We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchyWe prove a priori bound of solutions for H s (R) with s ≥ 5 4 and the local well-posedness for s ≥ 2. The method is a short time X s,b space, which is first developed by Ionescu-Kenig-Tataru [11] in the context of the KP-I equation. In addition, we use a weight on localized X s,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 have the fifth-order equation in the KdV hierarchy, ∂tu − ∂ 5x u − 30u 2 ∂xu + 20∂xu∂ 2 x u + 10u∂ 3 x u = 0 is globally well-posed in the energy space H 2 .
In this paper, we consider the cubic fourth-order nonlinear Schrödinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in H s (T) with −1/3 ≤ s < 0 for the Cauchy problem of the Wick ordered 4NLS. The other one is the non-squeezing property for the flow map of 4NLS in the symplectic phase space L 2 (T). To prove the former we used the ideas introduced in [36] and [27], and to prove the latter we used the ideas in [8].2010 Mathematics Subject Classification. 35Q55,70H15.
In this paper, we consider the fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. We prove the local well-posedness in H s (T), 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.2010 Mathematics Subject Classification. 35Q53, 37K10.
This paper concerns the initial-boundary value problem (IBVP) of the Kawahara equation posed on the right and left half-lines. We prove the local well-posedness in the low regularity Sobolev space. We introduce the Duhamel boundary forcing operator, which is introduced by Colliander -Kenig [11] in the context of Airy group operators, to construct solutions on the whole line. We also give the bilinear estimate in X s,b space for b < 1 2 , which is almost sharp compared to IVP of Kawahara equation [10,19].
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