School of Mathematical Sciences Academic Report [2025] No. 035  

(High-Level University Construction Series Report No. 1057)  

Lecture Title: Minimal Entropy Conditions for Scalar Conservation Laws

Speaker: Cao Gaowei (Associate Researcher, Institute of Precision Measurement Science and Technology Innovation, Chinese Academy of Sciences)  

Date & Time: 10:30–11:30 AM, May 14, 2025  

Venue: Room 501, Huixing Building  

Abstract: In 1989, Arnol’d and Kruzkov et al., posed an important open question on whether only one single convex entropy η(u) can enforce the uniqueness of the solution for one-dimensional scalar conservation laws with convex flux functions, which is called the “Minimal Entropy Conditions” by De Lellis-Otto-Westdickenberg(2004).

For these scalar conservation laws, we prove that a single entropy-entropy flux pair (η(u),q(u)) with η(u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in that satisfy the inequality: η(u)ₜ+q(u)ₓ⩽μ, controlled by some non-negative Radon measure μ (weaker than controlled by 0), in the distributional sense. Furthermore, we extend this result to the class of weak solutions in , based on the asymptotic behavior of the flux function f(u) and the entropy function η(u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness. This is a joint work with Professor Gui-Qiang Chen.

Biography: Cao Gaowei is an associate researcher at the Institute of Precision Measurement Science and Technology Innovation, Chinese Academy of Sciences. His main research focuses on hyperbolic conservation laws, Euler equations in fluid mechanics, and the mathematical theory of stochastic partial differential equations. He has made several academic visits to City University of Hong Kong, the University of Oxford, etc. He has presided over the National Natural Science Foundation Youth Project and the Interdisciplinary Project of the Chinese Academy of Sciences. He has published more than ten papers in domestic and international mathematical journals such as J. Differential Equations, AMS Quarterly of Applied Mathematics.

All faculty and students are welcome to attend!

Invited by: Li Xing

School of Mathematical Sciences  

May 12, 2025