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The transformation from Eulerian to Lagrangian coordinates for solutions with discontinuities

Hyperbolic P.D.E. Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1270)

Abstract

We demonstrate the equivalence of Eulerian and Lagrangian coordinates for weak, discontinuous solutions in one space dimension. This transformation also induces a one to one correspondence between the convex extensions, or "entropy functions" of the system of conservation laws in either coordinate system. Such entropy functions are of interest in the theory of compensated compactness, and our results imply the equivalence of some of the elements of that theory in either coordinate system.

As an application, we translate a large-data existence result of DiPerna for the Euler equations of isentropic gas dynamics into a similar theorem for the Lagrangian equations.

More detailed proofs of these results will appear elsewhere [11].

Keywords

  • Shock Wave
  • Weak Solution
  • Radon Measure
  • Entropy Function
  • Young Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1987 Springer-Verlag

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Wagner, D.H. (1987). The transformation from Eulerian to Lagrangian coordinates for solutions with discontinuities. In: Carasso, C., Serre, D., Raviart, PA. (eds) Nonlinear Hyperbolic Problems. Lecture Notes in Mathematics, vol 1270. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078327

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  • DOI: https://doi.org/10.1007/BFb0078327

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18200-9

  • Online ISBN: 978-3-540-47805-8

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