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http://hdl.handle.net/10397/95575
Title: | Boundary layers and stabilization of the singular Keller-Segel system | Authors: | Peng, H Wang, ZA Zhao, K Zhu, C |
Issue Date: | Oct-2018 | Source: | Kinetic and related models, Oct. 2018, v. 11, no. 5, p. 1085-1123 | Abstract: | The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε → 0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-" estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L1- estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications. | Keywords: | Chemotaxis Vanishing diffusion limit Boundary layer Effective viscous flux Weighted energy estimates Large-time behavior |
Publisher: | American Institute of Mathematical Sciences (AIMS Press) | Journal: | Kinetic and related models | ISSN: | 1937-5093 | EISSN: | 1937-5077 | DOI: | 10.3934/krm.2018042 | Rights: | © American Institute of Mathematical Sciences This article has been published in a revised form in Kinetic & Related Models [http://dx.doi.org/10.3934/krm.2018042]. This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative works. |
Appears in Collections: | Journal/Magazine Article |
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