Exact solutions of the Navier-Stokes equations via Leray’s scheme

Okamoto, Hisashi

doi:10.1007/bf03167263

Cited by 28 publications

(25 citation statements)

References 26 publications

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“…Hence we obtain vak (Ya") <-for any k and m with k > Nm > m > M. (8) On the other hand, the definition of Ya yields -e/4 < 0 = vak (Y ak for L> Nk > k > Nom,, > M. Here the inequality in the right-hand side is given by (8). This is a contradiction.…”

Section: Vß(y) + 2yvp(y) = -Wß(y)'vß(y) + Vß(y)2 >mentioning

confidence: 66%

“…In the interesting paper [8], Okamoto obtained so many exact solutions of backward and forward self-similar solutions (see, also [7]). Here, forward self-similar solutions are solutions in the form:…”

Section: Introductionmentioning

confidence: 99%

“…Their solutions are not explicitly written by elementary functions or solutions of ordinary differential equations. In the section 12 in [8] he defined a forward self-similar stagnation flow of (NS) in the following manner. In (L), putting…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Existence of a forward self-similar stagnation flow of the Navier-Stokes equations

Ishikawa

Matsui

2000

Japan J. Indust. Appl. Math.

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Section: Vß(y) + 2yvp(y) = -Wß(y)'vß(y) + Vß(y)2 >mentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence of a forward self-similar stagnation flow of the Navier-Stokes equations

Ishikawa

Matsui

2000

Japan J. Indust. Appl. Math.

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“…There is a large amount of literature on the mathematical problems of the NavierStokes equations: articles include [21,15,1,20,17,8,28,16,26,22,35] and books [36,5,9,14,34]. For applied mathematical interests, see e.g.…”

Section: /2mentioning

confidence: 99%

Study of the Navier–Stokes regularity problem with critical norms

Ohkitani¹

2016

Fluid Dyn. Res.

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Abstract. We study the basic problems of regularity of the Navier-Stokes equations. The blowup criteria on the basis of the critical H 1/2 -norm, is bounded from above by a logarithmic function, Robinson, Sadowski and Silva (2012). Assuming that the Cauchy-Schwarz inequality for the H 1/2 -norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy. We carry out a simple asymptotic analysis to determine the time evolution of the energy. This generalises the (already ruled-out) self-similar blowup ansatz. Some numerical results are also presented, which support the above-mentioned replacement. We carry out a similar analysis for the four-dimensional Navier-Stokes equations.

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“…By choosing spatial and temporal units suitably, we have taken the prefactor as 1/2 in front of the Laplacian, so that applications of probabilistic methods (with a standard Brownian motion) will be simplified. There are lots of publications on the mathematical problems of the Navier-Stokes equations, including [6,60,10,11,48,17,55,34,51,12,29,21,52,58] on pure analysis side and [53,13,39,36,43,57,14,47,3,45,46] on applied mathematical aspects.…”

Section: Introductionmentioning

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.

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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.

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Exact solutions of the Navier-Stokes equations via Leray’s scheme

Cited by 28 publications

References 26 publications

Existence of a forward self-similar stagnation flow of the Navier-Stokes equations

Existence of a forward self-similar stagnation flow of the Navier-Stokes equations

Study of the Navier–Stokes regularity problem with critical norms

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

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