Low regularity ill-posedness 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 Alfv\'{e}n 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 multi-speed waves, mathematically, whether one can prove shock formation for three dimensional (3D) MHD is still open. The multiple-speed nature of the MHD system makes it fascinating and challenging. In this paper, we report our recent progress in answering the above question. For 3D ideal compressible MHD, we construct planar symmetric examples of shock formation allowing the presence of all characteristic waves with multiple wave speeds. This also answers a question raised by Majda on conservation law in 1984. Building on our construction, we further prove that the Cauchy problem for 3D ideal MHD is $H^2$ ill-posed. And this is caused by the shock formation. In particular, when the magnetic field is absent, we also provide a desired low-regularity ill-posedness 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 $7\times 7$ non-strictly 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.