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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 $u$ 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 $t$, there exists a position $x(t)$ and a frequency $N(t)$ such that the solution $u(t)$ is localised in space in the region $\{ x: x = x(t) + O(N(t)^{-1}) \}$ and in frequency in the region $\{ \xi: |\xi| \sim N(t) \}$, 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 $E_{crit}$, below which solutions exist globally (and obey scattering bounds), and above which solutions can blow up.

Now consider a solution $u$ 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 $u(t)$ is not localised in frequency at some time t; then one can decompose $u(t)$ into a high frequency component $u_{hi}(t)$ and a low frequency component $u_{lo}(t)$, both of which have strictly smaller energy than $E_{crit}$, and which are widely separated from each other in frequency space.  By hypothesis, each of $u_{hi}$ and $u_{lo}$ 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 $u_{hi}+u_{lo}$ approximately obeys the nonlinear equation; with a suitable perturbation theory, this implies that $u_{hi}+u_{lo}$ is close to $u$.  But then $u$ 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.