Beltrami structure in hydrodynamics and magneto-hydrodynamics

Abstract

We investigate Beltrami structure in hydrodynamics and magneto-hydrodynamics. Beltrami fields, for which the velocity and the vorticity (curl of the velocity) are everywhere collinear, can exhibit chaotic behavior of trajectories in three dimension. Meanwhile, these fields arise in the incompressible Euler and Navier-Stokes equations, and in magnetic relaxation situation of the magneto-hydrodynamics (MHD) equations. In addition, more complex structure of Beltrami fields can appear in the Hall-magnetohydrodynamics (Hall-MHD) equations.

Our focus lies on solutions having Beltrami structure in each aforementioned circumstance. At first, we examine steady solutions of ideal systems where both viscosity and magnetic resistivity are neglected. Then, we look into time-evolutionary solutions with Beltrami structure in each system. In particular, we describe the family of solutions having Beltrami structure in the Hall-MHD equations, which have not been explored in mathematics to the best of our knowledge. Then, we show that near the specific solutions, the incompressible Navier-Stokes, MHD and Hall-MHD equations are well-posed globally in time.