Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables

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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.