The Euler equations as a differential inclusion

Lellis, Camillo De; Székelyhidi, László

doi:10.4007/annals.2009.170.1417

Cited by 520 publications

(632 citation statements)

References 29 publications

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“…Furthermore, it is of a different nature than the examples constructed, e.g., by Shnirelman (8) or De Lellis and Székelyhidi (9). On the other hand, it is similar to nonuniqueness of solutions of the Navier-Stokes equations defined in unbounded domains of the higher-dimensional Euclidean space (cf.…”

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confidence: 82%

Euler and Navier–Stokes equations on the hyperbolic plane

Khesin

Misiołek

2012

Proc. Natl. Acad. Sci. U.S.A.

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We show that nonuniqueness of the Leray-Hopf solutions of the Navier-Stokes equation on the hyperbolic plane H 2 observed by Chan and Czubak is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on H n whenever n ≥ 3. We also describe the corresponding general Hamiltonian framework of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.harmonic forms | steady flows | ill-posedness | Dirichlet problem | Dodziuk's theorem C onsider the initial value problem for the Navier-Stokes equations on a complete n-dimensional Riemannian manifold Mvð0; xÞ = v 0 ðxÞ:The symbol ∇ denotes the covariant derivative and L = Δ − 2r, where Δ is the Laplacian on vector fields and r is the Ricci curvature of M. Dropping the linear term Lv from the first equation in Eq. 1 yields the Euler equations of hydrodynamics,[3]Most of the work on well-posedness of the Navier-Stokes equations has focused on the cases where M is either a domain in R n or the flat n-torus T n . In fundamental contributions J. Leray and E. Hopf established existence of an important class of weak solutions described as those divergence-free vector fields v inthat solve the Navier-Stokes equations in the sense of distributions and satisfyfor any 0 ≤ t < ∞ and where Def v = 1 2 ð∇v + ∇v T Þ is the so-called deformation tensor (ref. 2). When n = 2 using interpolation inequalities and energy estimates, it is possible to show that the Leray-Hopf solutions are unique and regular but the problem is in general open for n = 3 (e.g., refs. 3 and 4).There have also been studies on curved spaces, which with few exceptions have been confined to compact manifolds (possibly with boundary) (e.g., ref. 5 and references therein). In a recent paper (1) Chan and Czubak studied the Navier-Stokes equation on the hyperbolic plane H 2 and more general noncompact manifolds of negative curvature. In particular, they showed that in the former case the Cauchy problem (Eq. 1 and 2) admits nonunique Leray-Hopf solutions and in the latter a similar nonuniqueness holds for a modified Navier-Stokes equation using the results of Anderson (6) and Sullivan (7).Our goal in this paper is to provide a direct formulation of the nonuniqueness of the Leray-Hopf solutions on H 2 and explain that it relies on the specific form of the Hodge decomposition for 1-forms (or vector fields) in this case. We also show that no such phenomenon can occur in the hyperbolic space H n with n ≥ 3, thus answering the question raised in ref. 1. As a by-product, we describe the corresponding Hamiltonian setting of the Euler equations on complete Riemannian manifolds (in particular, hyperbolic spaces).We point out that this type of nonuniqueness cannot be found in the Euler equations. Furthermore, it is of a different nature than the examples constructed, e.g., by Shnirelman (8) or De Lellis and Székelyhidi (9). On the other hand, it is similar to nonuniqueness of solutions of the Navier-Stokes equations defined in unbounded domains of the higher-dimensional E...

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confidence: 82%

Euler and Navier–Stokes equations on the hyperbolic plane

Khesin

Misiołek

2012

Proc. Natl. Acad. Sci. U.S.A.

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“…For the incompressible Euler equations, therefore, admissible measure-valued solutions qualify. It is important though to emphasize the necessity of admissibility: Without this assumption, various examples are known where weakstrong uniqueness fails even for distributional solutions of incompressible Euler [Sch93,Shn97,DLS09,Wie11]. Also, uniqueness need not hold for admissible solutions in the absence of a strong solution, see [DLS10,SW12,Dan14].…”

Section: Introductionmentioning

confidence: 99%

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

2015

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ABSTRACT. We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.

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“…This question is important because CI solutions are important, many counter examples to natural conjectures in PDE have been achieved via CI [13,19,31,32]. Minimising functional I is the simplest problem that constrains oscillation in some slight way where we can hope to see the effect of the existence of exact minimisers of (1.1).…”

Section: Background and Statement Of Main Resultsmentioning

confidence: 99%

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Lorent¹

2008

ESAIM: COCV

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Abstract. Let K := SO (2) A1 ∪ SO (2) A2 . . . SO (2) AN where A1, A2, . . . , AN are matrices of nonzero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N -well problem with surface energy. Let p ∈ [1, 2], Ω ⊂ R 2 be a convex polytopal region. Defineand let AF denote the subspace of functions in W 2,2 (Ω) that satisfy the affine boundary condition Du = F on ∂Ω (in the sense of trace), where F ∈ K. We consider the scaling (with respect to ) of

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The Euler equations as a differential inclusion

Cited by 520 publications

References 29 publications

Euler and Navier–Stokes equations on the hyperbolic plane

Euler and Navier–Stokes equations on the hyperbolic plane

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

The regularisation of theN-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

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