Chirkunov, Yu A., Nazarenko, Sergey, Medvedev, S. B. and Grebenev, V. N. (2014) Invariant solutions for the nonlinear diffusion model of turbulence. Journal of Physics A: Mathematical and Theoretical, 47 (18). 185501. doi:10.1088/1751-8113/47/18/185501 ISSN 1751-8113.

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Abstract

We study Leith's model of turbulence represented by a nonlinear degenerate diffusion equation (Leith 1967 Phys. Fluids 10 1409–16, Connaughton and Nazarenko 2004 Phys. Rev. Lett. 92 044501–506). The model is constructed such that in the case of vanishing viscosity, there are two steady-state solutions: the Kolmogorov spectrum that corresponds to the cascade state and a thermodynamic equilibrium distribution. Using group analysis, we have obtained integral equations which describe all essentially different invariant solutions of the Leith equation with or without viscosity. The integral equations defining these solutions reveal new possibilities for analytical and numerical studies. In these equations, the presence of arbitrary constants allows one to solve them for different boundary conditions. We have proved existence and uniqueness for such boundary value problems.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Journal of Physics A: Mathematical and Theoretical
Publisher:	IOP Publishing Ltd
ISSN:	1751-8113
Official Date:	17 April 2014
Dates:	Date Event 17 April 2014 Published 21 March 2014 Accepted
Volume:	47
Number:	18
Article Number:	185501
DOI:	10.1088/1751-8113/47/18/185501
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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