Conservation of geometric structures for non-homogeneous inviscid incompressible fluids

TL;DR

This paper studies how geometric structures like striated and conormal regularity are preserved in solutions of non-homogeneous incompressible Euler equations across dimensions, providing lifespan bounds and regularity propagation results.

Contribution

It extends the understanding of geometric property conservation in non-homogeneous fluids, offering local-in-time results and explicit lifespan estimates in multiple dimensions.

Findings

Conservation of striated and conormal regularity in solutions.

Explicit lower bounds for solution lifespan.

Propagation of Hölder regularity in 2D and 3D domains.

Abstract

We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the 2-D structure of vortex patches. The results we get are only local in time, even in the dimension N=2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N=2 or 3, we investigate also propagation of H\"older regularity in the interior of a bounded domain.