【摘要】：在本文中,我們關(guān)注一類具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程.阻尼項(xiàng)是由阻礙流體的運(yùn)動產(chǎn)生的且描述各種各樣的物理現(xiàn)象,例如,阻力或者摩擦力,和一些耗散機(jī)制.我們首先考慮了帶乘性噪聲的具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程.最后,我們研究了帶跳噪聲的一類具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程.論文由四個部分組成,結(jié)構(gòu)如下:第一章,我們給出了本文的研究背景及一些基本不等式.第二章,通過經(jīng)典Faedo-Galerkin逼近和緊性方法,我們得到了具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程鞅解的存在性.在β 3且α 0和β = 3且α 1/2情況下,我們攻克一些困難(例如∫0t∫D |u · %絬|2dxds的估計)且得到了強(qiáng)解的存在性和唯一性.同時,當(dāng) 3且α 0和β= 3且α ≥ 1/2時,我們也得到了具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程的小時間大偏差原理.第三章,我們考慮了由乘性噪聲驅(qū)動下具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程.在β 3且α 0和β=3且α ≥ 1/2情況下,通過單調(diào)性方法,我們攻克一些困難(例如∫0t∫D|u·%絬u|2dxds的估計),并且證明了強(qiáng)解的存在性和唯一性.當(dāng)3 β≤5且α 0和β = 3且α≥1/2時,使用Krylov-Bogoliubov方法,我們克服一些困難(例如(?)的估計),并且證明了不變測度的存在性.在退化可加噪聲情況下,使用漸近強(qiáng)Feller性,我們同時得到了不變測度的唯一性.在β 3且α 0和β = 3且α ≥ 1/2情況下,通過克服一些困難(例如∫Dz-2(t)|v|2|%絭|v2ds的估計),我們證明了具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程生成隨機(jī)動力系統(tǒng)吸引子的存在性.我們在第三章中也證明了由乘性噪聲驅(qū)動下具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程的大偏差原理.當(dāng)3 ≤ β 5,使用弱收斂方法,我們引入不等式(?),并且證明了具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程的大偏差原理.第四章,我們考慮了由跳噪聲驅(qū)動的一類具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程.我們得到了函數(shù)g(s)是光滑的且滿足通過∫0t∫D|u|2|%絬|2dxds的估計,我們克服由跳噪聲造成的主要困難.在β 2且α 0和β = 2且α ≥ 1/2情況下,我們得到了強(qiáng)解的存在性和唯一性.這個主要結(jié)論應(yīng)用于各種各樣的隨機(jī)偏微分方程,例如,具有阻尼項(xiàng)的隨機(jī)三維Navier-Stokes方程,隨機(jī)馴服三維Navier-Stokes方程,隨機(jī)三維Brinkman-Forchheimer-eXtended Darcy模型.我們利用解的指數(shù)穩(wěn)定證明了唯一不變測度的存在性.
[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, 偽 _ 0 and 尾 = 3 and 偽 _ 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 偽 _ 0 and 尾 = 3 and 偽 _ 1 ~ (2). At the same time, when 3 and 偽 0 and 尾 = 3 and 偽 鈮，

本文編號：2434280