Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 3 / 3

Under the mass-critical spatial scalingthe $L^2$ norm is invariant and the gradient norm is multiplied by ${\lambda }_n$. Therefore chooseFinite-time blowup and the Blowup alternative for the nonlinear Schrödinger equation imply $\Vert \nabla u(t_n){\Vert }_2\to \infty$, so ${\lambda }_n\to 0$.

The energy has scaling degree two:where the second equality uses Energy conservation for the nonlinear Schrödinger equation.

The profile decomposition modulo translations says that every bounded sequence $(w_n)$ in $H^1({\mathbb{R}}^d)$ has, after passage to a subsequence,where the translation parameters are asymptotically orthogonal,the squared $L^2$ and gradient norms decouple,and the remainder vanishes in every subcritical norm:

Part 3 and Mass conservation for the nonlinear Schrödinger equation giveThe last relation and the energy formula implywhere the final equality follows from the Pohozaev identity for the mass-critical NLS ground state. Thus $(v_n)$ is a minimizing sequence for the Weinstein functional with the same normalization as $Q$.

Apply the profile decomposition from part 4. The Sharp Gagliardo-Nirenberg inequality bounds each profile by the product of its gradient energy and its mass to the power $2/d$. Since the total mass is exactly $\Vert Q{\Vert }_2^2$, any split into two nonzero profiles would make the limiting inequality strict. Hence precisely one profile carries all the mass and gradient energy. The norm decouplings then make the remainder converge strongly to zero in $H^1$. For suitable translations $x_n$,and the Sobolev embedding theorem gives the required strong convergence in $L^{2+4/d}$. This is the compactness of a mass-critical minimizing sequence.