You are currently browsing the tag archive for the ‘nonlinear wave equations’ tag.

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

\displaystyle \Box u = f(u) \ \ \ \ \ (1)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} is the unknown field, and {f: {\bf R}^m \rightarrow {\bf R}^m} 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

\displaystyle \Box u = \sin u

for a scalar field {u: {\bf R}^{1+d} \rightarrow {\bf R}}. Here {\Box = -\partial_t^2 + \Delta} 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 {f} to be a gradient {f = \nabla F} of a potential {F: {\bf R}^m \rightarrow {\bf R}}. 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 {u_0(x) = u(0,x)} and initial velocity {u_1(x) = \partial_t u(0,x)} are smooth and compactly supported, then from finite speed of propagation {u(t)} has uniformly bounded compact support for all {t} in a bounded interval. As the nonlinearity {f} is bounded, this immediately places {f(u)} in {L^\infty_t L^2_x} in any bounded time interval, which by the energy inequality gives an a priori {L^\infty_t H^1_x} bound on {u} in this time interval. Next, from the chain rule we have

\displaystyle \nabla f(u) = (\nabla_{{\bf R}^m} f)(u) \nabla u

which (from the assumption that {\nabla_{{\bf R}^m} f} is bounded) shows that {f(u)} is in {L^\infty_t H^1_x}, which by the energy inequality again now gives an a priori {L^\infty_t H^2_x} bound on {u}.

One might expect that one could keep iterating this and obtain a priori bounds on {u} in arbitrarily smooth norms. In low dimensions such as {d \leq 3}, this is a fairly easy task, since the above estimates and Sobolev embedding already place one in {L^\infty_t L^\infty_x}, and the nonlinear map {f} is easily verified to preserve the space {L^\infty_t H^k_x \cap L^\infty_t L^\infty_x} for any natural number {k}, 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 {L^\infty_x} bound is available. The main problem is that even a really nice nonlinearity such as {u \mapsto \sin u} is unbounded in higher Sobolev norms. The estimates

\displaystyle |\sin u| \leq |u|

and

\displaystyle |\nabla(\sin u)| \leq |\nabla u|

ensure that the map {u \mapsto \sin u} is bounded in low regularity spaces like {L^2_x} or {H^1_x}, but one already runs into trouble with the second derivative

\displaystyle \nabla^2(\sin u) = (\cos u) \nabla^2 u - (\sin u) \nabla u \nabla u

where there is a troublesome lower order term of size {O( |\nabla u|^2 )} which becomes difficult to control in higher dimensions, preventing the map {u \mapsto \sin u} to be bounded in {H^2_x}. Ultimately, the issue here is that when {u} is not controlled in {L^\infty}, the function {\sin u} can oscillate at a much higher frequency than {u}; for instance, if {u} is the one-dimensional wave {u = A \sin(kx)}for some {k > 0} and {A>1}, then {u} oscillates at frequency {k}, but the function {\sin(u)= \sin(A \sin(kx))} more or less oscillates at the larger frequency {Ak}.

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 {f} 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 {d \leq 9}. (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 {f}, 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 {d \geq 11} and {m \geq 2}, then there exists a nonlinearity {f: {\bf R}^m \rightarrow {\bf R}^m} with all derivatives bounded, and a solution {u} 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 {f}, and also needs some injectivity properties on the solution {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} (after making a symmetry reduction using an assumption of spherical symmetry to view {u} as a function of {1+1} variables rather than {1+d}) which restricts our counterexample to the {m \geq 2} case. Thus the model case of the higher-dimensional sine-Gordon equation {\Box u =\sin u} 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 {\Box u = \sin u} 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 {\Box u = \sin u}. The blowup ansatz we will use is as follows: for each frequency {N_j} in a sequence {1 < N_1 < N_2 < N_3 < \dots} of large quantities going to infinity, there will be a spacetime “cube” {Q_j = \{ (t,x): t \sim \frac{1}{N_j}; x = O(\frac{1}{N_j})\}} on which the solution {u} oscillates with “amplitude” {N_j^\alpha} and “frequency” {N_j}, where {\alpha>0} is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since {N_j^\alpha \rightarrow \infty} as {j \rightarrow \infty}, this will create a singularity at the spacetime origin {(0,0)}. To make this ansatz plausible, we wish to make the oscillation of {u} on {Q_j} driven primarily by the forcing term {\sin u} at {Q_{j-1}}. Thus, by Duhamel’s formula, we expect a relation roughly of the form

