The Euler Equations With Dissipation

Ильин, Алексей Андреевич

doi:10.1070/sm1993v074n02abeh003357

Cited by 21 publications

(30 citation statements)

References 29 publications

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Section: Bp06mentioning

confidence: 99%

“…The mathematical theory of damped Navier-Stokes and Euler equations is of a big current interest, see CV08,CVZ11,CR07,Il91,IlT06,IMT04 [4,5,6,10,11,12] and references therein. However, most part of these papers study either the case of bounded underlying domain (e.g., with periodic boundary conditions) or the case of finite energy solutions in the whole space R 2 and very few is known about the infiniteenergy solutions (for instance, starting with u 0 ∈ L ∞ (R 2 ) or from the proper uniformly local Sobolev space) which are natural from the physical point of view (for instance, for applications to the so-called uniform turbulence theory or/and statistical hydrodynamics, see VF80 [21] for the details).…”

Section: Bp06mentioning

confidence: 99%

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Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

Zelik

2013

J. Math. Fluid Mech.

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Abstract. We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field u0 ∈ L ∞ (R 2 ) is allowed and no assumptions on the spatial decay of solutions as |x| → ∞ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R 2 , we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity. We study the following damped Navier-Stokes system in the whole space x ∈ R 2 :where α is a positive parameter. These equations describe, for instance, a 2-dimensional viscous liquid moving on a rough surface and are used in geophysical models for large-scale processes in atmosphere and ocean. The term αu parameterizes the extra dissipation occurring in the planetary boundary layer (see, e.g., Ped79[15]; see also BP06[3] for the alternative source of damped Euler equations).The mathematical theory of damped Navier-Stokes and Euler equations is of a big current interest, see CV08,CVZ11,CR07,Il91,IlT06,IMT04 [4,5,6,10,11,12] and references therein. However, most part of these papers study either the case of bounded underlying domain (e.g., with periodic boundary conditions) or the 2000 Mathematics Subject Classification. 35Q30,35Q35.

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Section: Bp06mentioning

confidence: 99%

Section: Bp06mentioning

confidence: 99%

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

Zelik

2013

J. Math. Fluid Mech.

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“…Existence and uniqueness results for the stationary problem and also results on the stability of stationary solutions for (1.3) with ν = 0 can be found in [3], [32], [36]. Weak global attractors for this system, with ν = 0, were constructed in [21], [18] and [19] (see also [5]). An investigation of the bifurcation diagram and other related dynamical properties of the system (1.3) are reported in [6] (see also references therein).…”

Section: Introductionmentioning

confidence: 99%

Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations

Ilyin¹,

Miranville²,

Titi³

2004

Communications in Mathematical Sciences

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Abstract. We consider in this article the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficient ν → 0 + . In particular, we obtain upper bounds of the order ν −1 on the fractal and Hausdorff dimensions of the global attractor for the system on the torus T 2 , on the sphere S 2 and in a bounded domain. Furthermore, in the case of the torus, we establish a lower bound of the order ν −1 . This sharp estimate is remarkably smaller than the well established sharp bound for the dimension of the global attractor of the Navier-Stokes equations on the torus T 2 , which is of the order ν −4/3 . This means that the damping/friction term plays a significant role in reducing the number of degrees of freedom in this two-dimensional model. This, we believe, is done by dissipating the energy at the large spatial scales which is transferred to these scales via the inverse cascade mechanism. Finally, we remark that the system of equations studied here is related to the Stommel-Charney barotropic ocean circulation model of the gulf stream.

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“…The existence of weak attractors for the system of equations (1.4) in the case of σ > 0 and ν = 0 has been proved in [6].…”

Section: Attractor and Estimate Of Its Dimensionmentioning

confidence: 98%

Some dimensionless numbers and the associated inequalities describing the asymptotic behaviour of the solutions to the barotropic vorticity equation

Ilyin¹

1992

Russian Journal of Numerical Analysis and Mathematical Modelling

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The behaviour of the solutions of the barotropic vorticity equation on a sphere is analyzed for f -oo. It is proved that the inequality G < 0.59 is sufficient for the stationary solution to be a global attractor (of the dimension 0). Here the dimensionless number G = U/H/^V 2 ), where ||/|| is the L^ norm of baroclinic forcing, λ. is the first eigenvalue of the Laplacian operator on the sphere and V is the viscosity coefficient.For large numbers G the dimension of the global attractor has been estimated as 5.6G 2/3 (4.3 + ^lnG) 1/3 + 1 , dim,,«*/*: 15.8G 2/3 (4.3 + f lnG) 1/3 + 2 .The characteristic value of G for large-scale atmospheric processes is discussed.The stability of large-scale atmospheric processes has been intensely analyzed in the recent years (see ref.[3] and references therein). The best studied is the case when the large-scale dynamics is described by the barotropic vorticity equation on the rotating sphere. The sufficient conditions have been derived for the global asymptotic stability of the stationary regimes in terms of smallness of their gradients and the spectrum of the Hermitian part of the linearized operator [2,3]. The theorem of stability with respect to the spectrum of the linearized operator has been proved in [8]. All these sufficient conditions are, of course, not satisfied for the real quasi-stationary atmospheric processes and we shall demonstrate that they are not satisfied within a very large margin (see equations (2.1) and (2.2). The behaviour of solutions for the real parameter values is highly complicated. The concept of the invariant attractive set -the attractor -is essential in its analysis [1,10,16]. The dimension of the attractor in the appropriate phase space is one of the few geometric characteristics that we can estimate. An inequality is being derived here which guarantees the uniqueness of the stationary solution if it is satisfied. The same condition proves to be sufficient for this stationary solution to be a global attractor. In addition, the dimension of the attractor is estimated with the explicit numerical upper bounds of the respective constants. The appropriate estimates are given and partially proved in the Appendix. These are primarily integral inequalities for functions and families of functions on a sphere.Brought to you by |

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The Euler Equations With Dissipation

Cited by 21 publications

References 29 publications

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations

Some dimensionless numbers and the associated inequalities describing the asymptotic behaviour of the solutions to the barotropic vorticity equation

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