Global well-posedness for non-linear wave equations below the energy norm

Mark Keel (Caltech), Terence Tao (UCLA)

     www.math.ucla.edu/∼tao/preprints

Abstract

Traditionally, global well-posedness and regularity for the Cauchy problem for non-linear wave equations with large data has only been achieved when (a) there is a conserved Hamiltonian which is essentially positive definite, (b) the data has finite energy, and (c) the energy regularity H^{1} is sub-critical or critical.

Recent arguments of Bourgain have shown how to relax condition (b) and obtain global well-posedness results for data which has regularity somewhere between energy and critical (so it has infinite energy). His arguments rely on the non-linear portion of the equation to be smoother than the linear portion.

We develop new arguments which give similar results to those of Bourgain, but do not require any smoothing. In particular, we obtain new global well-posedness results for the 3-D MKG and 1-D wave map equations.

Introduction

• Consider the Cauchy problem for a general semilinear wave equation

□u = F (u, ∇u, u_{t})
u(x, 0) = f(x)
u_{t}(x, 0) = g(x)

where □ = \frac{∂^{2}}{∂t^{2}} −Δ is the wave operator, u is a scalar or vector field, and F is some smooth combination of u, ∇u, and u_{t}. Examples of these equations include the wave map to the sphere

□u = −u(|u_{t}|^{2} − |∇u|^{2})

or the Maxwell-Klein-Gordon equation in Coulomb gauge, which looks roughly like

□Φ = Φ∂Φ + Φ^{3}

(plus an additional elliptic equation for the scalar potential A_{0}, which I’ll ignore).

• A basic question in the study of these equations is whether the Cauchy problem is globally well-posed for certain classes of data. In other words, we seek to find conditions on f, g such that the solution u exists for all time and is unique, has the same regularity as the initial data, and depends continuously on the data f, g.

• Iteration methods usually give local well-posedness providing that f, g is in a certain regularity class, say in a Sobolev space H^{s} × H^{s−1}. Such methods are only available if s is above the critical regularity s_{c}, which is the unique regularity such that the Ḣ^{s} norm is dimensionless. (the dimensionality of u can be deduced from the equation). In the sub-critical case, this gives a time of existence which has some inverse relationship with H^{s}×H^{s−1} norm of the data. (Things are more subtle in the critical case, and the time of existence depends on how concentrated the data is at a point).

• However, iteration methods cannot by themselves give global well-posedness, unless the data is very small (in which case one can hope to exploit some decay properties of the wave equation). Indeed, it is easy to construct semi-linear wave equations which blow up in finite time for large data (basically they are variants of the ODE u̇ = u^{p}, which blows up in finite time for p > 1).

• The standard way to prevent blow-up and to ensure global well-posedness for large data is to exploit some sort of conservation law, such as a conserved Hamiltonian. This is a quantity H(u(t)), which roughly has the form

H(u(t)) ≈ ∫ \frac{1}{2}|∇u(t)|^{2} + \frac{1}{2}|u_{t}(t)|^{2} dx

(in practice, a potential energy term may also be added, or the derivatives may need to be replaced with covariant derivatives) which satisfies the conservation law

\frac{d}{dt}H(u(t)) = ∫ u_{t}(t)(□(u) − F (u, ∇u, u_{t})) dx.

Thus, if u solves the above Cauchy problem, then H(u(t)) is independent of time.

• If the initial data is in energy class H^{1} × L^{2}, then H(u(0)) should be finite, and so H(u(t)) stays finite in time. If there is enough positivity in the Hamiltonian, this should imply that the H^{1} norm of u does not blow up, and if the H^{1} norm is sub-critical, this is usually enough to imply global well-posedness for energy-class data. (For H^{1}-critical problems things are more subtle, and one must demonstrate nonconcentration before global well-posedness can be attained).

