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Regularity of the moments of the solution of a transport equation. (English) Zbl 0652.47031
Author’s abstract: Let $$u=u(x,v)$$ satisfy the transport equation $$u+v\cdot \partial_ xu=f$$, $$x\in {\mathbb R}^ N$$, $$r\in {\mathbb R}^ N$$, where $$f$$ belongs to some space of type $$L^ p(dx\otimes d\mu (v))$$ (where $$\mu$$ is a positive bounded measure on $${\mathbb R}^ N)$$. We study the resulting regularity of the moment $$\int u(x,v)\,d\mu (v)$$ (in terms of fractional Sobolev spaces, for example). Counterexamples are given in order to test the optimality of our results.
Reviewer: R.Weikard

##### MSC:
 35F05 Linear first-order PDEs 47F05 General theory of partial differential operators 82C70 Transport processes in time-dependent statistical mechanics 35B65 Smoothness and regularity of solutions to PDEs
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