On the local and global well-posedness theory for the KP-I equation
On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations.
On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients
On the local well-posedness of the KP equations
On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth.
On the long time behavior of KdV type equations
In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.
On the long time behaviour of solutions of a class of autonomous reaction-diffusion systems.
On the long-time behaviour of compressible fluid flows subjected to highly oscillating external forces
We show that the global-in-time solutions to the compressible Navier-Stokes equations driven by highly oscillating external forces stabilize to globally defined (on the whole real line) solutions of the same system with the driving force given by the integral mean of oscillations. Several stability results will be obtained.
On the long-time behaviour of solutions of the p-Laplacian parabolic system
Convergence of global solutions to stationary solutions for a class of degenerate parabolic systems related to the p-Laplacian operator is proved. A similar result is obtained for a variable exponent p. In the case of p constant, the convergence is proved to be , and in the variable exponent case, L² and -weak.
On the low frequency asymptotics in electromagnetic theory.
On the maximum modulus theorem for the Stokes system
On the maximum principle for principal curvatures
The paper contains the estimates from above of the principal curvatures of the solution to some curvature equations. A correction of the author's previous argument is presented.
On the modeling of the transport of particles in turbulent flows
We investigate different asymptotic regimes for Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.
On the modeling of the transport of particles in turbulent flows
We investigate different asymptotic regimes for Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.
On the natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva
On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent
We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.
On the Neumann problem for systems of elliptic equations involving homogeneous nonlinearities of a critical degree
We establish the existence of solutions for the Neumann problem for a system of two equations involving a homogeneous nonlinearity of a critical degree. The existence of a solution is obtained by a constrained minimization with the aid of P.-L. Lions' concentration-compactness principle.
On the Neumann problem of one-dimensional nonlinear thermoelasticity with time-independent external forces
On the Neumann problem with combined nonlinearities
We establish the existence of multiple solutions of an asymptotically linear Neumann problem. These solutions are obtained via the mountain-pass principle and a local minimization.
On the Neumann problem with L¹ data
We investigate the solvability of the linear Neumann problem (1.1) with L¹ data. The results are applied to obtain existence theorems for a semilinear Neumann problem.