• However, this leaves open the issue of what happens with data which is not in the energy class, so that H(u(t)) is infinite. In these cases one can still have local well-posedness (if the data is in some subcritical regularity class H^{s}), but it appears that the conservation of the Hamiltonian is unavailable to treat this case.

• Physically, data which is only in H^{s} for s < 1 still has finite frequency in the low frequencies, but may contain infinite energy in the high frequencies. There may be an infinite transfer of energy from the high frequencies to the low frequencies in finite time, which can cause blow-up.

• Recently, Bourgain discovered a way to demonstrate that this infinite transfer of energy cannot occur if s is sufficiently close to 1, providing that there is some smoothing effect arising from the non-linearity. To be more precise about this, let us rewrite the Cauchy problem in integral form as

u = S(t)(f, g) + □^{−1}F (u, ∇u, ∂_{t}u)

where S(t) is the free wave evolution with initial data f, g, and □^{−1} is the fundamental solution for the inhomogeneous wave equation.

• If f, g is only in the regularity class H^{s} × H^{s−1} for some s < 1, then the free evolution S(t)(f, g) is also only in the regularity class H^{s} (the wave equation is time-reversible with finite speed of propagation, and thus cannot possibly have any smoothing effects). However, the non-linear component □^{−1}F (u, ∇u, ∂_{t}u) may be smoother than H^{s}; for instance, it may remain in energy class H^{1} even if u is merely in H^{s}. This is because of the smoothing effects of the □^{−1} operator.

• The □ operator is second order, but is not elliptic, and can degenerate to first order in certain directions. (As a rough caricature, think of □ as (∂_{t} − ∂_{x})(∂_{t} + ∂_{x})). As such, □^{−1} ideally has two degrees of smoothing, but in most cases only smoothes out one degree of regularity at best. This can also be seen by the explicit formula

□^{−1}F (t) = ∫_{s<t} \frac{sin((t − s)\sqrt{−Δ}}{\sqrt{−Δ}}F (s) ds

for □^{−1}. In particular, if F depends only on u and not on the first-order derivatives of u (so F = F (u) rather than F = F (u, ∇u, u_{t})), then one expects (ideally) that □^{−1}F (u) is about one derivative smoother than u. Thus one expects to be able to control □^{−1}F (u) in the energy class even if u is only in some weaker space such as H^{1−ε}.

• When there are derivatives in the non-linearity, such smoothing is generally not available; for instance, the operator □^{−1}u_{t} can easily be seen to have essentially the same regularity as u in the worst-case scenario. There are some exceptional null-forms, e.g. P (u × ∇u), where P is the projection onto divergence-free operators, which do have this smoothing effect, but most non-linearities with derivatives do not have this smoothing.

• Bourgain produced a method which generates global well-posedness results for data in H^{s} for s slightly below 1, as long as the energy norm is sub-critical and there is smoothing in the non-linearity, in the sense that □^{−1}F (u, ∇u, u_{t}) is in H^{1} even when u is only in H^{1−ε}.

• This method was initially demonstrated for the nonlinear Schrödinger equation in two dimensions, and the non-linear Klein-Gordon equation in one dimension, but Kenig, Ponce, and Vega extended it to the non-linear wave equation □u = u^{3} in three dimensions, obtaining global well-posedness in H^{s} for all s > 3/4. (For comparison, the critical regularity here is 1/2).

Outline of Bourgain’s method

• We now demonstrate Bourgain’s method. For concreteness we shall use the equation

□u = u^{3}; u(0) = f

and ignore the initial velocity g. We let s be a number just slightly smaller than 1 (in fact, 3/4 < s < 1 will work).

• Bourgain’s method proceeds as follows. Let u be a solution to the Cauchy problem with initial data f in H^{s}; say ∥f∥_{H^{s}} = 1. From the local theory, it suffices to show that the H^{s} norm of the solution does not blow up in finite time. Accordingly, let us fix a large time T ≫ 1, and try to show that ∥u(t)∥_{H^{s}} stays finite for all 0 ≤ t ≤ T.

