Critical non-Sobolev regularity for continuity equations with rough velocity fields

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Abstract

We present a divergence free vector field in the Sobolev space H1 such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.

Original languageEnglish (US)
Pages (from-to)4739-4757
Number of pages19
JournalJournal of Differential Equations
Volume260
Issue number5
DOIs
StatePublished - Mar 5 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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