Matched asymptotic expansions and the Kaplun-Lagerstrom model

In a post I wrote a long time ago I described matched asymptotic expansions as being like black magic. Now I have understood some more about how to get from there to rigorous mathematics. My main guide in doing so has been Chapter 7 of the book ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. There they give an extensive treatment of a model problem by Kaplun and Lagerstrom. The ultimate source of this is work of Stokes on hydrodynamics around 1850. In his calculations he found some paradoxical phenomena. Roughly speaking, attempting to obtain an asymptotic expansion for a solution led to inconsistencies. These things remained a mystery for many years. A big step forward came in the work of Kaplun and Lagerstrom in 1957. There they introduced an ODE model which, while having no direct physical interpretation, provides a relatively simple mathematical context in which to understand these phenomena. It is this model problem which is treated in detail by Hastings and McLeod. The model is a boundary value problem for the equation . We look for a solution with and . The first two terms look like the expression for the Laplacian of a spherically symmetric function in dimensions and for this reason the motivation is strong to look at the cases and which are vaguely related to fluid flow around a cylinder and flow around a sphere, respectively. It turns out that the case is a lot harder to analyse than the case . When the problem has a unique solution for . We would like to understand what happens to this solution as . It is possible to find an asymptotic expansion in but it is not enough to use powers of when building the expansion. There occurs a so-called switchback term containing . This is a singular limit although the parameter in the equation only occurs in a lower order term. This happens because the equation is defined on a non-compact region.

Consider the case . In applying matched asymptotic expansions to this problem the first step is to do a straightforward (formal) expansion of the equation in powers of . This gives differential equations for the expansion coefficients. At order zero there is no problem solving the equation with the desired boundary conditions. At order one this changes and it is not possible to implement the desired boundary condition at infinity. This has to do with the fact that in the correct asymptotic expansion the second term is not of order but of order . This extra term is the switchback term. Up to this point all this is formal. One method of obtaining rigorous proofs for the asymptotics is to use GSPT, as done in two papers of Popovic and Szmolyan (J. Diff. Eq. 199, 290 and Nonlin. Anal. 59, 531). There is an introduction to this work in the book but I felt the need to go deeper and I looked at the original papers as well. To fit the notation of those papers I replace by . Reducing the equation to first order by introducing as a new variable leads to a non-autonomous system of two equations. Introducing as a new dependent variable and using it to eliminate from the right hand side of the equations in favour of leads to an autonomous system of three equations. This allows the original problem to be reformulated in the following geometric way. The -axis consists of steady states. The point is denoted by . The aim is to find a solution which starts at a point of the form and tends to as . A solution of this form for small and positive is to be found by perturbation of a corresponding solution in the case . For the centre manifold of is two-dimensional and given explicitly by . In the case it is more degenerate and has an additional zero eigenvalue. To prove the existence of the desired connecting orbit we may note that for this is equivalent to showing that the manifold consisting of solutions starting at points of the form and the manifold consisting of solutions converging to intersect. The first of these manifolds is obviously a deformation of a manifold for . We would like the corresponding statement for the second manifold. This is difficult to get because of the singularity of the limit. To overcome this is introduced as a new dynamical variable and a suitable blow-up is carried out near the -axis. In this way it is possible to get to a situation where there are two manifolds which exist for all and depend smoothly on . They intersect for and in fact do so transversely. It follows that they also intersect for small and positive. What I have said here only scratches the surface of this subject but it indicates the direction in which progress could be made and this is a fundamental insight.

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Breaking waves in Madrid

Last week I was at the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid. It was very large, with more than 2700 participants and countless parallel sessions. This kind of situation necessarily generates a somewhat hectic atmosphere and I do not really like going to that type of conference. I have heard the same thing from many other paople. There is nevertheless an advantage, namely the possibility of meeting many people. To do this effectively it is necessary to proceed systematically since it is easy to go for days without seeing a particular person of interest. This aspect was of particular importance for me since I am still at a relatively early stage in the process of entering the field of mathematical biology and I have few contacts there in comparison to my old field of mathematical relativity. In any case, the conference allowed me to meet a lot of interesting people and learn a lot of interesting things.

The first plenary talk, by Charles Fefferman, was on a subject related to a topic I was interested in many years ago. I learned that a lot has happened since I last thought about this. The attempt to model a body of fluid with a free surface leads to considerable mathematical difficulties. When I started working on dynamical models for this kind of situation few people seemed to be interested in proving theorems on the subject. The source of my interest in the subject was the influence of Jürgen Ehlers, who always had a clear vision of what were the important problems. In this way I found myself in the position of a pioneer in a certain research area. Being in that situation has the advantage of not being troubled by strong competition. On the other hand it can also mean that whatever you achieve can be largely ignored and it is not the best way to get wide recognition. Often finishing mathematical research directions gets more credit than starting them. This could no doubt be compensated by suitable advertizing but that was never my strong point. This is a configuration which I have often found myself in and in fact, comparing advantages and disadvantages, I do not feel I need to change it. Coming back to the fluids with free surface, this is now a hot topic and played a prominent role at the conference. When I was working on this the issue of local existence in the case of inviscid fluids was still open. A key step was the work of Sijue Wu on water waves. I learned from the talk of Fefferman that this has been extended in the meantime to global existence for small data. The question which is now the focus of interest is formation of singularities (i.e. breakdown of classical solutions) for large data. Instead of considering the breaking of one wave the idea is to consider two waves which are approaching each other while turning over until they meet. There are already analytical results on parts on this process by Fefferman and collaborators and they plan to extend this to a more global picture by using a computer-assisted proof. Another plenary was by Ingrid Daubechies, who talked about applications of image processing to art history. I must admit that beforehand the theme did not appear very attractive to me but in fact the talk was very entertaining and I am glad I went to hear it.

