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Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the argument that requires a slightly non-trivial fix. The issue does not really affect the subsequent literature, because the main result has since been reproven and extended by methods that avoid the gap (see in particular this subsequent paper of Tataru), but I have decided to describe the gap and its fix on this blog.
I will assume familiarity with the notation of my paper. In Section 10, some complicated spaces are constructed for each frequency scale , and then a further space is constructed for a given frequency envelope by the formula
where is the Littlewood-Paley projection of to frequency magnitudes . Then, given a spacetime slab , we define the restrictions
where the infimum is taken over all extensions of to the Minkowski spacetime ; similarly one defines
The gap in the paper is as follows: it was implicitly assumed that one could restrict (1) to the slab to obtain the equality
(This equality is implicitly used to establish the bound (36) in the paper.) Unfortunately, (1) only gives the lower bound, not the upper bound, and it is the upper bound which is needed here. The problem is that the extensions of that are optimal for computing are not necessarily the Littlewood-Paley projections of the extensions of that are optimal for computing .
To remedy the problem, one has to prove an upper bound of the form
the extension will then obey (5) (here we use Lemma 9 from my paper), but unfortunately is not guaranteed to obey (4) (the norm does control the norm, but a key point about frequency envelopes for the small energy regularity problem is that the coefficients , while bounded, are not necessarily summable).
This can be fixed as follows. For each we introduce a time cutoff supported on that equals on and obeys the usual derivative estimates in between (the time derivative of size for each ). Later we will prove the truncation estimate
Assuming this estimate, then if we set , then using Lemma 9 in my paper and (6), (7) (and the local stability of frequency envelopes) we have the required property (5). (There is a technical issue arising from the fact that is not necessarily Schwartz due to slow decay at temporal infinity, but by considering partial sums in the summation and taking limits we can check that is the strong limit of Schwartz functions, which suffices here; we omit the details for sake of exposition.) So the only issue is to establish (4), that is to say that
for all .
For this is immediate from (2). Now suppose that for some integer (the case when is treated similarly). Then we can split
The contribution of the term is acceptable by (6) and estimate (82) from my paper. The term sums to which is acceptable by (2). So it remains to control the norm of . By the triangle inequality and the fundamental theorem of calculus, we can bound
By hypothesis, . Using the first term in (79) of my paper and Bernstein’s inequality followed by (6) we have
and then we are done by summing the geometric series in .
It remains to prove the truncation estimate (7). This estimate is similar in spirit to the algebra estimates already in my paper, but unfortunately does not seem to follow immediately from these estimates as written, and so one has to repeat the somewhat lengthy decompositions and case checkings used to prove these estimates. We do this below the fold.
I’ve just uploaded to the arXiv a draft version of the final paper in my “heatwave” project, “Global regularity of wave maps VII. Control of delocalised or dispersed solutions“. This paper establishes the final ingredient needed to obtain global regularity and (uniform) scattering for wave maps into hyperbolic space, by showing that any sufficiently delocalised or dispersed wave map (an approximate superposition of two maps of lower energy that are widely separated in space, frequency, or time) can be controlled by wave maps of lesser energy.
This type of result is well understood for scalar semilinear equations, such as the nonlinear Schrodinger equation (NLS) or nonlinear wave equation (NLW). The main new difficulties here are that
- (a) the wave maps equation is an overdetermined system, rather than a scalar equation, rendering such basic operations as decomposing a scalar field into components, or superimposing those components to reassemble the solution, much more delicate and nonlinear;
- (b) the wave maps equation is no longer semilinear (in the sense that it can be viewed as a perturbation of the free wave equation) unless a gauge transform is first performed, but the gauge is itself nonlinear and thus interacts in a complicated way with the decompositions and superpositions in (a);
- (c) the function spaces required to control wave maps in two dimensions are extremely complicated and delicate compared to, say, the NLS theory (in which Strichartz spaces largely suffice), and the estimates are not as favourable. In particular, the “low-high” frequency interactions are non-negligible; the low frequency components of wave maps have a non-trivial “magnetic” effect on the high-frequency components. Furthermore, in contrast to the NLS and NLW settings, it takes substantial effort to show that the function spaces are “divisible”, which roughly means that a wave map only exhibits substantial nonlinear behaviour on a bounded number of time intervals and length scales.
