Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations

Shirikyan, Armen

doi:10.1016/j.jfa.2007.01.005

Cited by 18 publications

(19 citation statements)

References 19 publications

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“…However, the convergence will hold true on nice enough subsets -which suffices for our endeavour. These decomposability properties play a central role in the arguments of [Shi07,Shi17] and are discussed here in Appendix A.…”

Section: Setup Assumptions and Main Resultsmentioning

confidence: 99%

“…Most of the ideas for the next three results are present in different parts of [Shi17]; also see [Shi07]. We retrieve the key steps and repiece them in a way that is suitable for our endeavour.…”

Section: Approachability and Solid Controllabilitymentioning

confidence: 99%

“…Such results typically involve carefully studying smoothing properties of the associated Markov semigroup. The strategy here is different and instead relies on recent developments on the use of solid controllability in the study of mixing properties of random dynamical systems [AS05, AKSS07,Shi07,Shi17]. The simplicity of the argument can in itself justify the presentation of such an application.…”

Section: Introductionmentioning

confidence: 99%

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A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

Raquépas

2018

Ann. Henri Poincaré

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We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the formA encodes the harmonic part of the force and −F corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: A is dissipative, the pair (A, B) satisfies the Kalman condition, F grows sufficiently slowly at infinity (depending on the dimension d), and the vector fields in the equation of motion satisfy the weak Hörmander condition in at least one point of the phase space.Different sufficient conditions for the hypotheses of the main theorem to hold are given in more concrete terms throughout Sections 4 and 5. In the former, we give criteria for the dissipativity, Kalman and growth conditions in terms of more physical quantities for networks of oscillators based on [JPS17]. In the latter, we give a perturbative condition for the weak Hörmander condition to hold.

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Section: Setup Assumptions and Main Resultsmentioning

confidence: 99%

Section: Approachability and Solid Controllabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

Raquépas

2018

Ann. Henri Poincaré

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“…Again we deduce that P[π F u(t) ∈ E] = 0 from the fact that G t > 0. Finally, the fact that the density of π F u(t) is positive follows from the results of [30].…”

Section: Existence Of Densities With Non-degenerate Noise: Girsanov Amentioning

confidence: 83%

Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise

Debussche

Romito

2013

Probab. Theory Relat. Fields

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We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in [18]

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“…Uniqueness is still an open problem. However, Markov solutions have been constructed and ergodic properties have been proved (see [2]- [4], [7], [9], [11], [12], [22], [25], [26]). In [20], [21], a general form of the stochastic Navier-Stokes equations is derived from the assumptions that the fluid particles are subject to turbulent diffusion.…”

Section: Introductionmentioning

confidence: 99%

On the martingale problem associated to the $2D$ and $3D$ stochastic Navier–Stokes equations

Prato¹,

Debussche²

2008

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Probability theory. -On the martingale problem associated to the 2D and 3D stochastic Navier-Stokes equations, by GIUSEPPE DA PRATO and ARNAUD DEBUSSCHE.ABSTRACT. -We consider a Markov semigroup (P t ) t≥0 associated to the 2D and 3D NavierStokes equations. In the two-dimensional case P t is unique, whereas in the three-dimensional case (where uniqueness is not known) it is constructed as in [4] and [7].For d = 2, we specify a core, identify the abstract generator of (P t ) t≥0 with the differential Kolmogorov operator L on this core and prove existence and uniqueness for the corresponding martingale problem. In dimension 3, we are not able to prove a similar result and we explain the difficulties encountered. Nonetheless, we specify a core for the generator of the transformed semigroup (S t ) t≥0 , obtained by adding a suitable potential and then using the Feynman-Kac formula. Then we identify the abstract generator (S t ) t≥0 with a differential operator N on this core and prove uniqueness for the stopped martingale problem.

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Qualitative properties of stationary measures for three-dimensional Navier–Stokes equations

Cited by 18 publications

References 19 publications

A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

A Note on Harris’ Ergodic Theorem, Controllability and Perturbations of Harmonic Networks

Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise

On the martingale problem associated to the $2D$ and $3D$ stochastic Navier–Stokes equations

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