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

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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.

Original languageEnglish
Pages (from-to)463-557
Number of pages95
JournalJournal of Hyperbolic Differential Equations
Issue number3
StatePublished - 1 Sep 2018


  • complete integrability
  • local well-posedness
  • modified energy
  • nonlinear transformation
  • short time X s, b space
  • The fifth-order modified KdV equation


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