Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws

A system of hyperbolic conservation laws $${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x{\mathrm{\partial }}_{u}Q=0,Q=u_1^3\mathrm{/}3+u_1{u}_2^2,u=u(x,t)\in {\text{mR}}^2,$$ as well as its viscous regularization $${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_x{\mathrm{\partial }}_{u}Q=\text{calM}{\mathrm{\partial }}_x^{2}u,\text{calM}=\text{diag}({\mu }_1,{\mu }_2),{\mu }_1>0,{\mu }_2>0,$$ are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at $\pm \mathrm{\infty }$, the solution also contains undercompressive shocks and Jouguet waves.