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

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.

For real $h\in \mathcal{D}({\mathbb{R}}^d)$, differentiation under the integral givesThe ground-state equation $\Delta Q-Q+Q^{1+4/d}=0$ and integration by parts reduce this to

In particular, along the amplitude direction $h=Q$ the derivative is $-\Vert Q{\Vert }_2^2<0$. Hence $u_0=(1+\delta )Q$ has negative energy for every sufficiently small $\delta >0$, whilewhen $\delta$ is chosen small enough. The ground state has finite variance, so negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation shows that the corresponding solution blows up in finite time.

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.