Stability of vacuum for the Landau equation with moderately soft potentials

TL;DR

This paper proves the global stability of vacuum solutions for the Landau equation with moderately soft potentials, showing solutions remain regular and approach linear transport solutions over time.

Contribution

It is the first to establish vacuum stability for a long-range interaction collisional model, using weighted energy estimates and null structure techniques.

Findings

Solutions near vacuum remain regular globally in time.

Solutions approach linear transport equations as time goes to infinity.

Solutions generally do not tend to a Maxwellian at infinity.

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.