---
tag: QBTK36P203
kind: definition
name: Boundary layer
msc: ["34E15", 76D10]
defines: YCFPD0GE5G
uses: [G6J21TGRMM]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_perturbation
---

## Definition

Let $a < b$ and let $(y_\varepsilon)_{0 < \varepsilon < \varepsilon_0}$ be a family of functions $[a, b] \to \R^n$, for example the solutions of a [[singularly perturbed|G6J21TGRMM]] boundary value problem.
Assume there is a continuous function $y_0 \colon (a, b] \to \R^n$, the *outer limit*, such that $y_\varepsilon \to y_0$ uniformly on $[a + \eta, b]$ for every $\eta \in (0, b - a)$.
The family has a *boundary layer* at $a$ if the convergence is not uniform on $(a, b]$, or $y_\varepsilon(a)$ does not tend to $\lim_{x \to a^+} y_0(x)$.
Let $\delta(\varepsilon) > 0$ with $\delta(\varepsilon) \to 0$, and consider the rescaled functions $X \mapsto y_\varepsilon(a + \delta(\varepsilon) X)$.
If they converge to a nonconstant function, uniformly on compact subsets of $[0, \infty)$, then $\delta$ is a *thickness* of the layer.
A boundary layer at $b$ is defined in the same way.

## Remarks

Example: $y_\varepsilon(x) = (1 - e^{-x/\varepsilon}) / (1 - e^{-1/\varepsilon})$ on $[0, 1]$, the solution of $\varepsilon y'' + y' = 0$, $y(0) = 0$, $y(1) = 1$.
The outer limit is $y_0 = 1$, and on $[\eta, 1]$ the error is at most $e^{-\eta/\varepsilon} / (1 - e^{-1/\varepsilon}) \to 0$.
But $y_\varepsilon(0) = 0$, so there is a boundary layer at $0$.
With $\delta(\varepsilon) = \varepsilon$, $y_\varepsilon(\varepsilon X) \to 1 - e^{-X}$, so the thickness is $\varepsilon$.

Non-example: $y_\varepsilon(x) = \varepsilon + (1 - \varepsilon) e^{-x}$ converges uniformly on all of $[0, 1]$, so it has no boundary layer.

Layers can also sit inside the interval; these are called *interior layers*, or *shocks* when the solution jumps.

## Explanations

### L1 human

A boundary layer is where the simple approximation of the solution stops working.
It is a strip near the boundary whose width shrinks with the small parameter.