Aubin–Lions compactness with exact time exponent

Fix $\epsilon >0$. Compactness in [F1] makes the image in $H$ of the $V$ unit ball totally bounded. Take a finite $\epsilon \mathrm{/}2$-net in $H$ for this image; since the range $i(V)$ is dense in $H$ by [F1], approximate each net point within $\epsilon \mathrm{/}2$ by a point of $i(V)$. Let $E\subset i(V)$ be the span of these finitely many points. The $H$-orthogonal projection $P_E$ is the best approximation from $E$, so $\mathrm{\parallel }(I-P_E)iv{\mathrm{\parallel }}_H\le \epsilon \mathrm{\parallel }v{\mathrm{\parallel }}_V(v\in V)\mathrm{.}$ After orthonormalizing a basis, write $e_{\mathrm{\ell }}=i{v}_{\mathrm{\ell }}$ with $v_{\mathrm{\ell }}\in V$.

For each $j,\mathrm{\ell }$, put $a_{j\mathrm{\ell }}(t)=(u_j(t),e_{\mathrm{\ell }}{)}_H$. Its weak derivative is $⟨u_j^{\mathrm{\prime }}(t),v_{\mathrm{\ell }}⟩$ by [F2]. The uniform bounds and the continuous $V\to H$ inclusion give uniform $L^p$ bounds on $a_{j\mathrm{\ell }}$ and uniform $L^q$ bounds on their derivatives. Since $T<\mathrm{\infty }$, each coefficient is in $L^1(0,T)$ and its derivative is in $L^1(0,T)$; hence it is in scalar $W^{1,1}(0,T)$ and [F4] gives an absolutely continuous representative. For any such representative, some $s\in [0,T]$ has $\mathrm{\mid }{a}_{j\mathrm{\ell }}(s)\mathrm{\mid }\le T^{-1\mathrm{/}p}\mathrm{\parallel }{a}_{j\mathrm{\ell }}{\mathrm{\parallel }}_{L^p}$; otherwise its $L^p$ norm would exceed that bound. Its total variation is at most $T^{1-1\mathrm{/}q}\mathrm{\parallel }{a}_{j\mathrm{\ell }}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^q}$ (with $T^{1-1\mathrm{/}q}=T$ for $q=\mathrm{\infty }$). Thus all coefficients are uniformly bounded, and Hölder gives the common modulus $\mathrm{\mid }{a}_{j\mathrm{\ell }}(t)-a_{j\mathrm{\ell }}(s)\mathrm{\mid }\le C\mathrm{\mid }t-s{\mathrm{\mid }}^{1-1\mathrm{/}q}$ for $q<\mathrm{\infty }$, with $C\mathrm{\mid }t-s\mathrm{\mid }$ for $q=\mathrm{\infty }$.

The finitely many coefficient functions of $P_E{u}_j$ are uniformly bounded and share the modulus from step 1.2. Choose a finite time grid so the modulus between adjacent grid points is below $\epsilon$, and finite $\epsilon$-nets for their bounded coefficient vectors at each grid point. The piecewise-linear interpolants of the finitely many rounded grid values form a finite family; each coefficient vector is within $2\epsilon$ of its corresponding interpolant in the uniform norm, by the common modulus and the grid-point rounding error. Hence $\{P_E{u}_j\}$ is totally bounded in $L^p(0,T;H)$.

By step 1.1, $\mathrm{\parallel }{u}_j-P_E{u}_j{\mathrm{\parallel }}_{L^p(0,T;H)}\le \epsilon \mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{L^p(0,T;V)}\mathrm{.}$ For each desired accuracy, choose $\epsilon$ small and use step 2.1 to approximate the projected family by a finite net. The displayed remainder then shows that the full family is totally bounded in $L^p(0,T;H)$.

From total boundedness, [A1] selects a Cauchy subsequence; [F5] makes it convergent in $L^p(0,T;H)$. This proves relative compactness. If the $L^{\mathrm{\infty }}(H)$ bound is $M$, pairwise differences have $L^{\mathrm{\infty }}(H)$ norm at most $2M$. For $r<p$, finite-interval Hölder gives convergence in $L^r(H)$; for $r\ge p$, $\mathrm{\parallel }{u}_j-u_k{\mathrm{\parallel }}_{L^r(H)}^r\le (2M{)}^{r-p}\mathrm{\parallel }{u}_j-u_k{\mathrm{\parallel }}_{L^p(H)}^p\mathrm{.}$ Completeness in each $L^r(H)$ then gives convergence for every finite $r$. These limits equal the $L^p(H)$ limit since both convergences imply convergence in measure, whose limit is unique up to almost-everywhere equality.