Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We study the Navier-Stokes system with initial data belonging to sum of two weak-L-p spaces, which contains the sum of homogeneous function with different degrees. The domain Omega can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension n >= 2. We obtain the existence of local mild solutions in the same class 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 Holder-type inequality on the sum of Lorentz spaces.271171183Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [303365/2007-2]