Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior

Marco Di Francesco, Alexander Lorz, Peter Markowich

Research output: Contribution to journalArticlepeer-review

161 Scopus citations


We study the system ct + u · ∇c = ∇c -nf(c) nt + u · ∇n = ∇nm - ∇ · (n×(c) ∇c) ut + u·∇u + ∇P - η∇u + nφ/ = 0 ∇·u = 0. arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous-medium-like diffusion in the equation for the density n of the bacteria, motivated by a finite size effect. We prove that, under the constraint m ε (3/2, 2] for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case m = 2 we prove that solutions converge to constant states in the large-time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case m = 1. The case m = 2 is very special as we can provide a Lyapounov functional. We generalize our results to the three-dimensional case and obtain a smaller range of exponents m ε (m*, 2] with m* > 3/2, due to the use of classical Sobolev inequalities.

Original languageEnglish (US)
Pages (from-to)1437-1453
Number of pages17
JournalDiscrete and Continuous Dynamical Systems
Issue number4
StatePublished - Dec 2010
Externally publishedYes


  • Chemotaxis model
  • Nonlinear diffusion
  • Stokes equations

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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