Spectral theory and self-similar blowup in wave equations

By Roland Donninger

Abstract

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account of the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.

1. Introduction

Nonlinear wave equations play a fundamental role in many branches of the natural sciences and mathematics. Probably the most famous examples in physics are the Einstein equation of general relativity and the Yang–Mills equations of particle physics. What all of these fundamental equations have in common is the fact that they are energy-supercritical (in the case of Yang–Mills in spatial dimensions larger than four). This means that the known conserved quantities (most notably the energy) are not strong enough to control the evolution. As a result, the mathematical understanding of large-data evolutions is still embarrassingly poor. In many cases, however, there exist self-similar solutions and one may learn something about the general large-data behavior by looking at perturbations of these large but special solutions. This approach is promising because it allows one to employ perturbative techniques in a large-data regime that is otherwise inaccessible to rigorous mathematical analysis. Such a perturbative treatment involves a number of interesting spectral-theoretic aspects that are at the center of this article.

1.1. Wave maps

For the purpose of this exposition we will not discuss nonlinear wave equations in any kind of generality but rather focus on a particular example: the classical SU(2)-sigma model from particle physics, also known as the wave maps equation, which constitutes the simplest and prototypical example of a geometric wave equation. The methods we discuss, however, have a much broader scope and we mention applications to other problems in the end. In order to introduce the model, we consider maps , where denotes the -dimensional Minkowski space. Then is called a wave map if it satisfies the partial differential equation

Here, we employ standard relativistic notation with Einstein’s summation convention in force⁠Footnote1 and denotes the Euclidean inner product on . The wave maps equation arises as the Euler–Lagrange equation of the action functional

1

That is to say, we number the slots of a function on from to where the th slot holds the time variable. The partial derivative with respect to the th slot is denoted by and we write as well as for . Furthermore, indices that come in pairs of subscripts and superscripts get summed over implicitly. Greek (spacetime) indices run from to and latin (spatial) indices run from to .

under the constraint that for all . Note that without the constraint, the Euler–Lagrange equation associated to equation Equation 1.2 is the standard free wave equation . In this sense, wave maps are natural generalizations of solutions to the wave equation when the unknown takes values in the sphere. In place of Minkowski space and the three-sphere, one may also consider more general manifolds by adapting the functional Equation 1.2 accordingly. This shows that the wave maps action is a rich source for interesting and natural geometric wave equations. In this exposition, for the sake of concreteness, we restrict ourselves to maps from to . We remark in passing that in more traditional notation, equation Equation 1.1 would read

but in this form, the underlying geometric structure is severely obscured.

1.2. Corotational wave maps and singularity formation

The most basic question concerns the existence of smooth solutions to equation Equation 1.1. For the sake of simplicity, we further restrict our attention to the special class of corotational maps which are of the form

for an auxiliary function . This ansatz turns out to be compatible with the wave maps equation, i.e., when plugging it in, we obtain the single semilinear radial wave equation

The principal goal is to construct global solutions and since equation Equation 1.3 is a wave equation, the natural mathematical setting to approach this question is to study the Cauchy problem, i.e., we prescribe initial data and and try to construct a solution to equation Equation 1.3 with these data. However,

for any solves equation Equation 1.3 on , as one checks by a direct computation. At , exhibits a gradient blowup and hence, it is impossible to construct global smooth solutions for arbitrary data. Consequently, the goal is to understand the nature of this breakdown (or “loss of smoothness” or “singularity formation” or “blowup”) and its relevance for “generic” evolutions. More precisely, the question is whether can tell us something about more general large-data evolutions, even though it is just one particular solution. In other words, we are interested in stability properties of , i.e., we would like to understand all solutions that are close to . We remark that is a self-similar solution, i.e., it depends on the ratio only. The existence of self-similar solutions to equation Equation 1.3 was first proved in Reference 34 and the explicit example was found in Reference 37; see Reference 4 for higher dimensions. In fact, there are many more self-similar solutions to equation Equation 1.3 (see Reference 2) but they are all linearly unstable and hence less important for studying generic evolutions.

