Global regularity for a Birkhoff–Rott- approximation of the dynamics of vortex sheets of the 2D Euler equations

Bardos, Claude; Linshiz, Jasmine S.; Titi, Edriss S.

doi:10.1016/j.physd.2008.01.003

Cited by 22 publications

(36 citation statements)

References 70 publications

(177 reference statements)

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“…In addition, several of the α-models of turbulence have been tested against averaged empirical data collected from turbulent channels and pipes, for a wide range of Reynolds numbers (up to 17 × 10 6 ) [ [14][15][16]. The successful analytical, empirical and computational aspects (see for example [28,40,62,63] and references therein) of the alpha turbulence models have attracted numerous applications, see for example [5] for application to the quasi-geostrophic equations, [51] for application to Birkhoff-Rott approximation dynamics of vortex sheets of the 2D Euler equations, [60,66,67] for applications to incompressible magnetohydrodynamics equations. See also [53,54] for the α-regularization of the inviscid 3D Boussinesq equation.…”

Section: Application To Turbulence Modelsmentioning

confidence: 99%

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

Farhat¹,

Lunasin²,

Titi³

2019

Partial Differential Equations Arising From Physics and Geometry

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In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary that one has to collect coarse mesh measurements of all the state variables, that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information of the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; viz. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model. This paper is dedicated to the memory of Professor Abbas Bahri MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.

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Section: Application To Turbulence Modelsmentioning

confidence: 99%

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

Farhat¹,

Lunasin²,

Titi³

2019

Partial Differential Equations Arising From Physics and Geometry

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“…We only sketch the main steps of the proof, since it is in the spirit of [4,5] and [57,Chapter 8], which can be consulted for details of such a proof. In [5] we show the well-posedness of vortex sheet problem for Euler-α equations, and it contains various estimates involving the derivatives of the kernel Ψ α , and [57,Chapter 8] describes the proof of the original vortex patch problem in the Euler equations case.…”

Section: A1 Global Regularity Of Contour Dynamics-α Equationmentioning

confidence: 99%

On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations

Linshiz

Titi

2010

J Stat Phys

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We study the convergence rate of the solutions of the incompressible Euler-α, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter α approaches zero. First we show the convergence in H s , s > n/2 + 1, in the whole space, and that the smooth Euler-α solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.

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“…There is a large literature directly associated with Delort's Theorem. The convergence to a weak solution was extended to approximations obtained by vanishing viscosity, see [20], numerical approximations, see [15,25] and Euler-α, see [2]. The initial data class was extended to the limiting case p = 1, see [8,28], and an alternative proof using harmonic analysis was produced, see [9].…”

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confidence: 99%

Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

Iftimie

Filho

Lopes

2019

Communications in Partial Differential Equations

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In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data. We formulate our results for flows outside a finite number of smooth obstacles. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulations around connected components of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Our first result is existence of a solution for the weak velocity formulation with vortex sheet initial data for flow outside a finite number of smooth obstacles. The proof is a straightforward adaptation of Delort's original existence result and requires the usual sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation at the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs.

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Global regularity for a Birkhoff–Rott- approximation of the dynamics of vortex sheets of the 2D Euler equations

Cited by 22 publications

References 70 publications

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

A Data Assimilation Algorithm: the Paradigm of the 3D Leray-α Model of Turbulence

On the Convergence Rate of the Euler-α, an Inviscid Second-Grade Complex Fluid, Model to the Euler Equations

Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

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