Virial Identities, Blowup, and the Soliton-Stability Outlook

Intuition Beginner

The previous units handed you two opposite fates for a focusing wave. A small wave spreads out and lives forever; a wave with negative energy and finite spread collapses to a spike in finite time. What was missing was a single ruler that watches the collapse happen and proves it must finish in finite time. That ruler is the average width of the wave, the typical distance of its mass from the origin.

Track that width as time runs. There is an exact rule for how its acceleration is set by the energy and the self-pull of the wave. In the collapsing case the rule forces the width to curve downward like a thrown ball returning to earth: its acceleration is a fixed negative number. A width that always accelerates downward must reach zero in finite time. But a width is a distance and can never be negative. The only way out is that the wave stops existing before the width would have gone negative. That is the blowup: the construction runs out of room before it runs out of time.

Sitting exactly between spreading and collapse is one special standing wave that neither spreads nor collapses. It just rotates in place forever, keeping its shape. This is the soliton, and the amount of mass it carries is the exact tipping point. Put in less mass than the soliton and the wave is safe; put in more and collapse becomes possible. The soliton is the knife-edge, and the last question is whether that edge is stable: nudge the soliton a little, and does it stay near its shape, or does the nudge grow?

The one-sentence takeaway: an exact rule for the acceleration of the wave's width turns negative energy into guaranteed finite-time collapse, the soliton marks the precise mass threshold between safety and collapse, and the closing question is whether that threshold shape is stable to small nudges.

Visual Beginner

Picture a ball thrown straight up with a downward push that never lets up — not just gravity, but a constant negative acceleration baked in. Its height is the width of the wave. Because the acceleration is a fixed negative number, the height traces a downward parabola, and a downward parabola always comes back to zero in finite time. The catch is that this height is a width, and a width cannot pass below zero. So the wave must cease to exist at the moment the parabola would have crossed zero. That crossing time is the blowup time.

The companion picture is the three-panel mass sorter on the right. The soliton in the middle panel is the special standing wave: drop it in and it spins in place forever without changing its profile, the one shape the equation leaves untouched. Its mass is the dividing mark. Below that mark, in the top panel, every wave spreads and survives. Above it, in the bottom panel, the door to collapse is open. The whole drama of the focusing equation is reading off which side of that single mark your starting wave lands on.

Worked example Beginner

We watch the width of a collapsing wave and read off, with concrete numbers, that it must hit zero in finite time.

Step 1. Name the width. Let $V$ be the average squared distance of the wave's mass from the origin — a single positive number measuring how spread out the wave is at a given moment. A large $V$ means a wide, gentle wave; a small $V$ means a narrow, concentrated one. Collapse means $V$ heading toward zero.

Step 2. Use the acceleration rule. In the collapsing case the equation pins the acceleration of $V$ to a fixed negative number set by the energy: the acceleration equals $16$ times the energy. Take a wave with energy $E=-1$ (negative, the dangerous case). Then the acceleration of $V$ is $16\times (-1)=-16$, the same at every instant.

Step 3. Track the width like a thrown ball. Start with width $V=100$ and an initial rate of change $0$ (the wave momentarily at rest). With constant acceleration $-16$, the width follows $V(t)=100-8t^2$ (the $8$ is half the acceleration's size). At $t=0$ the width is $100$; the downward curve eats into it as time grows.

Step 4. Find when it would hit zero. Set $100-8t^2=0$. Then $t^2=100/8=12.5$, so $t=3.54$ (about). The width would reach zero at roughly $t=3.54$.

Step 5. Read the consequence. A width is a distance and cannot be negative, so the wave cannot survive to $t=3.54$. The solution must break down at or before that time. The negative energy, through the fixed negative acceleration, has forced a finite-time collapse — and given a concrete clock for it.

What this tells us: negative energy makes the width accelerate downward at a fixed rate, the width then traces a downward parabola, and since a width can never go negative, the wave must blow up by the time that parabola would cross zero. The energy alone hands you both the certainty of collapse and an upper bound on when it happens.

Check your understanding Beginner

Formal definition Intermediate+

Fix $n\ge 1$, a power $p>1$, and the focusing sign $\mu =-1$ in the NLS of 02.21.03, $$i{\partial }_{t}u+\Delta u=-|u{|}^{p-1}u,u(0)=u_0,$$ with the conserved mass $M[u]=\Vert u{\Vert }_{L^2}^2$ and energy $E[u]=\frac{1}{2}\Vert \nabla u{\Vert }_{L^2}^2-\frac{1}{p+1}\Vert u{\Vert }_{L^{p+1}}^{p+1}$ of 02.21.04. The objects below organise the dichotomy between global existence and finite-time blowup.

