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Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow \(p\)-Laplacian diffusion. (English) Zbl 1411.35064

Summary: This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow \(p\)-Laplacian diffusion \[ \begin{cases} n_t + u \cdot \nabla n = \nabla \cdot(| \nabla n |^{p - 2} \nabla n) - \nabla \cdot(n \chi(c) \nabla c), & x \in \Omega, t > 0, \\ c_t + u \cdot \nabla c = \Delta c - n f(c), & x \in \Omega, t > 0, \\ u_t +(u \cdot \nabla) u = \Delta u + \nabla P + n \nabla \Phi, & x \in \Omega, t > 0, \\ \nabla \cdot u = 0, & x \in \Omega, t > 0 \end{cases}. \] under homogeneous boundary conditions of Neumann type for \(n\) and \(c\), and of Dirichlet type for \(u\) in a bounded convex domain \(\Omega \subset \mathbb{R}^3\) with smooth boundary. Here, \(\Phi \in W^{1, \infty}(\Omega)\), \(0 < \chi \in C^2([0, \infty))\) and \(0 \leq f \in C^1([0, \infty))\) with \(f(0) = 0\). It is proved that if \(p > \frac{32}{15}\) and under appropriate structural assumptions on \(f\) and \(\chi\), for all sufficiently smooth initial data \((n_0, c_0, u_0)\) the model possesses at least one global weak solution.

MSC:

35D30 Weak solutions to PDEs
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
92C17 Cell movement (chemotaxis, etc.)
35Q30 Navier-Stokes equations
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