\displaystyle u(t,x) \approx \int \frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}} \sin(1_{Q_{j-1}} u(s)) (x)\ ds

on {Q_j}, where {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} is the usual free wave propagator, and {1_{Q_{j-1}}} is the indicator function of {Q_{j-1}}.

On {Q_{j-1}}, {u} oscillates with amplitude {N_{j-1}^\alpha} and frequency {N_{j-1}}, we expect the derivative {\nabla_{t,x} u} to be of size about {N_{j-1}^{\alpha+1}}, and so from the principle of stationary phase we expect {\sin(u)} to oscillate at frequency about {N_{j-1}^{\alpha+1}}. Since the wave propagator {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} preserves frequencies, and {u} is supposed to be of frequency {N_j} on {Q_j} we are thus led to the requirement

\displaystyle N_j \approx N_{j-1}^{\alpha+1}. \ \ \ \ \ (2)

 

Next, when restricted to frequencies of order {N_{j}}, the propagator {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} “behaves like” {N_{j}^{\frac{d-3}{2}} (s-t)^{\frac{d-1}{2}} A_{s-t}}, where {A_{s-t}} is the spherical averaging operator

\displaystyle A_{s-t} f(x) := \frac{1}{\omega_{d-1}} \int_{S^{d-1}} f(x + (s-t)\theta)\ d\theta

where {d\theta} is surface measure on the unit sphere {S^{d-1}}, and {\omega_{d-1}} is the volume of that sphere. In our setting, {s-t} is comparable to {1/N_{j-1}}, and so we have the informal approximation

\displaystyle u(t,x) \approx N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \int_{s \sim 1/N_{j-1}} A_{s-t} \sin(u(s))(x)\ ds

on {Q_j}.

Since {\sin(u(s))} is bounded, {A_{s-t} \sin(u(s))} is bounded as well. This gives a (non-rigorous) upper bound

\displaystyle u(t,x) \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}

which when combined with our ansatz that {u} has ampitude about {N_j^\alpha} on {Q_j}, gives the constraint

\displaystyle N_j^\alpha \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}

which on applying (2) gives the further constraint

\displaystyle \alpha(\alpha+1) \leq \frac{d-3}{2} (\alpha+1) - \frac{d-1}{2} - 1

which can be rearranged as

\displaystyle \left(\alpha - \frac{d-5}{4}\right)^2 \leq \frac{d^2-10d-7}{16}.

It is now clear that the optimal choice of {\alpha} is

\displaystyle \alpha = \frac{d-5}{4},

and this blowup ansatz is only self-consistent when

\displaystyle \frac{d^2-10d-7}{16} \geq 0

or equivalently if {d \geq 11}.

To turn this ansatz into an actual blowup example, we will construct {u} as the sum of various functions {u_j} that solve the wave equation with forcing term in {Q_{j+1}}, and which concentrate in {Q_j} with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that {\Box u} can be written in the form {f(u)} for some {f} with all derivatives bounded. For this one needs some injectivity properties of {u} (after imposing spherical symmetry to impose a dimensional reduction on the domain of {u} from {d+1} dimensions to {1+1}). 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 {{\bf R}^2} 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 {d=11}), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension {d}, any spherically symmetric function {u(t,x) = u(t,r)} of the form

\displaystyle u(t,r) = \left(\frac{1}{r} \partial_r\right)^{\frac{d-1}{2}} (g(t+r) + g(t-r))

for an arbitrary smooth function {g: {\bf R} \rightarrow {\bf R}^m}, will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity

\displaystyle \left( \partial_{tt} + \partial_{rr} + \frac{d-1}{r} \partial_r \right) \frac{1}{r} \partial_r = \frac{1}{r} \partial_r \left( \partial_{tt} + \partial_{rr} + \frac{d-3}{r} \partial_r \right).

