Exchange of conserved quantities, shock loci and Riemann problems (Q2761858)

The author considers systems of conservation laws in one space variable, of the form NEWLINE\[NEWLINEg(w)_t+ f(w)_x= 0,NEWLINE\]NEWLINE distinguishing the `conserved qualities' \(g\) from a convenient choice of dependent variables \(w\), the latter functions of \(x\), \(t\) in some open subset of \(\mathbb R^2\). The main results of this paper include sufficient conditions that a branch of the Hugoniot locus beginning at a fixed point \(\overline w\) continues to infinity after such an exchange of conserved quantities. This result is applied to the solvability of Riemann problems in the large. The author describes dynamical systems, the trajectories of which describe branches of the Hugoniot locus of a fixed point in phase space, at each stage of the homotopy. A test for the existence and nonexistence of critical points for this dynamical system is studied.