##
**On the Cauchy problem for Landau equations: Sequential stability, global existence.**
*(English)*
Zbl 0856.35020

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.

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.

Reviewer: J.Dolbeault (Paris)