This precise and relatively simple formula for {u} allows one to create “bespoke” solutions {u} that obey various unusual properties, without too much difficulty.

It is not clear to me what to conjecture for {d=10}. The blowup ansatz given above is a little inefficient, in that the frequency {N_{j+1}} component of the solution is only generated from a portion of the {N_j} 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 {d \leq 9}, 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 {d=10} case falls in all come down to some rather delicate numerology.

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear wave system, submitted to Analysis and PDE. This paper was inspired by a question asked of me by Sergiu Klainerman recently, regarding whether there were any analogues of my blowup example for Navier-Stokes type equations in the setting of nonlinear wave equations.

Recall that the defocusing nonlinear wave (NLW) equation reads

\displaystyle \Box u = |u|^{p-1} u \ \ \ \ \ (1)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}} is the unknown scalar field, {\Box = -\partial_t^2 + \Delta} is the d’Alambertian operator, and {p>1} is an exponent. We can generalise this equation to the defocusing nonlinear wave system

\displaystyle \Box u = (\nabla F)(u) \ \ \ \ \ (2)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} is now a system of scalar fields, and {F: {\bf R}^m \rightarrow {\bf R}} is a potential which is homogeneous of degree {p+1} and strictly positive away from the origin; the scalar equation corresponds to the case where {m=1} and {F(u) = \frac{1}{p+1} |u|^{p+1}}. We will be interested in smooth solutions {u} to (2). It is only natural to restrict to the smooth category when the potential {F} is also smooth; unfortunately, if one requires {F} to be homogeneous of order {p+1} all the way down to the origin, then {F} cannot be smooth unless it is identically zero or {p+1} is an odd integer. This is too restrictive for us, so we will only require that {F} be homogeneous away from the origin (e.g. outside the unit ball). In any event it is the behaviour of {F(u)} for large {u} which will be decisive in understanding regularity or blowup for the equation (2).

Formally, solutions to the equation (2) enjoy a conserved energy

\displaystyle E[u] = \int_{{\bf R}^d} \frac{1}{2} \|\partial_t u \|^2 + \frac{1}{2} \| \nabla_x u \|^2 + F(u)\ dx.

Using this conserved energy, it is possible to establish global regularity for the Cauchy problem (2) in the energy-subcritical case when {d \leq 2}, or when {d \geq 3} and {p < 1+\frac{4}{d-2}}. This means that for any smooth initial position {u_0: {\bf R}^d \rightarrow {\bf R}^m} and initial velocity {u_1: {\bf R}^d \rightarrow {\bf R}^m}, there exists a (unique) smooth global solution {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} to the equation (2) with {u(0,x) = u_0(x)} and {\partial_t u(0,x) = u_1(x)}. These classical global regularity results (essentially due to Jörgens) were famously extended to the energy-critical case when {d \geq 3} and {p = 1 + \frac{4}{d-2}} by Grillakis, Struwe, and Shatah-Struwe (though for various technical reasons, the global regularity component of these results was limited to the range {3 \leq d \leq 7}). A key tool used in the energy-critical theory is the Morawetz estimate

\displaystyle \int_0^T \int_{{\bf R}^d} \frac{|u(t,x)|^{p+1}}{|x|}\ dx dt \lesssim E[u]

which can be proven by manipulating the properties of the stress-energy tensor

\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + F(u))

(with the usual summation conventions involving the Minkowski metric {\eta_{\alpha \beta} dx^\alpha dx^\beta = -dt^2 + |dx|^2}) and in particular exploiting the divergence-free nature of this tensor: {\partial^\beta T_{\alpha \beta}} See for instance the text of Shatah-Struwe, or my own PDE book, for more details. The energy-critical regularity results have also been extended to slightly supercritical settings in which the potential grows by a logarithmic factor or so faster than the critical rate; see the results of myself and of Roy.

