Genuinely Nonlinear Systems of Two Conservation Laws

Abstract

The theory of solutions of genuinely nonlinear, strictly hyperbolic systems of two conservation laws will be developed in this chapter at a level of precision comparable to that for genuinely nonlinear scalar conservation laws, expounded in Chapter XI.

This will be achieved by exploiting the presence of coordinate systems of Riemann invariants and the induced rich family of entropy-entropy flux pairs. The principal tools in the investigation will be generalized characteristics and entropy estimates.

The analysis will reveal a close similarity in the structure of solutions of scalar conservation laws and pairs of conservation laws. Thus, as in the scalar case, jump discontinuities are generally generated by the collision of shocks and/or the focussing of compression waves, and are then resolved into wave fans approximated locally by the solution of associated Riemann problems.

The total variation of the trace of solutions along space-like curves is controlled by the total variation of the initial data, and spreading of rarefaction waves affects total variation, as in the scalar case.

The dissipative mechanisms encountered in the scalar case are work here at as well, and have similar effects on the large time behavior of solutions. Entropy dissipation induces O(t^−1/2) decay of solutions with initial data in L¹(−∞, ∞). When the initial data have compact support, the two characteristic families asymptotically decouple, the characteristics spread and form a single N-wave profile for each family. Finally, as in the scalar case, confinement of characteristics under periodic initial data induces O(_t⁻¹) decay in the total variation per period and formation of saw-toothed profiles, one for each characteristic family.