Logarithmically regularized inviscid models in borderline sobolev spaces

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 Hs is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the Hs-norm of the solution but by the \documentclass[12pt]{minimal}\begin{document}$H^{\widetilde{s}}$\end{document}Hs̃-norm for any \documentclass[12pt]{minimal}\begin{document}$\widetilde{s}>s$\end{document}s̃>s. This paper establishes the local well-posedness of the logarithmically regularized counterparts of these inviscid models in the borderline Sobolev spaces.