Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing

Abstract

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 $\Omega \subset {\mathbb{R}}^2$. For all $\alpha >0$ and all sufficiently regular $\Phi$, 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.

arXiv comment17 pages

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