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

Chae, Dongho; Shvydkoy, Roman

doi:10.1007/s00205-013-0630-z

Cited by 42 publications

(47 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These solutions emerge, for instance, in vortex line models of Kida's high-symmetry flows (see Pelz and others [1,8,9,10]), although previously self-similar blow-up has been observed as well, [7,2]. In a recent joint effort with D. Chae [3] (see also [4,6,11]) solutions of the form (3) have been ruled out under additional integrability condition, v ∈ L p (R N ) ∩ C , self-similar solutions are excluded provided v ∈ L 2 and the power bounds…”

Section: Introductionmentioning

confidence: 90%

See 1 more Smart Citation

A study of energy concentration and drain in incompressible fluids

Shvydkoy¹

2013

Nonlinearity

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if u is a regular solution on a time interval [0,where N is the dimension of the fluid, then the energy at the time T cannot concentrate on a set of Hausdorff dimension smaller than N − 2 r−1 . The same holds for solutions of the three-dimensional Navier-Stokes equation in the range 5/3 < r < 7/4. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T − t) 1−δ , for any δ > 0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.

show abstract

Section: Introductionmentioning

confidence: 90%

“…, without any L p -condition as previously considered in [3]. The full list of exclusions based on energy drain is stated in Corollary 3.2.…”

Section: Introductionmentioning

confidence: 99%

A study of energy concentration and drain in incompressible fluids

Shvydkoy¹

2013

Nonlinearity

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note also that t = 1 − e −τ . These equations have been used in many articles, including [7,25,38,8,9,24].…”

Section: Dynamic Scaling Transformations: Leray Equationsmentioning

confidence: 99%

Near-invariance under dynamic scaling for Navier–Stokes equations in critical spaces: a probabilistic approach to regularity problems

Ohkitani¹

2017

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. We make a detailed comparison between the Navier-Stokes equations and their dynamically-scaled counterpart, the so-called Leray equations. The Navier-Stokes equations are invariant under static scaling transforms, but are not generally invariant under dynamic scaling transforms. We will study how close they can be brought together using the critical dependent variables and discuss the implications on the regularity problems.Assuming that the Navier-Stokes equations written in the vector potential have a solution that blows up at t = 1, we derive the Leray equations by dynamic scaling. We observe: (1) The Leray equations have only one term extra on top of those of the Navier-Stokes equations. (2) We can recast the Navier-Stokes equations as a Wiener path integral and the Leray equations as another Ornstein-Uhlenbeck path integral. By the Maruyama-Girsanov theorem, both equations take the identical form modulo the Maruyama-Girsanov density, which is valid up to t = 2 √ 2 by the Novikov condition. (3) The global solution of the Leray equations is given by a finite-dimensional projection R of a functional of an Ornstein-Uhlenbeck process and a probability measure. If R remains smooth beyond t = 1 under an absolute continuous change of the probability measure, we can rule out finite-time blowup by contradiction. There are two cases: (A) R given by a finite number of Wiener integrals, and (B) otherwise. Ruling out blowup in (A) is straightforward. For (B), a condition based on a limit passage in the Picard iterations is identified for such a contradiction to come out. The whole argument equally holds in R d for any d ≥ 2.

show abstract

“…For such (v, p) we have the following scaling invariance v(x, t) = λ α v(λx, λ α+1 t) and p(x, t) = λ 2α v(λx, λ α+1 t) for all λ > 0, α ∈ R, and for all (x, t) ∈ R 3 ×(−∞, 0). The nonexistence of nontrivial selfsimilar blowing up solution under suitable assumption on the blow-up profile V was obtained in [4]. A discretely self-similar solution v is a solenoidal vector field, for which there exist…”

Section: The Euler Equationsmentioning

confidence: 99%

“…4 We consider the vorticity equation of (1.7), 18) and 0 ≤ σ(x) ≤ 1 for 1 < |x| < 2. For each R > 0, we define σ R (x) := σ |x| R…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Remarks on the asymptotically discretely self-similar solutions of the Navier–Stokes and the Euler equations

Chae

2015

Nonlinear Analysis

Self Cite

View full text Add to dashboard Cite

We study scenarios of self-similar type blow-up for the incompressible NavierStokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to C 1 (R; L 3 (R 3 ) ∩ C 2 (R 3 )). For the 3D Euler equations we show that the scenario of asymptotically discretely self-similar blow-up is excluded if the blow-up profile satisfies suitable integrability conditions. Mathematics Subject Classification(2000): 76B03, 35Q311 The main theorems Navier-Stokes equationsWe consider the Cauchy problem of the 3D Navier-stokes equations.where v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) is the velocity, p = p(x, t) is the pressure, and v 0 (x) is the initial data satisfying divv 0 = 0. We study the possibility of finite time blow-up of smooth solution of (NS). By translation of time we may assume that the solution is smooth for t < 0, and the blow-up happens at t = 0. We say a solution 1

show abstract

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

Cited by 42 publications

References 21 publications

A study of energy concentration and drain in incompressible fluids

A study of energy concentration and drain in incompressible fluids

Near-invariance under dynamic scaling for Navier–Stokes equations in critical spaces: a probabilistic approach to regularity problems

Remarks on the asymptotically discretely self-similar solutions of the Navier–Stokes and the Euler equations

Contact Info

Product

Resources

About