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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $(y_\\varepsilon)_{0 \u003c \\varepsilon \u003c \\varepsilon_0}$ be a family of functions $[a, b] \\to \\R^n$, for example the solutions of a [[singularly perturbed|G6J21TGRMM]] boundary value problem.\nAssume 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 $\\eta \\in (0, b - a)$.\nThe family has a *boundary layer* at $a$ if the convergence is not uniform on $(a, b]$, or $y_\\varepsilon(a)$ does not tend to $\\lim_{x \\to a^+} y_0(x)$.\nLet $\\delta(\\varepsilon) \u003e 0$ with $\\delta(\\varepsilon) \\to 0$, and consider the rescaled functions $X \\mapsto y_\\varepsilon(a + \\delta(\\varepsilon) X)$.\nIf they converge to a nonconstant function, uniformly on compact subsets of $[0, \\infty)$, then $\\delta$ is a *thickness* of the layer.\nA boundary layer at $b$ is defined in the same way.\n\n## Remarks\n\nExample: $y_\\varepsilon(x) = (1 - e^{-x/\\varepsilon}) / (1 - e^{-1/\\varepsilon})$ on $[0, 1]$, the solution of $\\varepsilon y'' + y' = 0$, $y(0) = 0$, $y(1) = 1$.\nThe outer limit is $y_0 = 1$, and on $[\\eta, 1]$ the error is at most $e^{-\\eta/\\varepsilon} / (1 - e^{-1/\\varepsilon}) \\to 0$.\nBut $y_\\varepsilon(0) = 0$, so there is a boundary layer at $0$.\nWith $\\delta(\\varepsilon) = \\varepsilon$, $y_\\varepsilon(\\varepsilon X) \\to 1 - e^{-X}$, so the thickness is $\\varepsilon$.\n\nNon-example: $y_\\varepsilon(x) = \\varepsilon + (1 - \\varepsilon) e^{-x}$ converges uniformly on all of $[0, 1]$, so it has no boundary layer.\n\nLayers can also sit inside the interval; these are called *interior layers*, or *shocks* when the solution jumps.\n\n## Explanations\n\n### L1 human\n\nA boundary layer is where the simple approximation of the solution stops working.\nIt is a strip near the boundary whose width shrinks with the small parameter.\n",
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