Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

Abstract

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations with weak data in the homogeneous spaces. We give a method which can be used to construct local mild solutions of the abstract Cauchy problem inĊσ,s,p andLq([0, T);Hs,p) by introducing the concept of both admissible quintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic type equations. For the small data, we prove that these results can be extended globally in time. We also study the regularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Ċσ,s,p. As an application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneous Sobolev spaces.