Vanishing Viscosity and Shock Formation in 1D Conservation Laws

Learn about vanishing viscosity and shock formation in one-dimensional conservation laws through this comprehensive lecture delivered by Cole Graham from the University of Wisconsin–Madison at the Fields Institute Winter School on Shocks and Implosions in Fluids. Explore the mathematical foundations of how viscous effects disappear in the limit and lead to shock formation in conservation law equations. Examine the theoretical framework connecting viscous solutions to inviscid limits, understanding how smooth solutions can develop discontinuities as viscosity approaches zero. Investigate the mechanisms behind shock wave formation and the mathematical tools used to analyze these phenomena in one-dimensional systems. Study the relationship between parabolic regularization and hyperbolic conservation laws, gaining insights into how physical viscosity affects solution behavior. Discover applications of vanishing viscosity methods in fluid dynamics and their role in understanding shock wave propagation. Analyze the mathematical rigor required to prove convergence results and establish well-posedness of solutions in the vanishing viscosity limit.