Progress in nonlinear wave equations

Terence Tao (UCLA)

Aug 24, 2005 Some nonlinear wave equation models, in increasing order of nonlinearity:

• Free wave equation □u = 0. (□ = ∂^{α}∂_{α} = −∂_{t}^{2}+Δ.)

• Semilinear wave equation (NLW) □u = ±|u|^{p−1}u. (+ is defocusing, − is focusing.)

• Wave maps equation (WM) (ϕ^{∗}∇)^{α}∇_{α}ϕ = 0, ϕ ∈ M. (−ve curvature is defocusing, +ve curvature is focusing.)

• Maxwell-Klein-Gordon equation (MKG) D^{α}D_{α}ϕ = 0; ∂^{α}F_{αβ} = Im(ϕ\overline{D_{β}ϕ}).

• Yang-Mills equation (YM) D^{α}F_{αβ} = 0

• Einstein equation Ric = 0.

These equations are compatible with either special and general relativity, and have a very strong geometric structure (Lagrangian/Hamiltonian formulations, conserved stressenergy tensor, gauge invariances and other symmetries, null structure in nonlinearity, etc.). Will focus primarily on the middle four equations. Won’t focus on details.

A major problem of interest is the Cauchy problem for smooth data: given smooth initial data (position and velocity) at t = 0, how does the solution evolve for later times? Typical results, in roughly increasing order of difficulty (from easy to impossible):

(1) Local existence and uniqueness of smooth solutions

(2) Global regularity and asymptotics for small, compactly supported data

(3) Blowup criteria (which norms become infinite when singularity occurs?)

(4) Global regularity for large data with subcritical defocusing conservation law

(5) Global regularity and asymptotics for small critical data

(6) Global regularity and asymptotics for large data with critical defocusing conservation law

(7) ? Blowup profiles

(8) ??? Large data asymptotics with subcritical defocusing conservation law (9) ?????? Global regularity for large data with super-

critical defocusing conservation law (e.g. Navier-Stokes regularity, cosmic censorship)

Much recent progress on items (4) and (5). Regularity problems are also closely connected with optimal lowregularity well-posedness questions.

The techniques we have can be divided into three categories:

• Perturbative techniques: Approximate the nonlinear equation by a simpler equation (usually the free wave equation; more recently, other linearized equation have been used). Deal with the error by iteration, bootstrap, continuity arguments, etc. Needs data or time interval to be small, or a certain spacetime norm to be finite (and hence locally small). Iteration requires a carefully chosen function space and a good set of linear and nonlinear estimates (e.g. energy estimates, dispersive estimates, Strichartz estimates, null form estimates, geometric decay and regularity estimates). Our understanding of perturbation theory is quite satisfactory.

• Non-perturbative techniques: Conservation laws (primarily energy conservation), monotonicity formulae (e.g. Morawetz inequalities), symmetries and transformations (including gauge transformations), normal forms. We need more non-perturbative techniques. (Hamiltonian dynamics? Variational methods? Stochastic methods? Compactness/penalization methods? Complete integrability?)

• Induction on energy: work on the border between the perturbative and non-perturbative regime! (Closest thing we have to a variational method.)

The role of scaling

• In all the model equations, there is a conserved energy, and also a conserved scaling. The scaling gives a relationship between the evolution of high-frequency data, and the evolution of low-frequency data. The conservation law controls how large the solution can be at both high and low frequencies. The precise relationship between the two strongly influences the evolution.

• In subcritical equations, the conservation law is stronger at high frequencies than at low frequencies. Since low frequencies evolve slowly, we thus expect the solution to behave linearly for short times and nonlinearly for large times.

• In critical equations, the conservation law is scale invariant, so high and low frequencies are equally nonlinear for a fixed amount of energy. We thus expect the solution to behave linearly for all time when the energy is small, but exhibit borderline nonlinear behaviour when the energy is large. Controlling this behaviour is just barely within current technology.

• In supercritical equations, the conservation law is weaker at high frequencies, which thus behave very nonlinearly even at small energies. One expects very unstable and turbulent behaviour, unless one imposes smallness in much stronger norms than the energy. For large data essentially no positive results known for nonlinear wave equations here.

• Note: for the Einstein equations, scaling does not seem to be the dominant obstruction to regularity; stability of the geometry (light rays, null surfaces) seems to be a more important issue.

