Archive for December, 2022

Matched asymptotic expansions and the Kaplun-Lagerstrom model

December 13, 2022

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 $y''+\frac{n-1}{r}y'+\epsilon yy'=0$. We look for a solution with $y(1)=0$ and $\lim_{r\to\infty}y(r)=1$. The first two terms look like the expression for the Laplacian of a spherically symmetric function in $n$ dimensions and for this reason the motivation is strong to look at the cases $n=2$ and $n=3$ which are vaguely related to fluid flow around a cylinder and flow around a sphere, respectively. It turns out that the case $n=2$ is a lot harder to analyse than the case $n=3$. When $n=3$ the problem has a unique solution for $\epsilon>0$. We would like to understand what happens to this solution as $\epsilon\to 0$. It is possible to find an asymptotic expansion in $\epsilon$ but it is not enough to use powers of $\epsilon$ when building the expansion. There occurs a so-called switchback term containing $\log\epsilon$. 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 $n=3$. In applying matched asymptotic expansions to this problem the first step is to do a straightforward (formal) expansion of the equation in powers of $\epsilon$. 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 $\epsilon$ but of order $\epsilon\log\epsilon$. 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 $y$ by $u$. Reducing the equation to first order by introducing $v=u'$ as a new variable leads to a non-autonomous system of two equations. Introducing $\eta=1/r$ as a new dependent variable and using it to eliminate $r$ from the right hand side of the equations in favour of $\eta$ leads to an autonomous system of three equations. This allows the original problem to be reformulated in the following geometric way. The $u$-axis consists of steady states. The point $(1,0,0)$ is denoted by $Q$. The aim is to find a solution which starts at a point of the form $(0,v,1)$ and tends to $Q$ as $r\to\infty$. A solution of this form for $\epsilon$ small and positive is to be found by perturbation of a corresponding solution in the case $\epsilon=0$. For $\epsilon>0$ the centre manifold of $Q$ is two-dimensional and given explicitly by $v=0$. In the case $\epsilon=0$ it is more degenerate and has an additional zero eigenvalue. To prove the existence of the desired connecting orbit we may note that for $\epsilon>0$ this is equivalent to showing that the manifold consisting of solutions starting at points of the form $(0,v,1)$ and the manifold consisting of solutions converging to $Q$ intersect. The first of these manifolds is obviously a deformation of a manifold for $\epsilon=0$. 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 $\epsilon$ is introduced as a new dynamical variable and a suitable blow-up is carried out near the $u$-axis. In this way it is possible to get to a situation where there are two manifolds which exist for all $\epsilon\ge 0$ and depend smoothly on $\epsilon$. They intersect for $\epsilon=0$ and in fact do so transversely. It follows that they also intersect for $\epsilon$ 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.