Abstract
We consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in Hs(), s > 2, via the energy method. The main tool is the short-time Fourier restriction norm method, which was first introduced in its current form by Ionescu, Kenig and Tataru [Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173(2) (2008) 265-304]. Besides, we use the frequency localized modified energy to control the high-low interaction component in the energy estimate. We remark that under the periodic setting, the integrable structure is very useful (but not necessary) to remove harmful terms in the nonlinearity and this work is the first low regularity well-posedness result for the fifth-order modified KdV equation.
| Original language | English |
|---|---|
| Pages (from-to) | 463-557 |
| Number of pages | 95 |
| Journal | Journal of Hyperbolic Differential Equations |
| Volume | 15 |
| Issue number | 3 |
| DOIs | |
| State | Published - 1 Sep 2018 |
Bibliographical note
Funding Information:The author would like to thank his advisor Soonsik Kwon for his helpful comments and encouragement through this research problem. Moreover, the author is grateful to Zihua Guo for his helpful advice to understand well the short time Xs,b structure under the periodic setting. Furthermore, the author is grateful to the anonymous referee(s) for reading the manuscript carefully and helpful suggestions and comments. C. K. is supported by FONDECYT de Postdoctorado 2017 Proyecto No. 3170067.
Publisher Copyright:
© 2018 World Scientific Publishing Company.
Keywords
- complete integrability
- local well-posedness
- modified energy
- nonlinear transformation
- short time X s, b space
- The fifth-order modified KdV equation