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  "body": "\n## Statement\n\nLet $X_0 \\subseteq X \\subseteq X_1$ be three [[Banach spaces|06GZ956657]] such that $X_0$ and $X_1$ are [[reflexive|NN8RWPC8WG]], the inclusion $X_0 \\to X$ is compact, and the inclusion $X \\to X_1$ is continuous.\nLet $T \u003e 0$ and $1 \u003c p, q \u003c \\infty$.\nLet $L^p(0, T; X_0)$ be the [[Lebesgue space|SASP14MA1Z]] of [[Bochner-integrable|MHSMQ097GP]] functions with values in $X_0$, and let $u'$ be the weak time derivative.\nLet $W = \\set{ u \\in L^p(0, T; X_0) : u' \\in L^q(0, T; X_1) }$.\nGive $W$ the norm $\\norm{u}_W = \\norm{u}_{L^p(0, T; X_0)} + \\norm{u'}_{L^q(0, T; X_1)}$.\nThen the inclusion of $W$ into $L^p(0, T; X)$ is compact.\n\n## Remarks\n\nProof idea: by Ehrling's lemma, for every $\\eta \u003e 0$ there is $C_\\eta$ with $\\norm{v}_X \\le \\eta \\norm{v}_{X_0} + C_\\eta \\norm{v}_{X_1}$ for all $v \\in X_0$.\nA bounded set in $W$ is bounded in $L^p(0, T; X_0)$ and equicontinuous in time with values in $X_1$, because of the bound on $u'$.\nThis gives a subsequence that converges in $L^p(0, T; X_1)$, and Ehrling's lemma upgrades the convergence to $L^p(0, T; X)$.\n\nExample: let $\\Omega \\subseteq \\R^n$ be a bounded Lipschitz domain, $X_0 = H^1(\\Omega)$, $X = L^2(\\Omega)$ and $X_1 = (H^1(\\Omega))'$, the dual space.\nThe [[Rellich–Kondrachov theorem|YM3JZ34PME]] gives the compact inclusion.\nSo a sequence of approximate solutions of a parabolic equation, bounded in $L^2(0, T; H^1(\\Omega))$ with time derivatives bounded in $L^2(0, T; (H^1)')$, has a subsequence that converges strongly in $L^2(0, T; L^2(\\Omega))$.\nThis strong convergence lets one pass to the limit in nonlinear terms, as in the construction of [[Leray weak solutions|DHHR5PZZ7E]] of the Navier–Stokes equations.\n\nBoth hypotheses are needed.\nIf $X_0 = X = X_1$ is an infinite-dimensional Hilbert space, the constant functions $u_k(t) = e_k$, for an orthonormal sequence $(e_k)$, are bounded in $W$ with $u_k' = 0$, but no subsequence converges in $L^p(0, T; X)$.\nIf the bound on $u'$ is dropped, the functions $u_k(t) = \\sin(kt) v$ with a fixed $v \\ne 0$ are bounded in $L^p(0, T; X_0)$, but they converge weakly to $0$ and not strongly.\n\n## Explanations\n\n### L1 human\n\nBounds in space alone or in time alone do not give compactness for functions of time and space.\nThe lemma says that a bound on spatial derivatives together with a weak bound on time derivatives is enough to extract a strongly convergent subsequence.\n\n### L2 human\n\nThe lemma is the standard compactness tool for evolution equations.\nOne builds approximate solutions, for example by the Galerkin method.\nEnergy estimates bound them in $L^2(0, T; H^1)$ and their time derivatives in a dual space.\nThe lemma then gives strong convergence in $L^2(0, T; L^2)$.\nIt plays for time-dependent problems the role that the Rellich–Kondrachov theorem plays for stationary ones.\n",
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