La ecuación de Navier-Stokes y multifractales
There is currently no general theorem on the existence and unicity of solutions tothe Navier-Stokes equation, which describes the flow of a viscous and incompressiblefluid. This is an open problem at the international level, known as the millenniumprize problem, for which the Clay Institute of Franc...
|Main Author:||Mercado Escalante, José Roberto|
There is currently no general theorem on the existence and unicity of solutions tothe Navier-Stokes equation, which describes the flow of a viscous and incompressiblefluid. This is an open problem at the international level, known as the millenniumprize problem, for which the Clay Institute of France is offering one million dollarssince may 2000.The purpose of this article is to present a brief revision of the most importantaspects of the evolution and current status of the problem. Our contribution is theanalytical description of turbulence, fully developed, through the resolution rates andthe features of multifractal processes, as a collection of generalized Cantor processes.We present four models for the distribution of velocity variations. The first one isbased on the life times and risk functions for the interaction between the vortices andtheir later fragmentation in ever smaller and more numerous vortices. The secondone 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 shapeparameters with the box dimension of the maximum spectrum as well as with thelocal dimensions and we described qualitatively the associated tree.The above-mentioned rates serve as support, not only for the description of athree-dimensional model of intermittent turbulence that generalizes the Kolmogorovparadigmatic result, but also for the energy transferred in each stage of the fractalizationprocess, and also for the number of characteristic exponents, which producesa higher level for Hausdorff’s dimension of the set of singularities of the solution.Keywords: Navier-Stokes, turbulence, intermittency, multifractals, velocity gradients.