关于与 R2 中随机强迫不可压缩粘性流相互作用的凯勒-西格尔模型

IF 2.4 2区 数学 Q1 MATHEMATICS
Lei Zhang, Bin Liu
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引用次数: 0

摘要

本文研究了与随机纳维-斯托克斯方程(简称 KS-SNS)耦合的 Keller-Segel 模型,该模型描述了在随机强迫的二维不可压缩粘性流中氧气和细菌密度的动态演化。我们的主要目标是研究 KS-SNS 系统全局解(概率意义上的强解和 PDE 意义上的弱解)的存在性和唯一性。我们引入了一种新的近似 KS-SNS 系统,该系统在 Hs(R2) 中具有适当的正则化和截止算子,并通过一些先验均匀边界和对近似方案的仔细分析证明了近似解的存在性。在适当的假设条件下,推导出了两类形式看似新颖的随机熵能不等式,它们与 Prohorov 定理和 Jakubowski-Skorokhod 定理一起使我们能够证明近似解序列收敛于全局马氏弱解。此外,当χ(⋅)≡const.>0 时,我们证明了解是路径上唯一的,因此根据山田-万端部定理,KS-SNS 系统承认一个唯一的全局路径弱解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the Keller-Segel models interacting with a stochastically forced incompressible viscous flow in R2

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 Hs(R2) 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()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.

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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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