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

TL;DR

This paper introduces two new functional inequalities applicable to bounded domains in two dimensions and demonstrates their utility in proving the long-term stabilization and smoothness of solutions to a complex chemotaxis-Navier-Stokes system with rotational flux.

Contribution

The paper develops novel functional inequalities and applies them to establish the eventual smoothness and stabilization of solutions to a generalized chemotaxis-Navier-Stokes system with rotational flux.

Findings

Proved two new functional inequalities for functions on bounded domains.

Applied inequalities to show solutions stabilize and become smooth over time.

Extended analysis to systems with general chemotactic sensitivity S.

Abstract

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.