Local well-posedness for the fifth-order KdV equations on T

Abstract

This paper is a continuation of the paper Low regularity Cauchy problem for the fifth-order modified KdV equations on T [7]. In this paper, we consider the fifth-order equation in the Korteweg-de Vries (KdV) hierarchy as following: {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, (t, x) is an element of R x T, u(0, x) = u(0)(x) is an element of H-s (T). We prove the local well-posedness of the fifth-order KdV equation for low regularity Sobolev initial data via the energy method. This paper follows almost same idea and argument as in [7]. Precisely, we use some conservation laws of the KdV Hamiltonians to observe the direction which the nonlinear solution evolves to. Besides, it is essential to use the short time X-s,X-b spaces to control the nonlinear terms due to high x low double right arrow high interaction component in the non-resonant nonlinear term. We also use the localized version of the modified energy in order to obtain the energy estimate. As an immediate result from a conservation law in the scaling sub-critical problem, we have the global well-posedness result in the energy space H-2. (C) 2016 Elsevier Inc. All rights reserved