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The Boltzmann equation, Besov spaces, and optimal time decay rates in $$\mathbb{R}_x^n$$. (English) Zbl 1293.35195
Summary: 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 $$n \geq 3$$, converge in large time to the global Maxwellian with the optimal decay rate of $$O(t^{- \frac{1}{2}(k + \varrho + \frac{n}{2} - \frac{n}{r})})$$ in the $$L_x^r(L_v^2)$$-norm for any $$2 \leq r \leq \infty$$. These results hold for any $$\varrho \in(0, n / 2]$$ as long as initially $$\| f_0 \|_{\dot{B}_2^{- \varrho, \infty} L_v^2} < \infty$$. In the hard potential case, we prove faster decay results in the sense that if $$\| \mathbf{P} f_0 \|_{\dot{B}_2^{- \varrho, \infty} L_v^2} < \infty$$ 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 the global Maxwellian in $$L_v^2(L_x^2)$$ with the optimal large time decay rate of $$O(t^{- \frac{1}{2} \varrho})$$.

##### MSC:
 35Q20 Boltzmann equations 35R11 Fractional partial differential equations 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 82C40 Kinetic theory of gases in time-dependent statistical mechanics 35B65 Smoothness and regularity of solutions to PDEs 26A33 Fractional derivatives and integrals
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