Solutions in the Large for Some Nonlinear Hyperbolic Conservation Laws of Gas Dynamics.

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 mass, momentum, 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. For this thesis, a particular energy function has been discovered for which there is a global weak solution for bounded measurable data having finite total variation. This energy function models an ideal gas, and is given by the formula e = - nv + (S/(,R)). The following general existence theorem is also obtained: let e(,(epsilon))(v,S) be any smooth one parameter family of energy functions such that at (epsilon) = 0 the energy is given by e(,0)(v,S) = - nv + (S/(,R)). It is proven that there exists a constant C independent of (epsilon), such that, if (epsilon) (.) (total variation of the initial data) < C, then there exists a global weak solution to the equations. As a corollary we obtain an existence theorem of T. P. Liu for polytropic gases. The main point in my argument is that the nonlinear functional which I use to make the Glimm Scheme converge, depends only on properties of the equations at (epsilon) = 0. For general n x n systems of conservation laws, my technique provides an alternate proof for the interaction estimates in Glimm's 1965 paper. The main result here is that certain interaction differences are bounded by (epsilon) as well as by the approaching waves.Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158007/1/8025783.pd