 Solution

Both Mass conservation for the nonlinear Schrödinger equation and Energy conservation for the nonlinear Schrödinger equation hold throughout the maximal lifespan. Since $\Vert u_0{\Vert }_2<\Vert Q{\Vert }_2$, the Sharp Gagliardo-Nirenberg inequality gives the uniform estimateThus the gradient norm stays bounded. The Blowup alternative for the nonlinear Schrödinger equation rules out a finite endpoint of the lifespan in either time direction, so the solution is global.