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\(L^{\infty}\) 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 \(s>5\).

MSC:

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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References:

[1] L. Arkeryd, On the Boltzmann equation,Arch. Rat. Mechs. Anal. 45:1-34 (1972). · Zbl 0245.76060
[2] T. Carleman,Problèmes mathématiques dans la théorie cinétique des gaz (Almqvist and Wiksell, Uppsala, 1957). · Zbl 0077.23401
[3] T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for infinite range forces, to appear inArch. Rat. Mechs. Anal. (1983). · Zbl 0503.76091
[4] C. Truesdell and R. G. Muncaster,Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980).
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