On a stochastic Leray-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>α</mml:mi></mml:math>model of Euler equations

Barbato, David; Bessaih, Hakima; Ferrario, Benedetta

doi:10.1016/j.spa.2013.07.002

Cited by 27 publications

(50 citation statements)

References 21 publications

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“…[1]), that is the first entry of the bilinear term P [( v · ∇) η] is not the unknown η itself but indeed v, which has one order more of regularity with respect to η (recall that if η ∈ H b p then v ∈ H b+1 p ). Therefore, even if η satisfies a nonlinear equation, the quadratic term ( v ·∇) η in (24) (with η = ∇ × v) behaves better than ( v · ∇) v in (1) and this makes the difference in the analysis of systems (24) and (1).…”

Section: The Vorticity Transport Equationmentioning

confidence: 99%

“…where the unknown are the velocity v and the pressure p; the data are the viscosity ν > 0, the deterministic forcing term f and the random one n. Working in a bounded three dimensional spatial domain with suitable boundary conditions, it is known that for initial velocity of finite energy and suitable forcing terms there exists a weak solution to (1) defined for any positive time, but uniqueness is an open problem. On the other side, more regular initial velocities provide existence and uniqueness of a solution, which is only local in time.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of the law for 3D stochastic hyperviscous fluids

Ferrario¹

2016

Electron. J. Probab.

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We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term −∆ ξ of the Navier-Stokes equations is substituted by (−∆) 1+c ξ. We investigate how big the correction term c has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as c > 1 2 , improving previous results obtained with c > 3 2 in a different setting in [5,14].MSC2010: 76M35, 60H15, 35Q30.

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Section: The Vorticity Transport Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Characterization of the law for 3D stochastic hyperviscous fluids

Ferrario¹

2016

Electron. J. Probab.

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“…In contrast to the Lagrangian Averaged Euler equations the Leray-α Euler equation, either in two or three dimensional cases, admits a global weak solution. The uniqueness of solution of the Lerayα Euler equations is an open problem for the deterministic case, however when adding a special multiplicative noise it was proved in the interesting paper [47] that the solution of the stochastic Euler-α is unique in law.…”

Section: Literature Reviewmentioning

confidence: 99%

Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

Razafimandimby

2017

Journal of Differential Equations

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In this paper we present a systematic study of a stochastic PDE with multiplicative noise modeling the motion of viscous and inviscid grade-two fluids on a bounded domain O of R 2. We aim to identify the minimal conditions on the boundary smoothness of the domain for the well-posedness and time regularity of the solution. In particular, we found out that the existence of a H 1 pOq weak martingale solution holds for any bounded Lipschitz domain O. When O is a convex polygon the solution u lives in the Sobolev space W 2,r pOq for some r ą 2 and rotpu´α∆uq is continuous in L 2 pOq with respect to the time variable. Moreover, pathwise uniqueness of solution holds. The existence result is new for the stochastic inviscid model and improves previous results for the viscous one. The time continuity result is new, even for the deterministic case when the domain O is a convex polygon.

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“…In particular, for the 2d Navier-Stokes equations, some ergodic properties are proved when adding a random perturbation, There is an extensive literature about the convergence of α-models to the Navier-Stokes equations, see for eg. [29,1,7,8,18,22,23,24,25]. However, only a few papers deal with the rate of convergence, see [10,15].…”

Section: Introductionmentioning

confidence: 99%

On the Rate of Convergence of the 2-D Stochastic Leray- $$\alpha $$ α Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise

Bessaih

Razafimandimby

2015

Appl Math Optim

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In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-α model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate, as α → 0, of the following error functionwhere u α and u are the solution of stochastic Leray-α model and the stochastic NavierStokes equations, respectively. We show that when properly localized the error function ε α converges in mean square as α → 0 and the convergence is of order O(α). We also prove that ε α converges in probability to zero with order at most O(α).Keywords Navier-Stokes equations · Leray-α model · Rate of convergence in mean square · Rate of convergence in probability · Turbulence models · Navier-Stokes-α Mathematics Subject Classification 60H15 · 76D05 · 60H35 B Hakima Bessaih

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On a stochastic Leray-αmodel of Euler equations

Cited by 27 publications

References 21 publications

Characterization of the law for 3D stochastic hyperviscous fluids

Characterization of the law for 3D stochastic hyperviscous fluids

Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

On the Rate of Convergence of the 2-D Stochastic Leray- $$\alpha $$ α Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise

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