报告题目：On the blow-up mechanism of smooth solutions to hyperbolic conservation laws

报 告 人：许刚教授（南京师范大学）

报告时间：2022年7月6日（星期三）下午3:45-4:45

报告地点：腾讯会议：342-882-399

邀 请人：吴笛副教授

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数学学院

2022年7月4日

报告摘要：In this talk, we discuss the problems of the blow-up mechanisms of smooth to the hyperbolic system. For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\patial_tu+F(u)\patial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\Bbb R)$, when at least one eigenvalue of $F(u)$ is genuinely nonlinear, it is well-known that on the finite blow-up time $T_{\varepsilon}$, the derivatives $\patial_{t,x}u$ blow up while the solution $u$  keeps to be small. For the 1D scalar equation or $2\times 2$ strictly hyperbolic system (corresponding to $n=1, 2$), if the smooth solution $u$ blows up in finite time, then the blow-up mechanism can be well understood (i.e., only the blow-up of $\patial_{t,x}u$ happens). In this talk, for the $n\times n$ ($n\ge 3$) strictly hyperbolic system with a class of large initial data, we are concerned with the blow-up mechanism of smooth solution $u$ on the finite blow-up time and the detailed singularity behaviours of $\p_{t,x}u$ near the blow-up point. Our results are based on the efficient decomposition of $u$ along the different characteristic directions, the suitable introduction of the modulated coordinates and the global weighted energy estimates.

报告人简介：许刚，南京师范大学数学科学学院教授，博士生导师。南京大学博士毕业，香港中文大学博士后。主要研究方向为有非线性偏微分方程及流体动力学中的数学问题等。特别关注于可压缩流体中的激波问题和真空问题，部分解决了超音速流经过一个锥形或楔形障碍物所产生的跨音速激波不稳定性的公开猜测，相关工作发表在Advances in Mathematics，Archive for Rational Mechanics and Analysis，SIAM Journal on Mathematical Analysis，Journal of Differential Equations 等学术刊物上，主持研究国家自然科学基金多项。