This book offers a detailed, rigorous, and self-contained presentation of the theory of hyperbolic conservation laws from the basic theory to the forefront of research. The text offers extensive examples, exercises with hints and answers and comprehensive appendices.

This book presents the theory of hyperbolic conservation laws from basic theory to the forefront of research.

The text treats the theory of scalar conservation laws in one dimension in detail, showing the stability of the Cauchy problem using front tracking. The extension to multidimensional scalar conservation laws is obtained using dimensional splitting. Inhomogeneous equations and equations with diffusive terms are included as well as a discussion of convergence rates, and coverage of the classical theory of Kruzkov and Kuznetsov. Systems of conservation laws in one dimension are treated in detail, starting with the solution of the Riemann problem.

The book includes detailed discussion of the recent proof of well-posedness of the Cauchy problem for one-dimensional hyperbolic conservation laws, and a chapter on traditional finite difference methods for hyperbolic conservation laws with error estimates and a section on measure valued solutions.

Only book on front tracking which covers these new results Very well suited for graduate students Includes supplementary material: sn.pub/extras

Autorentext

Helge Holden is professor of mathematics at the Norwegian University of Science and Technology and an adjunct professor at the Center of Mathematics for Applications, part of the University of Oslo. He has done extensive research in stochastic analysis, in particular in its application to flow in porous media. Nils Henrik Risebro is professor of mathematics at the University of Oslo. His research interests are mainly nonlinear partial differential equations and numerical methods for these. He has also worked on conservation laws with discontinuous coefficients in a variety of applications.

Inhalt
1 Introduction.- 1.1 Notes.- 2 Scalar Conservation Laws.- 2.1 Entropy Conditions.- 2.2 The Riemann Problem.- 2.3 Front Tracking.- 2.4 Existence and Uniqueness.- 2.5 Notes.- 3 A Short Course in Difference Methods.- 3.1 ConservativeMethods.- 3.2 Error Estimates.- 3.3 APriori Error Estimates.- 3.4 Measure-Valued Solutions.- 3.5 Notes.- 4 Multidimensional Scalar Conservation Laws.- 4.1 Dimensional SplittingMethods.- 4.2 Dimensional Splitting and Front Tracking.- 4.3 Convergence Rates.- 4.4 Operator Splitting: Diffusion.- 4.5 Operator Splitting: Source.- 4.6 Notes.- 5 The Riemann Problem for Systems.- 5.1 Hyperbolicity and Some Examples.- 5.2 Rarefaction Waves.- 5.3 The Hugoniot Locus: The Shock Curves.- 5.4 The Entropy Condition.- 5.5 The Solution of the Riemann Problem.- 5.6 Notes.- 6 Existence of Solutions of the Cauchy Problem.- 6.1 Front Tracking for Systems.- 6.2 Convergence.- 6.3 Notes.- 7 Well-Posedness of the Cauchy Problem.- 7.1 Stability.- 7.2 Uniqueness.- 7.3 Notes.- A Total Variation, Compactness, etc..- A.1 Notes.- B The Method of Vanishing Viscosity.- B.1 Notes.- C Answers and Hints.- References.