Abstract

We consider the spatially inhomogeneous Landau equation with Maxwellian and hard potentials (i.e, with $\gamma \in [0,1)$) on the whole space ${ℝ}^3$. 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 and the long-time behavior is governed by dispersion coming from the transport operator. This work builds upon the author’s earlier work on local existence of solutions to the Landau equation with hard potentials.

Our proof uses an energy based approach and exploits the null structure established by Luk in Annals of PDE 5:11 (2019). To be able to close our estimates, we have to couple the weighted energy estimates, which were established by the author in arXiv:1910.11866 (2019) with the null structure and devise new weighted norms that take this into account.

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