Global well-posedness of korteweg-de vries equation in ...

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 H1/4(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 Hs (s > −3/4) solution which enable us to apply the I-method to extend it to a global solution.