Propagation of Striated Regularity of Velocity for the Euler Equations

Hantaek Bae, James P. Kelliher

Published 2015-08-08Version 1

The well-posedness of the Euler equations in Holder spaces for short time in 3D goes back to the work of Gunther and Lichtenstein in the 1920s, the global-in-time 2D result is due to Wolibner in 1933. The work in 2D of Chemin and in higher dimensions of Gamblin and Saint Raymond, and of Danchin, in the 1990s established analogous results for vorticity possessing negative Holder space regularity only in directions given by a sufficient family of vector fields, which are themselves transported by the flow ("striated" regularity). We prove that the propagation of striated velocity in a positive Holder space also holds, by establishing the equivalence of striated regularity of vorticity and of velocity. We go on to show that the results of Chemin and Danchin, which rely heavily on paradifferential calculus, can be obtained by elementary methods inspired by the work of Ph. Serfati from the 1990s. Finally, we show in 2D and 3D that the velocity gradient is regular after being corrected by a regular matrix multiple of the vorticity.