PDE: Dispersive & Wave¶
Job: develop the nonlinear dispersive PDE frontier — dispersive / Strichartz estimates (Keel–Tao) for Schrödinger / wave / Klein–Gordon propagators, local and global well-posedness for NLS via Strichartz + fixed-point (Ginibre–Velo, Tsutsumi), and the Kenig–Merle concentration-compactness / rigidity method for energy-critical scattering — with the admissible-pair tables and the critical-exponent landscape.
Core¶
Dispersive estimates¶
The free Schrödinger propagator \(e^{it\Delta}\) on \(\mathbb{R}^n\) is the Fourier multiplier \(e^{-it|\xi|^2}\). Its dispersive character is the decay of the \(L^1\to L^\infty\) norm:
Theorem (dispersive estimate). For \(u_0\in L^1(\mathbb{R}^n)\), \(\|e^{it\Delta}u_0\|_{L^\infty}\le(4\pi|t|)^{-n/2}\|u_0\|_{L^1}\) for \(t\ne 0\). Similarly the wave propagator satisfies \(\|e^{\pm it|D|}u_0\|_{L^\infty}\le C|t|^{-(n-1)/2}\|u_0\|_{W^{N,1}}\) for sufficient \(N\), and the Klein–Gordon propagator \(\|e^{it\langle D\rangle}u_0\|_{L^\infty}\le C|t|^{-n/2}\|u_0\|_{W^{N,1}}\).
The decay \(|t|^{-n/2}\) (Schrödinger) / \(|t|^{-(n-1)/2}\) (wave) is the spreading of the wave packet — the dispersive estimate. The Strichartz estimates integrate the dispersive decay over time:
Theorem (Keel–Tao 1998, sharp Strichartz). For the Schrödinger propagator, a pair \((q,r)\) with \(2\le q,r\le\infty\) is admissible if \(\frac{2}{q}+\frac{n}{r}=\frac{n}{2}\) and \((q,r,n)\ne(2,\infty,2)\). For admissible \((q,r),(\tilde q,\tilde r)\), $$ |e^{it\Delta}u_0|{L^q_t L^r_x}\le C|u_0|F(s)\,ds\Bigr|},\qquad\Bigl|\int_0^t e^{i(t-s)\Delta{L^q_t L^r_x}\le C|F|, $$ where }_t L^{\tilde r'}_x\(\tilde q',\tilde r'\) are dual exponents. The estimate is sharp: the endpoint \((q,r)=(2,2n/(n-2))\) for \(n\ge 3\) is admissible and the estimate holds (Tao's endpoint theorem); the naive endpoint \((q,r)=(\infty,2)\) fails for Schrödinger. The analogous estimates hold for wave (admissibility \(\frac{2}{q}+\frac{n-1}{r}\le\frac{n-1}{2}\)) and Klein–Gordon.
| Equation | Admissibility condition | Dispersive decay | Endpoint (sharp) |
|---|---|---|---|
| Schrödinger \(e^{it\Delta}\) | \(\frac{2}{q}+\frac{n}{r}=\frac{n}{2}\), \((q,r,n)\ne(2,\infty,2)\) | $ | t |
| Wave $e^{\pm it | D | }$ | \(\frac{2}{q}+\frac{n-1}{r}\le\frac{n-1}{2}\), \((q,r,n)\ne(2,\infty,3)\) |
| Klein–Gordon \(e^{it\langle D\rangle}\) | \(\frac{2}{q}+\frac{n}{r}=\frac{n}{2}\) | $ | t |
Admissible pairs (concrete). Schrödinger \(n=3\): \((q,r)=(2,6),(4,3),(\infty,2)\) (and duals). Wave \(n=3\): \((q,r)=(2,\infty)\) (Strichartz / endpoint excluded for \(n=3\)), \((4,4),(\infty,2)\) (with \(r\ge 2\)). Schrödinger \(n=2\): \((q,r)=(2,\infty)\) excluded, \((4,4),(\infty,2)\). Schrödinger \(n=1\): \((q,r)=(4,\infty),(\infty,2)\) and the sharp \((q,r)=(2,\infty)\) is excluded.
NLS local and global well-posedness¶
The nonlinear Schrödinger equation (NLS) on \(\mathbb{R}^n\):
with the \(+\) sign defocusing, \(-\) focusing. The critical exponents:
| Regime | Critical exponent | Conserved quantity |
|---|---|---|
| Mass-critical | \(p=1+\frac{4}{n}\) | \(\|u\|_{L^2}\) (mass) |
| Energy-subcritical | \(p<1+\frac{4}{n-2}\) (\(n\ge 3\)) | $E(u)=\frac12\int |
| Energy-critical | \(p=1+\frac{4}{n-2}\) (\(n\ge 3\)) | \(E(u)\) |
Theorem (Ginibre–Velo 1979; Tsutsumi 1987, LWP). For \(p<1+\frac{4}{n-2}\) (energy-subcritical), the NLS is locally well-posed in \(H^s\) for \(s\ge s_c=\frac{n}{2}-\frac{2}{p-1}\) (scaling-critical Sobolev index), via Strichartz + Banach fixed-point on the Duhamel formulation $$ u(t)=e^{it\Delta}u_0-i\int_0^t e^{i(t-s)\Delta}|u|^{p-1}u(s)\,ds. $$
Theorem (GWP, subcritical). (i) For \(p<1+\frac{4}{n}\) (mass-subcritical), NLS is globally well-posed in \(L^2\) by mass conservation + LWP + persistence. (ii) For \(p<1+\frac{4}{n-2}\) (energy-subcritical), NLS is globally well-posed in \(H^1\) by mass + energy conservation (the defocusing case has \(E\ge\frac12\|\nabla u\|_{L^2}^2\), giving an a priori \(H^1\) bound; the focusing case needs the coercivity / ground-state threshold of Kenig–Merle).
