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I’ve just uploaded to the arXiv my paper Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity, submitted to Comm. PDE. This paper is in the same spirit as (though not directly related to) my previous paper on finite time blowup of supercritical NLW systems, and was inspired by a question posed to me some time ago by Jeffrey Rauch. Here, instead of looking at supercritical equations, we look at an extremely subcritical equation, namely a system of the form
where is the unknown field, and is the nonlinearity, which we assume to have all derivatives bounded. A typical example of such an equation is the higher-dimensional sine-Gordon equation
for a scalar field . Here is the d’Alembertian operator. We restrict attention here to classical (i.e. smooth) solutions to (1).
We do not assume any Hamiltonian structure, so we do not require to be a gradient of a potential . But even without such Hamiltonian structure, the equation (1) is very well behaved, with many a priori bounds available. For instance, if the initial position and initial velocity are smooth and compactly supported, then from finite speed of propagation has uniformly bounded compact support for all in a bounded interval. As the nonlinearity is bounded, this immediately places in in any bounded time interval, which by the energy inequality gives an a priori bound on in this time interval. Next, from the chain rule we have
which (from the assumption that is bounded) shows that is in , which by the energy inequality again now gives an a priori bound on .
One might expect that one could keep iterating this and obtain a priori bounds on in arbitrarily smooth norms. In low dimensions such as , this is a fairly easy task, since the above estimates and Sobolev embedding already place one in , and the nonlinear map is easily verified to preserve the space for any natural number , from which one obtains a priori bounds in any Sobolev space; from this and standard energy methods, one can then establish global regularity for this equation (that is to say, any smooth choice of initial data generates a global smooth solution). However, one starts running into trouble in higher dimensions, in which no bound is available. The main problem is that even a really nice nonlinearity such as is unbounded in higher Sobolev norms. The estimates
ensure that the map is bounded in low regularity spaces like or , but one already runs into trouble with the second derivative
where there is a troublesome lower order term of size which becomes difficult to control in higher dimensions, preventing the map to be bounded in . Ultimately, the issue here is that when is not controlled in , the function can oscillate at a much higher frequency than ; for instance, if is the one-dimensional wave for some and , then oscillates at frequency , but the function more or less oscillates at the larger frequency .
In medium dimensions, it is possible to use dispersive estimates for the wave equation (such as the famous Strichartz estimates) to overcome these problems. This line of inquiry was pursued (albeit for slightly different classes of nonlinearity than those considered here) by Heinz-von Wahl, Pecher (in a series of papers), Brenner, and Brenner-von Wahl; to cut a long story short, one of the conclusions of these papers was that one had global regularity for equations such as (1) in dimensions . (I reprove this result using modern Strichartz estimate and Littlewood-Paley techniques in an appendix to my paper. The references given also allow for some growth in the nonlinearity , but we will not detail the precise hypotheses used in these papers here.)
In my paper, I complement these positive results with an almost matching negative result:
Theorem 1 If and , then there exists a nonlinearity with all derivatives bounded, and a solution to (1) that is smooth at time zero, but develops a singularity in finite time.
The construction crucially relies on the ability to choose the nonlinearity , and also needs some injectivity properties on the solution (after making a symmetry reduction using an assumption of spherical symmetry to view as a function of variables rather than ) which restricts our counterexample to the case. Thus the model case of the higher-dimensional sine-Gordon equation is not covered by our arguments. Nevertheless (as with previous finite-time blowup results discussed on this blog), one can view this result as a barrier to trying to prove regularity for equations such as in eleven and higher dimensions, as any such argument must somehow use a property of that equation that is not applicable to the more general system (1).
Let us first give some back-of-the-envelope calculations suggesting why there could be finite time blowup in eleven and higher dimensions. For sake of this discussion let us restrict attention to the sine-Gordon equation . The blowup ansatz we will use is as follows: for each frequency in a sequence of large quantities going to infinity, there will be a spacetime “cube” on which the solution oscillates with “amplitude” and “frequency” , where is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since as , this will create a singularity at the spacetime origin . To make this ansatz plausible, we wish to make the oscillation of on driven primarily by the forcing term at . Thus, by Duhamel’s formula, we expect a relation roughly of the form
on , where is the usual free wave propagator, and is the indicator function of .
