Dynamics of Partial Differential Equations

Volume 14 (2017)

Number 2

Analyticity and dynamics of a Navier-Stokes-Keller-Segel system on bounded domains

Pages: 125 – 158

DOI: http://dx.doi.org/10.4310/DPDE.2017.v14.n2.a2

Authors

Vincent R. Martinez (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)

Kun Zhao (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)

Abstract

We consider a coupled chemotaxis-fluid model:\[\begin{cases}\partial_t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} + \nabla P = \nu \Delta \mathbf{u} - n \nabla \phi \; \textrm{,} \\\nabla \cdot \mathbf{u} = 0 \; \textrm{,} \\\partial_t n + \mathbf{u} \cdot \nabla n = D_n \Delta n - \nabla \cdot (n {\chi} (c) \nabla (c)) \; \textrm{,} \\\partial_t c + \mathbf{u} \cdot \nabla c = D_c \Delta c - nf(c) \; \textrm{,} \\\end{cases}\]describing the interplay of hydrodynamics and chemotaxis in bacterial suspensions, on bounded domains in $\mathbb{R}^d (d = 2, 3)$. In the first part of the paper, by assuming $\phi$, $\chi$ and $f$ are analytic functions, we show that solutions on periodic domains become instantaneously analytic with respect to spatial variables for rough data, due to the parabolic smoothing effect. The result holds for all space dimensions, $d \geq 2$, for short time when the initial data is large and for an arbitrarily long given time when initial data is small in a suitably strong regularity class. However, if the initial data belongs to a weaker regularity class, then the smoothing effect is shown to hold for only small data and large time. The second part of the paper is devoted to the study of long-time asymptotic behavior of classical solutions in two space dimensions when $\chi (c) \equiv 0 \equiv f(c)$ and $\phi = x_d$. By using $L^p$-based energy methods, it is shown that when $\nu \gt 0$ and $D_n \gt 0$, the equilibrium determined by the no-flow boundary condition for $\mathbf{u}$ and a naturally stabilizing boundary condition for $n$ is globally asymptotically stable regardless of the magnitude of initial data. In addition, by developing a novel energy method, we show that when $\nu = 0$ and $D_n \gt 0$, the naturally stabilizing boundary datum for $n$ is still globally asymptotically stable. These appear to be the first such results for the model on bounded domains with physical boundaries. In addition, the method for proving the last result can be of independent interest.

Keywords

Navier-Stokes-Keller-Segel system, initial-boundary value problem, spatial analyticity, Gevrey regularity, long-time behavior

2010 Mathematics Subject Classification

35B30, 35B40, 35B45, 35B65, 76D05, 92C15, 92C45

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Published 31 May 2017