Riassunto
Conservation laws arise from the modeling of physical processes through the following three steps: 1) The appropriate physical balance laws are derived for m-phy- t cal quantities, ul""'~ with u = (ul' ... ,u ) and u(x,t) defined m for x = (xl""'~) E RN (N = 1,2, or 3), t > 0 and with the values m u(x,t) lying in an open subset, G, of R , the state space. The state space G arises because physical quantities such as the density or total energy should always be positive; thus the values of u are often con strained to an open set G. 2) The flux functions appearing in these balance laws are idealized through prescribed nonlinear functions, F.(u), mapping G into J j = 1, ..• ,N while source terms are defined by S(u,x,t) with S a given smooth function of these arguments with values in Rm. In parti- lar, the detailed microscopic effects of diffusion and dissipation are ignored. 3) A generalized version of the principle of virtual work is applied (see Antman [1]). The formal result of applying the three steps (1)-(3) is that the m physical quantities u define a weak solution of an m x m system of conservation laws, o I + N(Wt'u + r W ·F.(u) + W·S(u,x,t))dxdt (1.1) R xR j=l Xj J for all W E C~(RN x R+), W(x,t) E Rm.
Contenuti
1. Introduction.- 1.1. Some Physical Examples of Systems of Conservation Laws.- 1.2. The Importance of Dissipative Mechanisms.- 1.3. The Common Structure of the Physical Systems of Conservation Laws and Friedrichs’ Theory of Symmetric Systems.- 1.4. Linear and Nonlinear Wave Propagation and the Theory of Nonlinear Simple Waves.- 1.5. Weakly Nonlinear Asymptotics — Nonlinear Geometric Optics.- 1.6. A Rigorous Justification of Weakly Nonlinear Asymptotics in a Special Case.- 1.7. Some Additional Applications of Weakly Nonlinear Asymptotics in the Modeling of Complex Systems.- Bibliography for Chapter 1.- 2. Smooth Solutions and the Equations of Incompressible Fluid Flow.- 2.1. The Local Existence of Smooth Solutions for Systems of Conservation Laws.- 2.2. A Continuation Principle for Smooth Solutions 46 2.3. Uniformly Local Sobolev Spaces.- 2.4. Compressible and Incompressible Fluid Flow.- 2.5. Equations for Low Mach Number Combustion.- Bibliography for Chapter 2.- 3. The Formation of Shock Waves in Smooth Solutions.- 3.1. Shock Formation for Scalar Laws in Several Space Variables.- 3.2. Shock Formation in Plane Wave Solutions of General m × m Systems.- 3.3. Detailed Results on Shock Formation for 2 × 2 Systems.- 3.4. Breakdown for a Quasi-Linear Wave Equation in 3-D.- 3.5. Some Open Problems Involving Shock Formation in Smooth Solutions.- Bibliography for Chapter 3.- 4. The Existence and Stability of Shock Fronts in Several Space Variables.- 4.1. Nonlinear Discontinuous Progressing Waves in Several Variables — Shock Front Initial Data.- 4.2. Some Theorems Guaranteeing the Existence of Shock Fronts.- 4.3. Linearization of Shock Fronts.- 4.4. An Introduction to Hyperbolic Mixed Problems.- 4.5. Quantitative Estimates for Linearized Shock Fronts.- 4.6. Some Open Problems in Multi-D Shock Wave Theory.- Bibliography for Chapter 4.
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