Time oscillations defeat compactness without derivative control

Since each interval $[0,T]$ contains exactly $n$ periods of $\sin (2\pi nt\mathrm{/}T)$, a change of variable gives $\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_{L^p(0,T;V)}^p=\mathrm{\parallel }v{\mathrm{\parallel }}_V^p\frac{T}{2\pi }{\int }_0^{2\pi }\mathrm{\mid }\sin s{\mathrm{\mid }}^{p}ds,$ independent of $n$. Thus the family is uniformly bounded in each stated $L^p(V)$.

For distinct positive integers $n\mathrm{\ne }m$, sine orthogonality over $[0,T]$ gives $\mathrm{\parallel }{u}_n-u_m{\mathrm{\parallel }}_{L^2(0,T;H)}^2=\mathrm{\parallel }iv{\mathrm{\parallel }}_H^2{\int }_0^T(\sin (2\pi nt\mathrm{/}T)-\sin (2\pi mt\mathrm{/}T){)}^{2}dt=T\mathrm{\parallel }iv{\mathrm{\parallel }}_H^2>0,$ where positivity follows from injectivity in [F1]. Every two distinct members therefore stay a fixed positive distance apart, so no subsequence is Cauchy and none converges strongly in the complete space $L^2(0,T;H)$.

Scalar integration by parts against each $w\in V$ shows that the pivot weak derivative is $u_n^{\mathrm{\prime }}(t)=(2\pi n\mathrm{/}T)\cos (2\pi nt\mathrm{/}T)J(iv)$. Its $L^q(V^{\ast })$ norm is $(2\pi n\mathrm{/}T)\mathrm{\parallel }J(iv){\mathrm{\parallel }}_{V^{\ast }}{(\frac{T}{2\pi }{\int }_0^{2\pi }\mathrm{\mid }\cos s{\mathrm{\mid }}^{q}ds)}^{1\mathrm{/}q}$ for finite $q$, and $(2\pi n\mathrm{/}T)\mathrm{\parallel }J(iv){\mathrm{\parallel }}_{V^{\ast }}$ for $q=\mathrm{\infty }$. Since $⟨J(iv),v⟩=\mathrm{\parallel }iv{\mathrm{\parallel }}_H^2>0$ by [F1], these norms grow linearly in $n$. Hence no positive exponent derivative bound is present.