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

Abstract

In this paper, we use the method of convex integration to construct infinitely many distributional solutions in H β for 0 < β 1 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.

Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

“…At this point is worth mentioning the ground-breaking work of De Lellis and Székelyhidi Jr. [11,12], where they show non-uniqueness of solutions in L 2 by the method of convex integration (see also the work of Wiedemann [34]). Very recently, using similar tools, Khor and Miao [25] use the method of convex integration to construct infinitely many distributional 3d solutions in H β for 0 < β << 1 which has an instantaneous gap loss of Sobolev regularity.…”