Large-amplitude internal fronts in two-fluid systems

Robin Ming Chen, Samuel Walsh, Miles H. Wheeler

Research output: Contribution to journalArticlepeer-review

3 Citations (SciVal)


In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are bounded above and below by flat horizontal walls and acted upon by gravity. We obtain continuous curves of solutions to this system that bifurcate from the trivial solution where the interface is flat. Following these families to the their extreme, the internal interface either overturns, comes into contact with the upper wall, or develops a highly degenerate “double stagnation” point. Our construction is made possible by a new abstract machinery for global continuation of monotone front-type solutions to elliptic equations posed on infinite cylinders. This theory is quite robust and, in particular, can treat fully nonlinear equations as well as quasilinear problems with transmission boundary conditions.

Original languageEnglish
Pages (from-to)1073-1083
Number of pages11
JournalComptes Rendus Mathematique
Issue number9
Publication statusPublished - 5 Jan 2021

Bibliographical note

Funding Information:
Funding. The research of RMC is supported in part by the NSF through DMS-1907584. The research of SW is supported in part by the NSF through DMS-1812436. A portion of this work was completed during a Research-in-Teams Program generously supported by the Erwin Schrödinger Institute for Mathematics and Physics, University of Vienna.

Publisher Copyright:
© 2020 Elsevier Masson SAS. All rights reserved.

Copyright 2021 Elsevier B.V., All rights reserved.


  • bifurcation
  • water waves
  • nonlinear elliptic equations

ASJC Scopus subject areas

  • Mathematics(all)


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