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


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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