Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

Abstract

We introduce Besov type function spaces, based on the weak L p -spaces instead of the standard L p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in $${\mathbb{R}}^n , n \geq 3$$ . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces.