Construction of solutions to the 3D Euler equations with initial data in $H^\beta$ for $\beta>0$

TL;DR

This paper employs convex integration to construct infinitely many weak solutions to the 3D Euler equations with initial data in fractional Sobolev spaces, demonstrating solutions with continuous energy despite low regularity.

Contribution

It introduces a method to build solutions with fractional Sobolev regularity, extending the understanding of solution existence and energy continuity for the Euler equations.

Findings

Constructed infinitely many solutions in $H^{eta}$ for $0<eta extless1$

Established energy continuity for solutions with small fractional derivatives

Differentiates from previous results with discontinuous energy at initial time

Abstract

In this paper, we use the method of convex integration to construct infinitely many distributional solutions in $H^{\beta}$ for $0<\beta\ll1$ to the initial value problem for the three-dimensional incompressible Euler equations. We show that if the initial data has any small fractional derivative in $L^2$, then we can construct solutions with some regularity, so that the corresponding $L^2$ energy is continuous in time. This is distinct from the $L^2$ existence result of E. Wiedemann, Ann. Inst. Henri Poincar\'e, Anal. Non Lin\'eaire 28, No. 5, 727--730 (2011; Zbl 1228.35172), where the energy is discontinuous at $0$.