The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space

Abstract

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb{R}}_x^n$ with $n\ge 3$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\varrho +\frac{n}{2}-\frac{n}{r})})$ in the $L_x^r(L_v^2)$-norm for any $2\le r\le \infty$. These results hold for any $\varrho \in [0,n/2]$ as long as initially $\Vert f_0{|}_{{\dot{B}}_2^{-\varrho ,\infty }{L}_v^2}<\infty$. In the hard potential case, we prove faster decay results in the sense that if $|\mathbf{P}{f}_0{\Vert }_{{\dot{B}}_2^{-\varrho ,\infty }{L}_v^2}<\infty$ and $|(\mathbf{I}-\mathbf{P})f_0{|}_{{\dot{B}}_2^{-\varrho +1,\infty }{L}_v^2}<\infty$ for $\varrho \in (n/2,(n+2)/2]$ then the solution decays to zero in $L_v^2(L_x^2)$ with the optimal large time decay rate of $O(t^{-1/2\varrho })$.

47 pages. Updated paper to incorporate referee suggestions. Substantially shortened. Previous version may also be useful since it contains additional details in some places. Accepted for publication in Advances in Mathematics

References in corpus (6)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
• The Vlasov-Poisson-Landau System in ${\mathbb{R}}_x^3$
• Global solutions to the Vlasov-Poisson-Landau System
• Decay of dissipative equations and negative Sobolev spaces
• Global well-posedness in spatially critical Besov space for the Boltzmann equation

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