Showing papers by "Jean Dolbeault published in 2001"

Large time asymptotics of nonlinear drift-diffusion systems with poisson coupling

Abstract: We study the asymptotic behavior as t→ +∞ of a system of densities of charged particles satisfying nonlinear drift-diffusion equations coupled by a damped Poisson equation for the drift-potential. In plasma physics applications the damping is caused by a spatio-temporal rescaling of an “unconfined” problem, which introduces a harmonic external potential of confinement. We present formal calculations (valid for smooth solutions) which extend the results known in the linear diffusion case to nonlinear diffusion of e.g. Fermi–Dirac or fast diffusion/porous media type.

On Singular Limits of Mean-Field Equations

Abstract: Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.

Time-dependent rescalings and lyapunov functionals for the vlasov–poisson and euler–poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics

Abstract: We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrodinger–Poisson system, nonlinear Schrodinger equation).

Stationary solutions, intermediate asymptotics and large-time behaviour of type II Streater's models

Abstract: In this paper, we study type II Streater’s models. These models describe the coupled evolution of the density of a cloud of particles in an external potential and a temperature, preserving the energy, with eventually a nonlocal Poisson coupling. We introduce an entropy and consider in a bounded domain, or in an unbounded domain with a confining external potential, the stationary solutions (with given mass and energy), for which we have existence and uniqueness results. The entropy is reinterpreted as a relative entropy which controls the convergence to the stationary solutions. We consider also the whole IR d space problems without exterior potential using time-dependent rescalings and show the existence of intermediate asymptotics in special cases.