Infinite Gain of Regularity for Dispersive Evolution Equations

Abstract

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 R ⁿ, 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 L ² with a weight function 1 + e ^σx . 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.

Supported in part by NSF Grants DMS 87-22331, DMS 89-20624 and AFOSR Grant DAAL-3-86-0074. In addition, the first author is supported by an Alfred P. Sloan Foundation Fellowship.