Global compatible no-slip shear evolution

Let $\tilde{F}$ be the stated smooth odd $2h$-periodic extension. Repeated integration by parts on the circle, with no boundary term because the integrands and derivatives are periodic, shows its Fourier coefficients decay faster than every inverse power of $n$. By [F2] and oddness, the sine coefficients are exactly $b_n$ and the sine series equals $F$ in $L^2(0,h)$; since both are continuous, the equality holds pointwise. Countable Choice [A1] is the Fourier-basis theorem's stated inherited assumption.

For nonnegative integers $p,q$, the $(p,q)$ time/space derivative of the $n$th term in $V$ is bounded uniformly for $t\ge 0$ and $0\le y\le h$ by $\mathrm{\mid }{b}_n\mathrm{\mid }{\nu }^p(n\pi \mathrm{/}h{)}^{2p+q}$. The rapid coefficient decay from step 1.1 makes these bounds summable by choosing a decay power larger than $2p+q+1$. Thus every differentiated series converges uniformly on $[0,\mathrm{\infty })\times [0,h]$, so termwise differentiation is valid to all orders. Each term satisfies $V_t=\nu V_{yy}$ and vanishes at $y=0,h$, while step 1.1 gives $V(0,y)=F(y)$.

The field $u=(V(y),0,0)$ is divergence-free because its first component is independent of $x$ and its other components vanish. Its convective derivative is $u\cdot \mathrm{\nabla }u=V{\mathrm{\partial }}_{x}u=0$ and $\mathrm{\Delta }u=(V_{yy},0,0)$. Therefore step 2.1 gives the full system [A2] with constant pressure $\pi =0$ and the stated wall and initial traces. The pressure is mean-zero.

Let $u$ be this solution and $v$ any other smooth solution with the same data and mean-zero pressure; put $w=v-u$. Subtraction gives $w_t+v\cdot \mathrm{\nabla }w+w\cdot \mathrm{\nabla }u+\mathrm{\nabla }({\pi }_v-{\pi }_u)=\nu \mathrm{\Delta }w$. Dot with $w$ and integrate on $\mathrm{\Omega }$. Periodic side integrals cancel; the transport term is $\int v\cdot \mathrm{\nabla }(\mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{/}2)=0$ because $\mathrm{div}v=0$ and $v\cdot n=0$ on the walls; pressure contributes $-\int ({\pi }_v-{\pi }_u)\mathrm{div}w=0$; integration by parts gives $-\nu \mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2$ since $w=0$ on the walls. The remaining term is bounded by $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }w{\mathrm{\parallel }}_2^2$, so $z^{\mathrm{\prime }}=(\mathrm{\parallel }w{\mathrm{\parallel }}_2^2{)}^{\mathrm{\prime }}\le 2\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{\mathrm{\infty }}z$. Since $z(0)=0$, multiplying by $e^{-2{\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_{\mathrm{\infty }}ds}$ proves $z=0$.

Parseval on the odd extension yields $\mathrm{\parallel }V(t){\mathrm{\parallel }}_{L^2(0,h)}^2=(h\mathrm{/}2){\sum }_{n\ge 1}\mathrm{\mid }{b}_n{\mathrm{\mid }}^2{e}^{-2\nu (n\pi \mathrm{/}h{)}^{2}t}\le e^{-2\nu (\pi \mathrm{/}h{)}^{2}t}\mathrm{\parallel }F{\mathrm{\parallel }}_{L^2(0,h)}^2$. Multiplying by the finite periodic side area $L_1{L}_2$ and taking square roots proves the asserted decay. The preceding steps establish global smooth existence, full-flow smooth uniqueness for the data, and the bound.