Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

Abstract

In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient is that we are able to show that quasilinear parabolic evolution equations generate a smooth semiflow on the trace spaces associated with maximal regularity, which are the natural phase spaces in this framework.