Extended crystal PDE’S stability, II: The extended crystal MHD-PDE’S

Abstract

This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability, and its relationship with an important class of PDE’s called extended crystals PDE’s in the sense introduced in [A. Prástaro, Extended crystal PDE’s (submitted for publication)]. In fact, their integral bordism groups can be considered as extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I [A. Prástaro, Extended crystal PDE’s stability, I: The general theory, Math. Comput. Modelling, 49 (9–10) (2009) 1759–1780] we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions, finite time instabilities do not occur ( stable extended crystal PDE’s). Here, we study in some detail, a new PDE encoding anisotropic incompressible magnetohydrodynamics. Stable extended crystal MHD-PDE’s are obtained, where in their smooth solutions, instabilities do not occur in finite time. These results are considered first for systems without a body energy source, and later, by also introducing a contribution from an energy source, in order to take into account nuclear energy production. A condition in order that solutions satisfy the second principle of thermodynamics is given.