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Precise \(L^ p\)-regularity for averages in transport equations. (Régularité \(L^ p\) précisée des moyennes dans les équations de transport.) (French) Zbl 0798.35025
The author studies the regularity for velocity averages of transport equations \((v\cdot \nabla_ x)f= g\). It is proved that if \(f\) and \(g\) are in \(L^ p(\mathbb{R}^ n\times \mathbb{R}^ n)\), \(1< p\leq 2\) and satisfy the transport equation, then, for any \(\theta\in C^ \infty_ 0(\mathbb{R}^ n)\), the velocity average function \(F(x)= \int_{\mathbb{R}^ n} f(x,v)\theta(v)dv\) belongs to the Sobolev space \(W^{s,p}(\mathbb{R}^ n)\) with \(s= p'\) and \(\| F\|_{W^{s,p}}\leq C(\| f\|_{L^ p}+ \| g\|_{L^ p})\), where \(C\) only depends on \(\theta\), \(n\), \(p\). Other results are given when \(g\) is of the form \((1- \Delta_ v)^{m/2} g\) or \((1- \Delta_ x)^{\tau/2}(1-\Delta_ v)^{m/2} g\). In all these cases, the author obtains estimates for \(F\) in a Sobolev space, improving previous results obtained in Besov spaces [R. J. DiPerna, P. L. Lions and Y. Meyer, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 3/4, 271-287 (1991; Zbl 0763.35014)]. Some other results are given in the nonstationary case.

MSC:
35B65 Smoothness and regularity of solutions to PDEs
42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
Citations:
Zbl 0763.35014
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