Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Abstract

Building on the recent work of C. De Lellis and L. Székelyhidi, 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 Hölder class $C_{t,x}^{1/5 - ε}$. 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 Hölder 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 Hölder exponent $1/3 - ε$.