Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension

Jiahang Che; Li Chen; Simone GÖttlich; Anamika Pandey; Jing Wang

doi:10.3934/cpaa.2017049

Article Contents

2017, Volume 16, Issue 3: 1013-1036. Doi: 10.3934/cpaa.2017049

This issue Previous Article Positive ground state solutions of a quadratically coupled schrödinger system Next Article Global existence of solutions to an attraction-repulsion chemotaxis model with growth

Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension

1.
Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany
2.
Weierstraß-Institut, 10117 Berlin, Germany
3.
Department of Mathematics, Shanghai Normal University, 200234

Received: March 2016

Revised: December 2016

Published: January 2018

Li Chen is partially supported by the National Natural Science Foundation of China (NSFC), No. 11271218. Jing Wang is supported by NSFC (No.11101286), and the Doctoral Discipline Foundation for Young Teachers in the Higher Education Institutions of Ministry of Education, No.20113127120007.

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Abstract

This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.

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Mathematics Subject Classification: 35Q92, 35C20, 92C17.

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Figure 1. Time evolution of the Keller-Segel, the aggregation and the approximate solution for $\varepsilon = 0.01$

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Table 1. Error bound comparison for different values of $\varepsilon$

$\varepsilon$	$ \sup\limits_{0\leq t\leq T}\\|\rho^{\varepsilon}-\rho_a\\|_{\infty} $
0.01	0.165216
0.005	0.112758
0.001	0.048119
0.0005	0.033776
0.0001	0.015221

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Table 2. Elapsed time in seconds for the Keller-Segel model and the approximate solution

$\varepsilon$	$ k_{fac} $	Keller-Segel	approximate solution
0.01	9	13.25	8.94
0.005	18	13.17	4.35
0.001	50	13.26	1.58
0.0005	100	13.18	0.78
0.0001	120	13.05	0.65

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Table 3. Grid convergence

h	E_h	e
0.002	0.165698	1.9280
0.001	0.165216
0.0005	0.0164966

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