Global existence of classical solutions to the Vlasov–Poisson–Boltzmann system with given magnetic field

This paper considers the Vlasov–Poisson–Boltzmann system with given magnetic field. The global existence of classical solutions was obtained when the initial data is a small perturbation around a global Maxwellian. The proof is based on the theory of compressible Navier–Stokes–Poisson equations with forcing and the macro–microdecomposition of the solution to the Boltzmann equation with respect to the local Maxwellian introduced in [T.-P. Liu, T. Yang, S.-H. Yu, Energy method for the Boltzmann equation, Physica D 188 (3–4) (2004) 178–192] and elaborated in [T. Yang, H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys. 47 (2006)]. The result shows that the existence of solutions is independent of the magnetic field.