Growth of the $H^s$-norm for the modified KdV equation

Abstract

We study the growth of the $H^s$-norm for solutions of the modified Korteweg-de Vries equation, corresponding to data in $H^s$ for noninteger values of $s$ in the case where global solutions exist. The presence of conservation laws and the local existence theory permit us to obtain upper "polynomial" bounds, for the $H^s$-norm of these solutions, with power depending on the distance to the closest integer to $s$.