Generalized solutions to a chemotaxis-Navier-Stokes system with arbitrary superlinear degradation

Abstract

In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional setting as below, \begin{eqnarray} \left\{ \begin{array}{llll} \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=\Delta n-\nabla \cdot(n\nabla c)+f(n), &&x\in\Omega,\,t>0,\\ \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=\Delta c - c+ n, &&x\in\Omega,\,t>0,\\ \displaystyle \mathbf{u}_{t}+\kappa(\mathbf{u}\cdot\nabla)\mathbf{u}=\Delta \mathbf{u} +\nabla P+ n\nabla\phi, &&x\in\Omega,\,t>0,\\ \displaystyle \nabla\cdot\mathbf{u}=0,&&x\in\Omega,\,t>0.\\ \end{array} \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at investigating generalized solvability for the model the without imposing a critical superlinear exponent restriction on the logistic source function $f$. Specifically, it is proven in the present work that there exists a triple of integrable functions $(n,c,\mathbf{u})$ solving the system globally in a generalized sense provided that $f\in C^1([0,\infty))$ satisfies $f(0)\ge0$ and $f(n)\le rn-\mu n^{\gamma}$ ($n\ge0$) with any $\gamma>1$. Our result indicates that persistent Dirac-type singularities can be ruled out in our model under the aforementioned mild assumption on $f$. After giving the existence result for the system, we also show that the generalized solution exhibits eventual smoothness as long as $\mu/r$ is sufficiently large.