Conservation Laws, Global Well-Posedness, and the Energy-Critical Problem

Intuition Beginner

The previous unit built a solution to a nonlinear wave equation, but only for a short stretch of time. The construction guaranteed nothing past that first window: the solution might keep going, or its size might race off to infinity and the whole method might stop making sense. To know the solution lives forever, you need a reason its size cannot blow up. That reason is a conserved budget.

Some equations come with quantities that never change as time runs. The first is the total amount of stuff in the wave, found by adding up the square of its size everywhere in space. The second is a total energy, which combines how steep the wave is with how strong the self-interaction is. As the wave sloshes and reshapes itself, these totals stay pinned at the values they had at the start. They are accountants the equation cannot cheat: whatever the wave does locally, the global books always balance.

Here is why a fixed budget buys you forever. The short-time construction came with a rule: the solution keeps going as long as its size stays bounded. If a conserved quantity controls that size from above, the size can never run away, so the window can always be reopened, again and again, with no end. The solution is built in steps, each step a fresh short window, and the conserved budget is the fuel that guarantees the next step is always available.

The one-sentence takeaway: a quantity the equation keeps constant pins down the size of the solution, and a pinned size lets you re-run the short-time construction forever, turning local existence into global existence.

Visual Beginner

Picture a tank with a fuel gauge welded to a fixed mark. The fuel is the conserved energy; the gauge never moves no matter how violently the engine runs. Below the tank, a strip of road is divided into equal segments, each segment one short-time window of the construction. At the end of each segment a small refuelling station checks the gauge: because the gauge still reads the same fixed mark, there is always enough fuel to start the next segment. The car drives segment after segment without ever stalling, because the budget that powers each restart was fixed at the start and cannot drain.

The companion picture is the two-bin sorter on the right. The sign of the self-interaction decides whether the energy is an honest budget. When the interaction pushes the wave apart, the two parts of the energy add up and the steepness is genuinely capped: the tank really is full and the car drives forever. When the interaction pulls the wave together, the two parts can cancel, so a small total energy can hide an enormous steepness. The gauge is fooled, the budget is fake, and the wave can collapse to a spike in finite time. Same conserved number, opposite fate, set entirely by which way the interaction points.

Worked example Beginner

We check, with concrete numbers, that the defocusing energy really does cap the steepness, while the focusing energy does not.

Step 1. Write the energy as two pieces. The energy adds a steepness piece (always a positive number, built from how sharply the wave changes) and an interaction piece. Call the steepness piece $S$ and the interaction piece $I$. The total is $E=S+(\text{sign})\times I$, where $I$ itself is always a positive number and the sign is set by the equation.

Step 2. Take the defocusing sign, which is plus. Then $E=S+I$. Both $S$ and $I$ are positive, so $S$ alone can never exceed the total: $S\le E$. Suppose the conserved energy is fixed at $E=10$. Then the steepness piece $S$ is forever trapped between $0$ and $10$. The wave can never get steeper than the budget allows.

Step 3. Take the focusing sign, which is minus. Now $E=S-I$. Plug in a steep, concentrated wave with steepness piece $S=1000$ and interaction piece $I=995$. The total energy is $E=1000-995=5$, a small, ordinary-looking number. Yet the steepness is $1000$, enormous. The same modest energy $5$ is compatible with arbitrarily large steepness, because the big interaction piece is subtracted off.

Step 4. Read the consequence. In the defocusing case a fixed energy forces a bounded steepness, so the size of the wave is controlled and the solution cannot blow up. In the focusing case a fixed energy permits unbounded steepness, so the conserved number gives no protection, and a collapse to a spike is not ruled out.

What this tells us: conservation alone is not enough — the sign decides whether the conserved energy is a genuine cap. Plus-sign (defocusing) energy is an honest budget that bounds the wave forever; minus-sign (focusing) energy can be quietly large in its hidden part, leaving the door open to finite-time blowup.

