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Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics

By J. Blake Temple


The constraints under which a gas at a certain state will evolve can be given by three partial differential equations which express the conservation of momentum, mass, and energy. In these equations, a particular gas is defined by specifying the constitutive relation e = e(u, S), where e = specific internal energy, v = specific volume, and S = specific entropy. The energy function e =-In u + (S/R) describes a polytropic gas for the exponent y = 1, and for this choice of e(V, S), global weak solutions for bounded measurable data having finite total variation were given by Nishida in [lo]. Here the following general existence theorem is obtained: let e,(v, S) be any smooth one parameter family of energy functions such that at E = 0 the energy is given by e&v, S) =-In v + (S/R). It is proven that there exists a constant C independent of E, such that, if E. (total variation of the initial data) < C, then there exists a global weak solution to the equations. Since any energy function can be connected to e&V, S) by a smooth parameterization, our results give an existence theorem for all the conservation laws of gas dynamics. As a corollary we obtain an existence theorem of Liu, Indiana Univ. Math. J. 26, No. 1 (1977) for polytropic gases. The main point in this argument is that the nonlinear functional used to make the Glimm Scheme converge, depends only on properties of the equations at E = 0. For general n x n systems of conservation laws, this technique provides an alternate proof for the interaction estimates in Glimm’s 1965 paper. The new result here is that certain interaction differences are bounded by E as well as by the approaching waves

Year: 2013
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