Vanishing Viscosity Solutions to a Family of Active Vector Equations for Modelling Magnetic Relaxation

Active vector equations can be regarded as a special type of differential equations in which the unknown vector-valued function is related to the velocity field via some constitutive laws. This kind of differential equations comes from many physical models such as the in compressible Euler equations or magnetohydrodynamics (MHD), all of them have great importance in practical applications ranging from fluid mechanics to geophysics, astrophysics, cosmology and engineering. The primary goal of this project is to achieve some fundamental results on a general class of viscous active vector equations in various regimes of parameters, which include the global in-time well-posedness and the convergence of solutions when the diffusive parameters vanish. Such proposed problem is fruitful in the sense that by establishing well-posedness under the smoothing effects on the magnetic and velocity vector fields, it provides approximate solutions for the related inviscid models that will help identify singular behaviours as those diffusive parameters vanish. The secondary goal of this project is to rigorously address a class of inviscid active vector equations arising from the viscous models that are just mentioned in the primary goal. Such problem is crucial but challenging because these inviscid models come from a topology preserving diffusion process known as magnetic relaxation, in which its physical and mathematical properties are not yet fully understood.

Funding Source: RGC - General Research Fund (GRF)