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 γ ∈ [0, 1)) on the whole space R 3 . We prove that if the initial data f in are close to the vacuum solution f 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, https://arxiv.org/abs/1910.11866], with the null-structure and devise new weighted norms that take this into account.

The Vlasov--Poisson--Landau system in the weakly collisional regime

“…(See also [5,4,44,41,98,97,96] for related works on stability of vacuum type results for collisionless models.) For collisional models, recent works using the commutating vector field method give -for the first time -stability of vacuum results for collisional models with a long range interaction, first for the Landau equation [69,29], and more recently for Boltzmann equation without angular cutoff [28].…”

“…This lets one prove transport bounds in the presence of collision, and in fact to take advantage of the coercivity given by collision. Such a commutating vector field method is inspired by related techniques for treating dispersion in nonlinear wave equations and other kinetic models [28,29,63,69,83]. See Sections 1.1.4 and 1.2.6.…”

Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations

“…We say that a solution f blows up at a time Maxwellian equilibrium state, solutions exist globally and converge to equilibrium [41]. Solutions are also known to exist when initial data are near vacuum in the cases of moderately soft potentials [59] and hard potentials [18]. Recently, several new studies concerning regularity and continuation criteria have appeared; these results are based on conditional assumptions on the hydrodynamic quantities, see [37,11,45,21,57].…”

Stability of vacuum for the Boltzmann Equation with moderately soft potentials

“…The stability of vacuum for Landau equation with hard potentials (i.e. γ ∈ [0, 1]) was considered by the author in [24].…”

“…Although the present result is comparable to that of Luk in [61] in the sense that the range of potentials considered here (γ + 2s ∈ (0, 2]) is an analogue of the moderately soft potentials for Landau (γ ∈ (−2, 0)), the Boltzmann kernel poses various technical difficulties which are not present in [61]. Another major difference is that while Luk uses maximum principle in conjunction with energy estimates in [61], we only use L 2 estimates, as in [24]. We also need to combine the vector field approach used by Smulevici in [64] with the space-time weights in our energy norm to get enough time decay for us to close the estimates.…”