Definition (variance / virial functional). For $u(t)\in H^1$ with $|x|u(t)\in L^2$ (finite variance), the variance is $$V(t)={\int }_{{\mathbb{R}}^n}|x{|}^2|u(t,x){|}^{2}dx\ge 0.$$ It measures the spatial spread of the mass about the origin.

Definition (virial identity). Along a sufficiently regular finite-variance solution, the variance is twice differentiable with $$V^{\prime \prime }(t)=8{\int }_{{\mathbb{R}}^n}|\nabla u{|}^{2}dx+\frac{4n\mu (p-1)}{p+1}{\int }_{{\mathbb{R}}^n}|u{|}^{p+1}dx,$$ the virial (Morawetz-type) identity. For $\mu =-1$ at or above the $L^2$-critical power $p\ge 1+4/n$ this rearranges to $V^{\prime \prime }(t)\le 16E[u_0]$.

Definition (pseudoconformal symmetry). At the $L^2$-critical power $p=1+4/n$ the NLS is invariant under the pseudoconformal transformation: if $u(t,x)$ solves NLS on $t>0$, so does $$(\mathcal{C}u)(t,x)=\frac{1}{|t{|}^{n/2}}\overline{u\left(\frac{1}{t},\frac{x}{t}\right)}{e}^{i|x{|}^2/(4t)}.$$ Applied to the soliton it produces an explicit blowup solution concentrating at $t\to 0$.

Definition (ground-state soliton). A standing wave is a solution $u(t,x)=e^{i\omega t}\varphi (x)$ with $\omega >0$ and $\varphi \in H^1$ real, positive, radial; substituting into NLS gives the soliton profile equation $$-\Delta \varphi +\omega \varphi ={\varphi }^p,\varphi >0.$$ The ground state $Q$ is the case $\omega =1$: the unique (up to translation) positive radial $H^1$ solution of $-\Delta Q+Q=Q^p$, equivalently the optimiser of the Gagliardo-Nirenberg inequality of 02.16.01. The general standing wave is recovered by scaling ${\varphi }_{\omega }(x)={\omega }^{1/(p-1)}Q(\sqrt{\omega }x)$, and the standing wave of NLS is $e^{i\omega t}{\varphi }_{\omega }$.

Definition (sharp Gagliardo-Nirenberg constant and mass threshold). At the $L^2$-critical power $p=1+4/n$ the sharp Gagliardo-Nirenberg inequality reads $$\Vert u{\Vert }_{L^{p+1}}^{p+1}\le C_{\mathrm{GN}}\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{4/n},C_{\mathrm{GN}}=\frac{p+1}{2\Vert Q{\Vert }_{L^2}^{4/n}},$$ with equality exactly at $u=Q$ (up to symmetries). The constant is fixed by the ground-state mass $\Vert Q{\Vert }_{L^2}^2$, and that mass is the threshold: $\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$ forces $E[u]\ge 0$ and global existence, while $\Vert u_0{\Vert }_{L^2}\ge \Vert Q{\Vert }_{L^2}$ permits blowup.

Definition (orbital stability). Write the orbit of $Q$ as $\mathcal{O}=\{e^{i\theta }Q(\cdot -y):\theta \in \mathbb{R},y\in {\mathbb{R}}^n\}$, the family generated by the phase and translation symmetries. The standing wave $e^{i\omega t}{\varphi }_{\omega }$ is orbitally stable if for every $\epsilon >0$ there is $\delta >0$ such that ${\inf }_{v\in {\mathcal{O}}_{\omega }}\Vert u_0-v{\Vert }_{H^1}<\delta$ implies ${\sup }_t{\inf }_{v\in {\mathcal{O}}_{\omega }}\Vert u(t)-v{\Vert }_{H^1}<\epsilon$; orbitally unstable otherwise. Stability is measured modulo the symmetry orbit because the symmetries move the soliton without changing its shape.

Counterexamples to common slips Intermediate+

• The virial identity needs finite variance, not merely $H^1$. The computation differentiates $\int |x{|}^2|u{|}^2$, which is finite only when $|x|u_0\in L^2$. Glassey's theorem is stated for finite-variance data; the variance-free blowup of general $H^1$ negative-energy data is a strictly harder theorem (Ogawa-Tsutsumi) using a localised virial weight, not the bare identity. Asserting Glassey blowup for arbitrary $H^1$ data without the variance hypothesis or a localisation is the standard gap.