2. The mode stability problem

If the self-similar solution has any relevance for generic large-data evolutions, it certainly must be stable under perturbations of the initial data. Thus, an important mathematical goal is to prove (or disprove) the stability of . The most elementary form of stability is mode stability. The formulation of the mode stability problem can be given purely on the level of the differential equation and requires no operator-theoretic framework.

2.1. Similarity coordinates

In order to introduce the mode stability problem, we start with the wave maps equation Equation 1.3 and switch to similarity coordinates

or

where is a parameter. Then satisfies equation Equation 1.3 if and only if satisfies

Observe the remarkable fact that equation Equation 2.2 is an autonomous equation, i.e., its coefficients do not depend on . This is, in fact, a decisive feature of the similarity coordinates Equation 2.1. Furthermore, the parameter does not show up in equation Equation 2.2. To begin with, we will consider equation Equation 2.2 in the coordinate range and , which corresponds to the backward lightcone of the point in the “physical” coordinates .

The blowup solution transforms into

We would like to understand the stability of the family . First, let us point out that is independent of whereas nearby members of the family move away from as increases. Indeed, if , develops a gradient blowup as , where is determined by . On the other hand, if , as . By these observations, it is expected that the -independent solution is unstable because a generic perturbation will push it towards a nearby member of the family. However, such a “push” can be compensated by adapting . Thus, the instability is “artificial” and caused by the free parameter in the definition of the similarity coordinates or, on a more fundamental level, by the time-translation invariance of the wave maps equation. In other words, stability of the blowup means that for any given (small) initial perturbation of , say, there exists a close to that makes the evolution in similarity coordinates with parameter converge to . This is very natural in view of the expectation that a perturbation of a blowup solution will, in general, change the blowup time.

2.2. Mode solutions

The most elementary stability analysis consists of looking for mode solutions. This means that one plugs in the ansatz

into equation Equation 2.2 and linearizes in . This yields the “spectral problem”

Clearly, if there are “admissible” mode solutions with , we expect the solution to be unstable. What exactly “admissible” in this context means can only be answered once one has set up the functional analytic framework to study the wave maps evolution. For now we will restrict ourselves to smooth solutions and we will see later that this is the correct class of functions. Furthermore, observe that equation Equation 2.3 has singular points at and and, therefore, it is expected that only for special values of there will be nontrivial solutions in . Another important fact is that equation Equation 2.3 does not constitute a standard eigenvalue problem because the spectral parameter appears in the coefficient of the derivative as well. This is easily traced back to the fact that the wave maps equation is second-order in time. Consequently, this issue is not present in analogous parabolic problems where the corresponding spectral analysis is therefore much simpler. Of course, the first-order term can always be removed but the corresponding transformation depends on itself. As a consequence, it turns out that equation Equation 2.3 is not a self-adjoint Sturm–Liouville problem in disguise where standard methods from mathematical physics would apply. We discuss this in more detail below.

We have already argued that we expect an “artificial” instability of . So how does this instability show up in the context of the spectral problem equation Equation 2.3? To see this, we differentiate the equation

with respect to and evaluate the result at . This yields

with

Observe that is a mode solution. Consequently, the function solves equation Equation 2.3 with and this is the mode solution that reflects the expected “artificial” instability. This observation naturally leads to the following definition.

Definition 2.1.

We say that the blowup solution is mode stableFootnote2 if the existence of a nontrivial that satisfies equation Equation 2.3 necessarily implies that or .

2

The experienced reader might think ahead and be worried about spectral multiplicities. It turns out that this is never an issue in the class of problems we consider here and, therefore, Definition 2.1 is the “correct” one. At this point we cannot even discuss multiplicities because we do not yet have a proper operator-theoretic framework.

In what follows, we somewhat imprecisely call an eigenvalue of equation Equation 2.3 if equation Equation 2.3 has a nontrivial solution in . Accordingly, we call such a solution an eigenfunction of equation Equation 2.3.

3. Solution of the mode stability problem

In this section, which is at the heart of the present exposition, we describe an approach to the mode stability problem that was developed in Irfan Glogić’s PhD thesis Reference 27 and first published in Reference 9Reference 10, building on earlier work Reference 8Reference 15Reference 22 and ideas in Reference 2Reference 3Reference 5. So far, it is the only known method that can rigorously deal with spectral problems like equation Equation 2.3.

