Conservation law with discontinuous flux (Q1890268)

The following problem is considered \[ \begin{aligned} u_t+f(u)_x&=0,\quad x>0,\;t>0,\\ u_t+g(u)_x&=0,\quad x<0,\;t>0,\\ u(x,0)&=u_0(x), \end{aligned} \] where \( f(u(0+,t))=g(u(0-,t))\); \(f,g\in C^1(\mathbb R)\) are strictly convex and of super-linear growth. An explicit formula for the solution of the problem above is given as well as an entropy condition that guarantees uniqueness of the solution obtained. Finally, the authors derive the solution for the Riemann problem.