• Negative energy is sufficient but not necessary for blowup. $E[u_0]<0$ forces $V^{\prime \prime }\le 16E[u_0]<0$ and hence collapse, but positive-energy data can also blow up (e.g. above the ground state with $\Vert \nabla u_0{\Vert }_{L^2}$ large). Reading "blowup $⟺E<0$" off Glassey's theorem inverts a one-way implication.

• Mass below $\Vert Q{\Vert }_{L^2}$ gives global existence only at the $L^2$-critical power. The threshold $\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$ is the mass-critical statement. At other powers the relevant threshold is an energy/kinetic-energy condition against the energy-critical ground state $W$ (Kenig-Merle), not the mass of $Q$. Transporting the mass threshold to the energy-critical problem confuses two distinct ground states.

• Orbital stability is not asymptotic stability. Orbital stability says the solution stays near the orbit of $Q$; it does not say the solution converges to a single soliton. The latter (asymptotic stability) requires dispersive decay of the radiation and is a separate, finer theorem. Equating the two overstates what the Weinstein/GSS energy method delivers.

Key theorem with proof Intermediate+

Theorem (virial identity and Glassey blowup). Let $\mu =-1$, $p\ge 1+4/n$, and $u_0\in H^1({\mathbb{R}}^n)$ with $|x|u_0\in L^2$ and $E[u_0]<0$. Let $u$ be the maximal-lifespan $H^1$ solution of NLS from 02.21.03 on $(-T_{-},T_{+})$. Then the variance $V(t)=\int |x{|}^2|u{|}^2$ satisfies the virial identity, $V^{\prime \prime }(t)\le 16E[u_0]<0$, and consequently $T_{+}<\infty$ and $T_{-}<\infty$: the solution blows up in finite time both forward and backward.

Proof. Step 1 (first derivative). For smooth, finite-variance, decaying $u$ (the general $H^1$ finite-variance case follows by the regularise-conserve-pass-to-limit argument of 02.21.04, using persistence of regularity 02.21.03), differentiate $V(t)=\int |x{|}^2|u{|}^2$ and substitute ${\partial }_{t}u=i\Delta u+i|u{|}^{p-1}u$ (the $\mu =-1$ equation): $$V^{\prime }(t)=\int |x{|}^2{\partial }_t|u{|}^2=2\mathrm{Re}\int |x{|}^2\bar{u}{\partial }_{t}u=2\mathrm{Re}\int |x{|}^2\bar{u}(i\Delta u+i|u{|}^{p-1}u).$$ The nonlinear term contributes $2\mathrm{Re}i\int |x{|}^2|u{|}^{p+1}=0$ (the integral is real). For the linear term, integrate by parts: $2\mathrm{Re}i\int |x{|}^2\bar{u}\Delta u=-2\mathrm{Re}i\int \nabla (|x{|}^2\bar{u})\cdot \nabla u=-4\mathrm{Im}\int (x\cdot \nabla u)\bar{u}$ after expanding $\nabla |x{|}^2=2x$ and discarding the real $\int |x{|}^2|\nabla u{|}^2$ piece. Thus $V^{\prime }(t)=4\mathrm{Im}\int \bar{u}(x\cdot \nabla u)dx$.

Step 2 (second derivative). Differentiate $V^{\prime }(t)=4\mathrm{Im}\int \bar{u}(x\cdot \nabla u)$ once more, again using the equation and integrating by parts. The Schrödinger flow turns the dilation generator $x\cdot \nabla$ into the commutator with $\Delta$, and a standard computation (the Morawetz/virial computation) yields $$V^{\prime \prime }(t)=8\int |\nabla u{|}^2-\frac{4n(p-1)}{p+1}\int |u{|}^{p+1},$$ the virial identity at $\mu =-1$.

Step 3 (energy bookkeeping). Recall $E[u_0]=\frac{1}{2}\int |\nabla u{|}^2-\frac{1}{p+1}\int |u{|}^{p+1}$, so $8\int |\nabla u{|}^2=16E[u_0]+\frac{8}{p+1}\int |u{|}^{p+1}$. Substituting, $$V^{\prime \prime }(t)=16E[u_0]+(\frac{8}{p+1}-\frac{4n(p-1)}{p+1})\int |u{|}^{p+1}=16E[u_0]-\frac{4(n(p-1)-2)}{p+1}\int |u{|}^{p+1}.$$ The coefficient $n(p-1)-2\ge 0$ precisely when $p\ge 1+2/n$, and in particular for $p\ge 1+4/n$ (the $L^2$-critical-and-above range). Hence the second term is $\le 0$ and $$V^{\prime \prime }(t)\le 16E[u_0].$$

