一、报告题目：Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the Viscous Saint-Venant System with Large Data

二、报告人：朱圣国 副教授

三、时 间：2025年11月16日（周日）9:00-12:00

四、报告地点：闻理园A4-305

报告摘要：We talk about the global-in-time well-posedness of classical  solutions  to the vacuum free boundary problem  of the one-dimensional viscous Saint-Venant system for laminar shallow water with large data.

Since the depth of the fluid vanishes on the moving boundary, the momentum equations become degenerate both in the time evolution and spatial dissipation, which may lead to singularities for the derivatives of the velocity u of the fluid and then makes it challenging to study classical solutions. By exploiting the intrinsic degenerate-singular structures of the viscous Saint-Venant system, we are able to identify two classes of admissible  initial depth profile and obtain the global well-posedness theory here:  \rho_0^a\in H^3 (1/3<a<1) vanishes as the distance to the moving boundary, which satisfies the BD entropy condition; while \rho_0\in H^3 vanishes as the distance to the moving boundary, which satisfies the physical vacuum boundary condition, but violates the BD entropy condition. One of the key ingredients of the analysis here is to establish some degenerate weighted estimates for the effective velocity v=u+ (\log\rho)_y (y is the Eulerian spatial coordinate) via its  transport properties, which enables one to obtain the upper bounds for the first order derivatives of the flow map \eta(t,x) with respect to the Lagrangian spatial coordinate x. Then the global-in-time regularity uniformly up to the vacuum boundary can be obtained by carrying out a series of singular or degenerate weighted energy estimates carefully designed for this system.

报告人简介：朱圣国，男，上海交通大学数学科学学院副教授、博导。2015年于上海交通大学获理学博士学位。毕业之后先后在香港中文大学、澳大利亚莫纳什大学、英国牛津大学做博士后。2020年返回上海交大任教。主要从事与流体力学及相对论相关的非线性偏微分方程的理论研究工作，在可压缩Navier-Stokes及Euler方程组的适定性和奇异性方面取得了系统性的研究进展。目前已在国际学术期刊上发表学术论文30余篇，并于2017年入选英国皇家学会”Newton International Fellow”; 2019年入选中组部国家海外高层次人才引进计划（青年项目）。

欢迎广大师生参加！联系人：胡燕波