Maximal Regularity

Abstract

In Chapter II we studied the solvability of the Cauchy problem for linear parabolic evolution equations by taking into consideration, in particular, the smoothing effects of analytic semigroups. The regularizing properties deduced from these effects have important implications for the qualitative theory of quasilinear parabolic equations. However, they have the disadvantage that, in general, the derivative of a solution is less regular than the right-hand side of the corresponding parabolic evolution equation. This ‘loss of regularity’ leads to some difficulties in the treatment of nonlinear evolution equations. For this reason we investigate in this chapter situations where such a loss of regularity does not occur, that is, cases of ‘maximal regularity’.