Step 4 (blowup). With $E[u_0]<0$, $V^{\prime \prime }(t)\le 16E[u_0]=:-c$ with $c>0$ for all $t$ in the lifespan. Integrating twice, $V(t)\le V(0)+V^{\prime }(0)t-\frac{c}{2}{t}^2$, a downward parabola in $t$. As $t$ increases this expression becomes negative at some finite $t_{\ast }\le \frac{V^{\prime }(0)+\sqrt{V^{\prime }(0{)}^2+2cV(0)}}{c}$. But $V(t)\ge 0$ always, so the solution cannot exist up to $t_{\ast }$: $T_{+}\le t_{\ast }<\infty$. The same argument run backward gives $T_{-}<\infty$. $\square$

Bridge. The virial identity is the foundational reason the focusing problem refuses the soft global theory of 02.21.04: it builds toward the blowup half of the dichotomy, and it appears again in the Kenig-Merle rigidity argument where a localised virial monotonicity rules out the minimal-energy critical element. This is exactly the contrapositive of the defocusing coercivity of 02.21.04 — there the energy bounded the gradient from above and forbade blowup, here the same energy, with the opposite sign, drives the variance to zero. The convexity argument generalises verbatim to the $L^2$-critical pseudoconformal blowup, and putting these together the focusing dichotomy is dual to the sign of $\mu$ read through the virial coefficient: the bridge is the sharp Gagliardo-Nirenberg inequality of 02.16.01, whose optimiser $Q$ is the exact threshold separating the variance-decreasing regime from the variance-spreading one.

Exercises Intermediate+

Advanced results Master

Theorem (sharp Gagliardo-Nirenberg constant; Weinstein 1983). At the $L^2$-critical power $p=1+4/n$, the best constant in $\Vert u{\Vert }_{L^{p+1}}^{p+1}\le C\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{4/n}$ is $C_{\mathrm{GN}}=(p+1)/(2\Vert Q{\Vert }_{L^2}^{4/n})$, attained exactly at $u=Q$ and its symmetry images, where $Q$ is the ground state of $-\Delta Q+Q=Q^p$ ^{[Weinstein 1983]}. The variational problem $\inf \{\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{4/n}/\Vert u{\Vert }_{L^{p+1}}^{p+1}\}$ is solved by concentration-compactness (the radial Strauss compactness of $H_{\mathrm{rad}}^1\hookrightarrow L^{p+1}$), and the Euler-Lagrange equation of the optimiser, after scaling out the multipliers, is exactly the ground-state equation. The consequence is the mass threshold: $E[u_0]\ge \frac{1}{2}(1-(\Vert u_0{\Vert }_{L^2}/\Vert Q{\Vert }_{L^2}{)}^{4/n})\Vert \nabla u_0{\Vert }_{L^2}^2$, so $\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$ forces $E\ge 0$ and, with mass conservation, a uniform ${\dot{H}}^1$ bound and global existence.

Theorem (uniqueness and non-degeneracy of the ground state; Kwong 1989). For $1<p<1+4/(n-2{)}_{+}$ the positive radial decaying solution $Q$ of $-\Delta Q+Q=Q^p$ is unique up to translation, and the linearised operator about it, $L_{+}=-\Delta +1-p{Q}^{p-1}$ acting on radial perturbations, is non-degenerate: $\ker L_{+}=\mathrm{span}\{{\partial }_{x_1}Q,\dots ,{\partial }_{x_n}Q\}$, exactly the translation directions, while $L_{-}=-\Delta +1-Q^{p-1}$ has kernel $\mathrm{span}\{Q\}$, the phase direction. This non-degeneracy is the spectral hypothesis the Grillakis-Shatah-Strauss and Weinstein stability theory requires, and it underlies the rigidity step of the Kenig-Merle program.

Theorem (orbital stability/instability dichotomy; Weinstein 1985, Grillakis-Shatah-Strauss 1987). The NLS ground-state standing wave $e^{i\omega t}{\varphi }_{\omega }$ is orbitally stable in $H^1$ if and only if $\frac{d}{d\omega }M[{\varphi }_{\omega }]>0$, equivalently $p<1+4/n$; it is orbitally unstable if $p\ge 1+4/n$ ^{[Weinstein 1985] [Grillakis-Shatah-Strauss 1987]}. Stability is proved by a Lyapunov functional: $E+\omega M$ has ${\varphi }_{\omega }$ as a constrained critical point, the second variation $L=\mathrm{diag}(L_{+},L_{-})$ is positive on the codimension-two subspace orthogonal to the symmetry directions when $d^{\prime \prime }(\omega )>0$, and a modulation argument tracking the phase and translation parameters confines the solution to a tube around the orbit. Instability for $p\ge 1+4/n$ is Berestycki-Cazenave's contradiction with the virial blowup: a perturbation lowering the energy below $E[{\varphi }_{\omega }]$ at fixed mass triggers the convexity argument and the solution leaves every tube ^{[Berestycki-Cazenave 1981]}.

