Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation

Abstract

We consider the spatially inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential, a quantum modification of the classical Landau equation for fermions. Mathematically, the Pauli exclusion principle manifests as an additional a priori $L^\infty$-bound for solutions. Using this bound, we propagate polynomial decay in velocity, yielding unconditional upper bounds on the local mass and energy densities, thereby ruling out the possibility of implosions in the hydrodynamic quantities. Combining this estimate with a modified mass spreading method that yields desaturation, we deduce the existence of global-in-time classical solutions for rough initial data with polynomial decay in velocity. This result stands in stark contrast to the theory for the classical Landau and Boltzmann equations, for which no comparable nonperturbative global existence result is known despite sustained effort. Our treatment is almost entirely self-contained, using only robust, generic estimates for linear kinetic equations. In particular, our proof of local existence, in contrast to prior works, more closely mirrors the theory for parabolic equations using weak solutions and simpler function spaces. It may provide a concise roadmap to organizing, adapting, and applying the various linear and nonlinear estimates to obtain well-posedness for kinetic equations.