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 language | English (US) |
|---|---|
| Pages (from-to) | 752-782 |
| Number of pages | 31 |
| Journal | Journal of Functional Analysis |
| Volume | 264 |
| Issue number | 3 |
| DOIs | |
| State | Published - 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: [email protected] (R.M. Chen), [email protected] (J.L. Marzuola), [email protected] (D. Spirn), [email protected] (J.D. Wright).
Keywords
- Gravity-capillary
- Ill-posedness
- Water waves