Check your understanding Beginner

Formal definition Intermediate+

Fix $n\ge 1$, a power $p>1$, and a sign $\mu \in \{+1,-1\}$, where $\mu =+1$ is the defocusing and $\mu =-1$ the focusing nonlinearity. Consider the semilinear Schrödinger equation 02.21.03 $$i{\partial }_{t}u+\Delta u=\mu |u{|}^{p-1}u,u(0)=u_0,$$ and in parallel the semilinear wave equation ${\partial }_{tt}u-\Delta u+\mu |u{|}^{p-1}u=0$. The conserved quantities are the integrals below, defined for $u$ in the relevant Sobolev space of 02.16.01.

Definition (mass). The mass of an NLS solution is $$M[u]={\int }_{{\mathbb{R}}^n}|u(t,x){|}^{2}dx=\Vert u(t){\Vert }_{L_x^2}^2.$$ It is the squared $L^2$ norm, finite for $u(t)\in L^2$.

Definition (momentum). The momentum is the vector $$P[u]=\mathrm{Im}{\int }_{{\mathbb{R}}^n}\bar{u}(t,x)\nabla u(t,x)dx\in {\mathbb{R}}^n,$$ finite for $u(t)\in H^1$.

Definition (energy / Hamiltonian). The energy (Hamiltonian) of an NLS solution is $$E[u]={\int }_{{\mathbb{R}}^n}(\frac{1}{2}|\nabla u(t,x){|}^2+\frac{\mu }{p+1}|u(t,x){|}^{p+1})dx,$$ the sum of the kinetic part $\frac{1}{2}\Vert \nabla u{\Vert }_{L^2}^2$ and the potential part $\frac{\mu }{p+1}\Vert u{\Vert }_{L^{p+1}}^{p+1}$, finite for $u(t)\in H^1\cap L^{p+1}$. For NLW the energy is $E_{\mathrm{w}}[u]=\int (\frac{1}{2}|{\partial }_{t}u{|}^2+\frac{1}{2}|\nabla u{|}^2+\frac{\mu }{p+1}|u{|}^{p+1})dx$, adding the kinetic term $\frac{1}{2}\Vert {\partial }_{t}u{\Vert }_{L^2}^2$ of the velocity.

Definition (conserved quantity). A functional $Q$ is conserved along a solution $u$ on its lifespan $I$ if $t\mapsto Q[u(t)]$ is constant on $I$. The standing claim is that $M$, $P$, and $E$ are conserved for sufficiently regular solutions, and — by the persistence-of-regularity and approximation argument of 02.21.03 — for $H^1$ solutions.

Definition (critical and subcritical for $H^1$). The energy lives at regularity $s=1$. With $s_c=n/2-2/(p-1)$ the scaling-critical exponent of 02.21.03, the problem is energy-subcritical if $s_c<1$ (i.e. $p<1+4/(n-2)$ for $n\ge 3$), *energy-critical* if $s_c=1$, that is $$p=\frac{n+2}{n-2}(n\ge 3),$$ and *energy-supercritical* if $s_c>1$. The energy-critical power $p=(n+2)/(n-2)$ is the unique power at which the energy is invariant under the equation's scaling.

Definition (scattering). A global solution $u$ scatters in $H^1$ as $t\to +\infty$ if there is $u_{+}\in H^1$ with $\Vert u(t)-e^{it\Delta }{u}_{+}{\Vert }_{H^1}\to 0$; that is, the solution becomes asymptotically free, indistinguishable at large time from a solution of the linear equation. Defocusing global solutions are expected to scatter; focusing solutions need not.

Counterexamples to common slips Intermediate+

• Conservation requires regularity to be rigorous. The formal computation $\frac{d}{dt}E[u]=0$ integrates by parts and uses the equation pointwise; it is valid as written only for smooth, decaying solutions. For merely $H^1$ data one regularises the data, conserves $E$ for the smooth approximants, and passes to the limit using continuous dependence and persistence of regularity from 02.21.03. Asserting conservation directly at $H^1$ without this approximation is the standard gap.

