Infinite Gain of Regularity for Dispersive Evolution Equations

We say that an evolution equation has an infinite gain of regularity if its solutions are C∞ for t > 0, for initial data with only a finite amount of smoothness. An equation need not be hypoelliptic for this to happen provided the initial data vanish at spatial infinity. For instance, for the Schrödinger equation in Rn, this is clear from the explicit solution formula if the initial data decay faster than any polynomial. For the Korteweg-deVries equation, T. Kato [4], motivated by work of A. Cohen, showed that the solutions are C∞ for any data in L2 with a weight function 1 + eσx. While the proof of Kato appears to depend on special a priori estimates, some of its mystery has been resolved by the recent results of finite regularity for various other nonlinear dispersive equations due to Constantin and Saut [1], Ponce [5] and others [3]. However, all of them require growth conditions on the nonlinear term.