Juggling these three difficulties together led to an unusually large length for this paper (124 pages, and this is after taking some shortcuts, see below).
Last month, Sterbenz and Tataru managed, by a slightly different argument, to also establish global regularity and (non-uniform) scattering for wave maps into compact targets (and thence also to hyperbolic space targets, by a lifting argument). Their argument is significantly shorter (a net length of about 100 pages, compared to about 300 pages for my heatwave project) as it relies on a clever shortcut. In my approach, I seek to control all components of the wave map at once, as well as the nonlinear interactions between those components, in order to show that a delocalised wave map can be controlled by wave maps of lesser energy. In contrast, Sterbenz and Tataru focus on just the finest scale at which nontrivial blowup behaviour occurs; it turns out that the small energy theory and finite speed of propagation, together with a regularising effect arising from the Morawetz estimate, are enough to show that this behaviour is controlled by harmonic maps, and so blowup cannot occur below the critical energy. This approach requires substantially less perturbation theory, and thus largely eliminates the need to develop a nonlinear theory of decomposition and superposition alluded to in (a) above (developing this theory, and meshing it with (b) and (c), occupies the bulk of the current paper). On the other hand, the approach in my papers provides more information on the solution, in particular providing certain spacetime “scattering” bounds on the solution that depend only on the energy, as opposed to a “non-uniform” scattering result in which the scattering norms are finite but potentially unbounded.
Nevertheless, my arguments are much more complicated (though I do feel that the machinery set up to disassemble and reassemble maps into manifolds should be useful for other applications), and in the course of this project, I found that I had not quite set up the material in the earlier papers in a way which was perfectly suited for this last (and longest paper). Because of this, this final paper proved to be far more difficult to write than it ought to have been with the correct framework. At some point in the future, when it becomes clearer exactly what that framework is, I am thinking of collecting and reorganising all this material into a reasonably self-contained book (as opposed to being spread out over a half-dozen papers totaling hundreds of pages in length). But this would take a significant amount of effort, and this project has already distracted me from my other tasks for several months now. As such, I have decided to compromise somewhat and release only a draft version of this paper here, with some of the arguments only sketched rather than given out in full, and continuing to use the existing framework provided by the preceding papers as much as possible, rather than to overhaul the entire series of papers. This is not the most satisfactory outcome – and in particular, I do not consider these papers ready for publication at this stage – but all of the important mathematical material in the arguments should be present here for those who are interested. I do hope though that the various technical components of the theory (particularly the points (a), (b), (c) mentioned above) will be simplified in the future (and the results generalised to other targets), at which point I may begin the process of converting these papers into a publication-quality monograph.
I’ve just uploaded to the arXiv my paper “Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions“, to be submitted with the rest of the “heatwave” project to establish global regularity (and scattering) for energy-critical wave maps into hyperbolic space. Initially, this paper was intended to cap off the project by showing that if global regularity failed, then a special minimal energy blowup solution must exist, which enjoys a certain almost periodicity property modulo the symmetries of the equation. However, the argument was more technical than I anticipated, and so I am splitting the paper into a relatively short high-level paper (this one) that reduces the problem to four smaller propositions, and a much longer technical paper which establishes those propositions, by developing a substantial amount of perturbation theory for wave maps. I am pretty sure though that this process will not iterate any further, and paper VII will be my final paper in this series (and which I hope to finish by the end of this summer). It is also worth noting that a number of papers establishing similar results (though with slightly different hypotheses and conclusions) will shortly appear by Sterbenz-Tataru and Krieger-Schlag.
Almost periodic minimal energy blowup solutions have been constructed for a variety of critical equations, such as the nonlinear Schrodinger equation (NLS) and the nonlinear wave equation (NLW). The formal definition of almost periodicity is that the orbit of the solution stays in a precompact subset of the energy space once one quotients out by the non-compact symmetries of the equation (namely, translation and dilation). Another (more informal) way of saying this is that for every time , there exists a position and a frequency such that the solution is localised in space in the region and in frequency in the region , with the solution decaying in energy away from these regions of space and frequency. Model examples of almost periodic solutions include traveling waves (in which N(t) is fixed, and x(t) moves at constant velocity) and self-similar solutions (in which x(t) is fixed, and N(t) blows up in finite time at some power law rate).
