Given
$v_k \rightarrow v \ \ \ \text{weakly star in } L^2(0,T;W^{1,\infty}(\mathbb{R}^n)) $
$\eta _k(t,\cdot) \rightarrow \eta(t,\cdot) \ \ \ \text{in } C_\text{loc}(\mathbb{R}^n)\ $ uniformly in $t,$
it should follow that
$v_k(t, \eta_k(t,x)) \rightharpoonup v(t,\eta(t,x)) \ \ \ \text{in } L^2(0,T)\ \ \ \text{for any } x\in \mathbb{R}^n$,
but I can't see why this holds true. For $\psi \in L^2(0,T)$ I tried writing
$\int_0^T(v_k(t, \eta_k(t,x)) - v(t,\eta(t,x)))\psi(t)dt = $
$\int_0^T(v_k(t, \eta_k(t,x)) - v(t,\eta_k(t,x)))\psi(t)dt + \int_0^T(v(t, \eta_k(t,x)) - v(t,\eta(t,x)))\psi(t)dt $ .
In the second term I don't understand how we can obtain some statement about every $x$, since we only have convergence in $W^{1,\infty}$ in $x$. And in the other integral I still don't see, how we can handly the composition of two (weakly) convergent sequences.
Of course we can also split up the term on the left-hand side the other way around, but this leads to similar problems.
Can someone give me a hint here? Thanks in advance for any help.