On the vanishing viscosity limit of a chemotaxis model

Hua Chen; Jian-Meng Li; Kelei Wang

doi:10.3934/dcds.2020101

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2020, Volume 40, Issue 3: 1963-1987. Doi: 10.3934/dcds.2020101

This issue Previous Article Uniform attractors for non-autonomous plate equations with

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On the vanishing viscosity limit of a chemotaxis model

School of Mathematics and Statistics, Wuhan University, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China

^* Corresponding author: Kelei Wang
^* Corresponding author: Kelei Wang

Received: September 1, 2019

Published online: March 1, 2020

H. Chen and K. Wang were supported by the NSFC grant no. 11631011.

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Abstract

A vanishing viscosity problem for the Patlak-Keller-Segel model is studied in this paper. This is a parabolic-parabolic system in a bounded domain $ \Omega\subset \mathbb{R}^n $, with a vanishing viscosity $ \varepsilon\to 0 $. We show that if the initial value lies in $ W^{1, p} $ with $ p>\max\{2, n\} $, then there exists a unique solution $ (u_\varepsilon, v_\varepsilon) $ with its lifespan independent of $ \varepsilon $. Furthermore, as $ \varepsilon\rightarrow 0 $, $ (u_\varepsilon, v_\varepsilon) $ converges to the solution $ (u, v) $ of the limiting system in a suitable sense.

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Mathematics Subject Classification: Primary: 35K51, 35Q92; Secondary: 35M12.

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