The Riemann function, singular entropies, and the structure of oscillations in systems of two conservation laws (Q1417406)

The author studies the strictly hyperbolic system of two conservation laws, \[ u_t+a(u,v)_x=0, \quad v_t+b(u,v)_x=0 \] with characteristic speed \(\lambda_1<\lambda_2\), right eigenvectors \(r_1\), \(r_2\) and left eigenvectors \(l_1\), \(l_2\). The eigenvectors are normalized so that \(r_i\cdot l_j=\delta_{ij}\). The flux is denoted by \(F(U)=(a(u,v),b(u,v))\), where \(U=(u,v)\) is the conserved variable. The entropy \(\eta (U)\) with corresponding entropy flux \(q(U)\) plays a central role in this paper. The function \(\eta (U)\) is an entropy if every smooth solution of the considered hyperbolic system satisfies the additional conservation law \(\partial_t\eta (U)+\partial_x q (U)=0\). Entropy pairs \(\eta (U)-q(U)\) satisfy the equation \(\nabla q(U)=\nabla \eta (U)\cdot \nabla F(U)\) and describe the nonlinear structure of the system under consideration. The author introduces singular entropies and making use of the connection with the fundamental solution of the entropy equation, and its adjoint operator derives a new formula describing the coupling of oscillations between the two characteristic fields in the considered systems of conservation laws.