Global existence of small solutions to semilinear Schrödinger equations with gauge invariance (Q1913415)

Summary: We present a global existence theorem for semilinear Schrödinger equations \[ \partial_t u- i \Delta u= F(u, \nabla u),\quad \text{in} \quad (0, \infty)\times \mathbb{R}^n,\quad u(0, x)= u_0(x),\quad \text{in} \quad \mathbb{R}^N, \] satisfying gauge invariance. Combining the local existence results and a priori estimates, we construct global solutions with small initial data.