Global Well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\mathbb{R})$

Abstract

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\mathbb{R})$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\mathbb{R})$. The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at $s=-3/4$ by constructing some special resolution spaces in order to avoid some 'logarithmic divergence' from the high-high interactions. Our local solution has almost the same properties as those for $H^s(s>-3/4)$ solution which enable us to apply the I-method to extend it to a global solution.

arXiv comment20 pages, 0 figure

References in corpus (2)

• Decay estimates for a class of wave equations
• Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Cited by in corpus (4)

• A para-differential renormalization technique for nonlinear dispersive equations
• The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces
• A higher dispersion KdV equation on the half-line
• Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$