Rough solutions of the fifth-order KdV equations

2012/05/04 by Zihua Guo, Guo, Zihua, Chulkwang Kwak +3
Mathematics · Physics and Astronomy · #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1205.0850

openalex publication_date 2012/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy u_t + u_xxxxx + c₁u_x u_xx + c₂u u_x = 0 x,t ∈ \R We 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 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 is globally well-posed in the energy space H².

Related

• Local well-posedness for the fifth-order KdV equations on \mathbbT
• Low regularity Cauchy problem for the fifth-order modified KdV equations on \mathbbT
• Energy solutions for the fifth-order modified Korteweg de-Vries equations
• On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map
• Global well-posedness for the fifth-order KdV equation in H^-1(ℝ)