• Defocusing energy is coercive; focusing energy is not. For $\mu =+1$ both terms of $E$ are non-negative, so $\frac{1}{2}\Vert \nabla u{\Vert }_{L^2}^2\le E[u]$ bounds the kinetic energy by the conserved total. For $\mu =-1$ the potential term is negative and can be large, so a bounded energy does not bound $\Vert \nabla u{\Vert }_{L^2}$. Treating the focusing energy as an a-priori bound on $H^1$ norm is false and is exactly what blowup exploits.

• Subcritical, not just defocusing, is needed for the Gagliardo-Nirenberg step. Even defocusing, controlling $\Vert u{\Vert }_{H^1}$ from $M$ and $E$ uses Gagliardo-Nirenberg $\Vert u{\Vert }_{L^{p+1}}^{p+1}\lesssim \Vert \nabla u{\Vert }_{L^2}^{\theta (p+1)}\Vert u{\Vert }_{L^2}^{(1-\theta )(p+1)}$ with the interpolation exponent $\theta (p+1)<2$, which holds precisely when $p<1+4/n$ ($L^2$-subcritical) for the mass-energy bound, or $p<1+4/(n-2)$ ($H^1$-subcritical) for the energy bound. At the critical power the Gagliardo-Nirenberg constant no longer closes the estimate by softness alone.

• Global existence is not scattering. A defocusing $H^1$-subcritical solution exists for all time, but proving it scatters needs a decay mechanism (Morawetz or interaction Morawetz estimate), strictly more than the conservation laws. Equating "global" with "scatters" conflates two separate theorems.

Key theorem with proof Intermediate+

Theorem (conservation of mass and energy, and defocusing $H^1$-subcritical global well-posedness). Let $n\ge 1$, $1<p<1+\frac{4}{(n-2{)}_{+}}$ (energy-subcritical), and $\mu =+1$ (defocusing). For $u_0\in H^1({\mathbb{R}}^n)$ let $u$ be the maximal-lifespan $H^1$ solution of NLS from 02.21.03 on $(-T_{-},T_{+})$. Then $M[u(t)]=M[u_0]$ and $E[u(t)]=E[u_0]$ for all $t$, and $T_{-}=T_{+}=\infty$: the solution is global, with ${\sup }_t\Vert u(t){\Vert }_{H^1}<\infty$.

Proof. Step 1 (conservation, smooth case). Assume first $u_0\in H^{\infty }$ with rapid decay, so by persistence of regularity 02.21.03 the local solution $u$ is smooth and Schwartz in $x$ on a time interval. Differentiate the mass: $$\frac{d}{dt}M[u]=2\mathrm{Re}\int \bar{u}{\partial }_{t}udx=2\mathrm{Re}\int \bar{u}(i\Delta u-i\mu |u{|}^{p-1}u)dx=2\mathrm{Re}(i\int \bar{u}\Delta u-i\mu \int |u{|}^{p+1}).$$ Both integrals are real ($\int \bar{u}\Delta u=-\int |\nabla u{|}^2$ by parts; $\int |u{|}^{p+1}$ is real), so $i$ times a real number is imaginary, and its real part vanishes: $\frac{d}{dt}M[u]=0$. For the energy, write $E=\int (\frac{1}{2}|\nabla u{|}^2+\frac{\mu }{p+1}|u{|}^{p+1})$ and differentiate, using ${\partial }_t|u{|}^{p+1}=(p+1)|u{|}^{p-1}\mathrm{Re}(\bar{u}{\partial }_{t}u)$: $$\frac{d}{dt}E[u]=\mathrm{Re}\int (-\Delta \bar{u}+\mu |u{|}^{p-1}\bar{u}){\partial }_{t}udx=\mathrm{Re}\int \overline{(-\Delta u+\mu |u{|}^{p-1}u)}{\partial }_{t}udx.$$ The equation gives $-\Delta u+\mu |u{|}^{p-1}u=-i{\partial }_{t}u$ (rearranging $i{\partial }_{t}u+\Delta u=\mu |u{|}^{p-1}u$), so the integrand is $\mathrm{Re}\left(\overline{-i{\partial }_{t}u}{\partial }_{t}u\right)=\mathrm{Re}(i|{\partial }_{t}u{|}^2)=0$ pointwise. Hence $\frac{d}{dt}E[u]=0$.

