Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials

Abstract

Although there recently have been extensive studies on theperturbation theory of the angular non-cutoff Boltzmann equation(cf. [4] and [17]), it remains mathematicallyunknown when there is a self-consistent Lorentz force coupled withthe Maxwell equations in the nonrelativistic approximation. In thepaper, for perturbative initial data with suitable regularity andintegrability, we establish the large time stability of solutionsto the Cauchy problem of the Vlasov-Maxwell-Boltzmann system withphysical angular non-cutoff intermolecular collisions including theinverse power law potentials, and also obtain as a byproduct theconvergence rates of solutions. The proof is based on a newtime-velocity weighted energy method with two key technical parts:one is to introduce the exponentially weighted estimates into thenon-cutoff Boltzmann operator and the other to design a delicatetemporal energy $X(t)$-norm to obtain its uniform bound. The resultalso extends the case of the hard sphere model considered by Guo[Invent. Math. 153(3): 593--630 (2003)] to the general collisionpotentials.