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I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear Schrödinger system, submitted to Analysis and PDE. This paper is an analogue of a recent paper of mine in which I constructed a supercritical defocusing nonlinear wave (NLW) system which exhibited smooth solutions that developed singularities in finite time. Here, we achieve essentially the same conclusion for the (inhomogeneous) supercritical defocusing nonlinear Schrödinger (NLS) equation
where is now a system of scalar fields, is a potential which is strictly positive and homogeneous of degree (and invariant under phase rotations ), and is a smooth compactly supported forcing term, needed for technical reasons.
To oversimplify somewhat, the equation (1) is known to be globally regular in the energy-subcritical case when , or when and ; global regularity is also known (but is significantly more difficult to establish) in the energy-critical case when and . (This is an oversimplification for a number of reasons, in particular in higher dimensions one only knows global well-posedness instead of global regularity. See this previous post for some exploration of this issue in the context of nonlinear wave equations.) The main result of this paper is to show that global regularity can break down in the remaining energy-supercritical case when and , at least when the target dimension is allowed to be sufficiently large depending on the spatial dimension (I did not try to achieve the optimal value of here, but the argument gives a value of that grows quadratically in ). Unfortunately, this result does not directly impact the most interesting case of the defocusing scalar NLS equation
in which ; however it does establish a rigorous barrier to any attempt to prove global regularity for the scalar NLS equation, in that such an attempt needs to crucially use some property of the scalar NLS that is not shared by the more general systems in (1). For instance, any approach that is primarily based on the conservation laws of mass, momentum, and energy (which are common to both (1) and (2)) will not be sufficient to establish global regularity of supercritical defocusing scalar NLS.
The method of proof in this paper is broadly similar to that in the previous paper for NLW, but with a number of additional technical complications. Both proofs begin by reducing matters to constructing a discretely self-similar solution. In the case of NLW, this solution lived on a forward light cone and obeyed a self-similarity
The ability to restrict to a light cone arose from the finite speed of propagation properties of NLW. For NLS, the solution will instead live on the domain
and obey a parabolic self-similarity
and solve the homogeneous version of (1). (The inhomogeneity emerges when one truncates the self-similar solution so that the initial data is compactly supported in space.) A key technical point is that has to be smooth everywhere in , including the boundary component . This unfortunately rules out many of the existing constructions of self-similar solutions, which typically will have some sort of singularity at the spatial origin.
The remaining steps of the argument can broadly be described as quantifier elimination: one systematically eliminates each of the degrees of freedom of the problem in turn by locating the necessary and sufficient conditions required of the remaining degrees of freedom in order for the constraints of a particular degree of freedom to be satisfiable. The first such degree of freedom to eliminate is the potential function . The task here is to determine what constraints must exist on a putative solution in order for there to exist a (positive, homogeneous, smooth away from origin) potential obeying the homogeneous NLS equation
Firstly, the requirement that be homogeneous implies the Euler identity
(where denotes the standard real inner product on ), while the requirement that be phase invariant similarly yields the variant identity
so if one defines the potential energy field to be , we obtain from the chain rule the equations
Conversely, it turns out (roughly speaking) that if one can locate fields and obeying the above equations (as well as some other technical regularity and non-degeneracy conditions), then one can find an with all the required properties. The first of these equations can be thought of as a definition of the potential energy field , and the other three equations are basically disguised versions of the conservation laws of mass, energy, and momentum respectively. The construction of relies on a classical extension theorem of Seeley that is a relative of the Whitney extension theorem.
