Nonlinear Conservation Laws and Applications

1. Front Matter

   Pages i-xi

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2. General Survey Lectures

   1. Front Matter

      Pages 1-1

      PDF

   2. Open Questions in the Theory of One Dimensional Hyperbolic Conservation Laws

      • Alberto Bressan

      Pages 1-22

   3. Multidimensional Conservation Laws: Overview, Problems, and Perspective

      • Gui-Qiang G. Chen

      Pages 23-72

   4. Mathematical Analysis of Fluids in Motion

      • Eduard Feireisl

      Pages 73-100

   5. Selected Topics in Approximate Solutions of Nonlinear Conservation Laws. High-Resolution Central Schemes

      • Eitan Tadmor

      Pages 101-122

   6. Stability and Dynamics of Viscous Shock Waves

      • Kevin Zumbrun

      Pages 123-167

3. Specialized Research Lectures

   1. Front Matter

      Pages 168-168

      PDF

   2. Mathematical Aspects of A Model for Granular Flow

      • Debora Amadori, Wen Shen

      Pages 169-179

   3. The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems

      • Luigi Ambrosio

      Pages 181-193

   4. Existence and Uniqueness Results for the Continuity Equation and Applications to the Chromatography System

      • Luigi Ambrosio, Gianluca Crippa, Alessio Figalli, Laura V. Spinolo

      Pages 195-204

   5. Finite Energy Weak Solutions to the Quantum Hydrodynamics System

      • Paolo Antonelli, Pierangelo Marcati

      Pages 205-216

   6. The Monge Problem in Geodesic Spaces

      • Stefano Bianchini, Fabio Cavalletti

      Pages 217-233

   7. Existence of a Unique Solution to a Nonlinear Moving-Boundary Problem of Mixed Type Arising in Modeling Blood Flow

      • Sunčica Čanić, Andro Mikelić, Tae-Beom Kim, Giovanna Guidoboni

      Pages 235-256

   8. Transonic Flows and Isometric Embeddings

      • Gui-Qiang G. Chen, Marshall Slemrod, Dehua Wang

      Pages 257-266

   9. Well Posedness and Control in Models Based on Conservation Laws

      • Rinaldo M. Colombo

      Pages 267-277

   10. Homogenization of Nonlinear Partial Differential Equations in the Context of Ergodic Algebras: Recent Results and Open Problems

       • Hermano Frid, Jean Silva

       Pages 279-291

   11. Conservation Laws at A Node

       • Mauro Garavello

       Pages 293-302

   12. Nonlinear Hyperbolic Surface Waves

       • John K. Hunter

       Pages 303-314

   13. Vacuum in Gas and Fluid Dynamics

       • Juhi Jang, Nader Masmoudi

       Pages 315-329

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.

• Hyperbolic conservation laws
• fluid motion
• multidimensional conservation laws
• nonlinear conservation laws
• viscous shock waves
• fluid- and aerodynamics

• University Park, USA

  Alberto Bressan

• , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, Oxford, Montserrat

  Gui-Qiang G. Chen

• , School of Mathematics, University of Minnesota, Minneapolis, USA

  Marta Lewicka

• , Department of Mathematics, University of Pittsburgh, Pittsburgh, USA

  Dehua Wang

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