Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness

Abstract

We consider the question of well-posedness for the incompressible Euler equations in generalized function spaces of the type $B^{s,\psi}_{p,q}(\mathbb{R}^d)$ and $F^{s,\psi}_{p,q}(\mathbb{R}^d)$ where $\psi$ is a slowly varying function in the Karamata sense and $s=d/p+1$. We prove that if $\psi$ grows fast enough, then there is a local in time solution to the Euler equations. We also establish a BKM-type criterion that allows us to obtain global existence in two dimensions.