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.)