Abstract
This paper is a continuation of the paper Low regularity Cauchy problem for the fifth-order modified KdV equations on T [7]. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as following:. (∂tu-∂x5u-30u2∂xu+20∂xu∂x2u+10u∂x3u=0,(t,x)∈R×T,u(0,x)=u0(x)∈Hs(T).We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev initial data via the energy method. This paper follows almost same idea and argument as in [7]. Precisely, we use some conservation laws of the KdV Hamiltonians to observe the direction which the nonlinear solution evolves to. Besides, it is essential to use the short time Xs,b spaces to control the nonlinear terms due to high × low ⇒ high interaction component in the non-resonant nonlinear term. We also use the localized version of the modified energy in order to obtain the energy estimate.As an immediate result from a conservation law in the scaling sub-critical problem, we have the global well-posedness result in the energy space H2.
| Original language | English |
|---|---|
| Pages (from-to) | 7683-7737 |
| Number of pages | 55 |
| Journal | Journal of Differential Equations |
| Volume | 260 |
| Issue number | 10 |
| DOIs | |
| State | Published - 15 May 2016 |
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 structure under the periodic setting. C.K. is partially supported by NRF (Korea) grant 2015R1D1A1A01058832 .
Publisher Copyright:
© 2016 Elsevier Inc.
Keywords
- Complete integrability
- Energy method
- Local well-posedness
- Modified energy
- The fifth-order KdV equation
- X space