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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
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by Philippe LeFloch and Jean-Claude Nédélec PDF
Trans. Amer. Math. Soc. 308 (1988), 667-683 Request permission


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.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 308 (1988), 667-683
  • MSC: Primary 35L65
  • DOI:
  • MathSciNet review: 951622