advances in nonlinear pdes · 2014-09-01 · advances!in!nonlinear!pdes! september!385,2014!!! 3!...

Advances in Nonlinear PDEs !in#honor#of##Nina#N.#Uraltseva##

!!

Book$of$abstracts$St.!Petersburg,!!2014!!!

Advances in Nonlinear PDEs September 3-‐5, 2014

1

St.!Petersburg!Steklov!Ins7tute!of!Mathema7cs!

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Advances in Nonlinear PDEs !

Interna2onal#Conference#

in#honor#of#the#outstanding#mathema2cian#

Nina#N.#Uraltseva##and#on#the#occasion#of#her#80th#anniversary#

St.!Petersburg,!!September!3@5,!2014!!!

Supported!by:!


2

!!

Organizing(Commi,ee:(

Darya%Apushkinskaya%Alexander%Mikhailov%Alexander%Nazarov%Sergey%Repin%Ta;ana%Vinogradova%Nadya%Zalesskaya%


3

Cauchy-Dirichlet problem for quasilinear parabolic systems with a nonsmooth in time

principal matrix

Arina Arkhipova

St. Petersburg State University, Russia [email protected]

We consider quasilinear parabolic systems with nonsmooth in time principal matrices. Only boundedness of these matrices is assumed in time variable. We prove partial regularity up to the parabolic boundary of a cylinder. To prove the result, we apply the method of modified A-‐caloric approximation assuming that the matrix A is not the constant one but depends on the time variable. The talk reports on results obtained jointly with J.Stara and O.John.


4

Semilinear elliptic problems with a Hardy potential

Catherine Bandle

University of Basel, Switzerland [email protected]

We consider problems of the type

€

Δu +V (x)u = up in a bounded domain in

€

n where

€

V is a Hardy potential

€

µδ 2(x)

and

€

δ(x) is the distance

from a point

€

x to the boundary of the domain. We are interested in the existence of positive

solutions, and the interplay between the nonlinearity and the boundary singularity. If

€

0 < p <1 the nonlinearity gives rise to dead cores and if

€

p >1 to boundary blowup. We give a fairly complete picture of the radial solutions and use those solutions as upper and lower solutions for general domains.

The talk reports on results obtained in collaboration with V.Moroz (Swansea), W.Reichel (KIT Karslruhe) and M.A.Pozio (La Sapienza Rome)


5

Stochastic counterparts of the Cauchy problem for quasilinear systems of parabolic

equations

Yana Belopolskaya

St. Petersburg State University for Architecture and Civil Engineering, Russia [email protected]

We consider the Cauchy problem for two types

of quasilinear parabolic systems, namely systems with diagonal principal parts and systems with nondiagonal principal parts.

For systems of the first type, investigated in the famous monograph [1] we construct probabilistic representations for classical, generalized and viscosity solutions of the Cauchy problem and use them to investigate the PDE system solution. The probabilistic counterpart is constructed in terms of stochastic differential equations for corresponding Markov processes and their multiplicative operator functionals.

For systems of the second type studied in a number of papers started from [2] we construct a probabilistic representation of a generalized solution of the Cauchy problem for a PDE system of the second type in terms of time reversed stochastic flows, generated by solutions of corresponding stochastic differential equations.

Partial financial support of RFBR Grant 12-‐01-‐00457-‐a and Minoobrnauki project 1.370.2011 is gratefully acknowledged.

References [1] O. Ladyzenskaya, V. Solonnikov, N. Uraltzeva Linear and quasilinear equations of parabolic type 1967, Nauka. [2] H. Amann Dynamic theory of quasilinear parabolic systems, Mathematische Zeitschrift (1989), Vol. 202, Issue 2, pp 219-‐250.


6

Non-uniqueness of hydrodynamic equations from molecular dynamics

Stamatis Dostoglou

University of Missouri-‐Columbia, USA [email protected]

Hydrodynamic equations can be obtained at the limit of microscopic Newtonian equations as the number of molecules increases while the length scale changes accordingly. For some examples of molecule systems, with reasonable initial conditions and interaction potential, the resulting hydrodynamic equations for certain scaling turn out to be quite easy to analyze and show how microscopic fluctuations can lead to apparent macroscopic non-‐uniqueness with respect to their macroscopic initial conditions. The contents of the talk are part of an ongoing project to obtain rigorous Reynolds equations and are based on work currently done with Jianfei Xue.


