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Transactions of the American Mathematical Society

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Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

Authors: Philippe LeFloch and Jean-Claude Nédélec
Journal: Trans. Amer. Math. Soc. 308 (1988), 667-683
MSC: Primary 35L65
MathSciNet review: 951622
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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

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

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.

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