Theorem (closing outlook: the focusing dichotomy and its frontier). Assembling the chapter: for $L^2$-subcritical focusing NLS ($p<1+4/n$) every $H^1$ solution is global (Proposition 4 of 02.21.04) and the ground state is orbitally stable; at the $L^2$-critical power the ground-state mass $\Vert Q{\Vert }_{L^2}$ is the sharp threshold, below it global and above it the pseudoconformal blowup realises minimal-mass collapse; in the energy-(super)critical range the relevant threshold becomes the energy/kinetic-energy condition against the Aubin-Talenti ground state $W$ of 02.21.04, and the Kenig-Merle concentration-compactness/rigidity scheme — driven by the localised virial monotonicity of the Key Theorem — yields the scattering-versus-blowup dichotomy below the ground-state energy. The frontier (minimal-mass and minimal-energy blowup profiles, log-log blowup rates of Merle-Raphaël, soliton resolution) is the live edge of the subject.

Synthesis. The virial identity is the central insight that closes this chapter: it is the foundational reason the focusing energy of 02.21.04, coercive in the defocusing case, becomes a collapse engine in the focusing case, and the convexity of the variance is exactly the contrapositive of the blowup criterion of 02.21.03 — a bounded ${\dot{H}}^1$ norm forbids blowup, and here the variance forces ${\dot{H}}^1$ to diverge. Putting these together, the sharp Gagliardo-Nirenberg constant of 02.16.01 is dual to the ground state $Q$: the optimiser of the interpolation inequality is the soliton, the inequality's constant is fixed by $\Vert Q{\Vert }_{L^2}$, and that mass is the threshold the Beginner-tier sorter read off. This is exactly the same criticality arithmetic that, in 02.21.03, sorted the local theory by $s_c=n/2-2/(p-1)$ and, in 02.21.04, decided defocusing global existence — here the mass-critical power $p=1+4/n$ is where the virial coefficient $n(p-1)-4$ changes sign, the pseudoconformal symmetry becomes exact, the sharp constant's optimiser appears, and the stability index $d^{\prime \prime }(\omega )$ crosses zero, all at once. What looks from a distance like four unrelated phenomena — virial blowup, the sharp interpolation constant, the explicit pseudoconformal solution, and orbital (in)stability — is one ground-state geometry read at a single scaling weight, and the bridge to the frontier is the localised virial monotonicity that the Kenig-Merle rigidity theorem turns into a uniqueness statement for the critical element.

Full proof set Master

Proposition 1 (virial identity for finite-variance $H^1$ solutions). Let $\mu =-1$, $1<p<1+4/(n-2{)}_{+}$, $u_0\in H^1$ with $|x|u_0\in L^2$. Then $V(t)=\int |x{|}^2|u{|}^2$ is $C^2$ on the lifespan with $V^{\prime \prime }(t)=8\int |\nabla u{|}^2-\frac{4n(p-1)}{p+1}\int |u{|}^{p+1}$.

Proof. For $u_0\in H^{\infty }$ with $⟨x{⟩}^2{u}_0\in L^2$, persistence of regularity and a weighted-norm propagation estimate 02.21.03 keep $|x|u(t)\in L^2$ on the lifespan, and the pointwise computations of the Key Theorem (Steps 1-2) give the identity for smooth solutions. For general finite-variance $u_0\in H^1$, approximate $u_0^{(k)}\to u_0$ in $H^1$ with $|x|(u_0^{(k)}-u_0)\to 0$ in $L^2$ and $u_0^{(k)}\in H^{\infty }$. Continuous dependence 02.21.03 gives $u^{(k)}\to u$ in $C(J;H^1)$ on compact $J$, and the weighted estimate gives $|x|u^{(k)}\to |x|u$ in $C(J;L^2)$; the functionals $\int |\nabla u{|}^2$ and $\int |u{|}^{p+1}$ are continuous on $H^1$ via 02.16.01, so the identity passes to the limit. $\square$

Proposition 2 (Glassey blowup, $E<0$). Under the hypotheses of Proposition 1 with $p\ge 1+4/n$ and $E[u_0]<0$, the lifespan is finite in both time directions.

