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

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  • We study the system

    $ c_t+u \cdot \nabla c = \Delta c- nf(c) $
    $ n_t + u \cdot \nabla n = \Delta n^m- \nabla \cdot (n \chi(c)\nabla c) $
    $ u_t + u \cdot \nabla u + \nabla P - \eta\Delta u + n \nabla \phi=0 $
    $\nabla \cdot 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 $m\in(3/2, 2]$ 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 $m\in$( m*$,2]$ with m*>3/2, due to the use of classical Sobolev inequalities.

    Mathematics Subject Classification: Primary: 35K55, 35Q92, 35Q35, 92C17; Secondary: 35K57, 76N10, 35Q30.


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