  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.
