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 formation, including the role of characteristics and the breakdown of classical solutions. Study analytical techniques for handling the vanishing viscosity limit and its implications for understanding physical phenomena in fluid dynamics. Gain insights into the mathematical tools used to analyze these singular limits and their applications in modeling real-world systems where viscous effects become negligible. This lecture forms part of the broader thematic program on shocks and singularities in nonlinear evolution equations, providing essential background for understanding complex fluid behavior in various physical and life science contexts.