James Peirce
Department of Mathematics
University of California, Davis
Davis, CA 95616, USA
jpeirce@math.ucdavis.edu
Abstract:
We present a proof of the existence and uniqueness of smooth-in-time
solutions for the averaged Euler equations with free-slip boundary
conditions. We formulate the problem as a vortex method; namely
we write a differential equation for the material velocity field as
a convolution of the initial vorticity distribution with an integral
kernal. The proof rests on showing that the convolution integral
is smooth as a function of the Lagrangian flow map. Standard Picard
iteration then yield local well-posedness. In 2D, global well-posedness
follows from a priori energy control.