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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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