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For the mass-critical focusing nonlinear Schrödinger equation, let $u\in C([0,T_{+});H^1({\mathbb{R}}^d))$ be the maximal forward solution in the standard local $H^1$ theory. Either $T_{+}=\infty$, orBecause Mass conservation for the nonlinear Schrödinger equation fixes $\Vert u(t){\Vert }_2$, this is equivalent to divergence of $\Vert \nabla u(t){\Vert }_2$.

The proof uses a local existence time bounded below for data in a bounded H1 norm ball. If a sequence $t_n\uparrow T_{+}$ had uniformly bounded H1 norms, restart the equation at $t_n$ with that common time interval. Uniqueness joins the restarted solution to the old one and extends it beyond $T_{+}$, contradicting maximality.

The dependence of lifespan on the norm is a hypothesis of this argument. Other critical solution theories may use a spacetime control norm instead, so the stated H1 norm conclusion is attached here to the mass-critical equation and its $H^1$-subcritical local theory.^[ref]

The mass-critical focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$ isIts scaling $u(t,x)\mapsto {\lambda }^{d/2}u({\lambda }^{2}t,\lambda x)$ preserves the Lebesgue space norm $\Vert u{\Vert }_{L^2}$, which is the conserved mass.

Nonlinear Schrödinger blowup analysis asks whether a solution remains in its local well-posedness class or concentrates so strongly that continuation fails in finite time. For the mass-critical focusing nonlinear Schrödinger equation in the standard $H^1$ theory, finite-time failure forces the gradient L2 norm to diverge, even though Mass conservation for the nonlinear Schrödinger equation keeps the total squared L2 norm constant.

Concentration and total size are different. The mass-preserving scaling $u_{\lambda }(x)={\lambda }^{d/2}u(\lambda x)$ leaves $\Vert u_{\lambda }{\Vert }_2=\Vert u{\Vert }_2$ while $\Vert \nabla u_{\lambda }{\Vert }_2=\lambda \Vert \nabla u{\Vert }_2$. Thus a fixed amount of mass can occupy smaller spatial scales with larger gradients.

The Sharp Gagliardo-Nirenberg inequality gives coercivity below the NLS ground-state mass, while a virial identity can force finite-time failure for suitable negative-energy, finite-variance data. Near the threshold, profile decomposition modulo translations separates a concentrating profile from dispersive remainder. These arguments track the structure of a solution rather than merely the maximum of a plotted wave.

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.

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.

Suppose that $T=\infty$. Since $s_c<1$ and $d\ge 3$,The Radial Sobolev inequality and Mass conservation for the nonlinear Schrödinger equation giveBecause $(p-1)/2<2$, Young inequality and a sufficiently large fixed $R$ absorb this term into the left side of the estimate from part 5. The annular term is at most $C(u_0)R^{-2}$. Using the uniform lower bound $\Vert \nabla u(t){\Vert }_2\ge c(u_0)$ from part 2 and enlarging $R$ once more yields

Since $I_{{\psi }_R}^{\prime }=2V_{{\psi }_R}$, two integrations giveThe right-hand side is negative for large $t$, contradicting $I_{{\psi }_R}(t)=\int {\psi }_R|u{|}^2\ge 0$. Hence the maximal forward lifespan is finite: $T<\infty$.

Interpolate between $L^2$ and ${\dot{H}}^1$ and then use the preceding Sobolev embedding theorem:Mass conservation for the nonlinear Schrödinger equation fixes $\Vert u(t){\Vert }_2=\Vert u_0{\Vert }_2$, while Energy conservation for the nonlinear Schrödinger equation and $E(u_0)\le 0$ implySince $\sigma (p+1)>2$, rearrangement givesuniformly on the maximal lifespan.