2017
DOI: 10.1063/1.4996545
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Rigorous results in space-periodic two-dimensional turbulence

Abstract: We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a uniqu… Show more

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Cited by 10 publications
(10 citation statements)
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“…Rigorous results on ergodicity are available for 2-D Navier–Stokes on a doubly periodic domain (flat torus) with added regular-in-space noise proportional to the square root of the viscosity . In this setting there exists a unique stationary measure (Kuksin & Shirikyan 2012, 2017). As , one obtains a stationary measure for the 2-D Euler equations, but it is not expected to be unique (Kuksin & Shirikyan 2017).…”
Section: Introductionmentioning
confidence: 99%
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“…Rigorous results on ergodicity are available for 2-D Navier–Stokes on a doubly periodic domain (flat torus) with added regular-in-space noise proportional to the square root of the viscosity . In this setting there exists a unique stationary measure (Kuksin & Shirikyan 2012, 2017). As , one obtains a stationary measure for the 2-D Euler equations, but it is not expected to be unique (Kuksin & Shirikyan 2017).…”
Section: Introductionmentioning
confidence: 99%
“…In this setting there exists a unique stationary measure (Kuksin & Shirikyan 2012, 2017). As , one obtains a stationary measure for the 2-D Euler equations, but it is not expected to be unique (Kuksin & Shirikyan 2017).…”
Section: Introductionmentioning
confidence: 99%
“…We are motivated by finite dimensional models for the inviscid stochastic 2-d Navier-Stokes equations written in vorticity form (see [33], [45,Lecture 39])…”
Section: Introductionmentioning
confidence: 99%
“…in which K = ∇ ⊥ ∆ −1 is the Biot-Savart operator, η(t, x) is a noise, and the viscosity parameter ν → 0. An unsolved issue here targets at studying the vanishing noise limit of stationary measures of the 2-d stochastic Navier-Stokes system (see open problem 3 in the last section of the survey [33]). The difficulty there is that one has to put a rather restrictive hypothesis, namely the unperturbed dynamics has to be globally asymptotically stable.…”
mentioning
confidence: 99%
“…Такая добавка дает возможность использования диссипационно-флуктуационных соотношений для построения оператора отклика системы на малые внешние воздействия [8]. Галин наличие устойчивой эргодической стационарной меры при использовании стохастической регуляризации было строго доказано для двумерных уравнений несжимаемой вязкой жидкости [9] и двуслойной квазигеострофической модели атмосферы [10,11]. В настоящей работе мы формулируем и исследуем динамико-стохастическую параметризацию одного из процессов подсеточного масштаба -формирование балла неконвективной облачности, играющего фундаментальную роль в формировании радиационных притоков тепла.…”
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