\(L^{\infty}\) estimates for the space-homogeneous Boltzmann equation. (English) Zbl 0584.35090

The author studies the boundedness of solutions f of the initial-value problem for the space-homogeneous Boltzmann equation for inverse kth power forces, when \(k>5\), and under angular cutoff. The main result is that if the initial value is \(f_ 0\geq 0\) with \((1+| v|^ 2)f_ 0\in L^ 1,\) and \((1+| v|)^ sf_ 0\in L^{\infty}\) for some \(s>2\), then \((1+| v|)^{s'}f_ t\in L^{\infty}\) for \(t>0\) and \( \sup_{v,t}(1+| v|)^{s'}f(v,t)<\infty\) for any s’\(\leq s\) when \(s\leq 5\), and any \(s'<s\) if \(s>5\).


35Q99 Partial differential equations of mathematical physics and other areas of application
35G25 Initial value problems for nonlinear higher-order PDEs
35B35 Stability in context of PDEs
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