H\"older Continuous Euler Flows in Three Dimensions with Compact Support in Time

TL;DR

This paper constructs three-dimensional Euler flows with compact temporal support and H"older continuity, advancing understanding of weak solutions and energy dissipation in fluid dynamics.

Contribution

It demonstrates the existence of compactly supported, H"older continuous Euler solutions, and relates these to smooth solutions, providing insights into Onsager's conjecture.

Findings

Existence of global weak solutions with compact support in time.

Smooth solutions coincide with H"older solutions on certain intervals.

Conjecture linking H"older regularity to energy dissipation and Onsager's conjecture.

Abstract

Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the H\"{o}lder class $C_{t,x}^{1/5 - \epsilon}$. By slightly modifying the proof, we show that every smooth solution to incompressible Euler on $(-2, 2) \times {\mathbb T}^3$ coincides on $(-1, 1) \times {\mathbb T}^3$ with some H\"{o}lder continuous solution that is constant outside $(-3/2, 3/2) \times {\mathbb T}^3$. We also propose a conjecture related to our main result that would imply Onsager's conjecture that there exist energy dissipating solutions to Euler whose velocity fields have H\"{o}lder exponent $1/3 - \epsilon$.