Two new functional inequalities and their application to the eventual smoothness of solutions to a chemotaxis-Navier-Stokes system with rotational flux

We prove two new functional inequalities of the forms$$\int_G \varphi (\psi - \overline{\psi}) \leq \frac{1}{a}\int_G \psi \ln \left(\frac{\;\psi\;}{ \overline{\psi}}\right) + \frac{a}{4\beta_0} \left\{ \int_G \psi \right\}\int_G|\nabla \varphi|^2$$ and $$\int_G \psi \ln \left(\frac{\;\psi\;}{ \overline{\psi}}\right) \leq \frac{1}{\beta_0}\left\{ \int_G \psi \right\}\int_G |\nabla \ln(\psi)|^2$$ for any finitely connected, bounded $C^2$-domain $G \subseteq \mathbb{R}^2$, a constant $\beta_0 > 0$, any $a > 0$ and sufficiently regular functions $\varphi$, $\psi$. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties for certain generalized solutions to the chemotaxis-Navier-Stokes system$$\left\{\;\; \begin{aligned} n_t + u \cdot \nabla n &\;\;=\;\; \Delta n - \nabla \cdot (nS(x,n,c) \nabla c), \\ c_t + u\cdot \nabla c &\;\;=\;\; \Delta c - n f(c), \\ u_t + (u\cdot \nabla) u &\;\;=\;\; \Delta u + \nabla P + n \nabla \phi, \;\;\;\;\;\; \nabla \cdot u = 0, \end{aligned} \right.$$ on a smooth, bounded, convex domain $\Omega \subseteq \mathbb{R}^2$ with no-flux boundary conditions for $n$ and $c$ as well as a Dirichlet boundary condition for $u$. We further allow for a general chemotactic sensitivity $S$ attaining values in $\mathbb{R}^{2\times 2}$ as opposed to a scalar one

Depleting the signal: Analysis of chemotaxis-consumption models—A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures