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Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at $t=0$ and $t= T$), and $f \in C^1(0,T \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ with weak time derivative $w' \in L^2(0,T;H^{-1}(\Omega)$ (i.e. it satisfies $\int_0^T w(t)\varphi'(t) = -\int_0^T w'(t)\varphi(t)$ for all such $\varphi$.)

How do I show that $$\int_0^T \langle w', f\varphi \rangle_{H^{-1}, H^1} = \int_0^T \langle fw', \varphi \rangle_{H^{-1}, H^1}$$

I tried writing the pairing as a functional and used RRT but to no avail. Appreciate any help..

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up vote 1 down vote accepted

I asked this on Mathoverflow, and the equality is holds by definition. It's just how we interpret the functional $fw'$.

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