具阻尼项的随机三维Navier-Stokes方程的适定性和动力学
发布时间：2019-03-04 12:45
【摘要】：在本文中,我们关注一类具有阻尼项的随机三维Navier-Stokes方程.阻尼项是由阻碍流体的运动产生的且描述各种各样的物理现象,例如,阻力或者摩擦力,和一些耗散机制.我们首先考虑了带乘性噪声的具有阻尼项的随机三维Navier-Stokes方程.最后,我们研究了带跳噪声的一类具有阻尼项的随机三维Navier-Stokes方程.论文由四个部分组成,结构如下:第一章,我们给出了本文的研究背景及一些基本不等式.第二章,通过经典Faedo-Galerkin逼近和紧性方法,我们得到了具有阻尼项的随机三维Navier-Stokes方程鞅解的存在性.在β 3且α 0和β = 3且α 1/2情况下,我们攻克一些困难(例如∫0t∫D |u · %絬|2dxds的估计)且得到了强解的存在性和唯一性.同时,当 3且α 0和β= 3且α ≥ 1/2时,我们也得到了具有阻尼项的随机三维Navier-Stokes方程的小时间大偏差原理.第三章,我们考虑了由乘性噪声驱动下具有阻尼项的随机三维Navier-Stokes方程.在β 3且α 0和β=3且α ≥ 1/2情况下,通过单调性方法,我们攻克一些困难(例如∫0t∫D|u·%絬u|2dxds的估计),并且证明了强解的存在性和唯一性.当3 β≤5且α 0和β = 3且α≥1/2时,使用Krylov-Bogoliubov方法,我们克服一些困难(例如(?)的估计),并且证明了不变测度的存在性.在退化可加噪声情况下,使用渐近强Feller性,我们同时得到了不变测度的唯一性.在β 3且α 0和β = 3且α ≥ 1/2情况下,通过克服一些困难(例如∫Dz-2(t)|v|2|%絭|v2ds的估计),我们证明了具有阻尼项的随机三维Navier-Stokes方程生成随机动力系统吸引子的存在性.我们在第三章中也证明了由乘性噪声驱动下具有阻尼项的随机三维Navier-Stokes方程的大偏差原理.当3 ≤ β 5,使用弱收敛方法,我们引入不等式(?),并且证明了具有阻尼项的随机三维Navier-Stokes方程的大偏差原理.第四章,我们考虑了由跳噪声驱动的一类具有阻尼项的随机三维Navier-Stokes方程.我们得到了函数g(s)是光滑的且满足通过∫0t∫D|u|2|%絬|2dxds的估计,我们克服由跳噪声造成的主要困难.在β 2且α 0和β = 2且α ≥ 1/2情况下,我们得到了强解的存在性和唯一性.这个主要结论应用于各种各样的随机偏微分方程,例如,具有阻尼项的随机三维Navier-Stokes方程,随机驯服三维Navier-Stokes方程,随机三维Brinkman-Forchheimer-eXtended Darcy模型.我们利用解的指数稳定证明了唯一不变测度的存在性.
[Abstract]:In this paper, we focus on a class of stochastic three-dimensional Navier-Stokes equations with damping terms. Damping terms are produced by obstructing the movement of the fluid and describe a variety of physical phenomena, such as drag or friction, and some dissipation mechanisms. In this paper, we first consider the stochastic three-dimensional Navier-Stokes equation with damping term with multiplicative noise. Finally, we study a class of stochastic three-dimensional Navier-Stokes equations with damping terms with jump noise. This paper consists of four parts. The structure is as follows: in chapter 1, we give the research background and some basic inequalities of this paper. In chapter 2, by means of classical Faedo-Galerkin approximation and compactness method, we obtain the existence of martingale solutions for stochastic three-dimensional Navier-Stokes equation with damping term. In the case of 尾 _ 3, 伪 _ 0 and 尾 = 3 and 伪 _ 1 ~ (2), we overcome some difficulties (for example, the estimate of 2dxds) and obtain the existence and uniqueness of the strong solution in the case of 尾 _ 3 and 伪 _ 0 and 尾 = 3 and 伪 _ 1 ~ (2). At the same time, when 3 and 伪 0 and 尾 = 3 and 伪 鈮，
本文编号：2434280
[Abstract]:In this paper, we focus on a class of stochastic three-dimensional Navier-Stokes equations with damping terms. Damping terms are produced by obstructing the movement of the fluid and describe a variety of physical phenomena, such as drag or friction, and some dissipation mechanisms. In this paper, we first consider the stochastic three-dimensional Navier-Stokes equation with damping term with multiplicative noise. Finally, we study a class of stochastic three-dimensional Navier-Stokes equations with damping terms with jump noise. This paper consists of four parts. The structure is as follows: in chapter 1, we give the research background and some basic inequalities of this paper. In chapter 2, by means of classical Faedo-Galerkin approximation and compactness method, we obtain the existence of martingale solutions for stochastic three-dimensional Navier-Stokes equation with damping term. In the case of 尾 _ 3, 伪 _ 0 and 尾 = 3 and 伪 _ 1 ~ (2), we overcome some difficulties (for example, the estimate of 2dxds) and obtain the existence and uniqueness of the strong solution in the case of 尾 _ 3 and 伪 _ 0 and 尾 = 3 and 伪 _ 1 ~ (2). At the same time, when 3 and 伪 0 and 尾 = 3 and 伪 鈮，
本文编号：2434280
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