On the Cauchy problem for Landau equations: Sequential stability, global existence
type: Item
instance of: scholarly article
title: On the Cauchy problem for Landau equations: Sequential stability, global existence
zbMATH Open document ID: 0856.35020
published in: Advances in Differential Equations
publication date: 1997-02-19T00:00:00Z
review text: This paper deals with the non-homogeneous Landau (or Fokker-Planck) equation \[ {\partial f\over \partial t}+ v\cdot \nabla_x f= Q(f, f),\quad t\geq 0,\quad x\in \mathbb{R}^N,\quad v\in \mathbb{R}^N, \] \((N\geq 1, f\geq 0)\) with \[ Q(f, f)= \sum^N_{i, j= 1} {\partial\over \partial v_i} \Biggl\{ \int_{\mathbb{R}^N} dv_* a_{ij}(v- v_*)\Biggl[ f(v_*) {\partial f(v)\over \partial v_j}- f(v) {\partial f(v_*)\over \partial v_{*,j}}\Biggr]\Biggr\}, \] where \((a_{ij})\) is symmetric, nonnegative, even in \(z\) and such that \(\sum_{i, j} a_{ij}(z) z_i z_j= 0\) for almost every \(z\). The main result of the paper is an existence result of renormalized solutions with a defect measure for the Cauchy problem under technical assumptions that include the Coulomb case. The result relies on a compactness property (based on a priori estimates) previously obtained by P.-L. Lions and on a detailed study of the collision operator \(Q\), which provide a weak stability result for sequences of solutions. The author also introduces the notion of quasi-renormalized solutions i.e. renormalized solutions with a defect measure such that it vanishes as the parameter of the renormalization tends to zero, and gives two sufficient conditions for a solution to be quasi-renormalized. The long time behaviour is also studied using a Boltzmann's H-theorem, which is established in the paper: Renormalized solutions with a defect measure converge for large times -- up to the extraction of a subsequence -- to a Maxwellian function (but no asymptotic uniqueness result holds since the conservation of the energy is not established as in Boltzman's equation). This paper uses various techniques of the renormalized solutions that have been introduced by R. J. DiPerna and P.-L. Lions. The notion of renormalized solution is generalized and adapted to the Landau equation and a very careful analysis of the collision operator is provided.
reviewed by: Jean Dolbeault
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 82D10
zbMATH DE Number: 896500
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: quasi-renormalized solutions
zbMATH Keywords: defect measure
zbMATH Keywords: Landau equation
author: Cédric Villani
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 896500