Well-posedness theory for hyperbolic conservation laws

The paper presents a well-posedness theory for the initial value problem for a general system of hyperbolic conservation laws. We will start with the refinement of Glimm's existence theory and discuss the principle of nonlinear superposition through wave tracing. Our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the L₁(x) distance between the two solutions and is time-decreasing. Moreover, the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition. It consists of a linear term measuring the L₁(x) distance, a quadratic term measuring the coupling of waves and distance, and a generalized entropy functional. © 1999 John Wiley & Sons, Inc.