Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion
The authors analyze the longtime behavior of solutions to the Keller-Segel-Stokes model \[ \begin{aligned} n_{t}+u\cdot \nabla n & =\Delta (n^m) -\nabla\cdot(n\chi(c)\nabla c),\cr c_{t}+u\cdot \nabla c & =\Delta c- nf(c),\cr u_t+\nabla p & =\eta \Delta u-n\nabla \phi,\cr \nabla\cdot u & =0, \end{aligned} \] in a bounded smooth domain \(\Omega\subset \mathbb{R}^2\), where \(n\) denotes the bacterial density, \(c\) the oxygen concentration and \(u\) the velocity field. Moreover, \(m>1\) is the adiabatic exponent, \(\chi\) the chemostatic sensitivity, \(f\) the oxygen consumption rate, \(p\) the pressure, \(\eta\) the viscosity, and \(\nabla \phi\) the gravitation force of the fluid. The following boundary conditions are assumed \[ \partial_\nu n|_{\partial \Omega}=\partial_\nu c|_{\partial \Omega}=u|_{\partial \Omega}=0, \] the initial densities \(n_0\), \(c_0\) are non-negative and \[ n_0\in L^\infty(\Omega),\qquad c_0\in W^{1,\infty}(\Omega),\qquad u_0\in D(A^\theta), \quad \theta>\frac{1}{2}, \] where \(A^\theta\) denotes the fractional power space associated with the Stokes operator. Under suitable assumptions on \(\chi, f\) and \(\phi\), the existence and boundedness of global weak solutions was previously shown in [\textit{Y. Tao}, and \textit{M. Winkler}, Discrete Contin. Dyn. Syst. 32, No. 5, 1901--1914 (2012; Zbl 1276.35105)]. The uniqueness of solutions remained open, and hence, the authors consider the trajectory attractor of the system, which is a suitable concept for problems without uniqueness. The existence of the trajectory attractor is proven, as well as the existence of a global attractor in a generalized sense.
scientific article; zbMATH DE number 6836834