7

Normal equations and nonlocal stabilization by feedback control for equations of Navier-

Stokes type

Andrey Fursikov

Moscow State University, Russia [email protected]

We study so-‐called parabolic equations of

normal type to understand better properties of equations of Navier-‐Stokes type.

By definition semilinear parabolic equation is normal parabolic equation (NPE) if its nonlinear term defined by operator

€

Bsatisfies the condition:

€

∀v∈H1 vector

€

B(v) is collinear to

€

v . In other words solutions of NPE does not satisfies energy estimate “in the most degree".

For Burgers and 3D Helmholtz equations we derive normal parabolic equations (NPE), which nonlinear terms

€

B(v) are orthogonal projections of nonlinear terms for corresponding original equations on the straight line generated by the vector

€

v. The

structure of dynamical flow corresponding to these NPE will be described.

For NPE corresponding to Burgers equation we construct nonlocal stabilization to zero of solutions by starting, impulse, or distributed feedback controls supported in an arbitrary fixed sub domain of the spatial domain. The last result is applied to nonlocal stabilization of solutions for Burgers equations.


8

On attractors of m-Hessian evolutions

Nina Ivochkina


Let

€

Ω be a bounded domain in

€

n,

€

Q =Ω× (0;∞) ,

€

u∈C2,1(Q ) ,

€

uxx be the Hesse matrix of

€

u in space variables. We denote by

€

Tp[u] = Tp (uxx ) ,

€

1≤ p ≤ n $1 the

€

p-‐trace of

€

uxx and introduce

€

p -‐Hessian evolution operator by

€

Ep[u] := utTp−1[u]+Tp[u]. We investigate asymptotic behavior of

solutions of the following initial boundary value problems:

€

Em[u] = f , u ∂ 'QT= φ, 1≤ m ≤ n, (1)

where

€

∂'QT = Ω× 0{ }{ }∪ ∂Ω× [0;T]{ } . In particular, we have proved Theorem 1. Let

€

f ≥ν > 0 ,

€

f ∈C2,1(Q T ) for all

€

T ∈[0;∞) ,

€

φ ∈C2,1(Q T ),

€

φ = 0 on

€

∂Ω × [0;∞) ,

€

∂Ω∈C2. Assume that

€

limt→∞ f (x, t) = f (x) and there exists a solution

€

u ∈C2(Ω ) to the Dirichlet problem

€

Tm[u] = f , u ∂Ω = 0. Then all solutions

€

u∈C2,1(Ω × [0;∞)) to the problem (1) tend uniformly in

€

C to the function

€

u (x) , when

€

t →∞ . It is of interest the following non existence

theorem.


9

Theorem 2. Assume that there are points

€

x0, x1∈Ω such that

€

φxx (x0,0) is

€

(m −1) -‐positive matrix, while

€

φxx (x1,0) is not

€

(m −1) -‐positive. Then there are no solutions in

€

C2,1(Q T ) to the problem (1), whatever

€

f > 0 ,

€

∂Ω ,

€

T > 0 ,

€

φ had been.

Eventually, we formulate the existence theorem assuming sufficiently smooth data in (1). Theorem 3. Let

€

f ≥ν > 0 ,

€

∂Ω is (

€

m −1)-‐convex hypersurface,

€

φ(x,0)∈Km−1(Ω ). Assume that compatibility conditions are satisfied. Then there exists a unique in

€

C2,1(Q T ) solution to the problem (1). The work is supported by the RFBR grant No.

12-‐01-‐00439 and by the grants Sci. Schools RF 1771.2014.1, and by the St. Petersburg State University grant 6.38.670.2013.


10

Analyticity of the free boundary in the thin obstacle problem

Herbert Koch

University of Bonn, Germany [email protected]‐bonn.de

We prove analyticity of the regular part of the boundary of the contact set for the thin obstacle problem. A key step is a change of variables which reduces the problem to a Monge-‐Ampere type equation, which in the relevant regime becomes a fully nonlinear variant of a Grushin-‐type operator. We prove analyticity of solutions to this fully nonlinear equation under non degeneracy assumptions. This talk is based on results obtained in collaboration with A.Petrosyan and W.Shi.


11

On uniqueness in the water wave theory

Vladimir Kozlov

Linköping University, Sweden [email protected]

This talk concerns the classical free-‐boundary

problem that describes two-‐dimensional steady gravity waves on water of finite depth. I will discuss some uniqueness results.


