scholarly journals Long‐time asymptotic behavior of the fifth‐order modified KdV equation in low regularity spaces

2021 ◽  
Author(s):  
Nan Liu ◽  
Mingjuan Chen ◽  
Boling Guo
Keyword(s):  
Asymptotic Behavior ◽  
Kdv Equation ◽  
Low Regularity ◽  
Modified Kdv Equation ◽  
Long Time ◽  
Fifth Order ◽  
Time Asymptotic Behavior

Low regularity Cauchy problem for the fifth-order modified KdV equations on 𝕋

2018 ◽  
Vol 15 (03) ◽  
pp. 463-557 ◽  
Author(s):  
Chulkwang Kwak
Keyword(s):  
Kdv Equation ◽  
Main Tool ◽  
Well Posedness ◽  
Current Form ◽  
Low Regularity ◽  
Integrable Structure ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
Modified Kdv Equation ◽  
Fifth Order

We consider the fifth-order modified Korteweg–de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in [Formula: see text], [Formula: see text], 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.

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