Intuitively, the reason almost periodic minimal energy blowup solutions ought to exist in the absence of global regularity is as follows. It is known (for any of the equations mentioned above) that global regularity (and scattering) holds at sufficiently small energies. Thus, if global regularity fails at high energies, there must exist a critical energy , below which solutions exist globally (and obey scattering bounds), and above which solutions can blow up.
Now consider a solution at the critical energy which blows up (actually, for technical reasons, we instead consider a sequence of solutions approaching this critical energy which come increasingly close to blowing up, but let’s ignore this for now). We claim that this solution must be localised in both space and frequency at every time, thus giving the desired almost periodic minimal energy blowup solution. Indeed, suppose is not localised in frequency at some time t; then one can decompose into a high frequency component and a low frequency component , both of which have strictly smaller energy than , and which are widely separated from each other in frequency space. By hypothesis, each of and can then be extended to global solutions, which should remain widely separated in frequency (because the linear analogues of these equations are constant-coefficient and thus preserve frequency support). Assuming that interactions between very high and very low frequencies are negligible, this implies that the superposition approximately obeys the nonlinear equation; with a suitable perturbation theory, this implies that is close to . But then is not blowing up, a contradiction. The situation with spatial localisation is similar, but is somewhat more complicated due to the fact that spatial support, in contrast to frequency support, is not preserved by the linear evolution, let alone the nonlinear evolution.
As mentioned before, this type of scheme has been successfully implemented on a number of equations such as NLS and NLW. However, there are two main obstacles in establishing it for wave maps. The first is that the wave maps equation is not a scalar equation: the unknown field takes values in a target manifold (specifically, in a hyperbolic space) rather than in a Euclidean space. As a consequence, it is not obvious how one would perform operations such as “decompose the solution into low frequency and high frequency components”, or the inverse operation “superimpose the low frequency and high frequency components to reconstitute the solution”. Another way of viewing the problem is that the various component fields of the solution have to obey a number of important compatibility conditions which can be disrupted by an overly simple-minded approach to decomposition or reconstitution of solutions.
The second problem is that the interaction between very high and very low frequencies for wave maps turns out to not be entirely negligible: the high frequencies do have a negligible impact on the evolution of the low frequencies, but the low frequencies can “rotate” the high frequencies by acting as a sort of magnetic field (or more precisely, a connection) for the evolution of those high frequencies. So the combined evolution of the high and low frequencies is not well approximated by a naive superposition of the separate evolutions of these frequency components.
I have just uploaded to the arXiv the third installment of my “heatwave” project, entitled “Global regularity of wave maps V. Large data local well-posedness in the energy class“. This (rather technical) paper establishes another of the key ingredients necessary to establish the global existence of smooth wave maps from 2+1-dimensional spacetime to hyperbolic space . Specifically, a large data local well-posedness result is established, constructing a local solution from any initial data with finite (but possibly quite large) energy, and furthermore that the solution depends continuously on the initial data in the energy topology. (This topology was constructed in my previous paper.) Once one has this result, the only remaining task is to show a “Palais-Smale property” for wave maps, in that if singularities form in the wave maps equation, then there exists a non-trivial minimal-energy blowup solution, whose orbit is almost periodic modulo the symmetries of the equation. I anticipate this to the most difficult component of the whole project, and is the subject of the fourth (and hopefully final) installment of this series.
This local result is closely related to the small energy global regularity theory developed in recent years by myself, by Krieger, and by Tataru. In particular, the complicated function spaces used in that paper (which ultimately originate from a precursor paper of Tataru). The main new difficulties here are to extend the small energy theory to large energy (by localising time suitably), and to establish continuous dependence on the data (i.e. two solutions which are initially close in the energy topology, need to stay close in that topology). The former difficulty is in principle manageable by exploiting finite speed of propagation (exploiting the fact (arising from the monotone convergence theorem) that large energy data becomes small energy data at sufficiently small spatial scales), but for technical reasons (having to do with my choice of gauge) I was not able to do this and had to deal with the large energy case directly (and in any case, a genuinely large energy theory is going to be needed to construct the minimal energy blowup solution in the next paper). The latter difficulty is in principle resolvable by adapting the existence theory to differences of solutions, rather than to individual solutions, but the nonlinear choice of gauge adds a rather tedious amount of complexity to the task of making this rigorous. (It may be that simpler gauges, such as the Coulomb gauge, may be usable here, at least in the case of the hyperbolic plane (cf. the work of Krieger), but such gauges cause additional analytic problems as they do not renormalise the nonlinearity as strongly as the caloric gauge. The paper of Tataru establishes these goals, but assumes an isometric embedding of the target manifold into a Euclidean space, which is unfortunately not available for hyperbolic space targets.)