The defocusing critical NLW

• The first critical equation to have a satisfactory theory was the defocusing energy-critical NLW □u = |u|^{\frac{4}{n−2}}u, where n ≥ 3 and u : R^{1+n} → R. Worked out by many people (Rauch, Struwe, Grillakis, Shatah, Bahouri, Gerard, Nakanishi, …).

• Conserved energy is ∫R^{n} \frac{1}{2}|∇u|^{2}+\frac{1}{2}|∂_{t}u|^{2}+\frac{n−2}{n+2}|u|^{(n+2)/(n−2)} dx, and is invariant under the scaling u(t, x) ↦ \frac{1}{λ^{(n−2)/2}}u(\frac{t}{λ}, \frac{x}{λ}).

• Perturbative theory (Strichartz estimates) shows that for large energy data, one has regularity as long as the potential energy does not concentrate, and asymptotics as long as the potential energy decays.

• Non-perturbative theory (Morawetz estimates) shows that the potential energy does not concentrate, and decays at infinity.

• In the focusing case □u = −|u|^{\frac{4}{n−2}}u, blowup can occur for large energy. Still open: asymptotic description of blowup profile (e.g. for radial data).

Wave maps

• After NLW, the next easiest model to analyze has been the wave map equation (ϕ^{∗}∇)^{α}∂_{α}ϕ = 0, where ϕ : R^{1+n} → M^{m} takes values in an m-dimensional Riemannian manifold. Here ∇ is the Levi-Civita connection on M, and ϕ^{∗}∇ is the pullback of that connection to R^{1+n}. Negative curvature M (e.g. H^{m}) is analogous to defocusing NLW, positive curvature (e.g. S^{m}) analogous to focusing NLW.

• This equation is subcritical for n = 1, critical for n = 2, and supercritical for n ≥ 3. Global regularity known in n = 1 (Ladyshenskaya, Shubov, Shatah, Gu) and blowup examples known for n ≥ 3 (Shatah, Tahvildar-Zadeh, Cazenave), though global solutions still exist for certain small data (Choquet-Bruhat, Sideris). The n = 1 case is also completely integrable. Now we focus on the critical theory for n = 2, which is still only partially complete.

• With equivariant U(1) symmetry, global existence for large energy data with negative curvature targets is known (Shatah, Struwe, Tahvildar-Zadeh, Grillakis), as one can use arclength co-ordinates to essentially transform the problem to the critical NLW.

• With radial symmetry, global existence for large energy data with both negative and positive curvature targets is known (Christodoulou, Tahvildar-Zadeh, Struwe) using the method of characteristics and exploiting some identities related to the n = 1 complete integrability.

• Without symmetry, perturbation theory (X^{s,b} estimates) was used to obtain a sequence of refinements for a blowup criterion (Klainerman, Machedon, Selberg, Tataru), culminating in a scale-invariant Besov norm blowup criterion. By combining these methods with a gauge renormalization, an energy blowup criterion was developed: small energy implies regularity, and blowup only occurs when energy concentrates (Tao, Klainerman, Rodnianski, Nahmod, Uhlenbeck, Stefanov, Shatah, Struwe, Tataru, Krieger).

• A new difficulty: the energy space H^{1}(R^{2}) does not control L^{∞}(R^{2}) and is not closed under multiplication. This makes the space unstable with respect to co-ordinate changes. Thus the choice of co-ordinate system is important.

• Geometric insight (Klainerman-Rodnianski, Nahmod-Uhlenbeck-Stefanov, Shatah-Struwe): work at the level of the derivative field ∂_{α}ϕ and use an orthonormal frame bundle rather than a co-ordinate system. This introduces a O(m)-gauge invariance. Choose the gauge to make the nonlinear terms as small as possible. Key observation: curvature of connection ϕ^{∗}∇ is quadratic in the fields ∂_{α}ϕ rather than linear, and thus smaller than what one naively expects. Use gauge to pass from control of curvature to control of connection.

• Three useful gauges known. Microlocal gauge: constructed using Littlewood-Paley theory. Complicated, only seems to work for small energies. Coulomb gauge: simple, but not as effective in low dimensions; also some technical issues with large energies. Caloric gauge: constructed using harmonic map heat flow. Intrinsic; works at small energies and also large energies if curvature is negative. More nonlinear than the other gauges.

