Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

Chae, Dongho

doi:10.1007/s00220-007-0249-8

Cited by 73 publications

(84 citation statements)

References 24 publications

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“…For the 3D Navier-Stokes this type of possibility leading to a self-similar singularity was first considered by Leray in [20], and its nonexistence was proved in [25], and the result was later refined by the authors in [34,24]. For the 3D Euler equations similar nonexistence result has been recently obtained by the author of this article in [3]. More refined notion of 'asymptotically self-similar singularity' is considered by the authors in [14], in the context of the nonlinear scalar heat equation, and also by physicists including the authors of [15,28] in the context of 3D Euler equations.…”

Section: Introductionsupporting

confidence: 55%

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Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Chae

2007

Math. Ann.

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In this paper we rule out the possibility of asymptotically selfsimilar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T . For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm,Ḃ 0 1,∞ (R 3 ). For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z, r)) for some q ∈ [2, ∞), where the ball of radius r is shrinking toward a possible singularity point z at the order of √ T − t as t approaches to T . In the L q (R 3 ) convergence case with q ∈ [3, ∞) we present a simple alternative proof of the similar result in [16]. *

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Section: Introductionsupporting

confidence: 55%

“…Remark 1.8 We note that the above corollary can be also deduced by a different reasoning from the above, based on the results of [25,34] combined with simple scaling argument, which is done in [4].…”

Section: T )mentioning

confidence: 93%

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Chae

2007

Math. Ann.

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“…The condition in terms of vorticity that excludes a non-trivial blowup stated and proved in [6] involves a requirement on decay at infinity in the sense that all L p -norms for 0 < p < p 0 are finite. In this section we will eliminate solutions under a much weaker condition.…”

Section: Exclusions Based On Vorticitymentioning

confidence: 99%

“…], see also [20] for 'pseudo self-similar solutions'). A study of self-similar blow-up in the settings adopted here was undertaken by the first author in a series of works [6,7,5,4]. The main two results obtained are the following…”

Section: Introductionmentioning

confidence: 99%

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

Chae

Shvydkoy

2013

Arch Rational Mech Anal

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Abstract. The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L p -condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables u ∈ L p and u ∼ 1 t α/(1+α) , then the blow-up does not occur provided α > N/2 or −1 < α ≤ N/p. This includes the L 3 case natural for the Navier-Stokes equations. For α = N/2 we exclude profiles with an asymptotic power bounds of the form |y| −N −1+δ |u(y)| |y| 1−δ . Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.

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“…The fact ξ has to have some roughness in order for blowup to be possible indicates some necessary complexity of the underlying geometric support of blowup. Moreover, single-scale self-similar behavior is impossible ( [25]). …”

Section: The Blowup Problemmentioning

confidence: 99%

On the Euler equations of incompressible fluids

Constantin

2007

Bull. Amer. Math. Soc.

204

212

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Abstract. Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

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Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

Cited by 73 publications

References 24 publications

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

On the Euler equations of incompressible fluids

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