Abstract
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 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 derivative of the vector potential in 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.
- Received 12 December 2018
- Revised 19 November 2019
- Corrected 28 January 2021
DOI:https://doi.org/10.1103/PhysRevE.101.013104
©2020 American Physical Society
Physics Subject Headings (PhySH)
Corrections
28 January 2021
Correction: Section 5 contained text and mathematical errors and has been fixed. Specifically, in the first paragraph of Sec. 5a, sentence 3 has been modified and sentence 4 deleted. In the second paragraph, sentences 1 and 3 have been modified. In Sec. 5b, sentence 6 has been modified.