Compactness of H^r-bounded families in H^{r-1} and locally uniform H^{s-1} subsequences of Galerkin solutions

(Compact embedding (a) by deterministic index selection.) Let $(v_j)$ be bounded by $R$ in $H^r$. Put $S_0=\mathbb{N}$. For each $n\ge 1$ let $M_n$ be the least integer $M\ge 0$ for which $R{C}_r(1+M^2{)}^{-1\mathrm{/}2}<2^{-n}\mathrm{/}4$, with any $M_n=0$ if $R=0$. On every infinite index subset $S\subseteq \mathbb{N}$, the sequence $(P_{M_n}{v}_j{)}_{j\in S}$ has range with compact closure in the finite-dimensional space $X_{M_n}$ by [L2], so its range admits a finite cover by open $H^{r-1}$-balls of radius $2^{-n}\mathrm{/}4$ centred at its own terms. Indeed compactness gives a finite cover by radius-$2^{-n}\mathrm{/}8$ balls, and choosing one range term from each ball that meets the range enlarges these to the required balls (only finite choices). At least one required ball contains projected terms for infinitely many $j\in S$. Starting with $S=S_{n-1}$, let ${\mathrm{\ell }}_n$ be the least index in $S_{n-1}$ whose centred ball has this property and define $S_n:=\{j\in S_{n-1}:\mathrm{\parallel }{P}_{M_n}(v_j-v_{{\mathrm{\ell }}_n}){\mathrm{\parallel }}_{H^{r-1}}<2^{-n}\mathrm{/}4\}$. This is an infinite nested subset, and for $j,l\in S_n$ the triangle inequality and the two tail bounds [L1] give $\mathrm{\parallel }{v}_j-v_l{\mathrm{\parallel }}_{H^{r-1}}<2^{-n}$. Let $k_n$ be the least index in $S_n$ greater than $k_{n-1}$, with $k_0=0$. Every such index exists. For $j,l\ge n$, $k_j,k_l\in S_n$, hence $(v_{k_n})$ is Cauchy in $H^{r-1}$ and converges by [L5]. All recursive selections use least natural-number indices; no dependent choice is needed. This proves (a).

(Equicontinuity (b).) For $t,t^{\mathrm{\prime }}\in (-T,T)$ the curve $s\mapsto u_N(s)$ is $C^1$ into the Banach space $H^{s-1}$, complete by [L5], so by [L4] $\mathrm{\parallel }{u}_N(t)-u_N(t^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^{s-1}}\le \mid {\int }_{t^{\mathrm{\prime }}}^t\mathrm{\parallel }{\mathrm{\partial }}_t{u}_N(s){\mathrm{\parallel }}_{H^{s-1}}ds\mid \le C_R(1+R)\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }$ by [L3]. This proves (b) with $C_0=C_R(1+R)$, a constant depending only on $A^{\mu },B,K,R,s,[-T,T]$.

(Locally uniform subsequence (c).) Enumerate $\mathcal{N}$ in increasing order and apply [L7] to that sequence with $X_0:=H^{s-1}({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ and $X_1:=H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$: $X_0$ is complete by [L5], the inclusion bound $\mathrm{\parallel }x{\mathrm{\parallel }}_{X_0}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{X_1}$ holds by [L1], property (ii) of [L7] is clause (a) with $r=s$ (legitimate since $s$ is an integer and $s\ge 5\ge 1$), and the family $u_N$ satisfies the two bounds of [L7] by the hypothesis and step 1.2. The extraction lemma therefore yields a strictly increasing sequence $N_j\in \mathcal{N}$, $N_j\to \mathrm{\infty }$ and a continuous $u:(-T,T)\to H^{s-1}$ with $u_{N_j}\to u$ uniformly on every compact subinterval of $(-T,T)$. This is (c).