Existence of entropy solutions for multidimensional conservation laws with $L^1$ boundary conditions

  • Carlo Bianca
  • Christian Dogbe


This paper deals with the construction of nonlinear boundary conditions for multidimensional conservation laws. Specifically by introducing a new type of entropy solution matching the boundary condition, the existence and uniqueness of a solution belonging to $L^\infty\cap \texttt{BV}$ is proved by using the Di Perna-Lions regularization method. The new entropy solution, which takes advantage by the entropy criterion introduced by Bardos-Le Roux-N\'ed\'elec for first-order quasilinear equations with boundary conditions, is based on a weaker assumption at the boundary.