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

Although there recently have been extensive studies on the pertur-bation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrabil-ity, we establish the large time stability of solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff inter-molecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy X(t)-norm to ob-tain its uniform bound. The result also extends the case of the hard sphere model considered by Guo [Invent. Math. 153(3): 593-630 (2003)] to the general collision potentials. © American Institute of Mathematical Sciences.