the base is a manifold with edge, and depending on thesecond order edge covariables ζ 6= 0. From the geometry of the conguration we have twohalf-axis variables r ∈ R+ and t ∈ R+ where r belongs to the inner cone inside B and t to thecorner which has the base B. The analysis of operators is very rich in detail; for instance,the corner symbol σ2(A) takes values in the edge calculus over B and depends on ζ in acorner-degenerate way for t→ 0 and has an exit behaviour for t→∞ where the corner exithas the singular cross section B. In any case the calculus remains manageable despite of itscomplexity, because it is iterative. In that sense, also the quantisation for higher cornersmay expected to have a similar structure.

22

Critical Coupling Constant for Magnetic Weyl Operators in 2d

Heinz Siedentop (München, Germany)

Coauthors: Thomas Maier

Key words: Hardy inequality, Kato inequality

Abstract. We show that a constant magnetic eld given as curl of A := B2 (−y, x)

neither increases nor decreases the value of the critical coupling constant γ of the no-pairWeyl operator Λ[σ · (p + A)− γ|x|−1]Λ where Λ := χ(0,∞)(σ · (p + A)).

23

The ∂-Neumann Laplacian on G-manifolds.

Peter Stollmann (Chemnitz, Germany)

Coauthors: Joe J. Perez

Key words: Analysis of PDEs, Complex Variables

Abstract. We study the ∂-Neumann Laplacian acting in spaces of dierential forms overnoncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry andcompact quotient. We also relate our results to those for an associated Laplace-Beltramioperator on functions. The main focus will lie on properties of the associated heat kerneland on generalized eigensolutions and their connection to spectra.

24

WELL-POSEDNESS OF EVOLUTIONARY INCLUSIONS

Sascha Trostor (Dresden, Germany)

Key words: Monotone operators, dierential inclusions

Abstract. We show well-posedness of dierential inclusions of the form

(u, f) ∈ ∂ν,0M0 +M1 +A,

where f is a given function, M0 andM1 are continuous, linear operators and A is a maximalmonotone relation on a Hilbert-space. The time derivative ∂ν,0 is established in a suitableHilbert-space, such that it is getting continuously invertible. This general setting includesa large class of partial dierential equations of the mathematical physics. Also, problemsincluding hysteresis-phenomena and switched dynamical systems are covered.

25

Approximate controllability of a class ofsemilinear degenerate systems

Wang Chunpeng (Jilin University, China)

Key words: approximate controllability, degenerate equation, weak degeneracy and strongdegeneracy

Abstract. In this talk we introduce the approximate controllability of a class of degeneratesemilinear systems. The equations may be weakly degenerate and strongly degenerate ona portion of the lateral boundary. It is shown that the control systems are approximatelycontrollable.

26

Stability of characteristic boundary layers for the compressibleNavier-Stokes equations with Navier-friction boundary

conditions

Wang Ya-Guang (Jiao- Tang Shanghai, China)

Abstract. In this talk, we study boundary layer solutions of the isentropic, compressibleNavier-Stokes equations with Navier-friction boundary conditions in the small viscosity limit.The Navier boundary condition expresses that the velocity on the boundary is proportionalto the tangential component of the stress, while the normal component of velocity is zero onthe boundary, which implies that the boundary is characteristic for the underlying inviscidproblem, the compressible Euler equations. By multi-scale analysis, we rst construct ahigh-order approximate solution that exhibits a boundary layer. As in the incompressiblecase the main contribution to the layer appears in the tangential velocity and is of width ofthe square root of the viscosity. Next we prove that the boundary layer is stable; that is, theapproximate solution stays close to the exact Navier-Stokes solution on a xed time intervalindependent of the viscosity. As an immediate corollary of the stability result, we show thatthe Navier-Stokes solution converges in L∞ in the small viscosity limit to the solution of thecompressible Euler equations with normal velocity equal to zero on the boundary.

27

A Hilbert Space Approach to Homogenization

Marcus Waurick (Dresden, Germany)

Key words: Homogenization, Hyperbolic Partial Dierential Equations

Abstract. Using a recent result concerning the structure of linear partial dierential equa-tions in classical mathematical physics([1]), we present a new way to view homogenization,i.e., limiting processes of partial dierential equations. For doing so a topology on so-calledmaterial laws is introduced. With those concepts homogenization results can be obtainedfor a wide class of hyperbolic-type partial dierential equations with memory term.

