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 -