Rough solutions of the fifth-order KdV equations

Abstract

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy {partial derivative(t)u + partial derivative(5)(x)u + c(1)partial derivative(x)u partial derivative(2)(x)u + c(2)u partial derivative(3)(x)u = 0, x, t is an element of R, u(0, x) = u(0)(x), u(0) is an element of H-s(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,X-b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,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 obtain that the fifth-order equation in the KdV hierarchy, partial derivative(t)u - partial derivative(5)(x)u - 30u(2)partial derivative(x)u + 20 partial derivative(x)u partial derivative(2)(x)u + 10u partial derivative(3)(x)u = 0 is globally well-posed in the energy space H-2. (C) 2013 Elsevier Inc. All rights reserved.