论文标题
两种新的功能不平等现象及其在旋转通量的趋化性 - 纳维尔 - 螺旋体系统中的最终平滑度中的应用
Two new functional inequalities and their application to the eventual smoothness of solutions to a chemotaxis-Navier-Stokes system with rotational flux
论文作者
论文摘要
我们证明了形式的两个新的功能不平等\ [ \int_gφ（ψ-\overlineψ）\ leq \ frac {1} {a} \int_gψ\ ln \ left（\ frac {\ frac {\;ψ\;} {\overlineψ} { ψ\ right \} \ int_g | \nablaφ|^2 \] 和 \[ \int_gψ\ ln \ left（\ frac {\;ψ\;} {\overlineψ} \ right）\ leq leq \ frac {1} {β_0} {β_0} \ left \ left \ left \ lest \ weft \ frac {\ int_g { \]对于任何有限连接的，有限的$ c^2 $ - domain $ g \ subseteq \ mathbb {r}^2 $，一个常数$β_0> 0 $，任何$ a> 0 $和足够的常规功能$φ$，$ψ$。 然后，我们通过证明长期稳定和最终的平滑性特性来说明它们的有用性 \左边\{\;\; \ 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} \正确的。 \]在平滑的，有限的，凸域$ω\ subseteq \ mathbb {r}^2 $，带有$ n $和$ c $的No-Flux边界条件，以及$ U $的Dirichlet边界条件。我们进一步允许在$ \ mathbb {r}^{2 \ times 2} $中获得一般的趋化敏感性$ s $，而不是标量。
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 $β_0 > 0$, any $a > 0$ and sufficiently regular functions $φ$, $ψ$. 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 $Ω\subseteq \mathbb{R}^2$ 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.
