On massless electron limit for a multispecies kinetic system with external magnetic field

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio $m_e/m_i$ tends to zero. Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satisfies an anisotropic drift-diffusion equation. To achieve this task, we overcome some specific technical issues of our model such as the strong effect of the magnetic field on electrons and the lack of regularity at the limit. With methods usually adapted to diffusion limit of collisional kinetic equations and including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model

Global estimates for nonlinear parabolic equations

We consider nonlinear parabolic equations of the type $$u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0),$$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.Comment: To appear in J. Evol. Equation

Weak solutions for some compressible multicomponent fluid models

The principle purpose of this work is to investigate a "viscous" version of a "simple" but still realistic bi-fluid model described in [Bresch, Desjardin, Ghidaglia, Grenier, Hillairet] whose "non-viscous" version is derived from physical considerations in \cite[Ishii, Hibiki]{ISHI} as a particular sample of a multifluid model with algebraic closure. The goal is to show existence of weak solutions for large initial data on an arbitrarily large time interval. We achieve this goal by transforming the model to an academic system which resembles to the compressible Navier-Stokes equations, with however two continuity equations and a momentum equation endowed with pressure of complicated structure dependent on two variable densities. The new "academic system" is then solved by an adaptation of the Lions--Feireisl approach for solving compressible Navier--Stokes equation, completed with several observations related to the DiPerna--Lions transport theory inspired by [Maltese, Michalek, Mucha, Novotny, Pokorny, Zatorska] and [Vasseur, Wen, Yu]. We also explain how these techniques can be generalized to a model of mixtures with more then two species. This is the first result on the existence of weak solutions for any realistic multifluid system

On the strong convergence of the gradient in nonlinear parabolic equations

We consider the Cauchy-Dirichlet Problem for a nonlinear parabolic equation with L1 data. We show how the concept of kinetic formulation for conservation laws [Lions, Perthame, Tamor 94] can be be used to give a new proof of the existence of renormalized solutions. To illustrate this approach, we also extend the method to the case where the equation involves an additional gradient term