Global well-posedness of Korteweg-de Vries equation in H-3/4(R)

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 in H-3/4 by constructing some special resolution spaces ill 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 1-method to extend it to a global Solution. (C) 2009 Elsevier Masson SAS. All rights reserved.