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

TL;DR

This paper establishes the existence of global generalized solutions for a chemotaxis-Navier-Stokes system with superlinear degradation, extending previous results by removing critical exponent restrictions on the logistic source function.

Contribution

It proves the existence of global generalized solutions for the chemotaxis-Navier-Stokes system with arbitrary superlinear degradation, without imposing critical exponent restrictions.

Findings

Existence of global generalized solutions under mild conditions on the source function.

Solutions exhibit eventual smoothness if the ratio of parameters is sufficiently large.

Persistent Dirac-type singularities are ruled out under the given assumptions.

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.