Proof. By Proposition 1 and the energy substitution of Key Theorem Step 3, $V^{\prime \prime }(t)=16E[u_0]-\frac{4(n(p-1)-2)}{p+1}\int |u{|}^{p+1}\le 16E[u_0]<0$ since $n(p-1)-2\ge 0$ for $p\ge 1+2/n$. Twice-integrating the bound $V^{\prime \prime }\le 16E[u_0]$ gives $0\le V(t)\le V(0)+V^{\prime }(0)t+8E[u_0]t^2$, a downward parabola with negative leading coefficient $8E[u_0]<0$; it is negative for $t$ large, contradicting $V\ge 0$ unless the solution ceases to exist first. Hence $T_{+}<\infty$, and the backward run gives $T_{-}<\infty$. $\square$

Proposition 3 (sharp Gagliardo-Nirenberg and the mass threshold). At $p=1+4/n$, $\Vert u{\Vert }_{L^{p+1}}^{p+1}\le \frac{p+1}{2\Vert Q{\Vert }_{L^2}^{4/n}}\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{4/n}$ with equality at $u=Q$; consequently $\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$ forces $E[u_0]\ge 0$ and global existence.

Proof. The functional $J(u)=\Vert \nabla u{\Vert }_{L^2}^2\Vert u{\Vert }_{L^2}^{4/n}/\Vert u{\Vert }_{L^{p+1}}^{p+1}$ is scaling- and dilation-invariant, so a minimising sequence may be taken radial and normalised; the Strauss radial embedding $H_{\mathrm{rad}}^1({\mathbb{R}}^n)\hookrightarrow \hookrightarrow L^{p+1}$ is compact for $2<p+1<2n/(n-2)$, yielding a minimiser $u_{\ast }$. Its Euler-Lagrange equation, after rescaling space and amplitude to clear the Lagrange multipliers, is $-\Delta Q+Q=Q^p$, so $u_{\ast }=Q$ up to symmetries and $1/\min J=C_{\mathrm{GN}}$. Evaluating $J(Q)=1/C_{\mathrm{GN}}$ and using the Pohozaev identities $\Vert \nabla Q{\Vert }_{L^2}^2=\frac{n(p-1)}{2(p+1)}\Vert Q{\Vert }_{L^{p+1}}^{p+1}$ and $\Vert Q{\Vert }_{L^2}^2=\frac{4-(n-2)(p-1)}{2(p+1)}\Vert Q{\Vert }_{L^{p+1}}^{p+1}$ fixes $C_{\mathrm{GN}}=(p+1)/(2\Vert Q{\Vert }_{L^2}^{4/n})$ at $p=1+4/n$. Inserting into $E$ gives $E[u_0]\ge \frac{1}{2}\Vert \nabla u_0{\Vert }_{L^2}^2(1-(\Vert u_0{\Vert }_{L^2}/\Vert Q{\Vert }_{L^2}{)}^{4/n})$, non-negative when $\Vert u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$; mass conservation then bounds $\Vert \nabla u(t){\Vert }_{L^2}$ uniformly and the iteration of 02.21.04 gives global existence. $\square$

Proposition 4 (pseudoconformal symmetry and explicit blowup). At $p=1+4/n$, if $u$ solves NLS then so does $\mathcal{C}u$, and applying $\mathcal{C}$ to $e^{it}Q$ yields a solution with $\Vert u(t){\Vert }_{L^2}=\Vert Q{\Vert }_{L^2}$ for all $t$ and $\Vert \nabla u(t){\Vert }_{L^2}\sim |t{|}^{-1}\to \infty$ as $t\to 0$.

Proof. The pseudoconformal identity is verified by direct substitution: writing $v(t,x)=|t{|}^{-n/2}\bar{u}(1/t,x/t)e^{i|x{|}^2/(4t)}$, the phase $e^{i|x{|}^2/(4t)}$ is the kernel that intertwines the dilation $x\mapsto x/t$ with the Schrödinger flow, and a computation using ${\partial }_t$ and $\Delta$ of the Gaussian phase shows $i{\partial }_{t}v+\Delta v=-|v{|}^{p-1}v$ exactly when $p=1+4/n$ (the power at which the amplitude weight $|t{|}^{-n/2}$ matches the nonlinear scaling). Mass invariance is the change of variables of Exercise 6; the gradient blowup is the $|t{|}^{-1}$ scaling of the dilated profile and phase, computed there. The solution exists on $t<0$, conserves the threshold mass, and concentrates into a point mass at $t=0$. $\square$

Proposition 5 (orbital stability for $p<1+4/n$). For $1<p<1+4/n$ the ground-state standing wave is orbitally stable in $H^1$.

