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

Abstract

We prove two new functional inequalities of the forms\[ \int_G φ(ψ- \overlineψ) \leq \frac{1}{a}\int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) + \frac{a}{4β_0} \left\{ \int_G ψ\right\}\int_G|\nabla φ|^2 \] and \[ \int_G ψ\ln \left(\frac{\;ψ\;}{ \overlineψ}\right) \leq \frac{1}{β_0}\left\{ \int_G ψ\right\}\int_G |\nabla \ln(ψ)|^2 \] 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 $\phi$, $\psi$. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties for certain generalized solutions to the chemotaxis-Navier-Stokes system\[ \left\{\;\; \begin{aligned} n_t + u \cdot \nabla n &\;\;=\;\; Δn - \nabla \cdot (nS(x,n,c) \nabla c), \\ c_t + u\cdot \nabla c &\;\;=\;\; Δc - n f(c), \\ u_t + (u\cdot \nabla) u &\;\;=\;\; Δu + \nabla P + n \nabla ϕ, \;\;\;\;\;\; \nabla \cdot u = 0, \end{aligned} \right. \] 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.