Stability of vacuum for the Landau equation with moderately soft potentials

Abstract

Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with $\gamma \in (-2,0)$) on the whole space $\mathbb R^3$. We prove that if the initial data $f_{\mathrm{in}}$ are close to the vacuum solution $f_{\mathrm{vac}} \equiv 0$ in an appropriate norm, then the solution $f$ remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction. Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as $t\to +\infty$. Furthermore, in general, solutions do not approach a traveling global Maxwellian as $t \to +\infty$. Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions.