Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912048855646208 |
|---|---|
| author | Fuest, Mario Winkler, Michael |
| author_facet | Fuest, Mario Winkler, Michael |
| contents | The chemotaxis-Navier-Stokes system \begin{equation*}\label{1} \left\{ \begin{array}{rcl} n_t+u\cdot\nabla n &=& Δ\big(n c^{-α} \big), \\[1mm] c_t+ u\cdot\nabla c &=& Δc -nc,\\[1mm] u_t + (u\cdot\nabla) u &=&Δu+\nabla P + n\nablaΦ, \qquad \nabla\cdot u=0, \end{array} \right. \end{equation*} modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain $Ω\subset \mathbb R^2$. For all $α> 0$ and all sufficiently regular $Φ$, we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier-Stokes system.
As a crucial new challenge, our analysis requires a priori estimates for $u$ at a point in the proof when knowledge about $n$ is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time $L^p$ estimates for solutions to the inhomogeneous Navier-Stokes equations merely depending on the space-time $L^2$ norm of the force term raised to an arbitrary small power. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing Fuest, Mario Winkler, Michael Analysis of PDEs 35K65 (primary), 35Q55, 35Q92, 92C17 (secondary) The chemotaxis-Navier-Stokes system \begin{equation*}\label{1} \left\{ \begin{array}{rcl} n_t+u\cdot\nabla n &=& Δ\big(n c^{-α} \big), \\[1mm] c_t+ u\cdot\nabla c &=& Δc -nc,\\[1mm] u_t + (u\cdot\nabla) u &=&Δu+\nabla P + n\nablaΦ, \qquad \nabla\cdot u=0, \end{array} \right. \end{equation*} modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain $Ω\subset \mathbb R^2$. For all $α> 0$ and all sufficiently regular $Φ$, we construct global classical solutions and thereby extend recent results for the fluid-free analogue to the system coupled to a Navier-Stokes system. As a crucial new challenge, our analysis requires a priori estimates for $u$ at a point in the proof when knowledge about $n$ is essentially limited to the observation that the mass is conserved. To overcome this problem, we also prove new uniform-in-time $L^p$ estimates for solutions to the inhomogeneous Navier-Stokes equations merely depending on the space-time $L^2$ norm of the force term raised to an arbitrary small power. |
| title | Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing |
| topic | Analysis of PDEs 35K65 (primary), 35Q55, 35Q92, 92C17 (secondary) |
| url | https://arxiv.org/abs/2401.09832 |