Theorem 3.1.

The blowup solution is mode stable.

The proof of mode stability proceeds by the following main steps.

We use Frobenius’s method to determine the local behavior of solutions to equation Equation 2.3 near the singular points and .

By a factorization procedure inspired by supersymmetric quantum mechanics we “remove” the eigenvalue . More precisely, we derive a “supersymmetric problem”, similar to equation Equation 2.3, that has the same eigenvalues as equation Equation 2.3 except for .

We prove that the supersymmetric problem has no eigenvalues in the closed complex right half-plane. To this end, we derive a recurrence relation for the coefficients of the power series of the admissible solution near and prove that the series necessarily diverges at if . This requires the interplay of techniques from the theory of difference equations and complex analysis.

3.1. Fuchsian classification

To begin with, we would like to understand better which problem we are actually facing. The term in equation Equation 2.3 involving the cosine turns out to be a rational function. Indeed, we have

and thus, equation Equation 2.3 has the (regular) singular points . By switching to the independent variable , the number of singular points can be reduced to four: , and . This means that equation Equation 2.3 is a Fuchsian differential equation of Heun type. The normal form for a Heun equation reads

where . Around each of the singular points there exist two linearly independent local solutions. The interesting question then is how local solutions around different singular points are related to each other. This is known as the connection problem and, unfortunately for Heun equations, this problem is widely open. If we had only three regular singular points we would be dealing with a hypergeometric differential equation for which the connection problem was solved in the 19th century. This indicates that the spectral problem we are dealing with is potentially hard.

3.2. Frobenius analysis

Now we turn to a more quantitative analysis and first recall Frobenius’ theory for Fuchsian equations of second order. These are equations over the complex numbers of the form

where and are given functions and is the unknown. In the following, we write .

Theorem 3.2.

Let and let be holomorphic. Suppose that the limits

exist and let satisfy , where

is the indicial polynomial. Let . Then there exists a holomorphic function with and such that , given by , satisfies equation Equation 3.1. Furthermore, if , there exists a holomorphic function with and such that is another solution of equation Equation 3.1 on . Finally, if , there exist and a holomorphic function with such that

is another solution of equation Equation 3.1 on .

Idea of proof.

The idea is to plug in a generalized power series ansatz and to determine and the coefficients by comparing powers of . The convergence of the corresponding series is then shown by a simple induction. The second solution can be obtained by the reduction of order ansatz. We remark in passing that even in the case , the term may be absent but this depends on the fine structure and needs to be analyzed on a case-by-case basis. We omit the details of the proof because Theorem 3.2 is a classical result that can be found in many textbooks; see, e.g., Reference 36 for a modern account.

Slightly rearranged, equation Equation 2.3 reads

with

and the indicial polynomial at reads with zeros and . As expected, there is only one smooth solution around and it behaves like . At , the indicial polynomial is given by with zeros and . Again, there is only one smooth solution around if (the cases require some extra care). Thus, our goal is to show that the local solution that is smooth around is necessarily nonsmooth at if (and ).

3.3. Supersymmetric removal

The case is special and we already know that this is an eigenvalue. In order to proceed, it is necessary to “remove” it. This can be achieved by a factorization procedure that has its origin in supersymmetric quantum mechanics (hence the name). In our setting, the procedure is as follows. First, we introduce an auxiliary function by , where we choose in such a way that the resulting equation for has no first-order derivative. Indeed, inserting the above ansatz into equation Equation 3.2 yields the condition

which is satisfied, e.g., by . Plugging the ansatz

into equation Equation 3.2 yields

Recall that the function solves equation Equation 3.2 with . Thus,

satisfies

Motivated by this, we rewrite equation Equation 3.3 as

This resembles a spectral problem for a Schrödinger operator with a ground state .

At this point we digress and reiterate that our mode stability problem cannot be reduced to studying the spectrum of the self-adjoint realization of the Schrödinger operator in equation Equation 3.4. The reason is that an admissible eigenfunction of equation Equation 2.3 transforms into a solution of equation Equation 3.4 that behaves like near . However, if , this function is not in