Abstract

Together with Haoyang Chen and Silu Yin, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three and two spatial dimensions (3D and 2D). For 3D, inspired by the works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for the 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave-speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, 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. In 2D, we also obtain the corresponding results. If time permits, we will also mention the counterpart low-regularity local well-posedness result for the admissible harmonic material based on a joint work with Haoyang Chen and Sifan Yu, and the recent progress joint with Sifan Yu on gravitational collapse for the Einstein-Euler system with no symmetry assumption.