Author: Hou, Qianqian
Title: Boundary layer problems in chemotaxis models
Degree: Ph.D.
Year: 2018
Subject: Hong Kong Polytechnic University -- Dissertations
Chemotaxis -- Mathematics
Department: Department of Applied Mathematics
Pages: xiii, 133 pages : illustrations
Language: English
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].
Access: open access

Files in This Item:
File Description SizeFormat 
991022165758303411.pdfFor PolyU Staff & Students636.21 kBAdobe PDFView/Open
32302.pdfFor All Users (Non-printable)644.78 kBAdobe PDFView/Open


Copyright Undertaking

As a bona fide Library user, I declare that:

  1. I will abide by the rules and legal ordinances governing copyright regarding the use of the Database.
  2. I will use the Database for the purpose of my research or private study only and not for circulation or further reproduction or any other purpose.
  3. I agree to indemnify and hold the University harmless from and against any loss, damage, cost, liability or expenses arising from copyright infringement or unauthorized usage.

By downloading any item(s) listed above, you acknowledge that you have read and understood the copyright undertaking as stated above, and agree to be bound by all of its terms.

Show full item record

Please use this identifier to cite or link to this item: https://theses.lib.polyu.edu.hk/handle/200/9669