Keywords
turbulence
intermittency
multifractals
velocity gradients
Navier-Stokes
turbulencia
intermitencia
multifractales
gradientes de velocidad
How to Cite
Abstract
There is currently no general theorem on the existence and unicity of solutions to the Navier-Stokes equation, which describes the flow of a viscous and incompressible fluid. This is an open problem at the international level, known as the millennium prize problem, for which the Clay Institute of France is offering one million dollars since may 2000.
The purpose of this article is to present a brief revision of the most important aspects of the evolution and current status of the problem. Our contribution is the analytical description of turbulence, fully developed, through the resolution rates and the features of multifractal processes, as a collection of generalized Cantor processes. We present four models for the distribution of velocity variations. The first one is based on the life times and risk functions for the interaction between the vortices and their later fragmentation in ever smaller and more numerous vortices. The second one is based on potentiated Bernoulli tests, and we found the number of features, the spectrum, and the structure function. We found the relationship of the shape
parameters with the box dimension of the maximum spectrum as well as with the local dimensions and we described qualitatively the associated tree.
The above-mentioned rates serve as support, not only for the description of a three-dimensional model of intermittent turbulence that generalizes the Kolmogorov paradigmatic result, but also for the energy transferred in each stage of the fractalization process, and also for the number of characteristic exponents, which produces a higher level for Hausdorff’s dimension of the set of singularities of the solution.
References
Andreev, A.; Kanto A.; Malo P. (2005) “Simple Approach for Distribution Selection in the Pearson System”, Helsingin Kauppakorkeakoulu, Helsinki Schools of Economics, Working Papers, W-388.
Benzi, R.; Biferale, L.; Paladin, G.; Vulpiani, A.; Vergassola, M. (1991) “Multifractality in the statistics of the velocity gradients in turbulence”, Phys. Rew. Lett. 67(17): 2299–2302.
Biferale, L. (1993) “Probability distribution functions in turbulent flows and shell models”, Phys. Fluids A 5(2): 428–435.
Chorin, A.J. (1994) Vorticity and Turbulence. Springer, New York.
Dressler, R. F. (1953) “Mathematical solution of the problem of roll-waves in inclined open
channels”. Commu. on Pure and Appl. Math. 6: 149–194. 70 j.r. mercado Rev.Mate.Teor.Aplic. (2008) 15(1)
Falconer, K. (1990) Fractal Geometry. John Wiley & Sons, Chichester.
Fefferman, C.L. (2000) “Existence and smoothness of the Navier-Stokes equation”. Princeton University, Department of Mathematics, Princeton, May 1.
Halsey, T.C.; Jensen, M.H.; Kadanoff, L. P.; Procaccia, I.; Shraiman, B.I. (1986) “Fractal measures and their singularities: the characterization of strange sets”, Phys. Rev A. 33: 1141–1151.
Hernández, A. (2005) Distribución Límite de los Extremos del Modelo T-student Truncado para Datos de Lluvia Diaria. Tesis de Doctorado, Univ. Simón Bolívar, Caracas.
Koroliuk, V.S. (1981) Manual de la Teoría de Probabilidades y Estadística Matemática. Ed. Mir, Moscú.
Hopf, E. (1950) “The partial differential equation ut + u.ux = uxx”, Comm. Pure and Appl. Math. 3: 201–230.
Lax, P.D. (1968) “Integrals of nonlinear equations of evolution and solitary waves”, Comn. Pure and Appl. Math. 21: 467–490.
Levi, E. (1989) El Agua según la Ciencia. Conacyt, Ed. Castell Mexicana, México.
Lieb, E.H. (1984) “On characteristic exponents in turbulence”, Commun. Math. Phys. 92: 473–480.
Meneveau, C.; Sreenivasan, K. (1987) “Simple multifractal cascade model for fully developed turbulence”, Phys. Rev. Lett. 59(13): 1424–1427.
Mercado, J.R.; Aldama, Á.A.; Brambila, F. (2004) “Sobre las soluciones de la ecuación de Navier-Stokes”, XVIII Congreso Nacional de Hidráulica, San Luis Potosí, SLP, 10 al 12 de nov. de 2004.
Mercado, J.R.; Brambila, F. (2001) “Problemas inversos en las ecuaciones de Fokker-Planck”, Aportaciones Matemáticas, Serie Comunicaciones 29, pp. 201–222.
Panton, R.L. (1984) Incompressible Flow. John Wiley & Sons, New York.
Rose, H A.; Sulem, P.L. (1978) “Fully developed turbulence and statistical mechanics”, J. Physique 39: 441–484.
Ruelle, D. (1982) “Large volume limit of the distribution of characteristic exponents in turbulence”, Commun. Math. Phys. 87: 287–302.
Scheffer, V. (1977) “Hausdorff measure and the Navier-Stokes equations”, Commun. Math. Phys. 55: 97–112.
Smoller, J. (1994) Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York.
Tarasov, V.E. (2005) “Fractional Fokker-Planck equation for fractal media”, Chaos 15.
Temam, R. (1984) Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam.
Waymire, E.C. (2005) “Probability & incomprensible Navier-Stokes equations: An overview of some recent developments”, Probability Surveys 2: 1–32.