The main technical difficulty that had to be overcome in the paper was that there were two different time variables t, s (one for the wave maps equation and one for the heat flow), and three types of PDE (hyperbolic, parabolic, and ODE) that one has to solve forward in t, forward in s, and backwards in s respectively. In order to close the argument in the large energy case, this necessitated a rather complicated iteration-type scheme, in which one solved for the caloric gauge, established parabolic regularity estimates for that gauge, propagated a “wave-tension field” by the heat flow, and then solved a wave maps type equation using that field as a forcing term. The argument can eventually be closed using mostly “off-the-shelf” function space estimates from previous papers, but is remarkably lengthy, especially when analysing differences of two solutions. (One drawback of using off-the-shelf estimates, though, is that one does not get particularly good control of the solution over extended periods of time; in particular, the spaces used here cannot detect the decay of the solution over extended periods of time (unlike, say, Strichartz spaces for ) and so will not be able to supply the long-time perturbation theory that will be needed in the next paper in this series. I believe I know how to re-engineer these spaces to achieve this, though, and the details should follow in the forthcoming paper.)
I have just uploaded to the arXiv the second installment of my “heatwave” project, entitled “Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class“. In the first installment of this project, I was able to establish the global existence of smooth wave maps from 2+1-dimensional spacetime to hyperbolic space from arbitrary smooth initial data, conditionally on five claims:
- A construction of an energy space for maps into hyperbolic space obeying a certain set of reasonable properties, such as compatibility with symmetries, approximability by smooth maps, and existence of a well-defined stress-energy tensor.
- A large data local well-posedness result for wave maps in the above energy space.
- The existence of an almost periodic “minimal-energy blowup solution” to the wave maps equation in the energy class, if this equation is such that singularities can form in finite time.
- The non-existence of any non-trivial degenerate maps into hyperbolic space in the energy class, where “degenerate” means that one of the partial derivatives of this map vanishes identically.
- The non-existence of any travelling or self-similar solution to the wave maps equation in the energy class.
In this paper, the second of four in this series (or, as the title suggests, the fourth in a series of six papers on wave maps, the first two of which can be found here and here), I verify Claims 1, 4, and 5. (The third paper in the series will tackle Claim 2, while the fourth paper will tackle Claim 3.) These claims are largely “elliptic” in nature (as opposed to the “hyperbolic” Claims 2, 3), but I will establish them by a “parabolic” method, relying very heavily on the harmonic map heat flow, and on the closely associated caloric gauge introduced in an earlier paper of mine. The results of paper can be viewed as nonlinear analogues of standard facts about the linear energy space , for instance the fact that smooth compactly supported functions are dense in that space, and that this space contains no non-trivial harmonic functions, or functions which are constant in one of the two spatial directions. The paper turned out a little longer than I had expected (77 pages) due to some surprisingly subtle technicalities, especially when excluding self-similar wave maps. On the other hand, the heat flow and caloric gauge machinery developed here will be reused in the last two papers in this series, hopefully keeping their length to under 100 pages as well.
A key stumbling block here, related to the critical (scale-invariant) nature of the energy space (or to the failure of the endpoint Sobolev embedding ) is that changing coordinates in hyperbolic space can be a non-uniformly-continuous operation in the energy space. Thus, for the purposes of making quantitative estimates in that space, it is preferable to work as covariantly (or co-ordinate free) manner as possible, or if one is to use co-ordinates, to pick them in some canonical manner which is optimally adapted to the tasks at hand. Ideally, one would work with directly with maps (as well as their velocity field ) without using any coordinates on , but then it becomes to perform basic analytical operations on such maps, such as taking the Fourier transform, or (even more elementarily) taking the difference of two maps in order to measure how distinct they are from each other.
