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

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

2008-10-20

arxiv.org/abs/0810.3445v3

J. Math. Pures Appl. 91 (2009) 583-597

Mathematics

Analysis of PDEs

20 pages, 0 figure

Scientific paper

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\R)$. 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.

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