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

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 $1/3 - \epsilon$.