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This book is a collection of lecture notes on Nonlinear Conservation Laws, Fluid Systems and Related Topics delivered at the 2007 Shanghai Mathematics Summer School held at Fudan University, China, by world's leading experts in the field. The volume comprises five chapters that cover a range of topics from mathematical theory and numerical approximation of both incompressible and compressible fluid flows, kinetic theory and conservation laws, to statistical theories for fluid systems. Researchers and graduate students who want to work in this field will benefit from this essential reference as

Conservation laws (Mathematics). --- Fluid dynamics -- Mathematics. --- Nonlinear theories. --- Fluid dynamics --- Conservation laws (Mathematics) --- Nonlinear theories --- Mathematics --- Mathematics.

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Systems of conservation laws arise naturally in physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws, describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject. The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation. Small waves are studied using geometrical optics methods. Finally, the initial-boundary problem is considered in depth. Throughout, the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations.

Conservation laws (Mathematics) --- Shock waves. --- Shock (Mechanics) --- Waves --- Differential equations, Hyperbolic

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Dinàmica de gasos --- Lleis de conservació (Matemàtica) --- Equacions diferencials hiperbòliques --- Solucions numèriques --- Equacions en derivades parcials --- Anàlisi numèrica --- Termodinàmica --- Aerodinàmica --- Gas dynamics. --- Conservation laws (Mathematics) --- Differential equations, Hyperbolic --- Gasdynamics --- Fluid dynamics --- Thermodynamics --- Numerical solutions.

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This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 130-31, 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.

Conservation laws (Mathematics). --- Nonlinear theories. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Conservation laws (Mathematics) --- Nonlinear problems --- Nonlinearity (Mathematics) --- Mathematics. --- Dynamics. --- Ergodic theory. --- Physics. --- Fluids. --- Dynamical Systems and Ergodic Theory. --- Mathematical Methods in Physics. --- Fluid- and Aerodynamics. --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics --- Math --- Science --- Mathematical analysis --- Differential equations, Hyperbolic --- Differentiable dynamical systems. --- Mathematical physics. --- Physical mathematics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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This monograph presents, in an attractive and self-contained form, techniques based on the L1 stability theory derived at the end of the 1990s by A. Bressan, T.-P. Liu and T. Yang that yield original error estimates for so-called well-balanced numerical schemes solving 1D hyperbolic systems of balance laws. Rigorous error estimates are presented for both scalar balance laws and a position-dependent relaxation system, in inertial approximation. Such estimates shed light on why those algorithms based on source terms handled like "local scatterers" can outperform other, more standard, numerical schemes. Two-dimensional Riemann problems for the linear wave equation are also solved, with discussion of the issues raised relating to the treatment of 2D balance laws. All of the material provided in this book is highly relevant for the understanding of well-balanced schemes and will contribute to future improvements.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Conservation laws (Mathematics) --- Mathematics. --- Partial differential equations. --- Mathematical physics. --- Numerical analysis. --- Partial Differential Equations. --- Numerical Analysis. --- Mathematical Applications in the Physical Sciences. --- Numerical and Computational Physics, Simulation. --- Differential equations, Hyperbolic --- Differential equations, partial. --- Mathematical analysis --- Partial differential equations --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Physical mathematics --- Physics