On the regularity of the flow map for the gravity-capillary equations

Robin Ming Chen, Jeremy L. Marzuola, Daniel Spirn, J. Douglas Wright

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We prove via explicitly constructed initial data that solutions to the gravity-capillary wave system in R3 representing a 2d air-water interface immediately fail to be C3 with respect to the initial data if the initial data (h0,ψ0)∈Hs+12⊗Hs for s<3. Similar results hold in R2 domains with a 1d interface. Furthermore, we discuss the related threshold for the pure gravity water wave system.

Original languageEnglish (US)
Pages (from-to)752-782
Number of pages31
JournalJournal of Functional Analysis
Volume264
Issue number3
DOIs
StatePublished - 2013

Bibliographical note

Funding Information:
✩ Acknowledgements: R.M.C. was supported in part by NSF grant DMS-0908663. J.L.M. was supported in part by an NSF Postdoctoral Fellowship at Columbia University and a Hausdorff Center Postdoctoral Fellowship at Universität Bonn. In addition, J.L.M. thanks the Courant Institute for graciously hosting him as a Visiting Academic during part of this research. D.S. was supported in part by NSF grants DMS-0707714 and DMS-0955687. J.D.W. was supported in part by NSF grants DMS-0807738, DMS-0908299 and DMS-1105635. The authors also thank Dave Ambrose, David Lannes and Fernando Reitich for helpful conversations along the way to completing this work. We also thank the anonymous referee for a careful reading of the draft and for assisting us in putting the results in a better context. We also thank Thomas Alazard, Nicolas Burq and Claude Zuilly for a careful reading of the draft and catching an error in an early version of the draft. * Corresponding author. E-mail addresses: mingchen@pitt.edu (R.M. Chen), marzuola@math.unc.edu (J.L. Marzuola), spirn@math.umn.edu (D. Spirn), jdoug@math.drexel.edu (J.D. Wright).

Keywords

  • Gravity-capillary
  • Ill-posedness
  • Water waves

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