Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 3

The NLS ground state is the positive radial solution ofIt is, up to phase, translation, and scaling, the optimizer of the Sharp Gagliardo-Nirenberg inequalityConsequentlyEquality in the sharp inequality occurs precisely for the symmetry orbit of $Q$.

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For every real $a$, the gauge transform $w_a=e^{ia\psi }w$ has the same $L^2$ norm as $w$. Since $\Vert w{\Vert }_2\le \Vert Q{\Vert }_2$, the variational bound from part 1 gives $E(w_a)\ge 0$. Expanding its gradient giveswhereThis quadratic polynomial is nonnegative for every $a\in \mathbb{R}$, so its discriminant is nonpositive. Thereforewhich is the desired estimate.

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If $\Vert u_0{\Vert }_2<\Vert Q{\Vert }_2$, the sharp variational estimate and the conservation laws giveThus the $H^1$ norm remains bounded on every finite time interval. The Blowup alternative for the nonlinear Schrödinger equation then prevents a finite endpoint of the maximal lifespan, so $T=\infty$.

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Let $(u_n)$ be bounded in $H_r^1({\mathbb{R}}^2)$. After passing to a subsequence it converges weakly in $H^1$, while the Rellich-Kondrachov compactness theorem gives strong $L^4$ convergence on every bounded ball. The Radial Sobolev inequality gives uniformlyThe same estimate applies to the weak limit. Choosing $R$ large and then using local compactness proves strong convergence in $L^4({\mathbb{R}}^2)$. Henceis compact.

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By the Blowup alternative for the nonlinear Schrödinger equation, choose $t_n\uparrow T$ with $\Vert \nabla u(t_n){\Vert }_2\to \infty$, and setThe mass-critical scaling gives

The sequence is bounded and radial in $H^1$, so part 4 yields, after extraction, weak $H^1$ and strong $L^4$ convergence to some $v$. The energy identity gives $\Vert v_n{\Vert }_4^4\to 2\Vert \nabla Q{\Vert }_2^2$. The Sharp Gagliardo-Nirenberg inequality and weak lower semicontinuity force equality in every limiting norm inequality. Hence $v_n\to v$ strongly in $H^1$, and the variational characterization identifiesfor some $\gamma \in \mathbb{R}$; radiality removes translations and the gradient normalization fixes the scale.

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Choose a smooth radial cutoff ${\theta }_R$ that vanishes on $|x|\le R$, equals one on $|x|\ge 2R$, and satisfies $|\nabla {\theta }_R|\le C/R$. The first localized virial identity and the estimate from part 2, applied to $w=u(t)$ and $\psi ={\theta }_R$, givebecause the mass and energy are conserved.

For fixed $R$, the strong profile convergence from part 5 and ${\lambda }_n\to 0$ implyIntegrating the flux estimate from $t$ to $t_n$ and then letting $n\to \infty$ yieldsuniformly for $0\le t<T$. Taking $R$ sufficiently large proves the claim.

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Change variables $x={\lambda }_{n}y$ and use the profiles from part 5:Since $v_n\to Q{e}^{i\gamma }$ in $L^2$, the difference made by replacing $|v_n{|}^2$ with $|Q{|}^2$ tends to zero against the bounded function $\varphi ({\lambda }_n\cdot )$. For each fixed $y$, continuity gives $\varphi ({\lambda }_{n}y)\to \varphi (0)$, and dominated convergence then givesThus the mass measures $|u(t_n,x){|}^{2}dx$ converge weakly to the Dirac delta function $\Vert Q{\Vert }_2^2{\delta }_0$.

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