This leaves the question of global regularity for the energy supercritical case when {d \geq 3} and {p > 1+\frac{4}{d-2}}. On the one hand, global smooth solutions are known for small data (if {F} vanishes to sufficiently high order at the origin, see e.g. the work of Lindblad and Sogge), and global weak solutions for large data were constructed long ago by Segal. On the other hand, the solution map, if it exists, is known to be extremely unstable, particularly at high frequencies; see for instance this paper of Lebeau, this paper of Christ, Colliander, and myself, this paper of Brenner and Kumlin, or this paper of Ibrahim, Majdoub, and Masmoudi for various formulations of this instability. In the case of the focusing NLW {-\partial_{tt} u + \Delta u = - |u|^{p-1} u}, one can easily create solutions that blow up in finite time by ODE constructions, for instance one can take {u(t,x) = c (1-t)^{-\frac{2}{p-1}}} with {c = (\frac{2(p+1)}{(p-1)^2})^{\frac{1}{p-1}}}, which blows up as {t} approaches {1}. However the situation in the defocusing supercritical case is less clear. The strongest positive results are of Kenig-Merle and Killip-Visan, which show (under some additional technical hypotheses) that global regularity for such equations holds under the additional assumption that the critical Sobolev norm of the solution stays bounded. Roughly speaking, this shows that “Type II blowup” cannot occur for (2).

Our main result is that finite time blowup can in fact occur, at least for three-dimensional systems where the number {m} of degrees of freedom is sufficiently large:

Theorem 1 Let {d=3}, {p > 5}, and {m \geq 76}. Then there exists a smooth potential {F: {\bf R}^m \rightarrow {\bf R}}, positive and homogeneous of degree {p+1} away from the origin, and a solution to (2) with smooth initial data that develops a singularity in finite time.

The rather large lower bound of {76} on {m} here is primarily due to our use of the Nash embedding theorem (which is the first time I have actually had to use this theorem in an application!). It can certainly be lowered, but unfortunately our methods do not seem to be able to bring {m} all the way down to {1}, so we do not directly exhibit finite time blowup for the scalar supercritical defocusing NLW. Nevertheless, this result presents a barrier to any attempt to prove global regularity for that equation, in that it must somehow use a property of the scalar equation which is not available for systems. It is likely that the methods can be adapted to higher dimensions than three, but we take advantage of some special structure to the equations in three dimensions (related to the strong Huygens principle) which does not seem to be available in higher dimensions.

The blowup will in fact be of discrete self-similar type in a backwards light cone, thus {u} will obey a relation of the form

\displaystyle u(e^S t, e^S x) = e^{-\frac{2}{p-1} S} u(t,x)

for some fixed {S>0} (the exponent {-\frac{2}{p-1}} is mandated by dimensional analysis considerations). It would be natural to consider continuously self-similar solutions (in which the above relation holds for all {S}, not just one {S}). And rough self-similar solutions have been constructed in the literature by perturbative methods (see this paper of Planchon, or this paper of Ribaud and Youssfi). However, it turns out that continuously self-similar solutions to a defocusing equation have to obey an additional monotonicity formula which causes them to not exist in three spatial dimensions; this argument is given in my paper. So we have to work just with discretely self-similar solutions.

Because of the discrete self-similarity, the finite time blowup solution will be “locally Type II” in the sense that scale-invariant norms inside the backwards light cone stay bounded as one approaches the singularity. But it will not be “globally Type II” in that scale-invariant norms stay bounded outside the light cone as well; indeed energy will leak from the light cone at every scale. This is consistent with the results of Kenig-Merle and Killip-Visan which preclude “globally Type II” blowup solutions to these equations in many cases.

We now sketch the arguments used to prove this theorem. Usually when studying the NLW, we think of the potential {F} (and the initial data {u_0,u_1}) as being given in advance, and then try to solve for {u} as an unknown field. However, in this problem we have the freedom to select {F}. So we can look at this problem from a “backwards” direction: we first choose the field {u}, and then fit the potential {F} (and the initial data) to match that field.

