School of Mathematical Sciences Academic Report [2025] No. 032  

(High-Level University Construction Series Report No. 1054)  

Lecture Title: Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the Viscous Saint-Venant System with Large Data

Speaker: Zhu Shengguo (Professor, Shanghai Jiao Tong University)  

Date & Time: 2:30–3:30 PM, May 9, 2025  

Venue: Room 501, Huixing Building  

Abstract: 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.

Biography: Zhu Shengguo, male, Associate Professor and Ph.D. Supervisor at the School of Mathematical Sciences, Shanghai Jiao Tong University. He received his Ph.D. in Science from Shanghai Jiao Tong University in 2015. After graduation, he worked as a postdoctoral fellow at The Chinese University of Hong Kong, Monash University in Australia, and the University of Oxford in the UK. He returned to Shanghai Jiao Tong University in 2020. His main research focuses on the theoretical study of nonlinear partial differential equations related to fluid mechanics and relativity. He has made systematic research progress in the well-posedness and singularity of the compressible Navier-Stokes and Euler equations. He has published more than 30 academic papers in international academic journals, including prestigious ones such as Transactions of the AMS, Advances in Mathematics, Arch. Ration. Mech. Anal., Ann. Inst. H. Poincare Anal. Non Lineaire, J. Math. Pures Appl. In 2017, he was selected as a Newton International Fellow of the Royal Society. In 2019, he was selected for the National Overseas High-level Talents Introduction Plan (Youth Project) of the Organization Department of the Central Committee. In 2020, he was selected for the Shanghai Overseas High-level Talents Introduction Plan. Currently, he presides over one Youth Scientist Project of the National Key R&D Program of the Ministry of Science and Technology and one Youth Project and one General Project of the National Natural Science Foundation of China.  

All faculty and students are welcome to attend!

Invited by: Duan Qin

School of Mathematical Sciences  

May 5, 2025