0. Introduction Conservation laws are first order systems of quasilinear partial differential equations in divergence form; they express the balance laws of continuum physics for media with "elastic" response, in which internal dissipation is neglected. The absence of internal dissipation is manifested in the emergence of solutions with jump discontinuities… (More)
A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus… (More)
We construct global BV solutions to the Cauchy problem for the damped p-system, under initial data with distinct end-states. The solution will be realized as a perturbation of its asymptotic profile, in which the specific volume satisfies the porous media equation and the velocity obeys the classical Darcy law for gas flow through a porous medium.
For a convex conservation law u t + f (u) x = 0, u(x, 0) = u 0 (x), −∞ < x < ∞, t > 0, bounded initial data u 0 (x), are considered that take on constant values u − to the left of a bounded interval, and u + to the right, with u − > u +. The solution of the initial value problem is shown to collapse in finite time to a single shock wave joining u − to u +.… (More)
A mathematical analysis of a new approach to solidification problems is presented. A free boundary arising from a phase transition is assumed to have finite thickness. The physics leads to a system of nonlinear parabolic differential equations. Existence and regularity of solutions are proved. Invariant regions of the solution space lead to physical… (More)
This is an expository paper discussing the regularity and large time behavior of admissible BV solutions of genuinely nonlinear, strictly hy-perbolic systems of two conservation laws. The approach will be via the theory of generalized characteristics. 1. Introduction As is well-known, the theory of the Cauchy problem for nonlinear hyperbolic systems of… (More)