Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0, )=u0 on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any , u0 L∞(Ω), f L∞(Q), a L∞((0,T)×∂Ω). In the L1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.