Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

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Abstract
We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form \[ ∂ ∂ t ( r u ) + ∂ ∂ x ( r f ( u ) ) = 0 , \frac {\partial } {{\partial t}}(ru) + \frac {\partial } {{\partial x}}(rf(u)) = 0, \] with initial data and boundary condition. The scalar function u = u ( x , t ) u = u(x,\,t) , x > 0 x > 0 , t > 0 t > 0 , is the unknown, the function f = f ( u ) f = f(u) is assumed to be strictly convex with inf f ( ⋅ ) = 0 f( \cdot ) = 0 and the weight function r = r ( x ) r = r(x) , x > 0 x > 0 , to be positive (for example, r ( x ) = x α r(x) = {x^\alpha } , with an arbitrary real α \alpha ). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux r ( ⋅ ) f ( u ( ⋅ , ⋅ ) ) 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.

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