Geometric solutions of the Riemann problem for the scalar conservation law

Abstract

For the Riemann problem $$ \left\{\begin{array}{l}u_t+(\Phi(u,x))_x=0,\\ u|_{t=0}=u_-+[u]\theta(x) \end{array}\right. $$ a new definition of the solution is proposed. We associate a Hamiltonian system with initial conservation law, and define the geometric solution as the result of the action of the phase flow on the initial curve. In the second part of this paper, we construct the equalization procedure, which allows us to juxtapose a geometric solution with a unique entropy solution under the condition that $\Phi$ does not depend on $x$. If $\Phi$ depends on $x$, then the equalization procedure allows us to construct a generalized solution of the original Riemann problem.

Keywords

• riemann problem
• conservation laws
• associated hamiltonian system