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

Abstract

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( v, S), where e = specific internal energy, v = specific volume, and S = specific entropy. The energy function e = −1n v + ( S R ) describes a polytropic gas for the exponent γ = 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 [10]. Here the following general existence theorem is obtained: let e ϵ ( v, S) be any smooth one parameter family of energy functions such that at ε = 0 the energy is given by e 0(v, S) = − 1n v + ( S R ) . It is proven that there exists a constant C independent of ε, such that, if ε · (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 ε 0( 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 ε = 0. For general n × 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 ε as well as by the approaching waves.