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Periodic pattern formation in the coupled chemotaxis-(Navier-)Stokes system with mixed nonhomogeneous boundary conditions. (English) Zbl 1458.92015

Summary: We consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that \((0,c,0,0)\) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

MSC:

92C17 Cell movement (chemotaxis, etc.)
92C15 Developmental biology, pattern formation
35M10 PDEs of mixed type
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