On , oscillates with amplitude and frequency , we expect the derivative to be of size about , and so from the principle of stationary phase we expect to oscillate at frequency about . Since the wave propagator preserves frequencies, and is supposed to be of frequency on we are thus led to the requirement
where is surface measure on the unit sphere , and is the volume of that sphere. In our setting, is comparable to , and so we have the informal approximation
Since is bounded, is bounded as well. This gives a (non-rigorous) upper bound
which when combined with our ansatz that has ampitude about on , gives the constraint
which on applying (2) gives the further constraint
which can be rearranged as
It is now clear that the optimal choice of is
and this blowup ansatz is only self-consistent when
or equivalently if .
To turn this ansatz into an actual blowup example, we will construct as the sum of various functions that solve the wave equation with forcing term in , and which concentrate in with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that can be written in the form for some with all derivatives bounded. For this one needs some injectivity properties of (after imposing spherical symmetry to impose a dimensional reduction on the domain of from dimensions to ). This requires one to construct some solutions to the free wave equation that have some unusual restrictions on the range (for instance, we will need a solution taking values in the plane that avoid one quadrant of that plane). In order to do this we take advantage of the very explicit nature of the fundamental solution to the wave equation in odd dimensions (such as ), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension , any spherically symmetric function of the form
for an arbitrary smooth function , will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity
This precise and relatively simple formula for allows one to create “bespoke” solutions that obey various unusual properties, without too much difficulty.
It is not clear to me what to conjecture for . The blowup ansatz given above is a little inefficient, in that the frequency component of the solution is only generated from a portion of the component, namely the portion close to a certain light cone. In particular, the solution does not saturate the Strichartz estimates that are used to establish the positive results for , which helps explain the slight gap between the positive and negative results. It may be that a more complicated ansatz could work to give a negative result in ten dimensions; conversely, it is also possible that one could use more advanced estimates than the Strichartz estimate (that somehow capture the “thinness” of the fundamental solution, and not just its dispersive properties) to stretch the positive results to ten dimensions. Which side the case falls in all come down to some rather delicate numerology.
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.)
where is a scalar function of one time and three spatial dimensions.
The evolution of this type of non-linear wave equation can be viewed as a “race” between the dispersive tendency of the linear wave equation
More precisely, solutions to (2) tend to decay in time as , as can be seen from the presence of the term in the explicit formula
for such solutions in terms of the initial position and initial velocity , where , , and dS is the area element of the sphere . (For this post I will ignore the technical issues regarding how smooth the solution has to be in order for the above formula to be valid.) On the other hand, solutions to (3) tend to blow up in finite time from data with positive initial position and initial velocity, even if this data is very small, as can be seen by the family of solutions
for , , and , where c is the positive constant . For T large, this gives a family of solutions which starts out very small at time zero, but still manages to go to infinity in finite time.
The equation (1) can be viewed as a combination of equations (2) and (3) and should thus inherit a mix of the behaviours of both its “parents”. As a general rule, when the initial data of solution is small, one expects the dispersion to “win” and send the solution to zero as , because the nonlinear effects are weak; conversely, when the initial data is large, one expects the nonlinear effects to “win” and cause blowup, or at least large amounts of instability. This division is particularly pronounced when p is large (since then the nonlinearity is very strong for large data and very weak for small data), but not so much for p small (for instance, when p=1, the equation becomes essentially linear, and one can easily show that blowup does not occur from reasonable data.)
The theorem of John formalises this intuition, with a remarkable threshold value for p:
Theorem. Let .
- If , then there exist solutions which are arbitrarily small (both in size and in support) and smooth at time zero, but which blow up in finite time.
- If , then for every initial data which is sufficiently small in size and support, and sufficiently smooth, one has a global solution (which goes to zero uniformly as ).
[At the critical threshold one also has blowup from arbitrarily small data, as was shown subsequently by Schaeffer.]
The ostensible purpose of this post is to try to explain why the curious exponent should make an appearance here, by sketching out the proof of part 1 of John’s theorem (I will not discuss part 2 here); but another reason I am writing this post is to illustrate how to make quick “back-of-the-envelope” calculations in harmonic analysis and PDE which can obtain the correct numerology for such a problem much faster than a fully rigorous approach. These calculations can be a little tricky to handle properly at first, but with practice they can be done very swiftly.