Global Well-Posedness for the Fifth-Order KdV Equation in \(H^{-1}(\pmb {\mathbb {R}})\)

Abstract

We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in \(H^{-1}(\mathbb {R})\). Global well-posedness in \(L^2({\mathbb {R}})\) was shown previously in [8] using the method of commuting flows. Since this method is insensitive to the ambient geometry, it cannot go beyond the sharp \( L^2\) threshold for the torus demonstrated in [3]. To prove our result, we introduce a new strategy that integrates dispersive effects into the method of commuting flows.