MAXIMUM-PRINCIPLES FOR A CLASS OF CONSERVATION-LAWS

We prove maximum principles for a class of conservation laws, u(t) + f(u)(x) = 0, and the corresponding regularized parabolic system, u(t) + f(u)(x) = epsilon u(xx). The class of conservation laws is determined by requiring the flux function f to be constant along certain coordinate directions in state space, The class includes models of multiphase flow in porous media, polymer flooding, and chemical chromatography, as well as gas dynamics. The maximum principle is first derived for the Cauchy problem for the parabolic equation and then for the Riemann problem of the hyperbolic equation. Finally, we conclude that the maximum principle also holds for approximate solutions to the hyperbolic equation generated by the Lax-Friedrichs, the Godunov, and the Glimm schemes. Hence the maximum principle also holds for weak solutions of the Cauchy problem for the hyperbolic equation, when these are limits of approximate solutions generated by such schemes.