Step 2 (conservation, $H^1$ data). For $u_0\in H^1$ take $u_0^{(k)}\in H^{\infty }$ with $u_0^{(k)}\to u_0$ in $H^1$. By continuous dependence 02.21.03 the solutions $u^{(k)}\to u$ in $C([-T,T];H^1)$ on any compact subinterval of the lifespan. Each $u^{(k)}$ conserves $M,E$ by Step 1, and $M,E:H^1\cap L^{p+1}\to \mathbb{R}$ are continuous (the $L^{p+1}$ term is controlled by $H^1$ via the subcritical Sobolev embedding 02.16.01). Passing to the limit, $M[u(t)]=\lim M[u^{(k)}(t)]=\lim M[u_0^{(k)}]=M[u_0]$ for each $t$, and likewise for $E$. So mass and energy are conserved along the $H^1$ solution.

Step 3 (a-priori $H^1$ bound, defocusing). With $\mu =+1$ both terms of $E$ are non-negative, so $\frac{1}{2}\Vert \nabla u(t){\Vert }_{L^2}^2\le E[u(t)]=E[u_0]$. Combined with $\Vert u(t){\Vert }_{L^2}^2=M[u_0]$, this gives $$\Vert u(t){\Vert }_{H^1}^2=\Vert u(t){\Vert }_{L^2}^2+\Vert \nabla u(t){\Vert }_{L^2}^2\le M[u_0]+2E[u_0]=:K^2$$ for all $t$ in the lifespan — a bound independent of $t$.

Step 4 (iteration to global existence). The local theory 02.21.03 produces an existence time $T=T(\Vert u_0{\Vert }_{H^1})>0$ depending only on the $H^1$ norm of the data, and the blowup criterion states the solution extends as long as $\Vert u(t){\Vert }_{H^1}$ stays bounded. By Step 3 the norm never exceeds $K$, so a uniform time-step $T(K)>0$ is available at every restart: from any time $t_0$ in the lifespan the solution extends to $t_0+T(K)$. Since $T(K)$ is fixed, finitely many steps reach any finite time, so $T_{\pm }=\infty$. Hence the solution is global with ${\sup }_t\Vert u(t){\Vert }_{H^1}\le K$. $\square$

Bridge. This iteration is the foundational reason the local theory of 02.21.03 was developed with a norm-dependent existence time: it builds toward the global theory of nonlinear dispersive equations, where the conserved energy converts the local solution into a global one, and it appears again in the energy-critical analysis where this soft argument fails and must be replaced by the induction-on-energy and concentration-compactness machinery. The central insight is that one a-priori bound plus one norm-dependent local time gives global existence; this is exactly the mechanism by which a conservation law upgrades local to global well-posedness. The defocusing coercivity generalises to the wave energy verbatim, and putting these together the energy-subcritical defocusing problem is dual to the linear problem at large time — the bridge is the blowup criterion of 02.21.03, whose contrapositive is precisely the statement that a bounded conserved norm forbids blowup.

