The mathematical modelling of MHD flows

The goal of this paper is to investigate the foundations of the mathematical modelling for turbolent MHD , with particular reference to incompressible fluids. A closely related issue involves the construction of phase-space approaches for ideal fluids (i.e., fluids descrbed as continuum media). Indeed, phase-space techniques are well known both in classical and quantum fluid dynamics. In fact, generally the fluid equations represent a mixture of hyperbolic and elliptic PDE's, which are extremely hard to study both analytically and numerically. This has motivated in the past efforts to replace them with other equations, possibly simpler to solve or mathematically more elegant. In this connection a particular viewpoint is represented by the class of so-called inverse problems, involving the search of a so-called inverse kinetic theory (IKT) able to yeald the complete set of fluid equations for the fluid fields, via a suitable correspondence principle. A basic consequence of the theory is the possibility of explicitly identifying the phase-space dynamical system which advances in time the complete set of fluid fields. An interesting open question, lying at the very heart of turbulence theory, is whether such a dynamical system is variational or not and in particular if it is Hamiltonian or not. The possible solution of the problem has potential important theoretical implications. Indeed, a fundamental issue is whether the usual concepts of chaotic dynamics can be applied to the interpretation of turbulence. The purpose of this note is to present an answer to this question.