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- Title
BOUNDARY LAYERS AND STABILIZATION OF THE SINGULAR KELLER-SEGEL SYSTEM.
- Authors
Peng, Hongyun; Wang, Zhi-An; Zhao, Kun; Zhu, Changjiang
- 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 "now plays a dual role in the transformed system by acting as the coeficients 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 δ → 0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the effective viscous ux" technique to establish the uniform-in-δ 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.
- Subjects
BOUNDARY layer (Aerodynamics); SINGULAR integrals; HOPF bifurcations; LOGARITHMIC integrals; CONSERVATION laws (Mathematics); COMPUTER simulation
- Publication
Kinetic & Related Models, 2018, Vol 11, Issue 5, p1085
- ISSN
1937-5093
- Publication type
Article
- DOI
10.3934/krm.2018042