Now that the potential is eliminated, the next degree of freedom to eliminate is the solution field . One can observe that the above equations involving and can be expressed instead in terms of and the Gram-type matrix of , which is a matrix consisting of the inner products where range amongst the differential operators
To eliminate , one thus needs to answer the question of what properties are required of a matrix for it to be the Gram-type matrix of a field . Amongst some obvious necessary conditions are that needs to be symmetric and positive semi-definite; there are also additional constraints coming from identities such as
Ideally one would like a theorem that asserts (for large enough) that as long as obeys all of the “obvious” constraints, then there exists a suitably non-degenerate map such that . In the case of NLW, the analogous claim was basically a consequence of the Nash embedding theorem (which can be viewed as a theorem about the solvability of the system of equations for a given positive definite symmetric set of fields ). However, the presence of the complex structure in the NLS case poses some significant technical challenges (note for instance that the naive complex version of the Nash embedding theorem is false, due to obstructions such as Liouville’s theorem that prevent a compact complex manifold from being embeddable holomorphically in ). Nevertheless, by adapting the proof of the Nash embedding theorem (in particular, the simplified proof of Gunther that avoids the need to use the Nash-Moser iteration scheme) we were able to obtain a partial complex analogue of the Nash embedding theorem that sufficed for our application; it required an artificial additional “curl-free” hypothesis on the Gram-type matrix , but fortunately this hypothesis ends up being automatic in our construction. Also, this version of the Nash embedding theorem is unable to prescribe the component of the Gram-type matrix , but fortunately this component is not used in any of the conservation laws and so the loss of this component does not cause any difficulty.
After applying the above-mentioned Nash-embedding theorem, the task is now to locate a matrix obeying all the hypotheses of that theorem, as well as the conservation laws for mass, momentum, and energy (after defining the potential energy field in terms of ). This is quite a lot of fields and constraints, but one can cut down significantly on the degrees of freedom by requiring that is spherically symmetric (in a tensorial sense) and also continuously self-similar (not just discretely self-similar). Note that this hypothesis is weaker than the assertion that the original field is spherically symmetric and continuously self-similar; indeed we do not know if non-trivial solutions of this type actually exist. These symmetry hypotheses reduce the number of independent components of the matrix to just six: , which now take as their domain the -dimensional space
One now has to construct these six fields, together with a potential energy field , that obey a number of constraints, notably some positive definiteness constraints as well as the aforementioned conservation laws for mass, momentum, and energy.
The field only arises in the equation for the potential (coming from Euler’s identity) and can easily be eliminated. Similarly, the field only makes an appearance in the current of the energy conservation law, and so can also be easily eliminated so long as the total energy is conserved. But in the energy-supercritical case, the total energy is infinite, and so it is relatively easy to eliminate the field from the problem also. This leaves us with the task of constructing just five fields obeying a number of positivity conditions, symmetry conditions, regularity conditions, and conservation laws for mass and momentum.
The potential field can effectively be absorbed into the angular stress field (after placing an appropriate counterbalancing term in the radial stress field so as not to disrupt the conservation laws), so we can also eliminate this field. The angular stress field is then only constrained through the momentum conservation law and a requirement of positivity; one can then eliminate this field by converting the momentum conservation law from an equality to an inequality. Finally, the radial stress field is also only constrained through a positive definiteness constraint and the momentum conservation inequality, so it can also be eliminated from the problem after some further modification of the momentum conservation inequality.
The task then reduces to locating just two fields that obey a mass conservation law
together with an additional inequality that is the remnant of the momentum conservation law. One can solve for the mass conservation law in terms of a single scalar field using the ansatz
so the problem has finally been simplified to the task of locating a single scalar field with some scaling and homogeneity properties that obeys a certain differential inequality relating to momentum conservation. This turns out to be possible by explicitly writing down a specific scalar field using some asymptotic parameters and cutoff functions.
Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.
A (real) manifold can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space . On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:
Theorem 1 (Whitney embedding theorem) Let be a compact manifold. Then there exists an embedding from to a Euclidean space .
In fact, if is -dimensional, one can take to equal , which is often best possible (easy examples include the circle which embeds into but not , or the Klein bottle that embeds into but not ). One can also relax the compactness hypothesis on to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take equal to .
A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:
Theorem 2 (Nash embedding theorem) Let be a compact Riemannian manifold. Then there exists a isometric embedding from to a Euclidean space .
In order to obtain the isometric embedding, the dimension has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound
is attained, which I believe is still the record for large . (In the converse direction, one cannot do better than , basically because this is the number of degrees of freedom in the Riemannian metric .) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).
I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of , but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in is that it allows one to freely exploit the very useful tool of pairing together two maps , to form a combined map that can be closer to an embedding or an isometric embedding than the original maps . This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of and then pairs them together to form a “better” embedding.
