TY - JOUR

T1 - Generalized solutions to a chemotaxis-Navier-Stokes system with arbitrary superlinear degradation

AU - Ding, Mengyao

AU - Lankeit, Johannes

PY - 2022/2

Y1 - 2022/2

N2 - 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.

AB - 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.

KW - math.AP

KW - 92C17, 35K55, 35A01, 35D99

KW - eventual smoothness

KW - fluid

KW - logistic source

KW - generalized solution

KW - chemotaxis

UR - http://www.scopus.com/inward/record.url?scp=85128987478&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2103.17199

DO - 10.48550/arXiv.2103.17199

M3 - Article

VL - 54

SP - 1022

EP - 1052

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 1

ER -