Minimal Entropy Conditions for Scalar Conservation Laws
数学专题报告
报告题目(Title)：Minimal Entropy Conditions for Scalar Conservation Laws
报告人(Speaker)：曹高伟 副研究员 中国科学院精密测量科学与技术创新研究院
地点(Place)：后主楼1225
时间(Time)：2024年9月12日星期四11:30-12:30
邀请人(Inviter)：袁迪凡
报告摘要
In 1989, Arnol’d and Kruzkov et al., posed an important open question on whether only one single convex entropy 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 with of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in that satisfy the inequality: , controlled by some non-negative Radon measure (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 and the entropy function 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.
