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arXiv · 1507.05104

Critical non Sobolev regularity for continuity equations with rough force fields

Pierre-Emmanuel Jabin
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Abstract

We present a divergence free vector field in the Sobolev space $H^1$ 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.

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Pierre-Emmanuel Jabin. 2015-07-17. Critical non Sobolev regularity for continuity equations with rough force fields. https://arxiv.org/abs/1507.05104

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