• Let N ≫ 1 be a large number to be chosen later. We divide the initial data f into two pieces f_{low} + f_{high}, where f_{low} consists of frequencies ≲ N and f_{high} consists of frequencies ≳ N. (This is easily done via the Fourier transform and some cutoff functions). The advantage of this decomposition is as follows. The function f_{low}, being band-limited, is infinitely smooth. In particular, it is in energy class (but with large energy):

∥f_{low}∥_{H^{1}} ≲ N^{1−s}.

Since the Hamiltonian H(f_{low}) is basically the square of the H^{1} norm, we thus have

H(f_{low}) ≲ N^{2(1−s)}.

• The remaining term f_{high} does not have finite energy. However, it is still bounded in H^{s}, ∥f_{high}∥_{H^{s}} ≲ 1, and if one measures it in even rougher spaces, e.g. L^{2}, then the norm is even smaller:

∥f_{high}∥_{L^{2}} ≲ N^{−s}∥f_{high}∥_{H^{s}} ≲ N^{−s}.

• We now split the Cauchy problem into two parts, u = u_{low} + u_{high}, where u_{low}, u_{high} evolve according to the equations

□u_{low} = u_{low}^{3}, u_{low}(0) = f_{low}

□u_{high} = 3u_{high}u_{low}^{2}+3u_{high}^{2}u_{low}+u_{high}^{3}, u_{high}(0) = f_{high}. One can easily verify that if u_{low} and u_{high} solve the above equations, then u = u_{low} + u_{high} solves the initial problem.

• Consider the u_{low} equation. This is exactly the same equation that we started with, except that now we have a finite Hamiltonian, and so this equation exists globally for all time. Indeed, H(u_{low}(t)) ≲ N^{2(1−s)} for all time.

• Now look at the u_{high} equation. All the terms in the non-linearity are first-order in u_{high} or better. Let us focus on the first-order term; the others are easier to handle. We rewrite this equation in integral form as

u_{high}(t) = S(t)f_{high} + □^{−1}(3u_{high}u_{low}^{2} + …).

If it were not for the non-linear term □^{−1}(3u_{high}u_{low}^{2}+ …) then this equation would clearly have global solutions (although they would not be in energy class, since f_{high} is not). To control the effect of this nonlinearity we use smoothing effects.

• The ”standard” estimate for this type of non-linearity is

∥□^{−1}(u_{high}u_{low}^{2})(t)∥_{H^{1}} ≲ ∥f_{high}∥_{H^{1}}∥f_{low}∥_{H^{1}}∥f_{low}∥_{H^{1}}+…

for t sufficiently small; this is proven using Strichartz estimates. This is basically the estimate used to show that the equation □u = u^{3} is locally well-posed in H^{1}. However, if one looks a little closer, one can relax the regularity requirements on the right-hand side, and one has a stronger estimate

∥□^{−1}(u_{high}u_{low}^{2})(t)∥_{H^{1}} ≲ ∥f_{high}∥_{L^{2}}∥f_{low}∥_{H^{1}}∥f_{low}∥_{H^{1}}+…

providing t is sufficiently small. (Because the H^{1} norm of f is so large, it turns out that the correct range of t is t = O(N^{−2(1−s)})). We thus see that u_{high}(t) differs from the free evolution S(t)f_{high} by something in energy class, at least for small times t:

∥u_{high}(t)−S(t)f_{high}∥_{H^{1}} ≲ N^{−s}N^{1−s}N^{1−s} for t = O(N^{−2(1−s)}).

The key point is that the energy of this non-linear effect is rather small if s is close to one; in fact, it is about N^{−1}. In particular, it is small when compared against the energy of the low frequencies u_{low}, which is like N^{s−1}.

