ON LeFLOCH'S SOLUTIONS TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR CONSERVATION LAWS

We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients Vx satisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that Vx is continuous almost everywhere, and a result concerning the traces of the sign of f′(Vx(t, ⋅)) is proven.