Sharp Global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\T$

Abstract

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{1/4}(\mathbb{R})$ endpoint for mKdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

submitted to JAMS

References in corpus (3)

• Global well-posedness for Schrödinger equations with derivative
• Global well-posedness for KdV in Sobolev Spaces of negative index
• Local and global well-posedness of wave maps on ${\mathbb{R}}^{1+1}$ for rough data

Cited by in corpus (18)

• Nonlinear stability of mKdV breathers
• A para-differential renormalization technique for nonlinear dispersive equations
• Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$
• Invariance of the white noise for KdV
• Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems
• Decoupled and unidirectional asymptotic models for the propagation of internal waves
• A higher dispersion KdV equation on the half-line
• Global rough solutions to the critical generalized KdV equation
• Expansion shock waves in regularised shallow water theory
• The low regularity global solutions for the critical generalized KdV equation
• Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$
• Loomis-Whitney-type inequalities and low regularity well-posedness of the periodic Zakharov-Kuznetsov equation
• Nearly linear dynamics of nonlinear dispersive waves
• Local and global well-posedness of dispersion generalized Benjamin-Ono equations on the circle
• Non-existence of solutions for the periodic cubic NLS below $L^2$
• Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces
• Generalized KdV equation subject to a stochastic perturbation
• Global control aspects for long waves in nonlinear dispersive media