• Still open: global regularity for large energies with negative curvature targets. But some progress. Caloric gauge allows for a perturbative large energy theory. Morawetz-type estimates are known (wave maps relax to harmonic maps on average when approaching a singularity). The induction-on-energy strategy of Bourgain (focus attention on a putative minimalenergy blowup solution) should in principle work, but will be very nonlinear and messy.

• For positive curvature, blowup for large data suspected numerically (Bizon, Chmaj, Tabor, Isenberg, Liebling) but not proven. Likely that blowup profile approaches one or more harmonic maps (as is the case with harmonic map heat flow), but nowhere near proving this with current technology.

Maxwell-Klein-Gordon (MKG)

• MKG has two fields, an electromagnetic potential field A_{α} : R^{1+n} → R (which induces a connection D_{α} := ∂_{α} + iA_{α}) and a charge field ϕ : R^{1+n} → C. The equations are D^{α}D_{α}ϕ = 0 and ∂^{α}F_{αβ} = Im(ϕ\overline{D_{β}}ϕ). Of the two equations, the first is more nonlinear than the second.

• Subcritical in n ≤ 3, critical in n = 4, supercritical for n > 4. Global regularity known for n ≤ 3 (Eardley, Moncrief, Klainerman, Machedon). Blowup suspected for n > 4. The n = 4 theory is still very primitive.

• U(1) gauge invariance. Conventional wisdom is to use Coulomb gauge divA = 0, though other gauges are also useful (Lorentz, temporal, Cronstrom). Unlike wave maps, curvature is not a quadratic quantity, and thus curvature and connection will have nontrivial size. However, curvature and connection will obey a wave (or Maxwell) equation.

• Blowup criterion for general n: H^{n/2−1+ε}(R^{n}) norm of solution must go to infinity. (Klainerman, Machedon, Tataru, Sterbenz, Selberg). Refined to a scaleinvariant Besov criterion for n ≥ 5 (Sterbenz). Obtained by perturbation theory (refined X^{s,b} space estimates), approximating MKG equations in Coulomb gauge by free wave equation.

• In n ≥ 6, small H^{n/2−1}(R^{n}) norm is now known to imply regularity (Rodnianski, Tao). Main idea: study the covariant wave equation D^{α}D_{α}ϕ = 0 directly rather than approximate it by free wave equation. In particular, obtain a good parametrix for the fundamental solution and derive dispersive estimates.

• Parametrix construction uses method of distorted plane waves. For each direction ω, apply a Cronstrom gauge in direction ω to eliminate the ω-null component of connection, and then propagate plane waves in direction ω. Full parametrix then obtained by principle of superposition. Dispersive estimates require method of stationary phase and need good control of the Cronstrom gauges as function of ω.

• In principle the method extends to n = 5 (needs endpoint dispersive estimates). The important n = 4 case (i.e. small energy implies regularity) needs new ideas (covariant X^{s,b} spaces?).

Yang-Mills

• Yang-Mills is a non-abelian version of Maxwell-Klein-Gordon, without the matter field ϕ. The connection A_{α} : R^{1+n} → 𝔤 takes values in a Lie algebra, and the equation is D^{α}F_{αβ} = 0.

• As with MKG, the n = 4 case is critical, n < 4 subcritical, n > 4 supercritical. Blowup solutions constructed for n = 5, 7, 9 (Cazenave, Shatah, Tahvildar-Zadeh)

• A non-abelian gauge group G. For small H^{n/2−1}(R^{n}) data, the Coulomb gauge divA = 0 is available (Uhlenbeck).

• Perturbative theory for YM is almost identical to MKG.

• Very recent result: small H^{n/2−1}(R^{n}) norm implies regularity for n ≥ 6 (Krieger, Sterbenz). Follows the Rodnianski-Tao approach; major new difficulty due to non-abelian gauge group, which makes Cronstrom gauge nonlinear. Establishing stability of the gauge in ω is a significant technical challenge.

• Future goal: small energy implies regularity for 4D Yang-Mills. Already known for Yang-Mills heat flow

and for elliptic Yang-Mills.

• Far future goal: blowup profile of large energy 4D Yang-Mills: bubbling off into instantons suspected.