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 \(\int_G \varphi (\psi - \overline{\psi }) \leq \frac{1}{a}\int_G \psi \ln (\frac{\;\psi \;}{ \overline{\psi }}) + \frac{a}{4\beta_0} \left\{ \int_G \psi \right\}\int_G|\nabla \varphi |^2\) and \(\int_G \psi \ln (\frac{\;\psi \;}{ \overline{\psi }}) \leq \frac{1}{\beta_0} \{ \int_G \psi \}\int_G |\nabla \ln (\psi )|^2\) for any finitely connected, bounded \(C^2\)-domain \(G \subseteq \mathbb{R}^2\), a constant \(\beta_0 \gt 0\), any \(a \gt 0\), 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 \(\{ n_t + u \cdot \nabla n = \Delta n - \nabla \cdot (nS(x,n,c) \nabla c); c_t + u\cdot \nabla c = \Delta c - n f(c); u_t + (u\cdot \nabla ) u = \Delta u + \nabla P+n \nabla \phi, \nabla \cdot u=0 \}\) 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.Keywordsfunctional inequalitiesvariational methodsTrudinger–Moser inequalityNavier–Stokeschemotaxisgeneralized solutionseventual smoothnessMSC codes35K5535A2335A1535J2035D3035Q9235Q3592C17