On nonlinear Schrödinger equations, II.H ^S-solutions and unconditional well-posedness

Abstract

We consider the nonlinear Schrödinger equation (NLS) (see below) with a general “potential”F(u), for which there are in general no conservation laws. The main assumption onF(u) is a growth rateO(|u| ^k) for large |u|, in addition to some smoothness depending on the problem considered. A uniqueness theorem is proved with minimal smoothness assumption onF andu, which is useful in eliminating the “auxiliary conditions” in many cases. A new local existence theorem forH ^S-solutions is proved using an auxiliary space of Lebesgue type (rather than Besov type); here the main assumption is thatk≤1+4/(m−2s) ifs<m/2,k<∞ ifs=m/2 (no assumption ifs>m/2). Moreover, a general existence theorem is proved for globalH ^S-solutions with small initial data, under the main additional condition thatF(u)=O(|u|^1+4/m) for small |u|; in particularF(u) need not be (quasi-) homogeneous or in the critical case. The results are valid for alls≥0 ifm≤6; there are some restrictions ifm≥7 and ifF(u) isnot a polynomial inu and$\bar u$.