Nonlinear Conservation Laws and Applications

This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 13--31, 2009. Hyperbolic conservation laws is a classical subject, which has experienced vigorous growth in recent years. The present collection provides a timely survey of the state of the art in this exciting field, and a comprehensive outlook on open problems. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for viscous systems, basic principles in the modeling of turbulent mixing, transonic flows past an obstacle and a fluid dynamic approach for isometric embedding in geometry, models of nonlinear elasticity, the Monge problem, and transport equations with rough coefficients. In addition, there are a number of papers devoted to applications. These include: models of blood flow, self-gravitating compressible fluids, granular flow, charge transport in fluids, and the modeling and control of traffic flow on networks.

Provides a comprehensive account of recent developments

Foreword.- Preface.- Open questions in the theory of one dimensionalhyperbolic conservation laws.- Multidimensional conservation laws: Overview,problems, and perspective.- Mathematical analysis of fluids in motion.- Selected topics in approximate solutions of nonlinear conservation laws.- High-resolution central schemes.- Stability and dynamics of viscous shock waves.- Mathematical aspects of a model for granular flow.- The flow associated to weakly differentiable vector fields: recent results and open problems.- Existence and uniqueness results for the continuity equation and applications to the chromatography system.- Finite energy weak solutions to the quantum hydrodynamics system.- The Monge problem in geodesic spaces.- Existence of a unique solution to a nonlinear moving-boundary problem of mixed type arising in modeling blood flow.- Transonic flows and isometric embeddings.- Well posedness and control in models based on conservation laws.-Homogenization of nonlinear partial differential equations in the context of ergodic algebras: Recent results and open problems.- Conservation laws at a node.- Nonlinear hyperbolic surface waves.- Vacuum in gas and fluid dynamics.- On radially symmetric solutions to conservation laws.- Charge transport in an incompressible fluid: New devices in computational electronics.- Localization and shear bands in high strain-rate plasticity.- Hyperbolic conservation laws on spacetimes.- Reduced theories in nonlinear elasticity.- Mathematical, physical and numerical principles essential for models of turbulent mixing.- On the Euler-Poisson equations of self-gravitating compressible fluids.- Viscous system of conservation laws: Singular limits.- A two-dimensional Riemann problem for scalar conservation laws.- Semi-hyperbolic waves in two-dimensional compressible Euler systems.- List of summer program participants.