Maximal regularity as a tool for partial differential equations

Abstract: In the last decades, a lot of progress has been made on the subject of maximal regularity. The property of maximal $L^p$ regularity is an a priori estimate and reads as follows: For A the negative generator of an analytic semigroup on a Banach space $X$, for $1<p<\infty$, for $0<T<=\infty$, does there exist $C_p\>0$ a constant such that for all $f\in L^p(0,T;X)$, there exists a unique solution $u\in L^p(0,T;D(A))\cap W^{1,p}(0,T;X)$ of the equation $\partial_t(u)+Au=f$, $u(0)=0$ with the estimate $$|\partial_t(u) |{L^p(0,T;X)} + |Au|{L^p(0,T;X)}\le C_p |f|_{L^p(0,T;X)} ?$$ It started with a paper by De Simon in 1964 in which the author proved maximal $L^p$ regularity for negative generators of analytic semigroups in Hilbert spaces. The next big step has been made by Dore and Venni in their 1987 paper on operators admitting bounded imaginary powers. The final result on maximal regularity was done by Weis in his 2001 paper where he gave a characterisation of negative generators of analytic semigroups in Banach spaces which have the maximal regularity property. These results are all about linear theory of unbounded operators in Banach spaces. I will then show how to use this property to find solutions or to prove uniqueness of solutions of semi linear partial differential equations of parabolic type such as the non linear heat equation and the Navier-Stokes system.