The initial-boundary value problem for a scalar conservation law

Abstract

We consider the initial-boundary value problem for a scalar conservation law. For nonlinear problems, it has turned out that classical initial boundary value problems are not well posed, requiring an extension of the concept of boundary conditions. Bardos, Le Roux and Nedelec (1979) proved existence and uniqueness of the solution for a scalar conservation law in several space variables with an initial condition and one generalized boundary condition. To prove existence they use the vanishing viscosity method and to prove uniqueness, they define an entropy condition also at the boundary. Le Roux (1977, 1979a, 1979b, 1981) also has proved existence and uniqueness of the initial-boundary value problem using the Godunov finite difference scheme for one first-order quasilinear hyperbolic equation in one dimension and the vanishing viscosity method for the first order in several dimensions.