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On measure-preserving selection of solutions of ODEs

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arxiv 2311.15661 v1 pith:ADSGB2YX submitted 2023-11-27 math.AP math.CA

Umberto Pappalettera This is my paper · ORCID
classification math.APmath.CA
keywords mathbbalphaselectionsolutionsconstructdivergence-freeeverylebesgue
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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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  1. Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

    math.AP 2025-04 conditional novelty 7.0 of 10

    Weak solutions of the Euler and Navier-Stokes equations exist for which two different particle trajectories start from the same point, in sharp regularity ranges, both with and without Brownian noise.

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