Stability of vacuum for the Landau equation with hard potentials

Abstract

We consider the spatially inhomogeneous Landau equation with Maxwellian and hard potentials (i.e with $\gamma \in [0,1)$) on the whole space ${\mathbb{R}}^3$. We prove that if the initial data $f_{\text{in}}$ are close to the vacuum solution $f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$ exists globally in time. This work builds up on the author's earlier work on local existence of solutions to Landau equation with hard potentials. Our proof uses $L^2$ estimates and exploits the null-structure established by Luk [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11]. To be able to close our estimates, we have to couple the weighted energy estimates, which were established by the author in a previous paper [Local existence for the Landau equation with hard potentials, arXiv:1910.11866], with the null-structure and devise new weighted norms that take this into account.

49 pages; comments welcome!

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