Global solution for the $3D$ quadratic Schr\"odinger equation of $Q(u, \bar{u})$ type

Abstract

We study a class of $3D$ quadratic Schr\"odinger equations as follows, $(\partial_t -i \Delta) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah, the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1\times 1 \rightarrow 0$ type interaction (the size of input frequency is "$1$" and the size of output frequency is "$0$"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1\times 0\rightarrow 1$ type interaction. The issue of strong $1\times 1\rightarrow 0$ type interaction makes the global existence problem very delicate. In this paper, we show that, as long as there are "$\epsilon$" derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i \Delta)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi. Instead of using vector fields, we consider this problem purely in Fourier space.