12

Common features of homogenization and of large scale limits in statistical mechanics

Stephan Luckhaus

Leipzig University, Germany [email protected]‐leipzig.de

We try to present large scale limits in equilibrium statistical mechanics from an analytic-‐functional analytic point of view. The results presented are joint with Roman Kotecky. We also try to point out the analogues with results in homogenization. The setting for both types of result is that of quasiconvex functionals.


13

Criteria for the Poincare-Hardy inequalities

Vladimir Maz’ya

University of Liverpool, UK and Linköping University, Sweden

[email protected] This is a survey of necessary and sufficient conditions for validity of various integral inequalities containing arbitrary weights (measures and distributions). These results have direct applications to the spectral theory of elliptic partial differential operators.


14

From Uraltseva to Zhikov, forty years of degenerate operators in Russia

Giuseppe Mingione

University of Parma, Italy [email protected]

In 1967 a seminal paper of Nina Ural'tseva appeared, featuring the proof of the

€

C1,β -‐nature of solutions to the degenerate equation

€

p -‐Laplacian equations. This is a cornerstone of modern nonlinear potential theory and the techniques introduced by Nina are nowadays classical. Several years later, Zhikov and a group of Russian mathematicians developed new models for strongly anisotropy materials, based on the

€

p -‐Laplacian operator, in order to study various questions in homogenisation, elasticity, Lavrentiev phenomenon. I will present a few new regularity results on such functionals.


15

The Dirichlet and Navier fractional Laplacians

Roberta Musina

Udine University, Italy [email protected]

This talk is based on results obtained jointly

with A.I. Nazarov. Let

€

Ω⊂n be a bounded and smooth domain. We denote by

€

λ j and

€

ϕ j the eigenvalues and

eigenfunctions of the Laplace operator

€

−Δ on

€

H01 (Ω) ,

respectively. For any real number

€

s > 0 we formally introduce the "Dirichlet" fractional Laplacian

€

F −Δ( )Ds u[ ](ξ) =|ξ |2s F[u](ξ) ,

where

€

F is the Fourier transform, and the "Navier" fractional Laplacian

€

(−ΔΩ)Ns u = λ j

s uϕ jΩ

∫⎛

⎝ ⎜

⎞

⎠ ⎟ ϕ jj

∑ .

In [MN1] and [MN2] we compare those two fractional Laplacians for arbitrary

€

s > 0. In this talk we focus our attention to the case

€

s∈ (0,1), when the domains of the quadratic forms

€

(−Δ)Ds u,u and

€

(−ΔΩ)Ns u,u

coincide with

€

˜ H s(Ω) = u∈H s({ n

€

) | supp(u)⊆ Ω } . The following facts are proved:

a. The operator

€

(−ΔΩ)Ns − (−Δ)D

s is positive definite and positivity preserving.

b. For any fixed

€

u∈ ˜ H s(Ω) one has

€

(−Δ)Ds u,u = inf

Ω'⊃Ω(−ΔΩ)N

s u,u


16

(the infimum is taken over the family of smooth bounded domains).

c. Assume

€

n ≥ 2 or

€

s <1/2 , and put

€

2s* =

2nn − 2s

. Then the "Dirichlet-Sobolev" and the "Navier-Sobolev" constants coincide, that is,

€

infu∈ ˜ H s (Ω), u≠0

(−Δ)Ds u,u

u 2s*

2 = infu∈ ˜ H s (Ω), u≠0

(−ΔΩ)Ns u,u

u 2s*

2 .

d. The "Dirichlet-Hardy" and the "Navier-Hardy" constants coincide as well:

€

infu∈ ˜ H s (Ω), u≠0

(−Δ)Ds u,u

| x |−s u2

2 = infu∈ ˜ H s (Ω), u≠0

(−ΔΩ)Ns u,u

| x |−s u2

2 .

References: [MN1] R. Musina, A.I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eqs, vol. 39 (2014), no. 9, 1780-‐-‐1790. [MN2] R. Musina, A.I. Nazarov, On fractional Laplacians-II, preprint (2014).


17

On the Cauchy problem for scalar balance laws in the class of Besicovitch almost

periodic functions

Evgeny Panov

Novgorod State University, Russia [email protected]

We study the Cauchy problem for a first order

quasilinear equation (a balance law) with a merely continuous flux vector and almost periodic (in Besicovitch sense) initial and source functions.

It is proved that the Kruzhkov entropy solution is almost periodic with respect to the spacial variables, it is unique (as a function with values in the Besicovitch space), and its spectrum is contained in the additive subgroup generated by the spectra of initial and source data. A precise nondegeneracy condition is also found for large-‐time decay property of entropy solutions.


