活动时间：2022-05-06 14:00

活动地点：腾讯会议：907 706 435

主讲人：尹思露

主讲人中文简介：

尹思露，杭州师范大学数学学院教师，博士毕业于复旦大学应用数学系。研究方向为非线性偏微分方程，特别是双曲型方程（组）解的适定性与激波形成。相关研究成果发表于SIAM、JDE等数学杂志。

活动内容摘要：

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 spatial dimensions (3D). Inspired by the recent 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 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 thefluid velocity. Our proofs for elastic waves and for MHD 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 the 3D elastic waves and the 3D ideal MHD equations into 6×6 and 7×7 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 singularevent, when a shock forms. This talk is based on joint works with Xinliang An and Haoyang Chen.

主持人：查冬兵