报告地点：行健楼学术活动室526
邀请人：丁冰冰副教授
摘要：We construct counterexamples to the local existence of low regularity solutions to elastic wave equations and to the ideal compressible MHD system in three and two spatial dimensions. Both elastic waves and MHD are physical systems with multiple wave propagation speeds. Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on scalar wave equation by showing that the Cauchy problem for elastic waves and IMHD systems are ii-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. 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 MHD, we also provide a desired low regularity ill-posedness result for the compressible Euler equations, and it is sharp with respect to the regularity of the fluid 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 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 earlist singular event, when a shock forms. This talk is based on joint works with Xinliang An (NUS) and Haoyang Chen (NUS).
报告人简介：尹思露，杭州师范大学数学学院副教授，博士毕业于复旦大学应用数学系。从事非线性波动方程（组）以及流体力学中的偏微分方程组解的整体适定性理论研究。在Amer. J. Math.、SIAM J. Math. Anal.、J. Differential Equations等杂志发表论文十余篇。主持国家自然科学基金1项、省部级科研项目3项，入选上海市青年科技英才扬帆计划。