I gave a talk on my recent work with Juliette Hell on the MAPK cascade in a session organized by Bernold Fiedler and Atsushi Mochizuki. I found the session very interesting and the highlight for me was Mochizuki’s talk on his work with Fiedler. The subject is how much information can be obtained about a network of chemical reactions by observing a few nodes, i.e. by observing a few concentrations. What I find particularly interesting are the direct connections to biological problems. Applied to the gene regulatory network of an ascidian (sea squirt) this theoretical approach suggests that the network known from experimental observations is incomplete and motivates searching for the missing links experimentally. Among the many other talks I heard at the conference, one which I found particularly impressive concerned the analysis of successive MRT pictures of patients with metastases in the lung. The speaker was using numerical simulations with these pictures as input to provide the surgeon with indications which of the many lesions present was likely to develop in a dangerous way and should therefore be removed. One point raised in the talk is that it is not really clear what information about the tissue is really contained in an MRT picture and that this could be an interesting mathematical problem in itself. In fact there was an encouragingly (from my point of view) large number of sessions and other individual talks at the conference on subjects related to mathematical biology.

The conference took place on the campus of the Universidad Autonoma somewhat outside the city. A bonus for me was hearing and seeing my first bee-eater for many years. It was quite far away (flying high) but it gave me real pleasure. I was grateful that the temperatures during the week were very moderate, so that I could enjoy walking through the streets of Madrid in the evening without feeling disturbed by heat or excessive sun.

Shock waves, part 3

In a previous post I wrote about shock waves in fluids, including the case that they are described by the Einstein-Euler equations for a self-gravitating fluid in general relativity. I mentioned there a result of Fredrik Ståhl and myself proving that smooth solutions of the Einstein-Euler system can lose regularity in the course of their time evolution. This was done in the framework of spacetimes with plane symmetry. Here I want to describe some complementary results which were recently obtained by Philippe LeFloch and myself. These new results concern the existence of global weak solutions in situations where shocks may be present. This work is done under the assumption of Gowdy symmetry, which is weaker than plane symmetry. It allows the presence of gravitational waves, which plane symmetry does not. It uses time coordinates different from the constant mean curvature (CMC) coordinate used in the work with Ståhl. This difference in the time coordinates makes it difficult to relate the results of the two papers directly. It would be interesting to adapt the results of either of these papers to the time coordinates of the other.

In the paper with LeFloch we use coordinates (areal, conformal) which have previously been used in analysing analogous problems for vacuum spacetimes or spacetimes where the matter content is described by collisionless kinetic theory. A big difference is the weak regularity. One effect of this is that while in the given context it has been possible to prove global existence theorems for the initial value problem, nothing is known about the uniqueness of the solutions in terms of initial data. It should, however, be noted that in the corresponding analytical framework uniqueness is not even known for a one-dimensional non-relativistic fluid without gravity. Another new element introduced by the use of weak solutions is that it is only possible to evolve in one time direction. This model is not reversible, a fact implemented mathematically by the imposition of entropy inequalities. One of the results obtained concerns a forever expanding cosmological model. The other one concerns a contracting model which ends in a singularity. The second is not a global existence result in the conventional sense but it can be thought of as saying that the solution can be extended until certain specific things happen (a big crunch singularity).

To finish this post I want to indicate the type of regularity of the solutions obtained. I only state this roughly – more precise information can be found in the paper. The energy density and momentum density of the fluid is integrable in space, with the norms locally bounded in time. The quantities parametrizing the spacetime metric have first order derivatives which are square integrable in space. These conditions allow for jump discontinuities in the energy density which is what comes up in shock waves. It also allows singularities of Dirac type in the metric, corresponding to what are often called impulsive gravitational waves.

Shock waves, part 2

In a previous post I mentioned the recent work of Demetrios Christodoulou on shock waves. On 08.12 I heard a talk by Christodoulou on this subject and since then he has explained some of the most important points of this work to me in more detail. Here I will present a little of what I learned. These results concern solutions of the relativistic Euler equations in Minkowski space. According to Christodoulou analogous results could be obtained in the non-relativistic case but no details of this have been published. The initial data is given on a hyperplane and is assumed to coincide outside a compact set with a constant state where the fluid is at rest with constant density. On the compact set the data is close to the constant state in a suitable sense.

