Statistical properties of stochastic 2D Navier-Stokes equations from linear models

Bessaih, Hakima; Ferrario, Benedetta

doi:10.3934/dcdsb.2016080

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…Additionally, we believe that establishing absolute continuity of the dynamics with respect to a Gaussian reference measure will open additional perspectives and approaches to analyzing these rough SPDEs. Finally, we mention that in [BF16] the authors prove a connection between a nonlinear problem and a linear problem.…”

Section: Introductionmentioning

confidence: 96%

The Gaussian Structure of the Singular Stochastic Burgers Equation

Mattingly¹,

Romito²,

Su³

2021

Preprint

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We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time t, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed as we are beyond the regime of classical solutions for much of the paper.LS dedicates this to his mother Deli Xie for everything she has done for the family.

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Section: Introductionmentioning

confidence: 96%

The Gaussian Structure of the Singular Stochastic Burgers Equation

Mattingly¹,

Romito²,

Su³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Additionally, we believe that establishing absolute continuity of the dynamics with respect to a Gaussian reference measure will open additional perspectives and approaches to analysing these rough SPDEs. Finally, we mention that in [BF16], the authors prove a connection between a nonlinear problem and a linear problem.…”

Section: Introductionmentioning

confidence: 99%

The Gaussian structure of the singular stochastic Burgers equation

Mattingly

Romito

2022

Forum of Mathematics, Sigma

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We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time t, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite-dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes, which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed, as we are beyond the regime of classical solutions for much of the paper.

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“…The 2D Euler equations with additive noise, possibly including friction, their corresponding stationary solutions and invariant measures had already been considered before. However, the space regularity of noise is such that solutions are function-valued, not distributions and invariant measures are supported on spaces of functions: we refer for instance to [11,19,8,12,21,44,9,43,10], and also to other related results in [52,53,32]. Many of those models and results are inspired by the open problem of turbulence (iii); in connection with this question and the previous references we also mention [14,46,37].…”

Section: Introductionmentioning

confidence: 99%

Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise

Flandoli,

Grotto,

Luo

2019

Preprint

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After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker-Planck equation. The regularity class of solutions investigated here does not allow energyor enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker-Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.

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Statistical properties of stochastic 2D Navier-Stokes equations from linear models

Cited by 4 publications

References 24 publications

The Gaussian Structure of the Singular Stochastic Burgers Equation

The Gaussian Structure of the Singular Stochastic Burgers Equation

The Gaussian structure of the singular stochastic Burgers equation

Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise

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