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arXiv · 2504.03578

A convex integration scheme for the continuity equation past the Sobolev embedding threshold

Maria Colombo·Roberto Colombo·Anuj Kumar
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Abstract

We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in $C_{t}W^{1,p}_x$ and nonunique solutions in $C_{t} L^{q}_x$ for any $p,q$ with $\frac{1}{p} + \frac{1}{q} > 1 + \frac{1}{d}- \delta$ for some $\delta>0$. This improves the previous bound, corresponding to $\delta=0$, or equivalently $q' > p^*$, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range.

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Maria Colombo, Roberto Colombo, Anuj Kumar. 2025-04-04. A convex integration scheme for the continuity equation past the Sobolev embedding threshold. https://arxiv.org/abs/2504.03578

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