Ill-posedness for Compressible Elastic Waves and Ideal MHD Systems
Time: 2026-09-17
Published By: He Liu
Speaker(s): Silu Yin (Hangzhou Normal University)
Time: 16:00-17:00 September 28, 2026
Venue: Room 77201, Jingchunyuan 78, BICMR
Both elastic waves and ideal MHD systems are physical systems with multiple wave propagation speeds. Inspired by the works of Christodoulou, we generalize Lindblad's ill-posedness results on scalar wave equation to the wave systems. We prove that the ill-posedness is caused by instantaneous shock formation, which characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in ideal MHD, we also provide a desired ill-posedness result for the compressible Euler equations, and the regularity is sharp with respect to the local well-posedness of the fluid velocity. Our proofs are 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 these non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them we give a complete description of solutions' dynamics up to the earliest singular event, when a shock forms.