The object of study is the maximal smooth solution evolving from the given data and its future boundary. This boundary will be non-empty exactly in the case when a shock is formed. Necessary and sufficient conditions are given for there to be a shock. Most of the results concern a fluid which is isentropic and irrotational. The two conditions are intimately connected and cannot be assumed independently of each other. These results do have consequences for the general case since a sufficiently large region exists where the extra conditions are satisfied. Here I will concentrate on the isentropic and irrotational case. A central point is that while the solution becomes singular at the future boundary it is actually smooth up to and including the boundary with respect to a non-standard differential structure. Key computations are done in coordinates which define this different kind of smoothness. One of these coordinates is constant on sound cones of the solution being considered. The condions for shock formation depend very much on the sign of the quantity at the constant state, where is the equation of state of the fluid. This is the same as the sign of the quantity introduced in the book. If this quantity actually vanishes on the constant state then there are no shocks. For an irrotational and isotropic flow the evolution equations of the fluid can be written as a quasilinear wave equation for a scalar function . In the case that is identically zero for a given equation of state this coincides with a geometric equation which is obtained as follows. Suppose that a timelike hypersurface in five-dimensional Minkowski space can be expressed as the graph of a function on . Suppose in other words that the hypersurface is of the form . Then the condition that this hypersurface has vanishing mean curvature is equivalent to the equation of motion for this particular type of fluid. The fluid is related to the Chaplygin gas which has been studied in cosmology in recent years. It has equation of state for a constant and satisfies . This type of fluid originally came up in aerodynamics in the early years of the twentieth century. Sergei Chaplygin, after whom this fluid is named, seems to have been quite a prominent figure since the town he grew up in is now named after him, as is a crater on the moon. For fluids under normal physical conditions but, as pointed out in the book, there are physical situations where the opposite sign occurs.

Shock waves

A partial differential equation is the requirement that partial derivatives up to order of some unknown function satisfy a certain algebraic relation. An evident prerequisite for this to make sense is that the derivatives of the function up to order exist. It turns out, however, that this apparently obvious statement is not true. In many cases it is possible to reformulate a PDE so as to make it meaningful to talk about solutions which have less derivatives than appear to be necessary at first sight. This leads to the concept of weak (or generalized) solutions, in contrast to classical solutions which have the derivatives needed for the straightforward interpretation of the equation.

In hydrodynamics, the study of fluids, weak solutions play an important role due to the phenomenon of shock waves. Solutions of the Euler equations which describe a fluid while neglecting viscosity have the tendency to develop discontinuities in the basic fluid variables (e.g. the density) even when starting from a perfectly smooth configuration. If the solution is to be used for modelling a physical situation beyond the time where a discontinuity appears it is necessary to use weak solutions. It turns out that weak solutions of the Euler equations can be used effectively in hydrodynamics and there are powerful methods for handling them numerically. At the same time it is very difficult to prove rigorous mathematical results about solutions of the Euler equations in the presence of shocks. Most of the theorems available in the literature concern situations which are reduced to an effective problem in one space dimension by means of a symmetry assumption (plane symmetry). Recently a notable exception to this appeared in the form of a book ‘The formation of shocks in 3-dimensional fluids‘ by Demetrios Christodoulou which treats the dynamics of solutions of the Euler equations up to the moment of shock formation without requiring symmetry assumptions. See also the recent review article of Christodoulou for background to this work and a concise history of mathematical developments in hydrodynamics (Bull. Amer. Math. Soc. 44, 581).

If viscosity is included in the description of fluids then the Euler equations are replaced by the Navier-Stokes equations. There is reason to suspect that in this case shock waves are smoothed out and a smooth initial configuration remains smooth in the course of the evolution, for all time. There are simple examples where this can be seen but there is still no global regularity result for Navier-Stokes (and no counterexample). The Clay Foundation has offered a prize of one million dollars for the solution of this problem in either direction. The fact that the prize has not yet been collected is a sign of the difficulty of the problem. For a discussion of this question and its broader mathematical significance I recommend the excellent account of Tao.

What effect does gravity have on the formation of shock waves? It is reasonable to suppose that the answer to this question is ‘almost none’. This intuition applies not only in Newtonian physics but also to the case of a fully relativistic description in the context of general relativity. The gravitational field curves space but in many situations the curvature produced should not affect the qualitative nature of the process of shock formation. A couple of years ago Fredrik Ståhl and I set out to confirm this idea rigorously. The project has been delayed by other things but now we have finished a manuscript and put it on the arXiv (Shock Waves in Plane Symmetric Spacetimes). The result is that there are plane symmetric solutions of the relativistic Euler equations coupled to the Einstein equations of general relativity which lose smoothness in an arbitrarily short time, depending on the initial size of the spatial derivatives of the fluid variables. The fact is used that the mechanism of breakdown bears a sufficiently close resemblance to that in a non-relativistic fluid without gravity. In this one-dimensional setting there are left-moving and right-moving waves in the fluid and gravity leads to a coupling between them. A central idea of the proof of our result is to use changes of variable so as to attain a sufficient amount of decoupling of the degrees of freedom corresponding to the two types of waves.