• Bourgain then iterates this scheme for about T N^{2(s−1)} steps, enough to get up to time T. The errors between u_{high} and the free evolution accumulate, to have a total energy of about

T N^{2(1−s)}N^{−s}N^{1−s}N^{1−s} = T N^{−1}N^{5(1−s)}.

This error can be absorbed into the low frequency evolution, which has energy about N^{1−s}, so long as

T N^{−1}N^{5(1−s)} ≪ N^{1−s}.

This is possible by choosing N large enough, as long as s > 3/4.

• Similar schemes work for other equations with smoothing in the non-linearity, but the numerology differs from equation to equation.

• Heuristically, the idea behind this kinds of schemes is that the non-linear effects of the very high frequencies |ξ| ≫ N are negligible, due to the smoothing effects of the non-linearity. So the high frequencies essentially behave like the free evolution, and it is only the low frequencies which are evolving in a nonlinear manner. But the low frequencies have finite (large) energy, so we have global existence.

Equations without smoothing

• Now let us consider an equation which does not have any smoothing. For simplicity, let us consider a cartoon equation such as

□u = u∂u; ∥f∥_{H^{s}} = 1.

(This is a highly oversimplified model of equations of Maxwell-Klein-Gordon and Yang-Mills type).

• There is a derivative in the non-linearity. Since □^{−1} only smoothes one derivative, we expect □^{−1}(u∂u) to have exactly the same regularity as u. This is just right for proving local well-posedness, but there is no regularity “left over” to apply Bourgain’s argument. So we seem stuck when it comes to proving global well-posedness in H^{s} for any s < 1.

• To see how to proceed, suppose as a toy model that we were trying to prove global well-posedness for the free wave equation □u = 0 in H^{s} for some s < 1, and that the H^{s} norm remained conserved. From energy conservation one could presumably show global wellposedness in H^{1}, with the H^{1} norm being conserved.

• To finish the argument, one would introduce an integration operator I which is a Fourier multiplier with symbol something like (1 + |ξ|)^{s−1}. This operator smoothes out H^{s} data to H^{1} data.

• Since I commutes with □, we see that if u solves the free wave equation, then Iu also solves the free wave equation. If u has H^{s} initial data, then Iu has H^{1} initial data, and thus we have global wellposedness for Iu with conserved H^{1} norm. Inverting the I operator we thus obtain H^{s} conservation, and global well-posedness for u.

• The fact that I commutes with the equation represents the fact that the free wave equation contains no frequency interactions; the high frequencies and low frequencies act independently of each other. Of course, for non-linear equations this is no longer the case, but one may hope to show that operators like I approximately commute with the equation. This turns out to be the case even when there is no smoothing in the equation, providing that I is chosen to be sufficiently “mild”.

• More precisely, let N ≫ 1 as before, and let I be a Fourier multiplier whose symbol is identically 1 for |ξ| ≲ N and is essentially (N/|ξ|)^{1−s} for |ξ| ≫ N, with a smooth interpolant in between. For N = 1 this is basically the same operator as considered previously, but for large N this operator is far milder, as it only does something non-trivial to the very high frequencies |ξ| ≫ N.

• Suppose the initial data f is in H^{s} with ∥f∥_{H^{s}} = 1, and we want to keep the H^{s} norm of a solution u under control up to some time T. By an a priori argument exploiting the local well-posedness theory we may assume that u is smooth.

• Since I is smoothing, If will be in H^{1}. However, the H^{1} norm of If is rather large, about N^{1−s}. In particular, the Hamiltonian H(If) will be quite large. To avoid this problem we shall scale the data by some large λ ≫ 1, exploiting some scale invariance f_{λ}(x) = λ^{−1}f(x/λ); u_{λ}(x, t) = λ^{−1}u(x/λ, t/λ) of the equation. By an appropriate λ (depending on N, s) we can make If_{λ} small in H^{1}, so that H(If_{λ}) is also small. (This is possible because the energy norm is sub-critical). The cost of this scaling is that now we need to get existence up to λT rather than just to T.

