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arXiv · 2311.15661

On measure-preserving selection of solutions of ODEs

Umberto Pappalettera
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Abstract

For every $k \in \mathbb{N}$ and $\alpha \in (0,1)$ we construct a divergence-free $u \in C^k([0,T],C^\alpha(\mathbb{T}^d,\mathbb{R}^d))$, $d \geq 2$, such that there is no measurable selection of solutions of the ODE $\dot{X}_t = u(t,X_t)$ that preserves the Lebesgue measure.

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Umberto Pappalettera. 2023-11-27. On measure-preserving selection of solutions of ODEs. https://doi.org/10.1090/proc%2F17097

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