Now, one cannot write down a completely arbitrary field {u} and hope to find a potential {F} obeying (2), as there are some constraints coming from the homogeneity of {F}. Namely, from the Euler identity

\displaystyle \langle u, (\nabla F)(u) \rangle = (p+1) F(u)

we see that {F(u)} can be recovered from (2) by the formula

\displaystyle F(u) = \frac{1}{p+1} \langle u, \Box u \rangle \ \ \ \ \ (3)

 

so the defocusing nature of {F} imposes a constraint

\displaystyle \langle u, \Box u \rangle > 0.

Furthermore, taking a derivative of (3) we obtain another constraining equation

\displaystyle \langle \partial_\alpha u, \Box u \rangle = \frac{1}{p+1} \partial_\alpha \langle u, \Box u \rangle

that does not explicitly involve the potential {F}. Actually, one can write this equation in the more familiar form

\displaystyle \partial^\beta T_{\alpha \beta} = 0

where {T_{\alpha \beta}} is the stress-energy tensor

\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + \frac{1}{p+1} \langle u, \Box u \rangle),

now written in a manner that does not explicitly involve {F}.

With this reformulation, this suggests a strategy for locating {u}: first one selects a stress-energy tensor {T_{\alpha \beta}} that is divergence-free and obeys suitable positive definiteness and self-similarity properties, and then locates a self-similar map {u} from the backwards light cone to {{\bf R}^m} that has that stress-energy tensor (one also needs the map {u} (or more precisely the direction component {u/\|u\|} of that map) injective up to the discrete self-similarity, in order to define {F(u)} consistently). If the stress-energy tensor was replaced by the simpler “energy tensor”

\displaystyle E_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle

then the question of constructing an (injective) map {u} with the specified energy tensor is precisely the embedding problem that was famously solved by Nash (viewing {E_{\alpha \beta}} as a Riemannian metric on the domain of {u}, which in this case is a backwards light cone quotiented by a discrete self-similarity to make it compact). It turns out that one can adapt the Nash embedding theorem to also work with the stress-energy tensor as well (as long as one also specifies the mass density {M = \|u\|^2}, and as long as a certain positive definiteness property, related to the positive semi-definiteness of Gram matrices, is obeyed). Here is where the dimension {76} shows up:

Proposition 2 Let {M} be a smooth compact Riemannian {4}-manifold, and let {m \geq 76}. Then {M} smoothly isometrically embeds into the sphere {S^{m-1}}.

Proof: The Nash embedding theorem (in the form given in this ICM lecture of Gunther) shows that {M} can be smoothly isometrically embedded into {{\bf R}^{19}}, and thus in {[-R,R]^{19}} for some large {R}. Using an irrational slope, the interval {[-R,R]} can be smoothly isometrically embedded into the {2}-torus {\frac{1}{\sqrt{38}} (S^1 \times S^1)}, and so {[-R,R]^{19}} and hence {M} can be smoothly embedded in {\frac{1}{\sqrt{38}} (S^1)^{38}}. But from Pythagoras’ theorem, {\frac{1}{\sqrt{38}} (S^1)^{38}} can be identified with a subset of {S^{m-1}} for any {m \geq 76}, and the claim follows. \Box

One can presumably improve upon the bound {76} by being more efficient with the embeddings (e.g. by modifying the proof of Nash embedding to embed directly into a round sphere), but I did not try to optimise the bound here.

The remaining task is to construct the stress-energy tensor {T_{\alpha \beta}}. One can reduce to tensors that are invariant with respect to rotations around the spatial origin, but this still leaves a fair amount of degrees of freedom (it turns out that there are four fields that need to be specified, which are denoted {M, E_{tt}, E_{tr}, E_{rr}} in my paper). However a small miracle occurs in three spatial dimensions, in that the divergence-free condition involves only two of the four degrees of freedom (or three out of four, depending on whether one considers a function that is even or odd in {r} to only be half a degree of freedom). This is easiest to illustrate with the scalar NLW (1). Assuming spherical symmetry, this equation becomes

\displaystyle - \partial_{tt} u + \partial_{rr} u + \frac{2}{r} \partial_r u = |u|^{p-1} u.

