Boundary layer problems in chemotaxis models

Abstract

This thesis is concerned with the zero-diffusion limit and boundary layers of a viscous hyperbolic system transformed via a Cole-Hopf transformation from a singular chemotactic system modeling numerous biological processes, such as traveling waves of bacterial chemotaxis[36], boundary movement of bacterial population in response to chemotaxis by Nossal (cf. [64]) and the initiation of tumor angiogenesis proposed in [41]. It was numerically found in [44] that when prescribed with Dirichlet boundary con-ditions, the considered system exhibits boundary layer phenomena at the boundaries in a bounded interval (0,1) as the chemical diffusion rate (denoted by ε > 0) is small, while the rigorous justifcation still remains open. The purpose of this thesis will be to develop some mathematical theories for the boundary layer solutions of chemotaxis models in one and multi-dimensions and hence to justify the numerical fndings of [44] with further development in multi-dimensions. We first show the existence of boundary layers (BLs) in one dimension, where outside the BLs the solution with ε > 0 converges to the one with ε = 0, but inside the BLs the convergence no longer holds. We then proceed to prove the stability of boundary layer solutions and identify its precise structure. Roughly speaking, we justify that the solution with ε > 0 converges to the solution with ε = 0 (outer layer) plus the (inner) boundary layer solutions with the optimal rate at order of O(ε¹/²), where the outer and inner layer solutions are well determined by explicit equations. For the multi-dimensional case, motivated from the study in one dimension, we first study the boundary layer problem for radial solutions in an annulus and show the existence of boundary layers. Then we study the system in a half-plane of R² subject to Dirichlet boundary conditions and prove the stability of boundary layer solutions with explicit outer and inner layer profles. Finally, we covert the result for the transformed system to the original pre-transformed chemotaxis system and discuss the biological implications of our results. Boundary layer formation in chemotaxis has been observed in the real experiment [78] and its theoretical study is just in its infant stage. This thesis develops the first theoretical results on the boundary layers of chemotaxis models and will pave the road for the further studies on the boundary layer theories of general/different chemotaxis models to explain the experimental observations of boundary layer phenomena of chemotaxis such as the one [78].