The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Mimura, Yoshifumi¹ 

[1] Univ Tokyo, Grad Sch Math Sci, Meguro Ku, 3-8-1 Komaba, Tokyo 1538914, Japan

关键词: Keller-Segel;    Chemotaxis;    Degenerate diffusion;    Wasserstein distance;    Gradient flows;   

DOI  :  10.1016/j.jde.2017.03.020

来源: Elsevier

【 摘 要 】

We prove the time-global existence of solutions of the degenerate Keller-Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value. What we deal with is the full parabolic parabolic system rather than the simplified parabolic elliptic system. Our approach is to formulate the problem as a gradient flow on the Wasserstein space. We first consider a time-discretized problem, in which the values of the solution are determined iteratively by solving a certain minimizing problem at each time step. Here we use a new minimizing scheme at each time level, which gives the time-discretized solutions favorable regularity properties. As a consequence, it becomes relatively easy to prove that the time-discretized solutions converge to a weak solution of the original system as the lime step size tends to zero. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free