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.
where is now a system of scalar fields, and is a potential which is homogeneous of degree and strictly positive away from the origin; the scalar equation corresponds to the case where and . We will be interested in smooth solutions to (2). It is only natural to restrict to the smooth category when the potential is also smooth; unfortunately, if one requires to be homogeneous of order all the way down to the origin, then cannot be smooth unless it is identically zero or is an odd integer. This is too restrictive for us, so we will only require that be homogeneous away from the origin (e.g. outside the unit ball). In any event it is the behaviour of for large which will be decisive in understanding regularity or blowup for the equation (2).
Formally, solutions to the equation (2) enjoy a conserved energy
Using this conserved energy, it is possible to establish global regularity for the Cauchy problem (2) in the energy-subcritical case when , or when and . This means that for any smooth initial position and initial velocity , there exists a (unique) smooth global solution to the equation (2) with and . These classical global regularity results (essentially due to Jörgens) were famously extended to the energy-critical case when and by Grillakis, Struwe, and Shatah-Struwe (though for various technical reasons, the global regularity component of these results was limited to the range ). A key tool used in the energy-critical theory is the Morawetz estimate
which can be proven by manipulating the properties of the stress-energy tensor
(with the usual summation conventions involving the Minkowski metric ) and in particular exploiting the divergence-free nature of this tensor: 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 and . On the one hand, global smooth solutions are known for small data (if 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 , one can easily create solutions that blow up in finite time by ODE constructions, for instance one can take with , which blows up as approaches . 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 of degrees of freedom is sufficiently large:
Theorem 1 Let , , and . Then there exists a smooth potential , positive and homogeneous of degree 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 on 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 all the way down to , 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 will obey a relation of the form
for some fixed (the exponent is mandated by dimensional analysis considerations). It would be natural to consider continuously self-similar solutions (in which the above relation holds for all , not just one ). 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 (and the initial data ) as being given in advance, and then try to solve for as an unknown field. However, in this problem we have the freedom to select . So we can look at this problem from a “backwards” direction: we first choose the field , and then fit the potential (and the initial data) to match that field.
we see that can be recovered from (2) by the formula
Furthermore, taking a derivative of (3) we obtain another constraining equation
that does not explicitly involve the potential . Actually, one can write this equation in the more familiar form
where is the stress-energy tensor
now written in a manner that does not explicitly involve .
With this reformulation, this suggests a strategy for locating : first one selects a stress-energy tensor that is divergence-free and obeys suitable positive definiteness and self-similarity properties, and then locates a self-similar map from the backwards light cone to that has that stress-energy tensor (one also needs the map (or more precisely the direction component of that map) injective up to the discrete self-similarity, in order to define consistently). If the stress-energy tensor was replaced by the simpler “energy tensor”
then the question of constructing an (injective) map with the specified energy tensor is precisely the embedding problem that was famously solved by Nash (viewing as a Riemannian metric on the domain of , 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 , 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 shows up:
Proposition 2 Let be a smooth compact Riemannian -manifold, and let . Then smoothly isometrically embeds into the sphere .
Proof: The Nash embedding theorem (in the form given in this ICM lecture of Gunther) shows that can be smoothly isometrically embedded into , and thus in for some large . Using an irrational slope, the interval can be smoothly isometrically embedded into the -torus , and so and hence can be smoothly embedded in . But from Pythagoras’ theorem, can be identified with a subset of for any , and the claim follows.
One can presumably improve upon the bound 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 . 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 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 to only be half a degree of freedom). This is easiest to illustrate with the scalar NLW (1). Assuming spherical symmetry, this equation becomes
Making the substitution , we can eliminate the lower order term completely to obtain
(This can be compared with the situation in higher dimensions, in which an undesirable zeroth order term shows up.) In particular, if one introduces the null energy density
and the potential energy density
then one can verify the equation
which can be viewed as a transport equation for with forcing term depending on (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 , one can solve this equation while giving and the right homogeneity (they have to be homogeneous of order , which is greater than 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 and will be concentrated near the boundary of the light cone, so this is how the solution will concentrate also.)