TY - JOUR
T1 - Low regularity Cauchy problem for the fifth-order modified KdV equations on
AU - Kwak, Chulkwang
N1 - 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.
PY - 2018/9/1
Y1 - 2018/9/1
N2 - 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.
AB - 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.
KW - complete integrability
KW - local well-posedness
KW - modified energy
KW - nonlinear transformation
KW - short time X s, b space
KW - The fifth-order modified KdV equation
UR - http://www.scopus.com/inward/record.url?scp=85054201993&partnerID=8YFLogxK
U2 - 10.1142/S0219891618500170
DO - 10.1142/S0219891618500170
M3 - Article
AN - SCOPUS:85054201993
SN - 0219-8916
VL - 15
SP - 463
EP - 557
JO - Journal of Hyperbolic Differential Equations
JF - Journal of Hyperbolic Differential Equations
IS - 3
ER -