Global weak solutions to a 3-dimensional degenerate and singular\n chemotaxis-Navier--Stokes system with logistic source

This paper considers the degenerate and singular chemotaxis-Navier--Stokes\nsystem with logistic term\n $n_t + u\\cdot\\nabla n =\\Delta n^m - \\chi\\nabla\\cdot(n\\nabla c) + \\kappa n\n-\\mu n^2$, $x \\in \\Omega,\\ t>0$,\n $c_t + u\\cdot\\nabla c = \\Delta c - nc$, $x \\in \\Omega,\\ t>0$,\n $u_t + (u\\cdot\\nabla)u = \\Delta u + \\nabla P + n\\nabla\\Phi, \\quad \\nabla\\cdot\nu = 0$, $x \\in \\Omega,\\ t>0$, where $\\Omega\\subset \\mathbb{R}^3$ is a bounded\ndomain and $\\chi,\\kappa \\ge 0$ and $m, \\mu >0$. In the above system without\nfluid environment Jin (J. Differential Equations, 2017) showed existence and\nboundedness of global weak solutions. On the other hand, in the above system\nwith $m=1$, Lankeit (Math.\\ Models Methods Appl. Sci., 2016) established global\nexistence of weak solutions. However, the above system with $m>0$ has not been\nstudied yet. The purpose of this talk is to establish global existence of weak\nsolutions in the chemotaxis-Navier--Stokes system with degenerate diffusion and\nlogistic term.\n