References

[1] R. Picard. A structural observation for linear material laws in classical mathematicalphysics. Mathematical Methods in the Applied Sciences, 32:17681803, 2009.

28

Dirichlet Problem for Semilinear Elliptic Equations withSingular Hardy Term on Manifolds with Edge Singularities

Wei Yawei (Nankai University, China)

Coauthors: Hua Chen, Xiaochun Liu

Key words: Dirichlet Problem, Elliptic equations, Hardy inequality, manifolds with edgesingularities

Abstract. This is the joint work with Professor Hua Chen and Professor Xiaochun LiufromWuhan University. In the talk, we introduce the weighted p-Sobolev spaces on manifoldwith edge singularities, and give the proofs for corresponding Sobolev inequality, Poincareinequality and Hardy inequality. Then we prove the existence of solutions for the Direchletproblem of semilinear elliptic equations with singular Hardy term on manifold with edgesingularities.

29

Some aspects of phase space analysis for hyperbolic systems

Jens Wirth (Stuttgart, Germany)

Coauthors: Michael Ruzhansky

Key words: hyperbolic systems, diagonalisation, representation of solutions, dispersive esti-mates

Abstract. In this talk we review some recent results obtained on dispersive type estimatesfor solutions to hyperbolic systems with time-dependent coecients based on diagonalisationof the full symbol of the operator and application of a mutli-dimensional van der Corputlemma. Such estimates are intimately connected to the structure of the representation ofsolutions in terms of Fourier integrals and main focus of the talk will be set on the relationbetween assumptions on the full symbol and structural properties of the appearing Fourierintegrals.Some of the aspects of the approach generalise to fully variable coecient pseudo-dierentialproblems in suitable symbol classes.

30

Boundary value problems for Tricomi-type operators

Ingo Witt (Göttingen, Germany)

Key words: PDEs of mixed type, degenerate pseudodierential operators

Abstract. In my talk, I shall be concerned with boundary value problems for partialdierential operators of Tricomi type. Among others, this means that such a dierentialoperator L is of the second order, is dened in a bounded domain Ω ⊂ R2, has real coecientsthat are C∞ up to the boundary, and that Ω is divided by an interface I across whichL changes type into an elliptic region Ω+ and a strictly hyperbolic region Ω−; bothnonempty. In addition, the discriminant of L must vanish exactly to the rst order on I.After a suitable change of coordinates, we may assume that L takes the form

L = A∂2x + 2xB∂x∂y + xC∂2y +D∂x + E∂y + F,

where A,B,C,D,E, F ∈ C∞(Ω;R). The main assumption then reads

AC − xB2 > 0 in Ω.

In particular, L is elliptic for x > 0 and strictly hyperbolic for x < 0, and it also followsthat characteristic curves hit the interface at x = 0 transversally.The investigations shall rely on a calculus for certain degenerate pseudodierential opera-tors, which I will present in this talk, that allows to reduce the problem under considerationto a problem in the interface I. The latter turns out to be cone-degenerate at both end-points of the interval I which, as a consequence, can be studied utilizing Schulze's conepseudodierential calculus.

31

Local stability of a 3-D steady supersonic centered rarefactionwave

Yin Huicheng (Nanjing University, China)

Abstract. As indicated in Section 111 of Courant-Friedrichs' book `"Supersonic Flow andShock Waves'", a supersonic ow around a sharp corner, one of the most important elemen-tary ows, is eected by a rarefaction wave. This talk is concerned with the local structuralstability problem on such a wave of three-dimensional steady full Euler system. More con-cretely, we establish the local existence and stability of a 3-D incomplete expansion centeredrarefaction wave when the perturbed supersonic incoming ow moves around a sharp cor-ner whose xed wall beyond the corner is of a perturbation with respect to some obliquehalf-plane.

32

Global existence in critical spaces for incompressibleviscoelastic uids

Zhang Ting (Zhejiang University, Hangzhou, China)

Abstract. We investigate local and global strong solutions for the incompressible viscoelas-tic system of OldroydB type. We obtain the existence and uniqueness of a solution in afunctional setting invariant by the scaling of the associated equations. More precisely, theinitial velocity has the same critical regularity index as for the incompressible NavierStokesequations, and one more derivative is needed for the deformation tensor. We point out asmoothing eect on the velocity and a L1−decay on the dierence between the deformationtensor and the identity matrix. Our result implies that the deformation tensor F has thesame regularity as the density of the compressible NavierStokes equations.

33