Stability of Global Maxwellian for Fully Nonlinear Fokker-Planck Equations

This paper considers the stability of solutions around a global Maxwellian to the fully non-linear Fokker-Planck equation in the whole space. This model preserves mass, momentum and energy at the same time, and its dissipation is much weaker than that in the simplified model considered in Liao et al. (J Stat Phys 173:222-241, 2018). To overcome the new difficulties, the macro-micro decomposition of the solution around the local Maxwellian and energy estimates introduced in Liu et al. (Physica D 188:178-192, 2004) and Yang and Zhao (J Math Phys 47:053301, 2006) for Boltzmann equation is used. That is, we reformulate the model into a fluid-type system coupled with an equation of the microscopic part. The a priori estimates of the solution could be obtained by the standard energy method. Especially, by careful computation, the viscosity and heat diffusion terms in the fluid-type system are derived from the microscopic part, which give the dissipative mechanism to the system.