TY - JOUR

T1 - Rough solutions of the fifth-order KdV equations

AU - Guo, Zihua

AU - Kwak, Chulkwang

AU - Kwon, Soonsik

N1 - Funding Information:
We are grateful to Didier Pilod for pointing out an error in the first draft, and to the anonymous referee for careful reading and improving the clearness of this paper. Z.G. is partially supported by NNSF of China (Nos. 11001003 , 11371037 ). S.K. is partially supported by TJ Park Science Fellowship and NRF (Korea) grant 2010-0024017 .

PY - 2013/12/1

Y1 - 2013/12/1

N2 - We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,x,t∈R,u(0,x)=u0(x),u0∈Hs(R). We prove a priori bound of solutions for Hs(R) with s≥54 and the local well-posedness for s≥2. The method is a short time Xs,b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,b structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we obtain that the fifth-order equation in the KdV hierarchy,∂tu-∂x5u-30u2∂xu+20∂xu∂x2u+10u∂x3u=0 is globally well-posed in the energy space H2.

AB - We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,x,t∈R,u(0,x)=u0(x),u0∈Hs(R). We prove a priori bound of solutions for Hs(R) with s≥54 and the local well-posedness for s≥2. The method is a short time Xs,b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,b structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we obtain that the fifth-order equation in the KdV hierarchy,∂tu-∂x5u-30u2∂xu+20∂xu∂x2u+10u∂x3u=0 is globally well-posed in the energy space H2.

KW - Fifth-order KdV equation

KW - KdV hierarchy

KW - Local well-posedness

KW - X space

UR - http://www.scopus.com/inward/record.url?scp=84883820205&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2013.08.010

DO - 10.1016/j.jfa.2013.08.010

M3 - Article

AN - SCOPUS:84883820205

VL - 265

SP - 2791

EP - 2829

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -