Explicit formula for scalar non‐linear conservation laws with boundary condition

We prove an uniqueness and existence theorem for the entropy weak solution of non‐linear hyperbolic conservation laws of the form\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{\partial }{{\partial t}}u + \frac{\partial }{{\partial x}}f\left(u \right) = 0 $$\end{document} , with initial data and boundary condition. The scalar function u = u ( x , t ), x > 0, t > 0, is the unknown; the function f = f ( u ) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ϵ ℝ\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{\partial }{{\partial t}}\left({x^\alpha u} \right) + \frac{\partial }{{\partial x}}\left({x^\alpha \frac{{u^2 }}{2}} \right) = 0 $$\end{document} . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f ( u (.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.