Global boundedness in a 3D quasilinear Keller-Segel-Stokes system with nonlinear sensitivity and indirect signal production (Q6105317)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^3\) and consider the Keller-Segel-Stokes system with nonlinear diffusion and indirect signal production \begin{align*} \partial_t n + \mathbf{u}\cdot \nabla n & = \mathrm{div}\left( \nabla n^m - \frac{n}{(1+n)^\alpha} \nabla v \right), \quad (t,x)\in (0,\infty)\times\Omega, \\ \partial_t v + \mathbf{u}\cdot \nabla v & = \Delta v - v + w, \quad (t,x)\in (0,\infty)\times\Omega, \\ \partial_t w + \mathbf{u}\cdot \nabla w & = \Delta w - w + n, \quad (t,x)\in (0,\infty)\times\Omega, \\ \partial_t \mathbf{u} & = \Delta\mathbf{u} + \nabla P + n \nabla\phi, \quad (t,x)\in (0,\infty)\times\Omega, \\ \mathrm{div}\,\mathbf{u} & = 0, \quad (t,x)\in (0,\infty)\times\Omega, \end{align*} supplemented with homogeneous Neumann boundary conditions for \(n\), \(v\), \(w\), and homogeneous Dirichlet boundary conditions for \(\mathbf{u}\). Assuming that \(m>0\), \(\alpha\in\mathbb{R}\), and \(\phi\in W^{2,\infty}(\Omega)\) with \(m+\alpha>10/9\), it is shown that there exists at least one global weak solution with components \((n,v,w,\mathbf{u})\) bounded in \(L^\infty(\Omega)\), \(W^{1,\infty}(\Omega)\), \(W^{1,\infty}(\Omega)\), and \(W^{2\beta,2}(\Omega,\mathbb{R}^3)\), \(\beta\in (3/4,1)\), respectively.