ﻻ يوجد ملخص باللغة العربية
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(mathbb{R}/mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $M^{2,p}_{s}(mathbb{R})$ for $s ge frac14$ and $2leq p < infty$. For $s < frac 14$,
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 contract
We prove that the cubic nonlinear Schrodinger equation (both focusing and defocusing) is globally well-posed in $H^s(mathbb R)$ for any regularity $s>-frac12$. Well-posedness has long been known for $sgeq 0$, see [51], but not previously for any $s<0
We prove global well-posedness for the $3D$ radial defocusing cubic wave equation with data in $H^{s} times H^{s-1}$, $1>s>{7/10}$.