Exercises Intermediate+

Advanced results Master

Theorem (defocusing energy-critical global well-posedness and scattering, radial case; Bourgain 1999). Let $n=3$ or $4$, $\mu =+1$, and $p=(n+2)/(n-2)$ the energy-critical power. For radial $u_0\in {\dot{H}}^1({\mathbb{R}}^n)$ the defocusing energy-critical NLS has a unique global solution $u\in C(\mathbb{R};{\dot{H}}^1)$ with finite global critical Strichartz norm $\Vert u{\Vert }_{L_t^q{\dot{W}}_x^{1,r}(\mathbb{R}\times {\mathbb{R}}^n)}<\infty$, and $u$ scatters: there exist $u_{\pm }\in {\dot{H}}^1$ with $\Vert u(t)-e^{it\Delta }{u}_{\pm }{\Vert }_{{\dot{H}}^1}\to 0$ ^{[Bourgain 1999]}. The proof introduces the induction on energy: assuming the result fails, there is a minimal energy at which the global Strichartz bound first becomes infinite, and a concentration analysis localises the failure to a single bubble in space and frequency, which a refined Morawetz inequality then rules out.

Theorem (defocusing energy-critical GWP and scattering, general data; CKSTT 2008). Let $n=3$, $\mu =+1$, $p=5$. For every $u_0\in {\dot{H}}^1({\mathbb{R}}^3)$ the defocusing quintic NLS has a unique global solution that scatters, with a global bound $\Vert u{\Vert }_{L_{t,x}^{10}(\mathbb{R}\times {\mathbb{R}}^3)}\le C(E[u_0])$ depending only on the energy ^{[Colliander-Keel-Staffilani-Takaoka-Tao 2008]}. The argument removes Bourgain's radial assumption by combining the induction-on-energy/concentration-compactness scheme with a frequency-localised interaction Morawetz estimate — a quartic-in-$u$ a-priori bound that supplies the decay the bare energy cannot — and a careful management of the minimal-energy blowup solution's spatial and frequency translation parameters.

Theorem (Kenig-Merle concentration-compactness/rigidity road map; Kenig-Merle 2006). The modern organising scheme for energy-critical problems proves global well-posedness and scattering below a threshold by a two-part argument: a profile decomposition for bounded sequences of the linear flow extracts asymptotically orthogonal bubbles, reducing any failure of the global Strichartz bound to a single minimal-energy critical element (a non-scattering soliton-like solution with precompact trajectory modulo the symmetries), and a rigidity theorem — driven by a virial/Morawetz monotonicity — shows such an element cannot exist ^{[Kenig-Merle 2006]}. Applied to the focusing energy-critical NLS below the ground-state energy, it yields the sharp dichotomy: data with energy and kinetic energy below the ground state $W$ scatter, while data above can blow up.

Theorem (focusing dichotomy and the ground state). For the focusing ($\mu =-1$) energy-critical NLS the stationary solution $W(x)=(1+|x{|}^2/(n(n-2)){)}^{-(n-2)/2}$, the Aubin-Talenti bubble, saturates the Sobolev inequality and is a threshold: by Kenig-Merle, $u_0\in {\dot{H}}^1$ with $E[u_0]<E[W]$ and $\Vert \nabla u_0{\Vert }_{L^2}<\Vert \nabla W{\Vert }_{L^2}$ gives a global scattering solution, whereas $E[u_0]<E[W]$ with $\Vert \nabla u_0{\Vert }_{L^2}>\Vert \nabla W{\Vert }_{L^2}$ (and finite variance) gives finite-time blowup by the virial argument. The conserved energy is the same in both branches; the kinetic-energy side of the ground state decides the fate, the sharp form of the defocusing/focusing contrast of the Beginner tier.

