Optimal time decay of the non cut-off Boltzmann equation in the whole space

Abstract

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\threed_x$ with $\DgE$. We use the existence theory of global in time nearby Maxwellian solutions from \cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption \cite{MR677262,MR2847536}. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}})$ in the $L^2_\vel(L^r_x)$-norm for any $2\le r\le \infty$.

31 pages, final version to appear in KRM

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• Global well-posedness of the Boltzmann equation with large amplitude initial data
• Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas
• The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials
• Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation
• Regularity of the Vlasov-Poisson-Boltzmann System without angular cutoff
• Smoothing Estimates of the Vlasov-Poisson-Landau System
• Non-cutoff Boltzmann equation with soft potentials in the whole space