Making the substitution {\phi := ru}, we can eliminate the lower order term {\frac{2}{r} \partial_r} completely to obtain

\displaystyle - \partial_{tt} \phi + \partial_{rr} \phi= \frac{1}{r^{p-1}} |\phi|^{p-1} \phi.

(This can be compared with the situation in higher dimensions, in which an undesirable zeroth order term {\frac{(d-1)(d-3)}{r^2} \phi} shows up.) In particular, if one introduces the null energy density

\displaystyle e_+ := \frac{1}{2} |\partial_t \phi + \partial_r \phi|^2

and the potential energy density

\displaystyle V := \frac{|\phi|^{p+1}}{(p+1) r^{p-1}}

then one can verify the equation

\displaystyle (\partial_t - \partial_r) e_+ + (\partial_t + \partial_r) V = - \frac{p-1}{r} V

which can be viewed as a transport equation for {e_+} with forcing term depending on {V} (or vice versa), and is thus quite easy to solve explicitly by choosing one of these fields and then solving for the other. As it turns out, once one is in the supercritical regime {p>5}, one can solve this equation while giving {e_+} and {V} the right homogeneity (they have to be homogeneous of order {-\frac{4}{p-1}}, which is greater than {-1} in the supercritical case) and positivity properties, and from this it is possible to prescribe all the other fields one needs to satisfy the conclusions of the main theorem. (It turns out that {e_+} and {V} will be concentrated near the boundary of the light cone, so this is how the solution {u} will concentrate also.)

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 {\Bbb R}^{1+2} to hyperbolic space \mathbf{H} = \mathbf{H}^m.  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 m=2 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 L^q_t L^r_x for q < \infty) 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.)

Today I’d like to discuss (part of) a cute and surprising theorem of Fritz John in the area of non-linear wave equations, and specifically for the equation

\partial_{tt} u - \Delta u = |u|^p (1)

where u: {\Bbb R} \times {\Bbb R}^3 \to {\Bbb R} 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

\partial_{tt} u - \Delta u = 0 (2)

and the positive feedback tendencies of the nonlinear ODE

\partial_{tt} u = |u|^p. (3)

More precisely, solutions to (2) tend to decay in time as t \to +\infty, as can be seen from the presence of the \frac{1}{t} term in the explicit formula

u(t,x) =  \frac{1}{4\pi t} \int_{|y-x|=t} \partial_t u(0,y)\ dS(y) + \partial_t[\frac{1}{4\pi t} \int_{|y-x|=t} u(0,y)\ dS(y)], (4)

for such solutions in terms of the initial position u(0,y) and initial velocity \partial_t u(0,y), where t > 0, x \in {\Bbb R}^3, and dS is the area element of the sphere \{ y \in {\Bbb R}^3: |y-x|=t \}. (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

u_T(t,x) := c (T-t)^{-2/(p-1)}

for T > 0, 0 < t < T, and x \in {\Bbb R}^3, where c is the positive constant c := (\frac{2(p+1)}{(p-1)^2})^{1/(p-1)}. 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 u(0,\cdot), \partial_t u(0,\cdot) of solution is small, one expects the dispersion to “win” and send the solution to zero as t \to \infty, 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 1 < p < \infty.

  1. If p < 1+\sqrt{2}, 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.
  2. If p > 1+\sqrt{2}, 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 t \to \infty).

[At the critical threshold p = 1 + \sqrt{2} 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 1+\sqrt{2} 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.

Read the rest of this entry »

Archives

RSS Google+ feed

  • An error has occurred; the feed is probably down. Try again later.
Follow

Get every new post delivered to your Inbox.

Join 5,929 other followers