On Certain Special Solutions of the \Phi-Equation of Statistical Hydrodynamics

Hopf, Eberhard; Titt, Edwin W.

doi:10.1512/iumj.1953.2.52029

Cited by 12 publications

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“…The other method deals directly with the probability measure of the velocity field and the corresponding Liouville equation is called the Hopf-Foias equation, see e.g. [1][2][3][4][5][6][7][8][9]. See also [10][11][12][13][14][15][16][17][18][19][20] for related works and [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] for mathematical works.…”

Section: Introductionmentioning

confidence: 99%

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

Ohkitani

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.

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Section: Introductionmentioning

confidence: 99%

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

Ohkitani

2020

Phys. Rev. E

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“…By introducing initial probability in a function space, and realizing the conservation of probability under the Navier-Stokes flow, Hopf derived a functional equation for the characteristic functional of the probability, around 1940, published in [89]. Initially, Hopf tried to find some near Gaussian solution [92], then he knew the work of Kolmogorov on the −5/3 law which is far from Gaussian probability, and verified by experiments. So the topic was not pursued much further.…”

Section: Kolmogorov Spectra and An Old Theory Of Hopfmentioning

confidence: 99%

Chaos in partial differential equations

Li¹

2002

The Legacy of the Inverse Scattering Transform in Applied Mathematics

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Abstract. This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n ≥ 2). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each category, we pick a representative to present the results. Then we review some initial results on 2D Euler equation.

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“…Hence in principle statistical theory of turbulence should be based on that footing. We will revisit the formulation of the Hopf functional differential equation (hereafter FDE) [2,3] which governs statistics of fluid turbulence. It is not our intention to give an exhaustive survey of the subject matter.…”

Section: Introductionmentioning

confidence: 99%

Remarks on the principles of statistical fluid mechanics

Ohkitani

2022

Phil. Trans. R. Soc. A.

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This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration, we review several functional integration approaches to the theory of turbulence. We note in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation. This article is part of the theme issue ‘’Scaling the turbulence edifice (part 1)’.

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On Certain Special Solutions of the \Phi-Equation of Statistical Hydrodynamics

Cited by 12 publications

References 0 publications

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

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

Chaos in partial differential equations

Remarks on the principles of statistical fluid mechanics

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