18

Homogenization of monotone operators with oscilating exponent of nonlinearity

Svetlana Pastukhova

Moscow Institute of Radioengineering, Electronic and Automation, Russia pas-‐[email protected]

We consider the Dirichlet problem for an

elliptic monotone-‐type equation under coerciveness and growth conditions with a variable exponent

€

pε (x).Here

€

pε (x) = p xε( ) and

€

p(y) is a measurable periodic function such that

€

1 < α ≤ p(y) ≤ β < ∞.When the small parameter

€

ε > 0 tends to zero, the exponent

€

pε (x)$ is highly oscillating and we need to homogenize the problem passing to the limit as

€

ε →0. Generally, we have here the Lavrent’ev phenomenon and solutions of two different types should be taken into consideration. These are so-‐called

€

W -‐ and

€

H -‐solutions considered, respectively, in the most broad and in the most narrow Sobolev spaces connected with the exponent

€

pε (x). For each type of solutions, we give the homogenization procedure and describe the limit problem.

This is a joint result with prof. V.V. Zhikov.


19

Viscous incompressible free-surface flow down an inclined perturbed plane

Konstantinas Pileckas

Vilnius University, Lithuania [email protected]

The stationary plane free boundary value problems for the Navier-‐Stokes equations is studied. The problem models the viscous fluid free-‐surface flow down a perturbed inclined plane. For sufficiently small data the solvability and uniqueness results are proved in Hölder spaces. The asymptotic behaviour of the solution is investigated. This is a joint work with V.A. Solonnikov.


20

On the regularity and stability of the free boundary of obstacle type heterogeneous

problems

José Francisco Rodrigues

CMAF, University of Lisbon, Portugal [email protected]

We extend some properties to the solutions of

free boundary problems of obstacle type with two phases for a class of heterogeneous quasilinear elliptic operators.

Under a natural non degeneracy assumption on the interface, corresponding to the zero level set, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgue measure with respect to the

€

L1-‐variation of the data, in a rather general framework.

For elliptic quasilinear equations that have solutions with integrable second order derivatives, we show that the characteristic functions of both phases are of bounded variations for non degenerating heterogeneous forces.

This extends recent results for the obstacle problem and is a first result on the regularity of the free boundary of the heterogeneous two phases problem.


21

Leray-Hopf solutions to Navier-Stokes equations with weakly converging initial data

Gregory Seregin

University of Oxford, UK and St. Petersburg Steklov Institute of Mathematics, Russia

[email protected]

The talk is addressed the question about convergence of a sequence of week Leray-‐Hopf solutions to the initial boundary value problem for the 3D Navier-‐Stokes equations provided that the corresponding initial data converge weakly to their limit.

Under certain rather mild assumptions, it is shown that the limit velocity field is a weak Leray-‐Hopf solution with the limit initial data.


22

Logarithmic interpolation-embedding inequalities on irregular domains

Tatyana Shaposhnikova

Royal Institute of Technology, Sweden [email protected]

Logarithmic interpolation-‐embedding inequalities of Brezis-‐Gallouet-‐Wainger type are proved for various classes of irregular domains, in particular, for power cusps and lambda-‐John domains. This is a joint work with Vladimir Maz'ya.


23

€

Lp -estimates of solutions of some problems of hydrodynamics and

magnetohydrodynamics

Vsevolod Solonnikov

St. Petersburg Steklov Institute of Mathematics, Russia

[email protected] We discuss

€

Lp -‐estimates of solutions of some initial-‐boundary value problems arising in the analysis of motion of an isolated liquid mass of viscous incompressible electrically conducting fluid.


24

Regularity of solutions to elliptic and parabolic systems with

€

L1 right hand side

Jana Stará

Charles University, Czech Republic [email protected]

The talk is devoted to extension of the results

of P. Baroni, J. Habermann for one nonlinear parabolic equation (published in 2012) to parabolic systems. We wanted to avoid stronger ellipticity assumption and we used an approach based on ideas of G. Stampacchia. This approach uses duality to define a type of very weak solution. As a consequence we restricted ourselves to the study of existence and regularity properties of solutions to linear parabolic systems with non smooth coefficients and right hand sides.

