Global Well-posedness of Korteweg-de Vries equation in H−3/4(R)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 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.Comment: 20 pages, 0 figur

Global dynamics below the ground state energy for the Zakharov system in the 3D radial case

We consider the global dynamics below the ground state energy for the Zakharov system in the 3D radial case. We obtain dichotomy between the scattering and the growup.Comment: 28 page