Suppression of chemotactic blowup by strong buoyancy in Stokes-Boussinesq flow with cold boundary (Q6581815)

The paper is focused on the Keller-Segel equation coupled to the Stokes flow via buoyancy force \N\[\N\left\{\begin{array}{l} \partial_t\rho+u\cdot \nabla \rho-\Delta \rho+\operatorname{div}(\rho\nabla(-\Delta)^{-1}\rho)=0,\,\,\,x\in\Omega,\\\N\partial_tu-\Delta u+\nabla p= g\rho e_z,\,\,\,\operatorname{div}u=0,\,\,\,x\in \Omega,\\\Nu(0,x)=u_0(x),\,\,\,\rho(0,x)=\rho_0(x),\,\,\,\rho_0(x)\ge 0,\\\Nu|_{\partial \Omega}=0,\,\,\,\rho|_{\partial \Omega}=0, \end{array}\right.\tag{1}\N\] \Nwhere \(\Omega\) is a smooth compact domain in \(\mathbb{R}^d\), with \(d=2\) or \(d=3\), and \(e_z\) is the unit vector \((0,1)\) when \(d=2\), or \((0,0,1)\) when \(d=3\). The number \(g\in\mathbb{R}^+\) is the Rayleigh number which represents the buoyancy strength, the operator \((-\Delta)^{-1}\) is the inverse homogeneous Dirichlet Laplacian corresponding to domain \(\Omega\), and \(\rho_0\ge 0\) is the initial density. The authors study the local well-posedness of strong solutions to problem \((1)\). Then they show the existence of a global-in-time strong solution for \((1)\), and quantify the quenching effect of the Stokes-Boussinesq flow with strong buoyancy on problem \((1)\). Finally, the authors prove some regularity properties of the strong solution of \((1)\).