The relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space

Abstract

We are concerned with the Cauchy problem for the relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. For perturbative initial data with suitable regularity and integrability, we obtain the global classical solutions and the large time decay rates. For the proof, a new structure of the linearized relativistic Fokker–Planck operator is observed so that the coercivity estimates are established, crucial for the optimal large time decay rates is the pivotal Sobolev inequalities and interpolation inequalities as well as the structure of the coupled system. Moreover, it is shown that the energy without any weights decays in time with the polynomial rates varying directly with the derivatives of the solution, and the lower order derivatives of the solution except the magnetic field are further proved to decay faster than the optimal ones which coincide with the linearized equations without electro and magnetic fields, such a new phenomenon occurs only in the case of the relativistic Fokker–Planck model.