论文标题
在Keller-Segel模型上，与随机强迫不可压缩的粘性流相互作用，$ \ Mathbb {r}^2 $
On the Keller-Segel models interacting with a stochastically forced incompressible viscous flow in $\mathbb{R}^2$
论文作者
论文摘要
本文考虑了与随机Navier-Stokes方程（简称KS-SNS）结合的Keller-Segel模型，该模型描述了在随机强迫的2D 2D不可压缩不可压缩的粘性流中演变的氧气和细菌密度的动力学。我们的主要目标是研究全球解决方案的存在和独特性（在概率意义上是强大的，在PDE意义上是弱的）对KS-SNS系统。引入了一个新颖的近似KS-SNS系统，具有适当的正则化和截止算子的$ h^s（\ mathbb {r}^2）$，并引入了近似解决方案的存在，一些先验统一的界限证明了近似解决方案的存在，并且对近似方案进行了仔细的分析。在适当的假设下，似乎是新形式的两种随机熵 - 能源不平等现象，它们与Prohorov Theorem和Jakubowski-Skorokhod定理一起，使我们能够表明近似解决方案的序列将近似解决方案融合到全球弱弱解决方案中。此外，当$χ（\ cdot）\ equiv \ textrm {const。}> 0 $时，我们证明解决方案是唯一的，因此，Yamada-Wantanabe Theorem KS-SNS系统承认了独特的全局路径弱解决方案。
This paper considers the Keller-Segel model coupled to stochastic Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of oxygen and bacteria densities evolving within a stochastically forced 2D incompressible viscous flow. Our main goal is to investigate the existence and uniqueness of global solutions (strong in the probabilistic sense and weak in the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with proper regularization and cut-off operators in $H^s(\mathbb{R}^2)$ is introduced, and the existence of approximate solution is proved by some a priori uniform bounds and a careful analysis on the approximation scheme. Under appropriate assumptions, two types of stochastic entropy-energy inequalities that seem to be new in their forms are derived, which together with the Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the sequence of approximate solutions converges to a global martingale weak solution. In addition, when $χ(\cdot)\equiv \textrm{const.}>0$, we prove that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem that the KS-SNS system admits a unique global pathwise weak solution.
