No-slip compactness and bounded-mode tails

For a real $C^1$ function $g$ on a compact interval $[a,b]$, put $G(t)={\int }_a^t{g}^{\mathrm{\prime }}(s)ds$. Continuity of $g^{\mathrm{\prime }}$ gives $\mathrm{\mid }[G(t+h)-G(t)]\mathrm{/}h-g^{\mathrm{\prime }}(t)\mathrm{\mid }\le {\sup }_{\mathrm{\mid }s-t\mathrm{\mid }\le \mathrm{\mid }h\mathrm{\mid }}\mathrm{\mid }{g}^{\mathrm{\prime }}(s)-g^{\mathrm{\prime }}(t)\mathrm{\mid }\to 0$ at interior points. Thus $(G-g{)}^{\mathrm{\prime }}=0$ everywhere inside the interval, and the mean value theorem and continuity at the endpoints give $g(b)-g(a)={\int }_a^b{g}^{\mathrm{\prime }}$. Apply this identity to $g(s)=w(x-sh)$, $0\le s\le 1$, componentwise for vector fields. For $w\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$, [F6] in the segment parameter then gives $\mathrm{\mid }w(x-h)-w(x){\mathrm{\mid }}^2\le \mathrm{\mid }h{\mathrm{\mid }}^2{\int }_0^1\mathrm{\mid }\mathrm{\nabla }w(x-sh){\mathrm{\mid }}^{2}ds\mathrm{.}$ Integrating in $x$, using translation invariance and [F4], yields $\mathrm{\parallel }w(\cdot -h)-w{\mathrm{\parallel }}_2\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2$. Density and the isometric zero extension [F3] extend this estimate to every $a\in V$.

Fix a nonnegative $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ with integral one and set ${\eta }_{ϵ}(x)={ϵ}^{-3}\eta (x\mathrm{/}ϵ)$. For each $x$, Cauchy–Schwarz with respect to the probability density ${\eta }_{ϵ}(h)$ gives $\mathrm{\mid }\int {\eta }_{ϵ}(h)(E_{0}a(x)-E_{0}a(x-h))dh{\mathrm{\mid }}^2\le \int {\eta }_{ϵ}(h)\mathrm{\mid }{E}_{0}a(x)-E_{0}a(x-h){\mathrm{\mid }}^{2}dh$. Integrating in $x$, applying [F4] to interchange the integrals, and using step 1.1 give $\mathrm{\parallel }{E}_{0}a-{\eta }_{ϵ}\ast E_{0}a{\mathrm{\parallel }}_2^2\le \int {\eta }_{ϵ}(h)\mathrm{\parallel }{E}_{0}a(\cdot -h)-E_{0}a{\mathrm{\parallel }}_2^{2}dh\le C_{\eta }^2{ϵ}^2\mathrm{\parallel }\mathrm{\nabla }a{\mathrm{\parallel }}_2^2\mathrm{.}$ For the $V$ unit ball this error is uniform and tends to zero with $ϵ$.

For fixed $ϵ$, all mollified zero extensions have support in one compact box. [F6] bounds their sup norms by $\mathrm{\parallel }{\eta }_{ϵ}{\mathrm{\parallel }}_2\mathrm{\parallel }a{\mathrm{\parallel }}_2$ and their first derivatives by $\mathrm{\parallel }\mathrm{\nabla }{\eta }_{ϵ}{\mathrm{\parallel }}_2\mathrm{\parallel }a{\mathrm{\parallel }}_2$, uniformly on the $V$ unit ball. Cover the box by a finite grid fine enough that the derivative bound changes every value by less than a prescribed tolerance; quantize the bounded vector values at grid points in a finite lattice. The resulting finitely many piecewise-constant fields approximate every mollified field in sup norm, hence in $L^2$ on the finite box. Step 2.1 transfers these finite nets to finite $L^2$ nets for the original $V$ unit ball, with arbitrarily small error. Thus that ball is totally bounded in $H$; completeness of $H$ from [F2] makes its closure compact.

For every $a$ in that compact closure, ${\mathrm{\Pi }}_{K}a\to a$ in $H$ by the $H$-basis property in [F2]. Given $ϵ>0$, take a finite $ϵ$-net of the compact set and choose $K$ so that $\mathrm{\parallel }(I-{\mathrm{\Pi }}_K)b\mathrm{\parallel }<ϵ$ at each net point $b$. Since $I-{\mathrm{\Pi }}_K$ is an orthogonal contraction, the triangle inequality gives ${\sup }_{\mathrm{\parallel }a{\mathrm{\parallel }}_V\le 1}\mathrm{\parallel }(I-{\mathrm{\Pi }}_K)a\mathrm{\parallel }\le 2ϵ$. These suprema decrease with $K$; taking them as ${\delta }_K$ proves ${\delta }_K\to 0$ and, by homogeneity, the asserted estimate.