Time mollification preserves the evolution-space derivative

Let $g_1,g_2\in L^2(0,T;V^{\ast })$ both satisfy [F1], and put $h=g_1-g_2$. For every $v\in V$ and $\varphi \in C_c^{\mathrm{\infty }}(0,T)$, $\int ⟨h(t),v⟩\varphi (t)dt=0$. Therefore every Bochner convolution of the zero extension, $h_{ϵ}(t)={\int }_0^{T}h(s){\eta }_{ϵ}(t-s)ds$, vanishes on any compact subinterval whose distance from $\{0,T\}$ exceeds $ϵ$: its scalar pairing against each $v$ is the zero distribution tested by $s\mapsto {\eta }_{ϵ}(t-s)$. By [F2], $h_{ϵ}\to h$ in local $L^2(V^{\ast })$, so $h=0$ almost everywhere.

Regard $u$ as a $V^{\ast }$-valued function through the continuous embedding $V\hookrightarrow V^{\ast }$ and set $G(t)={\int }_0^{t}g(s)ds$. The Bochner integral exists by [F3], and [F5] gives $g\in L^1(0,T;V^{\ast })$. For each fixed $v\in V$, [F4] and scalar integration by parts show that $G$ has weak derivative $g$ in the sense of [F1]. Thus $w:=u-G$ has zero distribution derivative. For any $\psi \in C_c^{\mathrm{\infty }}(0,T)$ with integral zero, its primitive is compactly supported in $(0,T)$, hence $\int w(t)\psi (t)dt=0$ in $V^{\ast }$. Fix $\eta \in C_c^{\mathrm{\infty }}(0,T)$ with integral one and put $c=\int w(t)\eta (t)dt$. Subtracting $(\int \psi )\eta$ shows $\int (w-c)\psi =0$ for every test $\psi$. The convolutions of $w-c$ are therefore zero on compact subintervals as in step 1.1; [F2] makes them converge to $w-c$ in local $L^2(V^{\ast })$. Hence $w=c$ almost everywhere, and $\tilde{u}(t):=c+G(t)$ is a continuous $V^{\ast }$ representative of $u$.

Reflect the $V$-valued $L^2$ class evenly to $(-T,2T)$ across $0$ and $T$, and reflect $g$ oddly. The continuous $V^{\ast }$ representative from step 2.1 has matching values at each seam. Integration by parts on the three intervals therefore shows that the reflected function has weak $V^{\ast }$ derivative equal to the odd reflection of $g$; the endpoint terms cancel, so there are no point masses. Extend both functions by zero beyond $(-T,2T)$. For $0<ϵ<T\mathrm{/}2$, the convolution on the resulting neighborhood of $[0,T]$ does not see the extension jumps at $-T$ or $2T$.

Let ${\eta }_{ϵ}$ be a nonnegative smooth mollifier of mass one, supported in $(-ϵ,ϵ)$, and convolve the reflected functions in time. On a neighborhood of $[0,T]$, $u_{ϵ}={\eta }_{ϵ}\ast u$ is smooth with values in $V$ by [F2]. Testing the reflected weak derivative identity against a fixed $v\in V$ and integrating by parts in the scalar variable gives $⟨u_{ϵ}^{\mathrm{\prime }}(t),v⟩=\int ⟨g(s),v⟩{\eta }_{ϵ}(t-s)ds=⟨({\eta }_{ϵ}\ast g)(t),v⟩$. The bounded inclusion $V\to V^{\ast }$ commutes with the Bochner integral by [F3], so $u_{ϵ}^{\mathrm{\prime }}={\eta }_{ϵ}\ast g$ in $V^{\ast }$.

Apply the $L^2$ approximation in [F2] to the reflected $V$-valued function and its $V^{\ast }$ derivative, then restrict to $(0,T)$. This gives $u_{ϵ}\to u$ in $L^2(V)$ and $u_{ϵ}^{\mathrm{\prime }}\to g$ in $L^2(V^{\ast })$ as $ϵ\downarrow 0$.

Choose any sequence ${ϵ}_n\downarrow 0$ small enough that each convolution is defined on a neighborhood of $[0,T]$, and set $u_n=u_{{ϵ}_n}$. Step 5.1 gives both asserted convergences, and step 4.1 gives $u_n^{\mathrm{\prime }}={\eta }_{{ϵ}_n}\ast g$. The uniqueness in step 1.1 identifies $g=u^{\mathrm{\prime }}$.