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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 regularfunctions $\varphi$, $\psi$.  We then illustrate their usefulness by proving long time stabilization andeventual smoothness properties for certain generalized solutions to thechemotaxis-Navier-Stokes system\[  \left\{\;\;  \begin{aligned}  n_t + u \cdot \nabla n &\;\;=\;\; \Delta n - \nabla \cdot (nS(x,n,c) \nablac), \\  c_t + u\cdot \nabla c &\;\;=\;\; \Delta c - n f(c), \\  u_t + (u\cdot \nabla) u &\;\;=\;\; \Delta u + \nabla P + n \nabla \phi,\;\;\;\;\;\; \nabla \cdot u = 0,  \end{aligned}  \right.  \] on a smooth, bounded, convex domain $\Omega \subseteq \mathbb{R}^2$ withno-flux boundary conditions for $n$ and $c$ as well as a Dirichlet boundarycondition 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.

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