On Elementary Interactions for Hyperbolic Conservation Laws

This is a survey of interactions of weak nonlinear waves in N × N systems of hyperbolic conservation laws. Recently a variety of surprising new phenomena have been observed, including strong nonlinear instability of solutions. This implies that further assumptions must be made to develop a Glimm–Lax existence and decay theory for N ≥ 3. As a first step towards such a theory, a systematic description of local interactions in terms of naturally occurring ‘flux coefficients’ is given. These coefficients characterize the nonlinearity of the system, and each has a physical interpretation in terms of wave interactions. There are two competing nonlinear interaction effects, namely decay and wave generation, which can be quantified using the flux coefficients. The imposition of physical assumptions leads to constraints on the coefficients, which in turn prevent blowup from occurring. Relevant existence and nonexistence results are briefly surveyed, and conditions under which different nonlinear phenomena dominate are described. An example of a compactly supported unstable solution is given, illustrating this phenomenon from the point of view of local interactions. Finally, the coefficients are calculated and interpreted for the Euler equations of gas dynamics.