Mathematical Analysis of Coupled Systems of Evolution Equations - Coupling Boundary Conditions and Block Structures

• Applicant: Professor Dr. Amru Hussein
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 508634462

Coupled evolution equations arise in a wide range of applications such as reaction-diffusion equations, geophysical flow equations, tumour growth models, thermoplastic beam and wave equations, transmission problems, and liquid crystals. The aim of this project is to develop mathematical tools for the analysis of such systems of coupled parabolic quasi-linear evolution equations in vector valued L^q_t-L^p_x-spaces focusing on1. coupling boundary conditions, and 2. couplings at the interior of domains having a block structure including mixed-order systems.The classical theory for elliptic boundary value problems considers operators with homogeneous leading terms of even order with compatibility conditions for the boundary conditions like the classical Lopatinskii-Shapiro or boundary conditions expressed via quadratic forms. However, there are situations which do not fit into this well-understood setting.First, there are boundary conditions which are not well-behaved under integration by parts or for which it is not sufficient to consider the leading part. Therefore, I propose to translate recent results on ODE systems which characterize spectral properties by the Cayley transforms of the boundary relations to the PDE setting. This should result in a much broader and more flexible Lopatinskii-Shapiro condition.Second, there are operators of mixed order where the analysis cannot be reduced to a homogeneous leading part. Instead, these systems can be understood by operator theoretical factorizations like to the well-known Schur factorization or the one introduced by Nagel. These encode the coupling of a block operator matrix in one single operator family. Thereby perturbation results for large but structured perturbation shall be proven.In both cases results on spectral properties like sectoriality, R-sectoriality and existence of a bounded H^\infty-calculus will be studied. The results shall then be applied to concrete models for instance in mathematical biology, thermoelasticity and mathematical fluid mechanics.

• DFG Programme: Research Grants