• When u solved the free wave equation, we saw that Iu also solved the free wave equation, since I commutes with □. For the non-linear wave equation □u = u∂u, this is not quite true:

□(Iu) = I(u∂u) ≠ (Iu)∂(Iu).

However, since I is fairly close to the identity, we expect that Iu approximately solves the correct equation. In particular, we expect H(Iu(t)) to be somewhat constant in time.

• To make this more rigorous, we differentiate H(Iu(t)):

\frac{d}{dt}H(Iu(t)) = ∫ Iu_{t}(□Iu − Iu∂Iu) dx.

since I commutes with □, and □u = u∂Iu, we are thus left with

\frac{d}{dt}H(Iu(t)) = ∫ Iu_{t}(I(u∂u) − (Iu)(I∂u)) dx.

From the fundamental theorem of calculus we thus have

H(Iu(1)) = H(Iu(0))+∫_{0}^{1} ∫ Iu_{t}(I(u∂u)−(Iu)(I∂u)) dxdt.

It is thus of interest to get good bounds on the commutator expression

I(u∂v) − (Iu)(I∂v),

whenever Iu, Iv are in the energy class. Since Iu_{t} is presumably only in L^{2}, we probably want L^{2} type bounds on this quantity. (In practice, one actually uses hyperbolic Sobolev spaces such as H^{0,−1/2+}, but we shall gloss over this complication).

• Standard estimates would then give

∥I(u∂v)−(Iu)(I∂v)∥_{L^{2}} ≲ ∥Iu(0)∥_{H^{1}}∥Iv(0)∥_{H^{1}}+…;

however this estimate is not good enough, as it means that the Hamiltonian might grow by about H(Iu(0))^{3/2} after time 1. This type of growth blows up vary rapidly. To do better we need to get some decay in N, e.g.

∥I(u∂v)−(Iu)(I∂v)∥_{L^{2}} ≲ N^{−1/2+}∥Iu(0)∥_{H^{1}}∥Iv(0)∥_{H^{1}}+….

This would show that, if the Hamiltonian of Iu started off small, that it only grows by about O(N^{−1/2+}) over a unit time interval. Iterating this, one can hope to keep the Hamiltonian under control for times O(N^{1/2−}). This can be made to exceed the target time of λN if s is sufficiently close to 1, by some numerology (which turns out, unsurprisingly, to be exactly the same numerology involved in Bourgain’s argument).

• In Bourgain’s argument, this additional type of decay was gained by lowering the regularity on the right hand side of the standard estimate. However, here we encounter a problem because regularity can only be lowered on one type of index. It is possible to lower the u regularity, say to

∥I(u∂v)−(Iu)(I∂v)∥_{L^{2}} ≲ ∥Iu(0)∥_{H^{1/2+}}∥Iv(0)∥_{H^{1}}+…

but one cannot lower the regularity on v. This is because of the derivative on v; Iv needs to be in H^{1} at least in order for I∂v, and then (Iu)(I∂v), etc., to be in L^{2}. Indeed one can easily construct counterexamples that show that any estimate with H^{1} replaced by a lower regularity on Iv is false, even when u is a bump function.

• Let’s see how this handicap obstructs us from getting the desired decay on our commutator estimate. We split u = u_{low} + u_{high}, where u_{low} and u_{high} are the portions of u at frequencies ≲ N and ≳ N respectively. Similarly split v = v_{low} + v_{high}. We then have four terms to consider:

∥I(u_{low}∂v_{low}) − (Iu_{low})(I∂v_{low})∥_{L^{2}}
∥I(u_{high}∂v_{low}) − (Iu_{high})(I∂v_{low})∥_{L^{2}}

∥I(u_{high}∂v_{high}) − (Iu_{high})(I∂v_{high})∥_{L^{2}} ∥I(u_{low}∂v_{high}) − (Iu_{low})(I∂v_{high})∥_{L^{2}}

• The first term is potentially troublesome, because the u_{low} and v_{low} are quite large, and have no decay in N. Fortunately, this term vanishes completely, because I is the identity on low frequencies. (Note that u_{low}∂v_{low} is the product of two low-frequency waves and is thus also low frequency).

