Compressible fluid flow and systems of conservation laws in several space variables
著者
書誌事項
Compressible fluid flow and systems of conservation laws in several space variables
（Applied mathematical sciences, v. 53）
Springer-Verlag, c1984
- us : pbk
- gw : pbk
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注記
Includes bibliographies and index
内容説明・目次
内容説明
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.
目次
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 x m Systems.- 3.3. Detailed Results on Shock Formation for 2 x 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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