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Advances in Nonlinear Pdes ! Book$Of$Abstracts$ ! ! Advances in Nonlinear PDEs ! in#honor#of##Nina#N.#Uraltseva## Book$of$abstracts$ St.!Petersburg,!!2014!!! ! ! Advances in Nonlinear PDEs September 3-5, 2014 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:! St.!Petersburg!Steklov!Ins7tute! of!Mathema7cs! 1 Advances in Nonlinear PDEs September 3-5, 2014 ! ! Organizing(Commi,ee:( Darya%Apushkinskaya% Alexander%Mikhailov% Alexander%Nazarov% Sergey%Repin% Ta;ana%Vinogradova% Nadya%Zalesskaya% 2 Advances in Nonlinear PDEs September 3-5, 2014 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. 3 Advances in Nonlinear PDEs September 3-5, 2014 Semilinear elliptic problems with a Hardy potential Catherine Bandle University of Basel, Switzerland [email protected] We consider problems of the type p Δu +V (x)u = u in a bounded domain in n where µ V is a Hardy potential and δ(x) is the distance δ 2(x) 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) 4 Advances in Nonlinear PDEs September 3-5, 2014 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. 5 Advances in Nonlinear PDEs September 3-5, 2014 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. 6 Advances in Nonlinear PDEs September 3-5, 2014 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 . The structure of dynamical flow corresponding to these NPE v 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. 7 Advances in Nonlinear PDEs September 3-5, 2014 On attractors of m-Hessian evolutions Nina Ivochkina St. Petersburg State University, Russia [email protected] Let Ω be a bounded domain in n, 2,1 Q = Ω × (0;∞), u ∈C (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 u and introduce p- ≤€ ≤ xx € Hessian evolution operator by € E [u]€:= u T [u]+€T [u]. € p t p −1 p We investigate € asymptotic behavior of € solutions € of the following € initial boundary € value problems: E [u] = f , u = φ, 1≤ m ≤ n, (1) € m ∂ 'QT where . In ∂'QT = {Ω × {0}} ∪{∂Ω × [0;T]} particular, we have proved € 2,1 Theorem 1. Let f ≥ν > 0, f ∈C (QT ) for all € 2,1 T ∈[0;∞), φ ∈C (QT ), φ = 0 on ∂Ω × [0;∞), 2 ∂Ω ∈C . Assume that limt → ∞ f (x,t) = f (x) and 2 there exists € a solution € u ∈C (Ω ) to the Dirichlet problem € € € € T [u] = f , u = 0. € € m ∂Ω Then all solutions u ∈C 2,1(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. € € € € 8 Advances in Nonlinear PDEs September 3-5, 2014 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 C 2,1(Q ) to the problem (1), whatever T € 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) ∈K m −1(Ω ). Assume that compatibility conditions are satisfied. Then there exists a unique in C 2,1(Q ) solution to the problem (1). T € € € € 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. 9 Advances in Nonlinear PDEs September 3-5, 2014 Analyticity of the free boundary in the thin obstacle problem Herbert Koch University of Bonn, Germany [email protected] 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. 10 Advances in Nonlinear PDEs September 3-5, 2014 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. 11 Advances in Nonlinear PDEs September 3-5, 2014 Common features of homogenization and of large scale limits in statistical mechanics Stephan Luckhaus Leipzig University, Germany [email protected] 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. 12 Advances in Nonlinear PDEs September 3-5, 2014 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. 13 Advances in Nonlinear PDEs September 3-5, 2014 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.
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