• The second term can be dealt with by the improved estimate above:

∥I(u_{high}∂v_{low}) − (Iu_{high})(I∂v_{low})∥_{L^{2}}
≲ ∥Iu_{high}(0)∥_{H^{1/2+}}∥Iv_{low}(0)∥_{H^{1}}.

We can now get the desired decay N^{−1/2+} since

∥Iu_{high}(0)∥_{H^{1/2+}} ≲ N^{−1/2+}∥Iu_{high}(0)∥_{H^{1}}.

This can also be used to deal with the third term.

• The fourth term, however, is problematic. We can lower the regularity on Iu_{low} to H^{1/2+} instead of H^{1}, but this does not earn us the decay in N, because Iu_{low} is low frequency rather than high frequency. (This is the lack of smoothing which blocks the original approach of Bourgain for this equation).

• To deal with this term we exploit the cancellation between the two terms in the commutator. Informally, we use the approximation

I(u_{low}∂v_{high})−(Iu_{low})(I∂v_{high}) = O((∇^{θ}Iu_{low})(∇^{−θ}I∂v_{high}))

for all 0 ≤ θ ≤ 1.

• This approximation is related to the fractional Leibnitz rule. Let us verify it in the special case u_{low}(x) = ce^{ix·ξ}, ∂v_{high} = c′e^{ix·η}, where |ξ| ≪ N and |η| ≫ N. The right-hand side works out to

(N/|ξ + η|)^{1−s}u_{low}∂v_{high} − (N/|η|)^{1−s}u_{low}∂v_{high}

and the left-hand side becomes

|ξ|^{θ}|η|^{−θ}(N/|η|)^{1−s}u_{low}∂v_{high}.

The claim then follows from the approximation

|ξ + η|^{s−1} − |η|^{s−1} = (\frac{|ξ|}{|η|})^{θ}|η|^{s−1}

which follows from the mean-value theorem and the fact that |η| ≫ |ξ|. (The general case is basically the same computation but goes through Plancherel’s theorem).

• This fractional Leibnitz rule allows us to borrow regularity from the low term u_{low} and place it on the high term v_{high}:

∥I(u_{low}∂v_{high}) − (Iu_{low})(I∂v_{high})∥_{L^{2}}

≲ ∥(∇^{1/2−}Iu_{low})(∇^{−1/2+}I∂v_{high})∥_{L^{2}}.

• Using the same estimate as before, this is bounded

by ∥∇^{1/2−}Iu_{low}∥_{H^{1/2+}}∥∇^{−1/2+}Iv_{high}∥_{H^{1}}.

• The first factor is basically the H^{1} norm of Iu_{low}, while the second term is basically the H^{1/2+} norm of Iv_{high}. We have effectively moved the H^{1} → H^{1/2+} gain of regularity from the u function to the v function. Since v is high frequency, we now get the desired decay in N.

• All terms arising from the derivative of the Hamiltonian H(u(t)) are now taken care of, and we have long-time control on the Hamiltonian. (This technique is reminiscent of some arguments of Bourgain and Staffilani in the context of growth of high Sobolev norms for the modified KdV equations).

Specific applications.

• We have applied the above scheme to several specific equations: 1D wave maps, Maxwell-Klein-Gordon, and the KdV equation.

• 1D wave maps are defined as maps ϕ : R^{1+1} → M to some smooth Riemannian manifold (M, g) which are critical points of the Lagrangian

∫ |∇u|_{g}^{2} − |u_{t}|_{g}^{2} dxdt.

In the special case when M is a sphere, the Euler-Lagrange equation takes the simple form

□u = −u(|u_{t}|^{2} − |∇u|^{2}).

