报告人：Peter Wittwer 教授（瑞士日内瓦大学）

报告时间：2020年11月27日周五晚20:00 – 21:00

报告地点：腾讯会议ID：991 111 323

报告题目：On a second order in time Hamilton-Jacobi equation

报告摘要：

We discuss some results for the Cauchy problem for the equation ü(x,t)="(u'(x,t)^2)," which is reminiscent of the Hamilton-Jacobi equation, except for being second order in time. This equation appears to be new. For analytic initial data, solutions exist at least for a short time by the Cauchy-Kowalevski theorem, but such solutions typically blow up in finite time. This blow up is in many ways similar to the blowup observed for the Hamilton Jacobi-equation (the derivative of solutions of the Hamilton-Jacobi equation satisfies the inviscid Burgers' equation which develops chocks), except that for the new equation first order derivatives diverge. It turns out that, since the new equation is second order in time, the list of possible blowup behaviors is richer than that of the Hamilton-Jacobi equation. We discuss these different blowup-profiles and comment on their stability. Unfortunately, at this point, we have not much to say concerning the original question, which is the question of the existence of some sort of generalized solutions global in time. Indeed, since first order derivatives diverge, one should not expect the concept of viscosity solutions (which was developed to discuss global in time solutions for the Hamilton-Jacobi equation), to carry over to the new equation in any straightforward way. Therefore, and since we were not able to find a liapunov functional for the new equation either, the question of the existence of solutions that are global in time remains wide open.

专家简介：

Peter Wittwer 教授在瑞士苏黎世联邦理工学院（ETH Zurich）获得硕士学位，在瑞士日内瓦大学获得博士学位，曾就职于瑞士日内瓦大学物理系，美国纽约大学 Courant 数学研究所，Rutgers大学以及国际知名的Swiss Re公司等，现任教于洛桑联邦理工学院（EPFL）。研究领域涉及动力系统与遍历理论，偏微分方程，数值分析，流体力学等，在外区域不可压缩流体的理论与数值计算等方面作了许多开创性的工作，研究成果发表在 Memoirs of the American Mathematical Society, SIAM Review, Comm. Math. Phy., Arch. Rational Mech. Anal., Math. Models & Methods in Appl. Sci., SIAM J. Math. Anal. 等国际知名杂志上，合作出版流体力学方程的经典著作一部。