Nonlinear Schrödinger blowup analysis

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.

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.

For the nonlinear Schrödinger equationwith real-valued $F$, a sufficiently regular, decaying solution with finite L2 norm conservesDifferentiating gives $M^{\prime }(t)=2\mathrm{Re}\int u_t\overline{u}$. Substitute $u_t=i\Delta u+iF(|u{|}^2)u$. Integration by parts makes $\int \Delta u\overline{u}=-\int |\nabla u{|}^2$ real, and $\int F(|u{|}^2)|u{|}^2$ is real as well, so multiplication by $i$ gives zero real part.

The real coefficient is essential to this calculation; phase covariance alone does not imply mass conservation. For an $H^1$ solution obtained by a well-posed approximation theory, the invariant extends from smooth solutions by continuity in the L2 norm.^[ref]

For the focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$,the energy functional isFor a sufficiently smooth solution with decay permitting differentiation under the integral sign and integration by parts, it is constant in time. Indeed,The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by $i$. The focusing sign makes the nonlinear contribution negative, so conserved energy alone need not bound the L2 norm of the gradient.^[ref]

For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in Sobolev space $H^1$ controls the nonlinear term through a suitable Sobolev embedding only in the applicable exponent range, or with additional integrability. An arbitrary weak solution must not be assumed to satisfy the identity solely because the smooth equation does.

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]

For a sufficiently regular solution of the mass-critical focusing nonlinear Schrödinger equation with finite variance, the variance and its first derivative areA second differentiation and integration by parts give the virial identityThese constants use Laplacian coefficient one in the evolution equation. Energy conservation makes the second derivative constant. If the energy is negative, the resulting quadratic expression for $I(t)$ eventually becomes negative, contradicting its definition as a nonnegative integral; this proves negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation under the finite-variance hypotheses.

The weighted moment condition is additional to an H1 norm bound. A localized virial identity replaces the unbounded weight by a smooth cutoff when only tail control is available.

For $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, a smooth real weight $\chi$ defines the localized mass $I_{\chi }(t)=\int \chi |u{|}^2$. For a sufficiently regular decaying solution and a weight with the required integrability,andThese formulas follow by substituting the nonlinear Schrödinger equation and integration by parts. The Hessian controls the gradient contribution; the fourth derivative term records variation of the cutoff.

Choose ${\chi }_R(x)=R^2\chi (x/R)$, with $\chi (x)=|x{|}^2$ near zero and constant outside a larger ball. The interior terms reproduce the virial identity, while the remaining terms are supported in the transition region or controlled by spatial tails. A bounded weight makes $I_{{\chi }_R}$ meaningful for finite mass without requiring finite variance. A blowup proof still needs estimates that control those tail errors.

For a finite-variance solution of the mass-critical focusing nonlinear Schrödinger equation, the virial identity givesIf $E(u_0)<0$, the right-hand side is a negative constant. The nonnegative variance would then become negative in finite time if the solution remained regular, so the solution blows up in finite time.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For $p=2+4/d$ and nonzero $u\in H^1({\mathbb{R}}^d)$, the mass-critical Weinstein functional isIt is invariant under multiplication by a nonzero scalar, spatial dilation, translation, and constant phase. Its positive minimizers are rescalings and translations of the NLS ground state.

For nonzero $u\in H^1({\mathbb{R}}^d)$, the Weinstein functional is invariant under amplitude multiplication and spatial dilation. Apply symmetric decreasing rearrangement to a minimizing sequence, then use those two invariances to arrangewith nonnegative radial, radially nonincreasing $u_n$. Rearrangement preserves the relevant Lp norms and does not increase the gradient L2 norm.

These monotone radial profiles have uniform pointwise bounds $|u_n(x)|\le C_d|x{|}^{-d/2}$, since their value on a ball is bounded by its average squared mass. For every $p>2$, this givesLocal Rellich-Kondrachov compactness theorem and this tail bound yield strong convergence in $L^{2+4/d}$ after a subsequence; this exponent is below $2^{\ast }=2d/(d-2)$ for $d\ge 3$, the upper range in the Sobolev embedding theorem; put $2^{\ast }=\infty$ for $d=1,2$. The argument also covers $d=1$ because monotone rearrangement, rather than radial symmetry alone, controls the tails.

The Gagliardo-Nirenberg interpolation inequality bounds the infimum of the Weinstein functional away from zero. The strong nonlinear-norm limit is therefore nonzero. Weak lower semicontinuity of the two normalized quadratic norms shows that the limit attains the infimum, giving a nonnegative minimizer.

Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

In dimension $d$, the mass-critical sharp inequality isFor nonzero $u$, equality holds exactly on the amplitude, phase, translation, and spatial-scaling orbit of the NLS ground state $Q$; $u=0$ also gives equality. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

The sharp inequality impliesThus energy controls the gradient whenever the mass is strictly below the mass of the NLS ground state.

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence, after passage to a subsequence, some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.

Every bounded sequence $(u_n)$ in $H^1({\mathbb{R}}^d)$ has, after passage to a subsequence, a decompositionwhere $|x_n^j-x_n^k|\to \infty$ for $j\ne k$, the $L^2$ and gradient norms decouple asymptotically, andfor every $2<p<2^{\ast }$, where $2^{\ast }=2d/(d-2)$ for $d\ge 3$ and $2^{\ast }=\infty$ for $d=1,2$.

This local phase change is a gauge transformation of the wave function. For real $a$ and a real continuously differentiable $\psi$, multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

For $\eta >0$, the trapped focusing cubic ground-state equation in two dimensions isA nonzero positive solution follows from minimizing the harmonic-oscillator quadratic form subject to a fixed $L^4$ norm and then rescaling by the positive Lagrange multiplier.

If $P_{\eta }$ solves the trapped focusing cubic ground-state equation, the ansatzsolves the two-dimensional mass-critical focusing nonlinear Schrödinger equation when

Let $\eta >0$ and let $P_{\eta }\in L^4({\mathbb{R}}^2)$ solve the trapped focusing cubic ground-state equation. In the Focusing Schrödinger lens ansatz, the initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ giveIndeed $b=-\lambda {\lambda }_t$ and ${\lambda }^2{b}_t+b^2=-\eta$. The positive $\eta$ keeps the scale nonzero for every real time.

A change of variables $y=x/\lambda (t)$ proves the critical spacetime bound exactly:Thus the solution expands at large times and has finite $L^4$ spacetime norm, rather than merely pointwise decay. This is the norm used by scattering from a finite Strichartz norm.^[ref]