Aubin–Lions lemma

Remarks

Proof idea: by Ehrling's lemma, for every $\eta > 0$ there is $C_\eta$ with $\norm{v}_X \le \eta \norm{v}_{X_0} + C_\eta \norm{v}_{X_1}$ for all $v \in X_0$. A 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 $u'$. This gives a subsequence that converges in $L^p(0, T; X_1)$, and Ehrling's lemma upgrades the convergence to $L^p(0, T; X)$.

Example: 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. The Rellich–Kondrachov theorem gives the compact inclusion. So 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 $L^2(0, T; L^2(\Omega))$. This strong convergence lets one pass to the limit in nonlinear terms, as in the construction of Leray weak solutions of the Navier–Stokes equations.

Both hypotheses are needed. If $X_0 = X = X_1$ 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 $L^p(0, T; X)$. If 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.

Going deeper

For a student

The lemma is the standard compactness tool for evolution equations. One builds approximate solutions, for example by the Galerkin method. Energy estimates bound them in $L^2(0, T; H^1)$ and their time derivatives in a dual space. The lemma then gives strong convergence in $L^2(0, T; L^2)$. It plays for time-dependent problems the role that the Rellich–Kondrachov theorem plays for stationary ones.