Systems of Conservation Laws: Two-Dimensional Riemann Problems: 38 (Progress in Nonlinear Differential Equations and Their Applications) - Tapa dura

This work is based on the lecture notes of the course M742: Topics in Partial Dif­ ferential Equations, which I taught in the Spring semester of 1997 at Indiana Univer­ sity. My main intention in this course was to give a concise introduction to solving two-dimensional compressibleEuler equations with Riemann data, which are special Cauchy data. This book covers new theoretical developments in the field over the past decade or so. Necessary knowledge of one-dimensional Riemann problems is reviewed and some popularnumerical schemes are presented. Multi-dimensional conservation laws are more physical and the time has come to study them. The theory onbasicone-dimensional conservation laws isfairly complete providing solid foundation for multi-dimensional problems. The rich theory on ellip­ tic and parabolic partial differential equations has great potential in applications to multi-dimensional conservation laws. And faster computers make itpossible to reveal numerically more details for theoretical pursuitin multi-dimensional problems. Overview and highlights Chapter 1is an overview ofthe issues that concern us inthisbook. It lists theEulersystemandrelatedmodelssuch as theunsteady transonic small disturbance, pressure-gradient, and pressureless systems. Itdescribes Mach re­ flection and the von Neumann paradox. In Chapters 2-4, which form Part I of the book, we briefly present the theory of one-dimensional conservation laws, which in­ cludes solutions to the Riemann problems for the Euler system and general strictly hyperbolic and genuinely nonlinearsystems, Glimm's scheme, and large-time asymp­ toties.

This work should serve as an introductory text for graduate students and researchers working in the important area of partial differential equations with a focus on problems involving conservation laws. The only requisite for the reader is a knowledge of the elementary theory of partial differential equations.

Key features of this work include:

* broad range of topics, from the classical treatment to recent results, dealing with solutions to 2D compressible Euler equations

* good review of basic concepts (1-D Riemann problems)

* concrete solutions presented, with many examples, over 100 illustrations, open problems, and numerical schemes

* numerous exercises, comprehensive bibliography and index

* appeal to a wide audience of applied mathematicians, graduate students, physicists, and engineers Written in a clear, accessible style, the book emphasizes more recent results that will prepare readers to meet modern challenges in the subject, that is, to carry out theoretical, numerical, and asymptotical analysis.