Maria Colombo

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math.AP2025

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

Maria Colombo, Roberto Colombo, Anuj Kumar

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}_…

math.AP2024

Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity

Elia BruÃ¨, Maria Colombo, Anuj Kumar

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $…

math.AP2024

Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Elia BruÃ¨, Maria Colombo, Anuj Kumar

Given a divergence-free vector field $u \in L_{t}^{\infty} W_{x}^{1, p} (R^{d})$ and a nonnegative initial datum $I ¨ _{0} \in L^{r}$ , the celebrated DiPerna--Lions theory establishe…