1206.0027 (Vedran Sohinger et al.)

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

Vedran Sohinger, Robert M. Strain

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, $\threed_x$ with $\DgE$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\ALTsig+\frac{\Ndim}{2}-\frac{\Ndim}{r})})$ in the $L^r_x(L^2_{\vel})$-norm for any $2\leq r\leq \infty$. These results hold for any $\ALTsig \in [0, \Ndim/2]$ as long as initially $\| f_0|_{\dot{B}^{-\ALTsig,\infty}_2 L^2_{\vel}} < \infty$. In the hard potential case, we prove faster decay results in the sense that if $|\FP f_0\|_{\dot{B}^{-\ALTsig,\infty}_2 L^2_{\vel}} < \infty$ and $|{\FI - \FP} f_0|_{\dot{B}^{-\ALTsig+1,\infty}_2 L^2_{\vel}} < \infty$ for $\ALTsig \in (\Ndim/2, (\Ndim+2)/2]$ then the solution decays to zero in $L^2_\vel(L^2_x)$ with the optimal large time decay rate of $O(t^{-1/2\ALTsig})$.

View original: http://arxiv.org/abs/1206.0027