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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.