On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces

Abstract

We study the Navier-Stokes system with initial data belonging to sum of twoweak-$L^{p}$ spaces, which contains the sum of homogeneous function withdifferent degrees. The domain $\Omega$ can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension $n\geq 2$. We obtain the existence of local mild solutions in the sameclass of initial data and moreover we show results about uniqueness,regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.