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 coordin
ate 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 th
e Cauchy problem for the parabolic equation and then for the Riemann p
roblem of the hyperbolic equation. Finally, we conclude that the maxim
um principle also holds for approximate solutions to the hyperbolic eq
uation generated by the Lax-Friedrichs, the Godunov, and the Glimm sch
emes. 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.