Generalized nonlinear heat and Navier-Stokes equations in supercritical function spaces

We consider the Cauchy problem of a generalized nonlinear heat equation in the context of supercritical Besov and Triebel-Lizorkin spaces. Starting with multiplication algebras we investigate to what extent one can weaken the conditions on the smoothness of the spaces to obtain unique mild and strong solutions of this equation. Furthermore, we apply the gained results to the by means of the so-called Leray projector reformulated generalized Navier-Stokes equations using its mapping properties in appropriate function spaces of Besov and Triebel-Lizorkin type.