Non-uniqueness of H\"older continuous solutions for Inhomogeneous Incompressible Euler flows

TL;DR

This paper demonstrates the existence of non-unique, H"older continuous weak solutions to the inhomogeneous incompressible Euler equations in three dimensions, including for the density, using convex integration techniques.

Contribution

It introduces the first construction of H"older continuous density solutions for inhomogeneous Euler flows, expanding the scope of non-uniqueness results.

Findings

Existence of non-unique H"older continuous solutions with density and velocity.

Construction of solutions for any ta<1/7.

Use of Mikado flows in convex integration framework.

Abstract

We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $\alpha<1/7$, both of them are $\alpha $-H\"older continuous and $(\varrho, u)$ is a weak solution to the underlying equations. The proof is based on typical convex integration techniques using Mikado flows as building blocks. As a main novelty with respect to the related literature, our result produces a H\"older continuous density.