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
and
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
Next, when restricted to frequencies of order , the propagator
“behaves like”
, where
is the spherical averaging operator
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
on .
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.
…
6 comments
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8 March, 2016 at 9:02 am
mustaliblog
Impressive. this just tickles my mathematical imagination
9 March, 2016 at 10:53 am
John Mangual
I don’t understand why you are so careful. No physicist in his right mind knows or cares what a
whos-a-whatsit space is.
11 March, 2016 at 7:20 pm
dcohen
line 11, it should be “d’Alembertian”
[Corrected, thanks – T.]
14 March, 2016 at 4:48 am
Michael
Start of second paragraph before Theorem 1 should be “In” not “Im”; In paragraph after Theorem 1 should be “u as” not “u mas”
Paper Proposition 3.2 starts with (iii) not (i), but late on page 17 you refer to 3.2(i)
Paper page 12, definition of $\hat\omega$ should have dx, not d\xi
[Corrected, thanks; the paper corrections will appear in the next revision of the ms. -T.]
29 June, 2016 at 6:44 am
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