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

2012/11/16 by Philip Isett, Isett, Philip · 3 citations
Mathematics · #Navier-Stokes equation solutions #Geometric Analysis and Curvature Flows #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1211.4065

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) × \mathbb T³ coincides on (-1, 1) × \mathbb T³ with some Hölder continuous solution that is constant outside (-3/2, 3/2) × \mathbb T³. 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.

Cited by

• Sharp non-uniqueness for the Boussinesq equation with fractional dissipation
• Admissible solutions of the 2D Onsager's conjecture
• Hölder continuous weak solutions of the 3D Boussinesq equation with thermal diffusion

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