Nonlinear quantum Fokker-Planck equation near equilibrium

We investigate a nonlinear quantum Fokker--Planck equation with self-consistent collision frequency, bulk velocity, and temperature.

In contrast to quantum Fokker--Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function.

The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case.

Its collision operator is also formally connected to the quantum Landau equation.

For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium.

The proof is based on a perturbative macro--micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument.

We further establish the propagation of nonnegativity and the fermionic Pauli upper bound.

Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.