# From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics

Volume 1

### Diogo Arsénio

Université Paris Diderot, France### Laure Saint-Raymond

École Normale Supérieure, Lyon, France

The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subject to self-induced electromagnetic forces. At the macroscopic scale, in the incompressible viscous fluid limit the evolution of the plasma is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing that may take on various forms depending on the number of species and on the strength of the interactions.

From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way.

The first part of this work is devoted to the systematic formal analysis of viscous hydrodynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature.

In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman–Enskog-type decomposition and regularity in the $v$ variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more).

The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known.