Logarithmically regularized inviscid models in borderline sobolev spaces

Abstract

Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation, and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H-s is referred to as a borderline Sobolev space if the L-infinity-norm of the gradient of the velocity is not bounded by the H-s-norm of the solution but by the H-(s) over tilde-norm for any (s) over tilde > s. This paper establishes the local well-posedness of the logarithmically regularized counterparts of these inviscid models in the borderline Sobolev spaces. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4725531]