Scattering and the energy-critical problem¶
The scattering question: does the solution \(u(t)\) asymptotically behave as a free Schrödinger solution, \(u(t)\sim e^{it\Delta}u_+\) as \(t\to\pm\infty\) for some scattering state \(u_+\in H^s\)? For subcritical equations, scattering is established for small data (and large data in defocusing / below threshold); for the critical equations, the problem is at the frontier.
Theorem (Kenig–Merle 2006, energy-critical NLS / wave). For the defocusing energy-critical NLS \(i\partial_t u+\Delta u=|u|^{4/(n-2)}u\) in \(n\ge 3\), and for the defocusing energy-critical wave, any finite-energy solution with \(E(u_0)<E(W)\) (where \(W\) is the ground state / soliton) and \(u_0\notin\{e^{i\theta}W(\cdot-x_0)\}\) is global and scatters (to a free solution). The proof uses the concentration-compactness / rigidity method: a minimal non-scattering solution is shown (via a compactness-rigidity argument and the virial / Morawetz identity) to be a soliton, contradicting the energy assumption.
| NLS regime | Critical exponent | Known global / scattering |
|---|---|---|
| Mass-critical (\(n=2\), \(p=3\)) | \(p=1+4/n\) | small data scattering; large data (focusing) threshold \(E< E(W)\) — Killip–Visan |
| Energy-critical defocusing (\(n=3\), \(p=5\)) | \(p=1+4/(n-2)\) | global + scattering, all \(E\) finite (Kenig–Merle 2006) |
| Energy-critical defocusing (\(n=4\), \(p=3\)) | \(p=1+4/2=3\) | global + scattering (Ryckman–Visan) |
| Energy-critical focusing (\(n=3\)) | \(p=5\) | global + scattering iff \(E<M\) threshold (Kenig–Merle) |
Conjecture (Duyckaerts–Kenig–Merle, soliton resolution). For the energy-critical wave / NLS, every finite-energy solution decomposes as \(u(t)=W_{soliton}(t)+u_{disp}(t)+o_{H^1}(1)\) as \(t\to+\infty\), where \(W_{soliton}\) is a soliton (ground state + Lorentz / Galilean transform) and \(u_{disp}\) scatters to a free solution. Proven in special cases (radial 3D wave, Duyckaerts–Kenig–Merle); open in general.
See it¶
The \(TT^\ast\) proof of the Strichartz estimate for the 3D Schrödinger, including the endpoint. The estimate \(\|e^{it\Delta}u_0\|_{L^q_t L^r_x}\le C\|u_0\|_{L^2}\) for admissible \((q,r)\) with \(\frac{2}{q}+\frac{3}{r}=\frac{3}{2}\) is proven by the \(TT^\ast\) argument: define \(T:L^2_x\to L^q_t L^r_x\) by \(Tf(t)=e^{it\Delta}f\). The desired estimate is \(\|T\|\le C\), equivalently \(\|TT^\ast\|\le C^2\) on \(L^{\tilde q'}_t L^{\tilde r'}_x\to L^q_t L^r_x\). The kernel of \(TT^\ast\) is \(K(t,s;x,y)=\int e^{it|\xi|^2}e^{-is|\eta|^2}e^{i(x\xi-y\eta)}\,d\xi\,d\eta=(4\pi i|t-s|)^{-n/2}e^{i|x-y|^2/(4(t-s))}\) (the free Schrödinger kernel), giving the dispersive estimate \(\|e^{i(t-s)\Delta}\|_{L^{\tilde r'}\to L^r}\le C|t-s|^{-(n/2)(1-2/r-2/\tilde r')}\) (the Hardy–Littlewood–Sobolev / fractional-integration exponent). The \(TT^\ast\) estimate then reduces to the Hardy–Littlewood–Sobolev (HLS) inequality in time: \(\int\int |t-s|^{-\alpha}F(s)G(t)\,ds\,dt\le C\|F\|_{L^{\tilde q'}}\|G\|_{L^{q'}}\) with \(\alpha=1-2/q-2/\tilde q\), which is exactly the HLS condition \(\alpha+(2/q)+(2/\tilde q)=2\) for \(q,\tilde q>2\). The endpoint \((q,r)=(2,2n/(n-2))\) is the borderline where \(q=2\) and the HLS condition is at the limit; the Keel–Tao theorem proves the endpoint by a bilinear interpolation / restricted-interpolation argument (the naive HLS fails at \(q=2\), but the bilinear structure of \(TT^\ast\) with the oscillatory kernel \(e^{i|x-y|^2/4(t-s)}\) recovers the estimate). This is the operational content of "the Strichartz estimate is a consequence of the dispersive decay via \(TT^\ast\) and the HLS inequality, with the Keel–Tao endpoint using the bilinear oscillatory structure to recover the borderline case."
