# A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks

• Herbert C. Kranzer
• Barbara Lee Keyfitz
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 27)

## 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 ij, 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.

## Keywords

Rarefaction Wave Riemann Problem Characteristic Speed Viscosity Approximation Left State
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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