A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks

Abstract

The system $$ \left\{ {\begin{array}{*{20}{c}} {{u_t} + {{\left( {{u^2} - v} \right)}_x} = 0} \\ {{v_t} + {{\left( {\frac{1}{3}{u^3} - u} \right)}_x} = 0} \\ \end{array} } \right. $$ (1) is an example of a strictly hyperbolic, genuinely nonlinear system of conservation laws. Usually the Riemann problem for such a system is well-posed: centered weak solutions consisting of combinations of simple waves and admissible jump discontinuities (shocks) exist and are unique for each set of values of the Riemann data [1–3]. The characteristic speeds λ1 and λ2 of system (1), however, do not conform to the usual pattern for strictly hyperbolic, genuinely nonlinear systems: although locally separated, they overlap globally (cf. Keyfitz [4] for a more general discussion of the significance of overlapping characteristic speeds). In particular, λ1 = u - 1 and λ2 = u + 1 are real and unequal at any particular point U = (u, v) of state space (as strict hyperbolicity requires), and λ2 - λ1 = 2 is even bounded away from zero globally, but λ1 at one point U 1 may be equal to λ2 at a different point U 2. The corresponding right eigenvectors r 1 = (1, u + l) and r 2 = (1, u - 1) of the gradient matrix for (1) display genuine nonlinearity, since r i ∙ ▽ λi > 0 for i = 1,2 but the two eigenvalues vary in the same direction: r i ∙ ▽ λj > 0 for i ≠ j, rather than the usual “opposite variation” r i ∙ ▽ λj < 0 familiar from (say) gas dynamics. As a result, classical global existence and uniqueness theorems [3,5] no longer apply.