Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$.
(Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$).
Let $v \in L^2(0,T;H^1_0(\Omega))$ with $\frac{\partial b(v)}{\partial t} \in L^2(0,T;H^{-1}(\Omega))$. Then $$\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v)).$$
This result follows by (the more general) Lemma 1.5 of the famous Quasilinear Elliptic-Parabolic Differential Equations paper by Alt and Luckhaus. The authors prove it by discretising in time.
Does anyone know an alternative proof (where no discretisation is used)? Using density of smooth functions does not help. Let us assume $b$ and $b^{-1}$ are differentiable if necessary.