Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form

$$\frac{\partial } {{\partial t}}(ru) + \frac{\partial } {{\partial x}}(rf(u)) = 0,$$

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 with inf $f( \cdot ) = 0$ and the weight function $r = r(x)$, $x > 0$, to be positive (for example, $r(x) = {x^\alpha }$, with an arbitrary real $\alpha$).

We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux $r( \cdot )f(u( \cdot , \cdot ))$ at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.