Uniform \(L^p\) Estimates for Solutions to the Inhomogeneous 2D Navier–Stokes Equations and Application to a Chemotaxis–Fluid System with Local Sensing

The chemotaxis-Navier–Stokes system \(\begin{aligned} \left\{ \begin{array}{rcl} n_t+u\cdot \nabla n & =& \Delta \big (n c^{-\alpha } \big ), \\ c_t+ u\cdot \nabla c & =& \Delta c -nc,\\ u_t + (u\cdot \nabla ) u & =& \Delta u+\nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill n_t+u·\mathrm{\nabla }n& =& \mathrm{\Delta }(n{c}^{-\alpha }),\hfill \\ \hfill c_t+u·\mathrm{\nabla }c& =& \mathrm{\Delta }c-nc,\hfill \\ \hfill u_t+(u·\mathrm{\nabla })u& =& \mathrm{\Delta }u+\mathrm{\nabla }P+n\mathrm{\nabla }\mathrm{\Phi },\mathrm{\nabla }·u=0,\hfill \end{array}\right)\end{array}$$ modelling the behavior of aerobic bacteria in a fluid drop, is considered in a smoothly bounded domain \(\Omega \subset \mathbb R^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ . For all \(\alpha > 0\) $\alpha >0$ and all sufficiently regular \(\Phi \) $\mathrm{\Phi }$ , 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 \(L^p\) $L^p$ estimates for solutions to the inhomogeneous Navier–Stokes equations merely depending on the space-time \(L^2\) $L^2$ norm of the force term raised to an arbitrary small power.