Non-uniqueness of Hölder continuous solutions for Inhomogeneous Incompressible Euler flows

TL;DR

The paper extends convex integration to the density-dependent incompressible Euler equations, proving non-uniqueness of Hölder continuous weak solutions on $\mathbb{T}^3$. It introduces a coupled density-velocity Euler-Reynolds framework and builds perturbations from Mikado blocks to achieve cancellations of Reynolds errors while maintaining a positive, Hölder continuous density with exponent up to $\alpha<\tfrac{1}{7}$. A dual collection of Mikado building blocks and a localized inverse-divergence apparatus enable control of transport-current errors and other interaction terms, yielding global-in-time weak solutions with the same initial data. This significantly broadens the non-uniqueness frontier to genuinely inhomogeneous flows, showing that density regularity can persist under convex integration and that the density can be made Hölder continuous alongside the velocity.

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 $α<1/7$, both of them are $α$-Hölder 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 Hölder continuous density.

Theorems & Definitions (42)

• Theorem 1.1
• Definition 2.1: Inhomogeneous Euler-Reynolds flows
• Remark 2.2
• Proposition 2.3: Inductive proposition
• Proposition 2.4: Bifurcating inductive proposition
• Definition 4.1: Mikado Density and Field
• Lemma 4.2
• Lemma 4.3: Geometric Lemma I
• Lemma 4.4: Geometric Lemma II
• Lemma 4.5