数学科学学院学术报告[2025] 032号

(高水平大学建设系列报告1054号)

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

报告人：朱圣国 教授（上海交通大学）

报告时间： 2025年05 月09日（周三）下午14:30—15:30

报告地点：汇星楼501

报告内容: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^\alpha\in H^3 (1/3<\alpha<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. Further, it is  shown that for arbitrarily large time, the solutions  obtained here are smooth (in Sobolev spaces) all the way up to  the moving boundary. Moreover, in contrast to the classical  theory,  the  L^\infty norm of u of the global classical 　solution  obtained here does not  decay to zero as time t goes to infinity.

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. 　It is worth pointing out that the result here seems to be the first  global existence theory of classical  solutions with large data  that is  independent of the BD entropy for such  degenerate systems, and the 　methodology developed here  can be applied to more general degenerate compressible Navier-Stokes equations.

报告人简历：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，其中包括Transactions of the AMS、Advances in Mathematics、Arch. Ration. Mech. Anal.、Ann. Inst. H. Poincare Anal. Non Lineaire、J. Math. Pures Appl. 等本领域权威杂志。 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）；2020年入选上海市海外高层次人才引进计划。 目前主持科技部国家重点研发计划青年科学家、基金委青年和面上项目各一项。

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邀请人：段琴

　　　 数学科学学院

　　　　　　　　　　　2025年5月5日