Mathematical analysis of experimental data and the existence of weak solutions in nonlinear equations

Abstract

The local emergency of Beltrami ows is a fundamental characteristic of the uid turbulence dynamics (Navier-Stokes equations), where the formation of singularities starting from smooth initial data, i.e. the breakdown of regularity in the solutions, can individuate the onset of the turbulent behaviour. This property of nonlinear interactions has been used as a basic ingredient in the formal proof of Onsager conjecture, about the existence of weak solutions of Euler equations which do not conserve kinetic energy of the ow. The breakdown from smooth to weak solutions and the energy dissipation phenomenon can be possibly found also in magnetohydrodynamics (MHD) when progressively increasing Reynolds and magnetic Reynolds numbers. Thus a deep study of these phenomena of local formation of strong correlations between the dynamical variables of the systems could give important elements for understanding which mathematical conditions characterise the singularity emergence in weak solutions of MHD ideal case. In order to deal with these problems a multidisciplinary approach, embedding experimental data analysis and mathematical rigorous study, is needed. In this thesis both approaches have been carried out. An ad hoc data analysis have been identi ed for investigating the dynamics described by particular nonlinear partial di erential equations that can generates wide modes cascades and thus turbulence (MHD equations and Hasegawa-Mima equation). In addition the problem of investigating the second order regularity of solutions to particular degenerate nonlinear elliptic equations has been discussed