On vacuum problems for different systems of conservation laws.

We prove that no vacuum state exists for the single-piston and double-piston problem for Navier-Stokes equations, if no vacuum state exists initially. We also conduct and evaluate some numerical experiments concerning the behavior of the sparse density and velocity in the Navier-Stokes equations, where the initial data is Riemann and a large velocity difference is applied initially in opposite directions, where for the Euler equations a vacuum is present for all t > 0. As the viscosity goes to zero, the velocity appears to approach the straight line between the two Euler equation velocity states at the limiting ends of the vacuum. Next, we construct the exact solution for the case of pressureless gases, when a vacuum state and a delta-shock interacts initially. We prove that this exact solution satisfies a family of entropies. Furthermore, this solution is the only one of many constructed for pressureless gas, satisfying an entropy.Ph.D.Aerospace engineeringApplied SciencesMathematicsPlasma physicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/124378/2/3138129.pd