Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions. (English) Zbl 1535.35008
Summary: In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.
MSC:
| 35A09 | Classical solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 35Q30 | Navier-Stokes equations |
| 92C17 | Cell movement (chemotaxis, etc.) |
Keywords:
chemotaxis-Navier-Stokes system; mixed nonhomogeneous boundary conditions; global classical solution; uniquenessReferences:
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