We consider the parabolic system

€

∂u∂t− div(ADu) = f in Q,

u = 0 on ∂pQ,

where

€

Q = (0,T) × Ω ,

€

Ω is a

€

C1 bounded domain in n and

€

∂pQ is the parabolic boundary of

€

Q. We obtain existence of very weak solutions for coefficient matrix

€

A that satisfies standard ellipticity and boundedness conditions and whose entries belong to Sarason space VMO

€

(Q) . We prove that for any right hand side

€

f ∈L1(Q) there exists unique very weak solution

€

u such that

€

u∈Lr(0,T;W01,q (Ω; n

€

)) for any

€

r, q ≥1;

€

2r + n

q ≥ n +1.


25

For "better" right hand side

€

f in Morrey space

€

L1,τ (Q) we prove that

€

Du∈Llocq,ν (Q); n with

€

q∈ 1, n+2n+1( ),

€

ν = n + 2 − q[n +1−τ(n + 2)]. Moreover, for coefficients that are

€

α -‐Hölder continuous in space variables we prove fractional differentiability of space gradient of

€

u , i.e.

€

Du∈Wlocη ,η / 2,q,τ with a positive number

€

η which depends on the parameters of the problem.


26

Properties of the free boundary in the optimal compliance problem

Eugene Stepanov

St. Petersburg Steklov Institute of Mathematics, Russia

[email protected]

We consider the problem of finding the optimal shape of a support of some elastic material under the given load so as to optimize the compliance of the latter. The topological properties of the optimal support (which may be viewed as the “free boundary”) as well as its regularity will be studied.


27

A chemotaxis-model with non-diffusing attractor

Angela Stevens

University of Münster, Germany [email protected]

Cell motion due to attractive chemicals is a

widespread phenomenon in developmental biology. A classical example is chemotaxis, which -‐ among others -‐ is modeled via the well known Keller-‐Segel cross-‐diffusion system.

In this talk a variant of this mathematical model is presented, namely a system where the attractive chemical is not diffusing. So an ODE instead of a reaction-‐diffusion equation is coupled to the nonlinear chemotaxis equation for the cells.

This model is analyzed w.r.t. the existence of global solutions and blow-‐up. The PDE-‐ODE system behaves very different in this respect if compared with the classical Keller-‐Segel model for chemotaxis.


28

Homogenization of periodic elliptic operators: error estimates in dependence of

the spectral parameter

Tatiana Suslina


We study matrix elliptic second order

differential operators

€

Aε ,

€

ε > 0 , in d or in a bounded domain

€

O ⊂d (with sufficiently smooth boundary) with the Dirichlet or Neumann boundary conditions. It is assumed that

€

Aε = b(D)*g(x /ε)b(D) , where a

€

(m × m) -‐matrix-‐valued function

€

g(x) is bounded, uniformly positive definite and periodic with respect to

some lattice;

€

b(D) = b jDjj=1

d∑ is

€

(m × n)-‐matrix first order differential operator. It is assumed that

€

m ≥ n and the symbol

€

b(ξ) = b jξ jj=1

d∑ has maximal

rank. We study the behavior of the resolvent

€

(Aε −ζ I)−1 for small

€

ε . It turns out that this resolvent converges to

€

(A0 −ζ I)−1 in the

€

L2-‐operator norm, as

€

ε →0 , where

€

A0 = b(D)*g0b(D) is the effective operator. We find twoparametric error estimates for the norm of the difference

€

(Aε −ζ I)−1 − (A0 −ζ I)−1

with respect to

€

ε and

€

ζ . Also, we find approximation for the resolvent

€

(Aε −ζ I)−1 in the

€

(L2 →H1)-‐operator norm with twoparametric error estimates. In this approximation, the corrector term is taken into account. The results can be applied to homogenization of parabolic initial boundary-‐value problems.


29

Free boundary problems for mechanical models of tumor growth

Juan Luis Vazquez

Autonomous University of Madrid, Spain [email protected]

Mathematical models of tumor growth, now

commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. We first formulate a free boundary model of Hele-‐Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit which leads to a Hele-‐Shaw type problem. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the limit problem. At variance with the classical Hele-‐Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics.

Using this theory as a basis, we go on to consider a more complex model including nutrients. Here, technical difficulties appear, that reduce the generality and detail of the description. We prove uniqueness for the system, a main mathematical difficulty.

Joint work with Benoit Perthame, from Paris, and Fernando Quiros, from Madrid.


30

On density of smooth functions in weighted Sobolev Spaces

Vasily Zhikov

Vladimir State University, Russia [email protected]

advances in nonlinear pdes · 2014-09-01 · advances!in!nonlinear!pdes! september!385,2014!!! 3!...

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