Diffusion, attraction and collapse

Michael P. Brenner, Peter Constantin, Leo P. Kadanoff, Alain Schenkel, Shankar C Venkataramani

Research output: Contribution to journalArticle

103 Citations (Scopus)

Abstract

We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2 < d < 10 there are two stable blowup modalities (self-similar and Burgers-like) and one stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.

Original languageEnglish (US)
Pages (from-to)1071-1098
Number of pages28
JournalNonlinearity
Volume12
Issue number4
DOIs
StatePublished - Jul 1999
Externally publishedYes

Fingerprint

Blow-up
attraction
Unstable
Parabolic-elliptic System
Blowing-up Solution
Finite-time Singularities
Radial Solutions
Chemotaxis
Phase Portrait
Systems of Partial Differential Equations
Unbounded Domain
blowing
Bacteria
Modality
Countable
Blow molding
partial differential equations
Correspondence
bacteria
Partial differential equations

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Brenner, M. P., Constantin, P., Kadanoff, L. P., Schenkel, A., & Venkataramani, S. C. (1999). Diffusion, attraction and collapse. Nonlinearity, 12(4), 1071-1098. https://doi.org/10.1088/0951-7715/12/4/320

Diffusion, attraction and collapse. / Brenner, Michael P.; Constantin, Peter; Kadanoff, Leo P.; Schenkel, Alain; Venkataramani, Shankar C.

In: Nonlinearity, Vol. 12, No. 4, 07.1999, p. 1071-1098.

Research output: Contribution to journalArticle

Brenner, MP, Constantin, P, Kadanoff, LP, Schenkel, A & Venkataramani, SC 1999, 'Diffusion, attraction and collapse', Nonlinearity, vol. 12, no. 4, pp. 1071-1098. https://doi.org/10.1088/0951-7715/12/4/320
Brenner MP, Constantin P, Kadanoff LP, Schenkel A, Venkataramani SC. Diffusion, attraction and collapse. Nonlinearity. 1999 Jul;12(4):1071-1098. https://doi.org/10.1088/0951-7715/12/4/320
Brenner, Michael P. ; Constantin, Peter ; Kadanoff, Leo P. ; Schenkel, Alain ; Venkataramani, Shankar C. / Diffusion, attraction and collapse. In: Nonlinearity. 1999 ; Vol. 12, No. 4. pp. 1071-1098.
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