A Stochastic Perturbation of Inviscid Flows

Iyer, Gautam

doi:10.1007/s00220-006-0058-5

Cited by 21 publications

(54 citation statements)

References 18 publications

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“…Further, as shown in [14] the displacements X − I and A − I are spatially C k+1,α (thus bounded in periodic domains) and hence the Itô integrals (and expectations) that arise here are all well defined. This said, we assume subsequently that all processes are spatially regular enough for our computations to be valid.…”

Section: Proof Of the Stochastic Representation Using The Itô Formulamentioning

confidence: 63%

“…We also remark that since the noise W t is spatially constant, the inverse map A t is as (spatially) regular as the flow map X t . We refer the reader to [14,15,17] for the details. Conversely, given a (global) solution u of (1.5) which is either spatially periodic or decays at infinity, standard theory shows that (1.1) has a global solution.…”

Section: Introductionmentioning

confidence: 99%

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A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations

Constantin

Iyer

2007

Comm Pure Appl Math

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135

288

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Abstract. In this paper we derive a probabilistic representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system, and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and LANS-alpha models.

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Section: Proof Of the Stochastic Representation Using The Itô Formulamentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations

Constantin

Iyer

2007

Comm Pure Appl Math

Self Cite

135

288

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“…Stochastic Weber variables p , = i , e, of the same form as ͑51͒ are shown to obey stochastic partial differential equations ͑PDEs͒ of the same form as ͑52͒. It is now enough to assume u e , u i C͓͑t 0 , t f ͔ , C k,␣ ͑⍀͒͒ for k Ն 2, because the integration-by-parts identity 11,12 can be employed to show that p e , p i C 2,␣ ͑⍀͒, just as for p i in the proof of Proposition 3.1.1.…”

Section: ͑67͒mentioning

confidence: 99%

Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

Eyink

2009

Journal of Mathematical Physics

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We prove that smooth solutions of non-ideal (viscous and resistive) incompressible magnetohydrodynamic equations satisfy a stochastic law of flux conservation. This property involves an ensemble of surfaces obtained from a given, fixed surface by advecting it backward in time under the plasma velocity perturbed with a random white-noise. It is shown that the magnetic flux through the fixed surface is equal to the average of the magnetic fluxes through the ensemble of surfaces at earlier times. This result is an analogue of the well-known Alfvén theorem of ideal MHD and is valid for any value of the magnetic Prandtl number. A second stochastic conservation law is shown to hold at unit Prandtl number, a random version of the generalized Kelvin theorem derived by Bekenstein-Oron for ideal MHD. These stochastic conservation laws are not only shown to be consequences of the non-ideal MHD equations, but are proved in fact to be equivalent to those equations. We derive similar results for two more refined hydromagnetic models, Hall magnetohydrodynamics and the two-fluid plasma model, still assuming incompressible velocities and isotropic transport coefficients.Finally, we use these results to discuss briefly the infinite-Reynolds-number limit of hydromagnetic turbulence and to support the conjecture that flux-conservation remains stochastic in that limit.

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“…One of the challenges would be to find the appropriate stochastic energy bounds. As a matter of fact this has actually been accomplished by [18,19] for the Weber formula approach of [7].…”

Section: Dmentioning

confidence: 99%