Global wellposedness of KdV in H−1(T,R)

Abstract

By the inverse method we show that the Korteweg–de Vries equation (KdV) $\partial {}_{t}v(x,t)=-{\partial }_x^{3}v(x,t)+6v(x,t)\partial {}_{x}v(x,t)$ $(x\in \mathbb{T},t\in \mathbb{R})$ is globally (in time) wellposed in the Sobolev space of distributions $H^{\beta }(\mathbb{T},\mathbb{R})$ for any $\beta \ge -1$