I’ve just uploaded to the arXiv a new paper, “Global regularity of wave maps III. Large energy from to hyperbolic spaces“, to be submitted when three other companion papers (“Global regularity of wave maps” IV, V, and VI) are finished. This project (which I had called “Heatwave”, due to the use of a heat flow to renormalise a wave equation) has a somewhat lengthy history to it, which I will now attempt to explain.
For the last nine years or so, I have been working on and off on the global regularity problem for wave maps . The wave map equation is a nonlinear generalisation of the wave equation in which the unknown field takes values in a Riemannian manifold rather than in a vector space (much as the concept of a harmonic map is a nonlinear generalisation of a harmonic function). This equation (also known as the nonlinear model) is one of the simplest examples of a geometric nonlinear wave equation, and is also arises as a simplified model of the Einstein equations (after making a U(1) symmetry assumption). The global regularity problem seeks to determine when smooth initial data for a wave map (i.e. an initial position and an initial velocity tangent to the position) necessarily leads to a smooth global solution.
The problem is particularly interesting in the energy-critical dimension d=2, in which the conserved energy becomes invariant under the scaling symmetry . (In the subcritical dimension d=1, global regularity is fairly easy to establish, and was first done by Gu and by Ladyzhenskaya-Shubov; in supercritical dimensions , examples of singularity formation are known, starting with the self-similar examples of Shatah.)
It is generally believed that in two dimensions, singularities can form when M is positively curved but that global regularity should persist when M is negatively curved, in analogy with known results (in particular, the landmark paper of Eells and Sampson) for the harmonic map heat flow (a parabolic cousin of the wave map equation). In particular, one should always have global regularity when the target is a hyperbolic space. There has been a large number of results supporting this conjecture; for instance, when the target is the sphere, examples of singularity formation have recently been constructed by Rodnianski-Sterbenz and by Krieger-Schlag-Tataru, while for suitably negatively curved manifolds such as hyperbolic space, global regularity was established assuming equivariant symmetry by Shatah and Tahvildar-Zadeh, and assuming spherical symmetry by Christodoulou and Tahvildar-Zadeh. I will not attempt to mention all the other results on this problem here, but see for instance one of these survey articles or books for further discussion.
For the past several years, my good friend and fellow Medalist Timothy Gowers has been devoting an enormous amount of effort towards editing (with the help of June Barrow-Green and Imre Leader) a forthcoming book, the Princeton Companion to Mathematics. This immense project is somewhat difficult to explain succinctly; a zeroth order approximation would be that it is an “Encyclopedia of mathematics”, but the aim is not to be a comprehensive technical reference, nor is it a repository of key mathematical definitions and theorems; it is neither Scholarpedia nor Wikipedia. Instead, the idea is to give a flavour of a subject or mathematical concept by means of motivating examples, questions, and so forth, somewhat analogous to the “What is a …?” series in the Notices of the AMS. Ideally, any interested reader with a basic mathematics undergraduate education could use this book to get a rough handle on what (say) Ricci flow is and why it is useful, or what questions mathematicians are trying to answer about (say) harmonic analysis, without getting into the technical details (which are abundantly available elsewhere). There are contributions from many, many mathematicians on a wide range of topics, from symplectic geometry to modular forms to the history and influence of mathematics; I myself have contributed or been otherwise involved in about a dozen articles.
If all goes well, the Companion should be finalised later this year and available around March 2008. As a sort of “advertising campaign” for this project (and with Tim’s approval), I plan to gradually release to this blog (at the rate of one or two a month) the various articles I contributed for the project over the last few years. [As part of this advertising, I might add that the Companion can already be pre-ordered on Amazon.]
I’ll inaugurate this series with my article on “wave maps”. This describes, in general terms, what a wave map is (it is a mathematical model for a vibrating membrane in a manifold), and its relationship with other concepts such as harmonic maps and general relativity. As it turns out, for editing reasons various articles solicited for the Companion had to be removed from the print edition (but possibly may survive in the on-line version of the Companion, though details are unclear at this point); Tim and I agreed that as wave maps were not the most crucial geometric PDE that needed to be covered for the Companion (there are other articles on the Einstein equations and Ricci flow, for instance), that this particular article would end up being one of the “deleted scenes”. As such, it seems like a logical choice for the first article to release here.