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

2017

Journal of Turbulence

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We consider the Navier-Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier-Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. ii) This in turn gives an additional characterisation of possible singularities for the Navier-Stokes equations. iii) Some numerical results for the two-dimensional Navier-Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open.

Section: Summary and Discussionmentioning

confidence: 99%

Analogue of the Cole-Hopf transform for the incompressible Navier–Stokes equations and its application

2017

Journal of Turbulence

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“…(See [18] for the 2D counterpart.) The dynamic scaling transformation for ψ(x, t) is defined by [22] ψ(x, t) = Ψ(ξ, τ )…”

Section: Navier-stokes and Leray Equationsmentioning

confidence: 99%

“…(We are using the Maruyama-Girsanov theorem as a pull-back to retrieve the Navier-Stokes equations from the Leray equations.) The expression (22) does make sense as a path-integral equation on the same time interval.…”

Section: Leray Equationsmentioning

confidence: 99%

“…By applying dynamic scaling to the Navier-Stokes equations written in the vector potential, it was found [22] that the Navier-Stokes and the Leray equations are just one drift term apart. Moving onto path-integral forms, it was also found that the equations are identical up to two Maruyama-Girsanov densities G. Our purpose here is to nail down the difference to just one G, by combining the above idea with the Cole-Hopf transform [23], followed by an application of the Feynman-Kac formula.…”

Section: Introductionmentioning

confidence: 99%

“…The theory of the Navier-Stokes equations [13] has a long history, see e.g. [3,6,7], or [22] for references cited therein. There have been substantial progress recently, including the results regarding behaviours of critical norms such as u L 3 or u Ḣ1/2.…”

Section: Introductionmentioning

confidence: 99%

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Cole–Hopf-Feynman–Kac formula and quasi-invariance for Navier–Stokes equations

J. Phys. A: Math. Theor.

2017

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We make a refined comparison between the Navier-Stokes equations and their dynamically-scaled Leray equations solely on the basis of their scaling property. Previously it was observed using the vector potentials that they differ only by one drift term [Ohkitani (2017)]. The Duhamel principle recasts the equations in path integral forms, which differ by two Maruyama-Girsanov densities. In this brief paper we simplify the concept of quasi-invariance (or, near-invariance) by combining the result with a Cole-Hopf transform and the Feynman-Kac formula. That way, as a multiplicative characterisation we can place those equations just one Maruyama-Girsanov density apart. Furthermore, as an additive characterisation we express the difference in terms of the Malliavin H-derivative.

Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling

2020

Phys. Rev. E

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We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale-invariance of the statistical Navier-Stokes equations in d-dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale-invariance is achieved for the characteristic functional of the d-th derivative of the vector potential in d-dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late stage evolution by combining statistical scale-invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.