Stability of the One‐Species Vlasov‐Poisson‐Boltzmann System With Given Magnetic Field

ABSTRACT In this paper, we focus on the one‐species Vlasov‐Poisson‐Boltzmann system under a given magnetic field with nonconstant background density in the whole space. We first give the existence of a stationary solution when the background density is a small perturbation of a constant state. Secondly, we establish the nonlinear stability of the Cauchy problem near the stationary solution in certain Sobolev spaces. The proof relies on macroscopic balance laws and several interactive energy functionals from [1], extending the results in [1] to the magnetized case and demonstrating that a constant magnetic field does not affect the nonlinear stability of the stationary solution.