• Global well-posedness for smooth solutions have been known for some time, and are due to several authors (Gu, Shatar, Ladyzhenskaya, Shubov, Zhou, and others). In fact it is fairly easy to get global well-posedness just for data in the energy class.

• However, the critical scaling regularity is H^{1/2}, so one expects local and global well-posedness down to this regularity. By adapting the above scheme, we have • Theorem (Keel, T.) The 1D wave map equation

is locally well-posed in H^{s} for s > 1/2 and globally well-posed for s > 3/4.

• Our original proof used the above scheme, combined with some very strong pointwise conservation laws due to Pohlmeyer that are special to the 1D wave map equation (they arise from the conformal invariance of the problem). It is possible to do without these laws, though, and just work through the Hamiltonian as described above. Due to the derivatives in the non-linearity, Bourgain’s original approach is unavailable.

• The critical regularity s = 1/2 is much more subtle. For instance it is known that the problem is ill-posed in H^{1/2} (T.); however, it is well-posed in the slightly more regular Besov space B_{2}^{1/2,1} (this result is due to Tataru). In two dimensions, the corresponding critical regularity is s = 1, so that energy is critical. In this case it is still an open problem whether one has global existence and uniqueness even for smooth data.

• The other equation we have worked with is the Maxwell-Klein-Gordon equation. In this case we have a U(1) field ϕ (i.e. a complex-valued field) together with a U(1) connection A_{α} (i.e. a purely imaginary vector field). Solutions to MKG are those fields ϕ, A_{α} which are critical points of the Lagrangian

∫ \frac{1}{4}F^{αβ}F_{αbeta} + \frac{1}{2}D_{α}ϕ\overline{D^{α}ϕ} dxdt

where D^{α} are the covariant derivatives ∂^{α} + A^{α} and F^{αβ} = [D^{α}, D^{β}] is the curvature of this connection. This equation comes of course with a gauge invariance. Assuming the Coulomb gauge ∂_{i}A^{i} = 0, the Euler-Lagrange equations basically simplify to something like

□ϕ = A^{α}∂_{α}ϕ + ϕ∂_{t}A_{0} + A^{α}A^{α}ϕ
□A^{i} = ϕ∂ϕ + A^{i}ϕ^{2}
ΔA^{0} = ϕ∂ϕ + A_{0}ϕ^{2}.

This is a more sophisticated form of the □u = u∂u cartoon considered earlier.

• Klainerman and Machedon have shown global wellposedness for this equation for data in the energy class (for smooth data this result is older, and due to Eardley and Montcrief), while Cuccagna has recently pushed the corresponding local well-posedness down to H^{s}, s > 3/4. The scaling regularity is s = 1/2, but this seems unattainable by current methods.

• Theorem (Keel, T.) The Maxwell-Klein-Gordon equations in Coulomb gauge are globally well posed in H^{s}, s > 5/6.

• The method used is basically as described above. There are additional difficulties coming from trying to invert the elliptic operator Δ for very low frequencies |ξ| ≪ 1 (reflecting the fact that the Coulomb gauge is non-local), but these technicalities are more of a nuisance than a true obstruction.

• More recently, we have adapted this scheme to semilinear evolution equations other than the wave equation. In particular, the Kortveg de Vries equation

u_{t} + u_{xxx} + uu_{x} = 0

in R^{1+1} is amenable to these methods. The L^{2} norm is conserved, and plays the role of the Hamiltonian for our purposes. Kenig, Ponce, and Vega have shown global well-posedness for L^{2} data, and local wellposedness in H^{s} for all s > −3/4. (Scaling is s = −3/2).

• Theorem (Colliander, Keel, Staffilani, Takaoka, T.) The KdV equation is globally well-posed for all s > −3/10.

• We plan to study other non-linear equations without smoothing effects, such as the non-linear Schrodinger equation with derivatives in the non-linearity, in future.