On the theory of Besov-Herz spaces and Euler equations

In this paper we consider Euler equations (E) for an incompressible ideal fluid filling the whole space a"e (n) for n ae<yen> 2. We prove local-in-time wellposedness and give a blow-up criterion for (E) in a new framework, namely Besov type spaces based on Herz spaces. Solutions are global-in-time when n = 2. Our results cover critical and supercritical cases of the regularity. For that, we develop properties and estimates in those spaces such as product and commutator-type estimates, interpolation, duality, among others. For the blow-up result, another ingredient is a logarithmic type inequality in our spaces.