Euler equations for incompressible ideal fluids

Bardos, Claude; Titi, Edriss S.

doi:10.1070/rm2007v062n03abeh004410

Cited by 128 publications

(163 citation statements)

References 79 publications

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“…By the Aubin-Lions lemma with spaces 12]) we have that the sequence w n has a strongly convergent subsequence (relabelled w n ) in L 2 (0, T ; H −1 (Ω)). More precisely, we have that 1 2 with −∆ the Laplacian with homogeneous Dirichlet boundary conditions in Ω. It is well known that…”

Section: Holdsmentioning

confidence: 99%

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Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Constantin

Vicol

2017

J Nonlinear Sci

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ABSTRACT. We prove that any weak space-time L 2 vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of R 2 satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that t − a.e. weak L 2 inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.

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Section: Holdsmentioning

confidence: 99%

“…There is a large literature concerned with related or similar conditional strong L 2 convergence results (a few examples are [1,2,4,5,9,10,19,20]). Some strong L 2 unconditional convergence results for short time do exist.…”

Section: Introductionmentioning

confidence: 99%

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Constantin

Vicol

2017

J Nonlinear Sci

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“…The adaptation to sufficiently regular forced cases, for example a trigonometric polynomial, is similar but more technical. 3 Furthermore, it is enough to prove the decay result up to a time T such that 1/N := T U/L < 1, where L and U are a typical length scale and velocity of the initial data. Extending the results to later times is easy (by propagation of regularity).…”

Section: Rate Of Decay Of the Fourier Coefficientsmentioning

confidence: 99%

“…More than a quarter of a millenium after the introduction by Leonhard Euler of the equations of incompressible fluid dynamics, the question of their well-posedness in three dimensions (3D) with sufficiently smooth initial data is still moot [1][2][3][4] (see also many papers in [5] and references therein). Even more vexing is the fact that switching to viscous flow for the solution of the Navier-Stokes equations (NSE) barely improves the situation in 3D [6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

Bardos

Frisch²,

Pauls

et al. 2009

Commun. Math. Phys.

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Abstract:We consider a modification of the three-dimensional Navier-Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e |k|/k d at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e −C(k/k d ) ln(|k|/k d ) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C = 1/ ln 2. The same behavior with a universal constant C is conjectured for the Navier-Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier-Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

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“…The models are used to examine different plasma phenomena such as Rayleigh-Timofeev hydrodynamic resonance, transformation of waves, magnetohydrodynamic instabilities. A survey concerning the state-of-the-art mathematical theory of the Euler equations for an incompressible homogeneous ideal fluid is made in [17]. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.…”

Section: Introductionmentioning

confidence: 99%

Model of formation of droplets during electric arc surfacing of functional coatings

Sarychev

Granovskii

Невский

et al. 2016

AIP Conference Proceedings

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Predictions of metal droplet formation in gas metal arc welding. II. Journal of Applied Physics 84, 3530 (1998) Abstract. The mathematical model was developed for the initial stage of formation of an electrode metal droplet in the process of arc welding. Its essence lies in the fact that the presence of a temperature gradient in the boundary layer of the molten metal causes thermo-capillary instability, which leads to the formation of electrode metal droplets. А system of equations including Navier-Stokes equations, heat conduction and Maxwell's equations was solved as well as the boundary conditions for the system electrodes-plasma. Dispersion equation for thermo-capillary waves in the linear approximation for the plane layer was received and analyzed. The values of critical wavelengths, at which thermocapillary instability appears in the nanometer wavelength range, were found. The parameters at which the mode of a finedroplet transfer of the material takes place were theoretically defined.

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Euler equations for incompressible ideal fluids

Abstract: This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.

Cited by 128 publications

References 79 publications

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

Model of formation of droplets during electric arc surfacing of functional coatings

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