Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications

TL;DR

This paper investigates the maximal $L^p$-regularity and $H^{ abla ext{infty}}$-calculus for block operator matrices, providing new perturbation results and applying them to various coupled evolution equations.

Contribution

It develops new results on sectoriality, $ ext{R}$-sectoriality, and $H^{ abla ext{infty}}$-calculus for diagonally dominant block operators, with applications to complex evolution problems.

Findings

Established perturbation results for large structured perturbations.

Analyzed maximal $L^p_t$-regularity for coupled systems.

Applied theory to models in liquid crystals, fluid dynamics, and biological systems.

Abstract

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part with $\mathsf{D}(\mathcal{A})=\mathsf{D}(A)\times \mathsf{D}(D)$ though with possibly large relative bound. For such operators the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.