Low regularity illposedness and shock formation for 3D ideal compressible MHD
Abstract
The study of magnetohydrodynamics (MHD) significantly boosts the understanding and development of solar physics, planetary dynamics and controlled nuclear fusion. Dynamical properties of the MHD system involve nonlinear interactions of waves with multiple travelling speeds (the fast and slow magnetosonic waves, the Alfvén wave and the entropy wave). One intriguing topic is the shock phenomena accompanied by the magnetic field, which have been affirmed by astronomical observations. However, permitting the residence of all above multispeed waves, mathematically, whether one can prove shock formation for 3D MHD is still open. The multiplespeed nature of the MHD system makes it fascinating and challenging. In this paper, we give an affirmative answer to the above question. For 3D ideal compressible MHD, we construct examples of shock formation allowing the presence of all characteristic waves with multiple wave speeds. Building on our construction, we further prove that the Cauchy problem for 3D ideal MHD is $H^2$ illposed. And this is caused by the instantaneous shock formation. In particular, when the magnetic field is absent, we also provide a desired lowregularity illposedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proof for 3D MHD is based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D ideal MHD equations into a $7\times 7$ nonstrictly hyperbolic system. Via detailed calculations, we reveal its hidden subtle structures. With them we give a complete description of MHD dynamics up to the earliest singular event, when a shock forms.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10647
 Bibcode:
 2021arXiv211010647A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Physics  Fluid Dynamics
 EPrint:
 65 pages