Proof. Let $S_{\omega }=E+\omega M$, so ${\varphi }_{\omega }$ is a critical point: $S_{\omega }^{\prime }({\varphi }_{\omega })=0$. The second variation $S_{\omega }^{\prime \prime }({\varphi }_{\omega })=L=\mathrm{diag}(L_{+},L_{-})$ with $L_{+}=-\Delta +\omega -p{\varphi }_{\omega }^{p-1}$, $L_{-}=-\Delta +\omega -{\varphi }_{\omega }^{p-1}$. By Kwong non-degeneracy $\ker L_{-}=\mathrm{span}\{{\varphi }_{\omega }\}$ and $\ker L_{+}=\mathrm{span}\{{\partial }_j{\varphi }_{\omega }\}$, and $L_{+}$ has exactly one negative eigenvalue. The Weinstein/GSS coercivity lemma states that when $d^{\prime \prime }(\omega )=\frac{d}{d\omega }\Vert {\varphi }_{\omega }{\Vert }_{L^2}^2>0$, the form $⟨L\eta ,\eta ⟩$ is bounded below by $c\Vert \eta {\Vert }_{H^1}^2$ on the subspace orthogonal to $\{{\varphi }_{\omega },i{\varphi }_{\omega },{\partial }_j{\varphi }_{\omega }\}$. Since $d^{\prime \prime }(\omega )>0⟺\sigma =2/(p-1)-n/2>0⟺p<1+4/n$, the form is coercive. A modulation decomposition $u(t)=e^{i\theta (t)}({\varphi }_{\omega }+\eta )(\cdot -y(t))$ with $\eta \perp$ the symmetry directions, combined with conservation of $E$ and $M$, then bounds $\Vert \eta (t){\Vert }_{H^1}$ by its initial size uniformly in $t$: the solution stays in an $\epsilon$-tube around the orbit. $\square$

Proposition 6 (instability for $p\ge 1+4/n$). For $p\ge 1+4/n$ the ground-state standing wave is orbitally unstable.

Proof. At $p\ge 1+4/n$, $\sigma \le 0$ so $d^{\prime \prime }(\omega )\le 0$ and the GSS coercivity fails; instability follows from the virial blowup of Proposition 2. Berestycki-Cazenave construct, for each $\delta >0$, data $u_0^{\delta }$ with $\Vert u_0^{\delta }-{\varphi }_{\omega }{\Vert }_{H^1}<\delta$, the same mass $M[u_0^{\delta }]=M[{\varphi }_{\omega }]$, finite variance, and energy $E[u_0^{\delta }]<E[{\varphi }_{\omega }]$; the scaling $u_0^{\delta }={\lambda }^{n/2}{\varphi }_{\omega }(\lambda \cdot )$ with $\lambda$ slightly $>1$ lowers the energy below the ground-state level at fixed mass for $p\ge 1+4/n$. Since $E[{\varphi }_{\omega }]=$ (a computation) gives $E[u_0^{\delta }]<0$ at the critical/supercritical power, Proposition 2 forces finite-time blowup, so the solution leaves every fixed tube around the orbit: orbital instability ^{[Berestycki-Cazenave 1981]}. $\square$

Connections Master

• The virial identity is the focusing counterpart of the conservation-and-coercivity machinery of 02.21.04: there the defocusing energy bounded $\Vert \nabla u{\Vert }_{L^2}$ from above and forbade blowup, here the same energy with $\mu =-1$ enters the variance with the opposite sign and drives $V^{\prime \prime }(t)\le 16E[u_0]$, so the blowup theory is the mirror image of the global-existence theory across the sign of the nonlinearity, and the Glassey criterion is the exact obstruction the defocusing iteration was able to rule out.

• The ground state $Q$ and the sharp Gagliardo-Nirenberg constant are the optimiser and best constant of the interpolation inequality of 02.16.01: the mass threshold $\Vert Q{\Vert }_{L^2}$, the energy coercivity below threshold, and the Weinstein stability index $d^{\prime \prime }(\omega )$ all read off the same variational problem whose Euler-Lagrange equation is the soliton profile, so the entire soliton half of this unit is the variational shadow of that unit's Gagliardo-Nirenberg theory specialised to its extremiser.

• The blowup-versus-global dichotomy runs on the local theory of 02.21.03: the maximal-lifespan solution, the blowup criterion phrased as divergence of the $H^s$ (or critical Strichartz) norm, and the persistence of regularity that licenses the virial computation are all inherited from that unit, so the finite-time blowup statement is the negation of the continuation criterion, and the pseudoconformal solution is an explicit witness that the critical-norm criterion cannot be relaxed to a bare ${\dot{H}}^{s_c}$ bound.