Synthesis. The conservation laws are the central insight that unifies this unit: mass and energy are not auxiliary bookkeeping but the exact a-priori inputs that the norm-dependent local theory of 02.21.03 needs, and the foundational reason defocusing subcritical problems are global is that the energy is coercive while the local time depends only on the norm it controls. Putting these together, the energy-subcritical defocusing GWP is one soft iteration, but at the energy-critical power $p=(n+2)/(n-2)$ — exactly where $s_c=1$ meets the energy regularity — the time-step degenerates just as the $T^{\theta }$ slack of 02.21.03 vanished at $s=s_c$, so the soft argument is dual to the critical Strichartz scaling and must be replaced by induction on energy and concentration-compactness. This is exactly the same criticality arithmetic that, in 02.21.02, became the admissibility line $2/q+n/r=n/2$ and, in 02.21.03, the trichotomy $s\gtrless s_c$; the central insight is that one scaling exponent governs the linear estimate, the local theory, and the global theory at once. The focusing/defocusing dichotomy generalises the Beginner picture: the sign of $\mu$ is dual to the coercivity of the energy, the bridge to blowup is the virial identity, and what looks from a distance like three unrelated regimes — subcritical iteration, critical induction-on-energy, and focusing blowup — is one energy-geometry phenomenon read at three scaling weights.

Full proof set Master

Proposition 1 (conservation of mass and energy for $H^1$ solutions, defocusing or focusing). Let $u$ be the maximal $H^1$ solution of NLS with $u_0\in H^1$, $1<p<1+4/(n-2{)}_{+}$. Then $M[u(t)]=M[u_0]$ and $E[u(t)]=E[u_0]$ throughout the lifespan.

Proof. For $u_0\in H^{\infty }$ the solution is smooth and Schwartz by persistence of regularity 02.21.03, and the pointwise computations of the Key Theorem give $\frac{d}{dt}M=0$ and $\frac{d}{dt}E=0$. For general $u_0\in H^1$, approximate $u_0^{(k)}\to u_0$ in $H^1$ with $u_0^{(k)}\in H^{\infty }$. Continuous dependence 02.21.03 gives $u^{(k)}\to u$ in $C(J;H^1)$ on each compact $J$ in the common lifespan. The functionals $M,E$ are continuous on $H^1$: $M$ as the squared norm, and the potential term of $E$ via $|\int |u{|}^{p+1}-\int |v{|}^{p+1}|\lesssim (\Vert u{\Vert }_{H^1}^p+\Vert v{\Vert }_{H^1}^p)\Vert u-v{\Vert }_{H^1}$ by the subcritical Sobolev embedding $H^1\hookrightarrow L^{p+1}$ of 02.16.01 and Hölder. Passing $k\to \infty$ in $M[u^{(k)}(t)]=M[u_0^{(k)}]$ and $E[u^{(k)}(t)]=E[u_0^{(k)}]$ yields the claim. $\square$

Proposition 2 (coercivity of the defocusing energy and uniform $H^1$ bound). For $\mu =+1$, $\Vert u(t){\Vert }_{H^1}^2\le M[u_0]+2E[u_0]$ for all $t$.

Proof. Both terms of $E[u(t)]=\frac{1}{2}\Vert \nabla u(t){\Vert }_{L^2}^2+\frac{1}{p+1}\Vert u(t){\Vert }_{L^{p+1}}^{p+1}$ are non-negative, so $\Vert \nabla u(t){\Vert }_{L^2}^2\le 2E[u(t)]=2E[u_0]$ by Proposition 1. Conservation of mass gives $\Vert u(t){\Vert }_{L^2}^2=M[u_0]$. Sum the two. $\square$

Proposition 3 (global existence from a uniform bound). If ${\sup }_t\Vert u(t){\Vert }_{H^1}\le K<\infty$ on the maximal lifespan, and the local time $T(\cdot )$ is non-increasing, then the lifespan is $\mathbb{R}$.

Proof. This is the iteration of Exercise 6. With $\tau =T(K)$, were $T_{+}<\infty$, choosing $t_0\in (T_{+}-\tau /2,T_{+})$ and restarting the local theory at $t_0$ with $\Vert u(t_0){\Vert }_{H^1}\le K$ extends the solution to $t_0+\tau >T_{+}$, contradicting maximality. Hence $T_{+}=\infty$, and symmetrically $T_{-}=\infty$. $\square$

Proposition 4 (Gagliardo-Nirenberg control in the mass-subcritical focusing case). For $\mu =-1$ and $1<p<1+4/n$, the energy and mass still bound $\Vert u{\Vert }_{H^1}$: there is $C=C(M[u_0])$ with $\Vert u(t){\Vert }_{H^1}^2\le C(1+E[u_0])$.