Check¶
Identify the scaling-critical Sobolev index \(s_c\) for the NLS and explain why well-posedness below \(s_c\) is impossible.
The NLS \(i\partial_t u+\Delta u=\pm|u|^{p-1}u\) is invariant under the scaling \(u(t,x)\mapsto u_\lambda(t,x)=\lambda^{2/(p-1)}u(\lambda^2 t,\lambda x)\) (with \(u_\lambda(0)=\lambda^{2/(p-1)}u_0(\lambda\cdot)\)). The \(H^s\) norm scales as \(\|u_\lambda(0)\|_{H^s}=\lambda^{2/(p-1)-n/2+s}\|u_0\|_{H^s}\), and the critical index is \(s_c=\frac{n}{2}-\frac{2}{p-1}\) (where the scaling exponent vanishes). For \(s>s_c\), small data LWP holds (subcritical); for \(s=s_c\), the problem is critical (LWP depends on the equation / data); for \(s<s_c\), LWP in \(H^s\) is impossible — the scaling produces a sequence of data with \(\|u_{0,\lambda}\|_{H^s}\to 0\) as \(\lambda\to\infty\) (if \(s<s_c\)) but the corresponding solutions have the same \(H^s\) norm at time \(\lambda^2 t\), contradicting the local well-posedness estimate \(\|u\|_{H^s}\le C\|u_0\|_{H^s}\) for small times. Hence \(s_c\) is the lower bound for LWP; the energy-critical NLS has \(p=1+4/(n-2)\), giving \(s_c=1\) (the energy space), and the mass-critical has \(s_c=0\) (the \(L^2\) space).
Why does the Kenig–Merle scattering theorem require the energy threshold \(E(u_0)<E(W)\), and what is \(W\)?
\(W\) is the ground state / soliton — the unique (up to symmetries) positive solution of the stationary NLS \(-\Delta W+|W|^{p-1}W=0\) (the elliptic equation for a static solution of the NLS). \(W\) has a specific energy \(E(W)\); solutions with \(E<E(W)\) are "below the soliton threshold." The necessity of the threshold: above \(E(W)\), the focusing energy-critical NLS has blow-up solutions (solutions that concentrate and become singular in finite time), so scattering cannot hold; the threshold \(E(W)\) is the energy of the borderline soliton. The sufficiency: the Kenig–Merle theorem proves that below \(E(W)\), no blow-up occurs and the solution scatters; the proof uses the concentration-compactness / rigidity (a minimal non-scattering solution must be a soliton by compactness, contradicting \(E<E(W)\) and the exclusion of the soliton). Hence the threshold is essential — it is the sharp energy below which scattering holds and above which blow-up can occur.
Pitfalls¶
- Strichartz endpoint is sharp (Keel–Tao): the naive endpoint \((q,r)=(\infty,2)\) fails for Schrödinger; the correct endpoint is \((q,r)=(2,2n/(n-2))\) via the bilinear argument. Citing the failed endpoint is the standard error.
- Admissibility differs per equation: Schrödinger \(\frac{2}{q}+\frac{n}{r}=\frac{n}{2}\), wave \(\frac{2}{q}+\frac{n-1}{r}\le\frac{n-1}{2}\) — do not transfer the condition between equations.
- Energy-critical scattering needs the ground-state threshold \(E<E(W)\): the theorem is sharp; above the threshold, blow-up occurs (focusing case), and scattering is false. The threshold is essential, not a technicality.
- Soliton resolution is conjectural in general: proven in special cases (radial 3D wave), open for general data / dimensions / equations. Cite the specific proven case, not the general conjecture as a theorem.
- Mass-critical vs energy-critical: the two critical regimes have different thresholds and different proofs (Killip–Visan for mass-critical, Kenig–Merle for energy-critical); conflating them is the standard error.
- Strichartz does not give scattering alone: it gives LWP / GWP; scattering requires the additional concentration-compactness / rigidity (Kenig–Merle) or the test-function / Morawetz method.
Refs¶
- Keel–Tao, Endpoint Strichartz estimates (AJM 1998)
- Tao, Nonlinear Dispersive Equations (CBMS 2006); Sogge, Hangzhou lectures on nonlinear wave equations
- Ginibre–Velo, On a class of nonlinear Schrödinger equations (JFA 1979/1985); Tsutsumi, \(L^2\)-solutions for NLS (1987)
- Kenig–Merle, Global well-posedness and scattering for the energy-critical wave / NLS (Annals / Invent. 2006); Duyckaerts–Kenig–Merle (soliton resolution, series)
Next¶
Elliptic Theory · Navier–Stokes · Calculus of Variations · PDE index · Foundations · Fourier analysis · Mathematical physics · Dynamical systems · FA / restriction & Strichartz