Abstracts :
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François HAMEL (Marseille)
Title : Rearrangement inequalities and applications to elliptic eigenvalue problems.
Abstract : The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of Rn. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
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Diego CORDOBA (Madrid)
Title : Rayleigh-Taylor breakdown for the Muskat problem.
Abstract : The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but breakdown in finite time.
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Marta SANZ-SOLE (Barcelone)
Title : A variational approach to large deviations for stochastic waves.
Abstract : Using Budhiraja and Dupuis' approach to large deviations, we shall establish such a principle for a non-linear stochastic wave equation in spatial dimension d=3 driven by a noise white in time and coloured in space. Firstly, we will derive a variational representation of the noise and then we will prove suitable convergences leading to the large deviation principle in Hölder norm.
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Robert DALANG (Lausanne)
Title : Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension k ≥ 1.
Abstract : click here.
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Arnaud DEBUSSCHE (Rennes)
Title : The nonlinear Schrödinger equation with white noise dispersion.
Abstract : click here.
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Gregory SEREGIN (Oxford)
Title : Liouville type theorems with applications to Navier-Stokes equations.
Abstract : In the talk, we are going to discuss fine properties of solutions to the heat equation with the divergence free drift which is a bounded in time function taking values in the space of functions with bounded mean oscillations. The problem is motivated by Liouville type theorems for the Navier-Stokes equations. Our approach is based on higher integrability and Harnack's inequality. Some results are almost sharp.
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Marek FILA (Bratislava)
Title : Homoclinic and heteroclinic orbits for a semilinear parabolic equation.
Abstract : We study the existence of connecting orbits for the Fujita equation ut=Δu+up with a critical or supercritical exponent p. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.
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Pavol QUITTNER (Bratislava)
Title : Liouville-type theorems and blow-up rate estimates for problems with nonlinear boundary conditions.
Abstract : We will review results on the blow-up rate of solutions of parabolic problems with nonlinear boundary conditions and compare these results and methods of their proofs with corresponding results and methods for the nonlinear heat equation. In particular, we will be interested in results based on Liouville-type theorems.
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Franco FLANDOLI (Pise)
Title : The interaction of noise with non-uniqueness and singularities of PDEs.
Abstract : It is well known that well posedness of ODEs is improved by white noise perturbations. The understanding of similar regularization phenomenona for PDEs is only at the beginning. A few examples that we start to understand will be discussed and compared. They include some inviscid linear and nonlinear models of interest in fluid dynamics, like transport equations, 2D point vortex motion, dyadic models of turbulence. Viscous cases where well posedness is an issue seem to be even more difficult; we shall remark on potential singularities of the 3D Navier-Stokes equations.
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Luigi MANCA (Marne-la-Vallée)
Title : Fokker-Planck equations for Kolmogorov operators on infinite dimensional spaces.
Abstract :