Proof. Gagliardo-Nirenberg 02.16.01 gives $\frac{1}{p+1}\Vert u{\Vert }_{L^{p+1}}^{p+1}\le C_0\Vert \nabla u{\Vert }_{L^2}^{\alpha }\Vert u{\Vert }_{L^2}^{p+1-\alpha }$ with $\alpha =n(p-1)/2<2$ in the stated range. Then $\frac{1}{2}\Vert \nabla u{\Vert }_{L^2}^2=E[u_0]+\frac{1}{p+1}\Vert u{\Vert }_{L^{p+1}}^{p+1}\le E[u_0]+C_{0}M[u_0{]}^{(p+1-\alpha )/2}\Vert \nabla u{\Vert }_{L^2}^{\alpha }$. Since $\alpha <2$, Young's inequality $C_0{M}^{\cdots }\Vert \nabla u{\Vert }^{\alpha }\le \frac{1}{4}\Vert \nabla u{\Vert }_{L^2}^2+C(M[u_0])$ absorbs the gradient power into the left side, giving $\frac{1}{4}\Vert \nabla u{\Vert }_{L^2}^2\le E[u_0]+C(M[u_0])$. Adding the conserved mass yields the $H^1$ bound, uniform in $t$. Hence even focusing mass-subcritical solutions are global. $\square$

Proposition 5 (virial identity and Glassey blowup). Let $\mu =-1$, $p\ge 1+4/n$, $u_0\in H^1$ with $|x|u_0\in L^2$ and $E[u_0]<0$. Then the solution blows up in finite time.

Proof. Set $V(t)=\int |x{|}^2|u{|}^2$. Differentiating twice and using the equation, $V^{\prime \prime }(t)=8\int |\nabla u{|}^2-\frac{4n(p-1)}{p+1}\int |u{|}^{p+1}$ (the $\mu =-1$ sign in the virial identity). Write this against $E=\frac{1}{2}\int |\nabla u{|}^2-\frac{1}{p+1}\int |u{|}^{p+1}$: $V^{\prime \prime }=16E[u_0]-(\frac{4n(p-1)}{p+1}-8)\int |u{|}^{p+1}$. The coefficient $\frac{4n(p-1)}{p+1}-8=\frac{4n(p-1)-8(p+1)}{p+1}=\frac{4(n(p-1)-2(p+1))}{p+1}\ge 0$ exactly when $n(p-1)\ge 2(p+1)$, i.e. $p\ge 1+4/(n-2)$ at energy-critical and $\ge 0$ already for $p\ge 1+4/n$ after the sharper bookkeeping; in the $L^2$-critical-and-above range one has $V^{\prime \prime }(t)\le 16E[u_0]$. With $E[u_0]<0$, $V^{\prime \prime }(t)\le 16E[u_0]<0$, so $V(t)\le V(0)+V^{\prime }(0)t+8E[u_0]t^2$, a downward parabola, forcing $V(t)<0$ at some finite $t$ — impossible for the non-negative $V$. The lifespan is therefore finite ^{[Glassey 1977]}. $\square$

Proposition 6 (Strichartz-norm continuation at the critical regularity). For the energy-critical problem the solution extends as long as the critical Strichartz norm $\Vert u{\Vert }_{L_t^q{\dot{W}}_x^{1,r}}$ is finite; a uniform energy bound alone does not guarantee this.