Historical & philosophical context Master

The virial identity for nonlinear Schrödinger equations entered the mathematics literature through Robert Glassey's 1977 Journal of Mathematical Physics paper On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations ^{[Glassey 1977]}, which made rigorous the convexity argument that V. E. Zakharov, S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov had used in the physics of optical self-focusing and Langmuir wave collapse in the late 1960s. The identity converts the conserved energy directly into a statement about the second derivative of the variance, and the negative-energy finite-variance hypothesis remains the cleanest sufficient condition for blowup. The sharp form of the Gagliardo-Nirenberg inequality and its role in the mass threshold is due to Michael Weinstein, whose 1983 Communications in Mathematical Physics paper Nonlinear Schrödinger equations and sharp interpolation estimates ^{[Weinstein 1983]} identified the ground state as the extremiser and fixed the constant by the soliton mass.

The stability theory has two roots. Weinstein's 1985 SIAM Journal on Mathematical Analysis paper Modulational stability of ground states of nonlinear Schrödinger equations ^{[Weinstein 1985]} proved orbital stability in the $L^2$-subcritical range by a Lyapunov-modulation method, and Manoussos Grillakis, Jalal Shatah, and Walter Strauss in their 1987 Journal of Functional Analysis paper Stability theory of solitary waves in the presence of symmetry I ^{[Grillakis-Shatah-Strauss 1987]} abstracted the convexity criterion $d^{\prime \prime }(\omega )\gtrless 0$ to a general Hamiltonian framework with symmetry, isolating the spectral hypotheses on the linearised operator. The matching instability for the critical and supercritical powers was given by Henri Berestycki and Thierry Cazenave in their 1981 Comptes Rendus note ^{[Berestycki-Cazenave 1981]}, who turned the virial blowup into a destabilisation of the standing wave, and the uniqueness and non-degeneracy of the ground state that the whole theory presupposes was settled by Man Kam Kwong's 1989 ODE-shooting analysis of $-\Delta Q+Q=Q^p$.

Bibliography Master

@article{Glassey1977,
  author  = {Glassey, Robert T.},
  title   = {On the blowing up of solutions to the {C}auchy problem for nonlinear {S}chr\"odinger equations},
  journal = {Journal of Mathematical Physics},
  volume  = {18},
  year    = {1977},
  pages   = {1794--1797}
}

@article{Weinstein1983,
  author  = {Weinstein, Michael I.},
  title   = {Nonlinear {S}chr\"odinger equations and sharp interpolation estimates},
  journal = {Communications in Mathematical Physics},
  volume  = {87},
  year    = {1983},
  pages   = {567--576}
}

@article{Weinstein1985,
  author  = {Weinstein, Michael I.},
  title   = {Modulational stability of ground states of nonlinear {S}chr\"odinger equations},
  journal = {SIAM Journal on Mathematical Analysis},
  volume  = {16},
  year    = {1985},
  pages   = {472--491}
}

@article{GSS1987,
  author  = {Grillakis, Manoussos and Shatah, Jalal and Strauss, Walter},
  title   = {Stability theory of solitary waves in the presence of symmetry. {I}},
  journal = {Journal of Functional Analysis},
  volume  = {74},
  year    = {1987},
  pages   = {160--197}
}

@article{BerestyckiCazenave1981,
  author  = {Berestycki, Henri and Cazenave, Thierry},
  title   = {Instabilit\'e des \'etats stationnaires dans les \'equations de {S}chr\"odinger et de {K}lein-{G}ordon non lin\'eaires},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences Paris, S\'erie I},
  volume  = {293},
  year    = {1981},
  pages   = {489--492}
}

@article{Kwong1989,
  author  = {Kwong, Man Kam},
  title   = {Uniqueness of positive solutions of {$\Delta u - u + u^p = 0$} in {$\mathbb{R}^n$}},
  journal = {Archive for Rational Mechanics and Analysis},
  volume  = {105},
  year    = {1989},
  pages   = {243--266}
}

@book{Cazenave2003,
  author    = {Cazenave, Thierry},
  title     = {Semilinear {S}chr\"odinger Equations},
  series    = {Courant Lecture Notes in Mathematics},
  volume    = {10},
  publisher = {American Mathematical Society},
  year      = {2003}
}

@book{Tao2006,
  author    = {Tao, Terence},
  title     = {Nonlinear Dispersive Equations: Local and Global Analysis},
  series    = {CBMS Regional Conference Series in Mathematics},
  volume    = {106},
  publisher = {American Mathematical Society},
  year      = {2006}
}