Nonlinear Schrödinger blowup analysis

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

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 sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

A virial identity describes the time derivatives of a spatial moment of a solution to a dispersive equation. For the mass-critical nonlinear Schrödinger equation, the second derivative of the variance is a constant multiple of the conserved energy.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable 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 $p=2+4/d$, 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.

Normalize a minimizing sequence so that its $L^2$ norm and gradient norm are fixed. Symmetric decreasing rearrangement does not increase the gradient norm and preserves every $L^p$ norm, while radial compactness prevents translation loss. A weakly convergent subsequence therefore converges strongly in the nonlinear $L^{2+4/d}$ norm and yields 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 isand equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$. 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 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 $2<p<2^{\ast }$.

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

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

The initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ selectThe scale grows linearly at infinity, yielding a finite critical spacetime norm for the corresponding lens solution.