Further regularity of solutions for almost cubic NLS equation

Abstract

This thesis consists of two major parts. In the rst one, we try to give the preliminary local well-posedness results for the ACNLS, and L2 -H1 regularity result which is an easy and straightforward consequence of the equation, since the norm of the gradientof a function can be estimated by di erence quotients. In the second part, we prove some regularity results for ACNLS. First, we prove Hs local well-posedness, where the continuous dependence is weakened; and an improvement of it by obtaining the continuous dependence with an additional condition. At the end, we prove local Xs;b local existence result using Banach xed point theorem, where the interval of existence is not taken to be maximal. The interval depends closely on the arguments of the high-low frequency decomposition.