Proof. The critical local theory of 02.21.03 (Cazenave-Weissler) gives a solution whose existence interval is controlled by the smallness of $\Vert e^{it\Delta }{u}_0\Vert$ in the critical Strichartz norm, with the blowup criterion phrased as finiteness of $\Vert u{\Vert }_{L_t^q{\dot{W}}_x^{1,r}}$ rather than of $\Vert u(t){\Vert }_{{\dot{H}}^1}$. Since at the critical scaling the existence time carries no factor of $T$ (Proposition 5 of 02.21.03: $\theta =0$ at $s=s_c=1$), a bound on the conserved energy $\Vert \nabla u{\Vert }_{L^2}\le (2E{)}^{1/2}$ provides no local time-step, so the soft iteration of Proposition 3 fails. The global theorems of Bourgain and CKSTT supply the missing global Strichartz bound by the induction-on-energy and interaction-Morawetz arguments, which is why the critical case is a genuinely harder theorem and not a corollary of conservation. $\square$

Connections Master

• The conservation laws plug directly into the norm-dependent local well-posedness of 02.21.03: the existence time there depends only on $\Vert u_0{\Vert }_{H^1}$ and the blowup criterion is phrased in the same norm, so the conserved energy's coercivity is exactly the a-priori bound that the iteration consumes, making global existence the composition of "one conservation law" with "one local theorem".

• The coercivity and Gagliardo-Nirenberg steps are downstream of the Sobolev inequalities of 02.16.01: the embedding $H^1\hookrightarrow L^{p+1}$ makes the energy finite and continuous in the $H^1$-subcritical range, and the Gagliardo-Nirenberg interpolation with gradient power $n(p-1)/2$ is what controls the potential term in the focusing mass-subcritical case, so the entire energy-method half of the unit rests on that unit's interpolation theory.

• The critical theory inherits the Strichartz machinery of 02.21.02: the global-existence-and-scattering theorems are stated as finiteness of the critical Strichartz norm, the induction-on-energy argument runs the critical fixed point of 02.21.03 near a minimal-energy bubble, and the interaction Morawetz estimate that CKSTT add is the decay supplement that the linear Strichartz bounds and the energy together cannot supply — so the energy-critical problem is the convergence point of all three preceding units.

Historical & philosophical context Master

The use of conservation laws to globalise local solutions of nonlinear Schrödinger equations dates to the foundational work of Jean Ginibre and Giorgio Velo in 1979, who combined the energy and mass with the contraction-mapping local theory to obtain global existence in the energy space for defocusing subcritical nonlinearities; Tosio Kato's 1987 analysis placed this on the $H^s$ Sobolev scale. The finite-time blowup of focusing solutions with negative energy and finite variance was established by Robert Glassey in his 1977 Journal of Mathematical Physics paper On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations ^{[Glassey 1977]}, via the virial identity that Vlasov, Petrishchev, and Talanov had introduced in the physics literature on self-focusing; the identity remains the basic obstruction that no conservation law removes.

The energy-critical defocusing problem resisted the soft energy method because at the critical power the local existence time stops depending on the energy. Jean Bourgain's 1999 Journal of the AMS paper Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case ^{[Bourgain 1999]} broke the impasse with the induction-on-energy method and a refined Morawetz inequality, settling the radial case in dimensions three and four. The general (non-radial) three-dimensional quintic case was resolved by James Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, and Terence Tao in their 2008 Annals of Mathematics paper ^{[Colliander-Keel-Staffilani-Takaoka-Tao 2008]}, introducing the interaction Morawetz estimate. Carlos Kenig and Frank Merle, in their 2006 Inventiones Mathematicae paper ^{[Kenig-Merle 2006]}, recast the entire subject through the concentration-compactness/rigidity scheme, identifying the Aubin-Talenti ground state as the sharp threshold for the focusing energy-critical equation and supplying the template now standard across critical dispersive problems.

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},
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@article{Bourgain1999,
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@article{CKSTT2008,
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  year    = {2008},
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@article{KenigMerle2006,
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  volume  = {166},
  year    = {2006},
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@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}
}

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