Rough transport noise selects the unique DiPerna-Lions solution in the zero-noise limit and yields a large deviations principle in the non-separable space L^∞_t L^p_x.
A convex integration scheme for the continuity equation past the Sobolev embedding threshold
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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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2025 1verdicts
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Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions
Rough transport noise selects the unique DiPerna-Lions solution in the zero-noise limit and yields a large deviations principle in the non-separable space L^∞_t L^p_x.