Holomorphic functional calculus and nonlinear evolution equations

17.07.2019, 14:00  –  Campus Golm, Haus 9, Raum 2.22
Institutskolloquium

Sylvie Monniaux (Université Aix-Marseille), Lutz Weis (Universität Karlsruhe)

Programm

14:00 Sylvie Monniaux (Université Aix-Marseille): Maximal regularity and Navier Stokes equations

15:00 Tea and Coffee Break

15:30 Lutz Weis (Universität Karlsruhe): The H^∞-calculus and its application to partial differential equations

Abstracts

Sylvie Monniaux (Université Aix-Marseille): Maximal regularity and Navier-Stokes equations
In this talk, we consider nonlinear evolution equations for which no loss of regularity occurs i.e., cases of ‘maximal regularity'. I will explain how to use maximal regularity results and bounded holomorphic functional calculus to prove uniqueness of some non linear partial differential equations.

Lutz Weis (Universität Karlsruhe): The H^∞-calculus and its application to partial differential equations
The holomorphic functional calculus for sectorial operators – commonly called H^∞-calculus-has found many applications to evolution equations. We will illustrate this with elliptic systems and the Navier Stokes equation. The reason behind the success of the H^∞-calculus is that it allows to extend many tools from classical harmonic analysis such as Littlewood-Paley decompositions and square function estimates to the more general setting of sectorial operators on L^p(U) spaces. It is used in the context of optimal regularity estimates and sharp convergence rates for their numerical approximation. Such results are useful since most sectorial partial differential operators which are of interest in analysis have a H^∞-calculus.