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Author Majda, A. author Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables [electronic resource] / by A. Majda New York, NY : Springer New York : Imprint: Springer, 1984 http://dx.doi.org/10.1007/978-1-4612-1116-7 172 p. online resource

SUMMARY

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 conยญ 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, ..โ{128}ข ,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 ). 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 ยทF.(u) + WยทS(u,x,t))dxdt (1.1) R xR j=l Xj J for all W E C̃(RN x R+), W(x,t) E Rm

CONTENT

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 Friedrichsโ{128}{153} Theory of Symmetric Systems -- 1.4. Linear and Nonlinear Wave Propagation and the Theory of Nonlinear Simple Waves -- 1.5. Weakly Nonlinear Asymptotics โ{128}{148} 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 ร{151} m Systems -- 3.3. Detailed Results on Shock Formation for 2 ร{151} 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 โ{128}{148} 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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