Defocusing cubic nonlinear Schrödinger equation in two dimensions ($i{u}_t+\Delta u-|u{|}^{2}u=0$)

The two-dimensional defocusing cubic nonlinear Schrödinger equation is mass-critical: the scaling $u(t,x)\mapsto {\lambda }^{-1}u(t/{\lambda }^2,x/\lambda )$ preserves the $L^2$ norm. The defocusing sign makes its nonlinear energy positive.

The harmonic-oscillator energy space isThe spatial moment prevents mass from escaping to infinity.

The embedding $\Sigma \hookrightarrow L^2({\mathbb{R}}^d)$ is compact. Rellich compactness controls a fixed ball, whilecontrols the tail uniformly.

Critical points of the trapped defocusing cubic energysatisfy $-\Delta v+|x{|}^{2}v-\omega v+v^3=0$.

If $v$ solves the trapped stationary equation, the rescaled quadratic-phase ansatzsolves the free mass-critical equation when the scaling, chirp, and phase parameters satisfy the associated lens-transform system.

In two dimensions, the endpoint spacetime estimate and its dual include

If a global cubic Schrödinger solution has finite $L_{t,x}^4$ norm on a time ray, the dual Strichartz estimate makes its interaction representation Cauchy in $L^2$. It therefore converges to scattering data $u_{\infty }$, and $u(t)-e^{it\Delta }{u}_{\infty }\to 0$ in $L^2$.