Infinitely many non-conservative solutions for the three-dimensional Euler equations with arbitrary initial data in $C^{1/3-\epsilon}$

TL;DR

This paper constructs infinitely many non-conservative solutions to the 3D Euler equations in certain Hölder spaces, demonstrating energy non-conservation for arbitrary initial data within a specific regularity class.

Contribution

It introduces a method to generate infinitely many solutions with prescribed initial data that violate energy conservation in the context of 3D Euler equations.

Findings

Existence of infinitely many solutions in $C^{eta}_{x,t}$ that do not conserve energy.

Solutions can be constructed for any initial data in $C^{areta}$ with $eta<areta<1/3$.

Limited control over energy increase for times greater than 1.

Abstract

Let $0<\beta<\bar\beta<1/3$. We construct infinitely many distributional solutions in $C^{\beta}_{x,t}$ to the three-dimensional Euler equations that do not conserve the energy, for a given initial data in $C^{\bar\beta}$. We also show that there is some limited control on the increase in the energy for $t>1$.