This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
- $ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $.
- $ g \in {C^{b}}(\Bbb{R}^{n}) $.
- $ G: [0,\infty) \to \Bbb{R} $ is defined by $ \displaystyle G(s) \stackrel{\text{df}}{=} \int_{0}^{s} g(\sigma) ~ \mathrm{d}{\sigma} $.
- $ F: \Bbb{C} \to \Bbb{R} $ is defined by $ F(z) \stackrel{\text{df}}{=} \dfrac{G \! \left( |z|^{2} \right)}{2} $.
Then the following question may be asked:
Question: Is it true that the derivative of the map $$ \left\{ \begin{matrix} (0,\infty) & \to & \Bbb{R} \\ t & \mapsto & \displaystyle \int_{\Bbb{R}^{n}} F(u(\bullet,t)) ~ \mathrm{d}{\mu} \end{matrix} \right\} $$ exists and is equal to $$ \left\{ \begin{matrix} (0,\infty) & \to & \Bbb{R} \\ t & \mapsto & \operatorname{Re} \! \left( \left\langle \dot{u}(\bullet,t),u(\bullet,t) ~ g \! \left( |u(\bullet,t)|^{2} \right) \right\rangle_{{L^{2}}(\Bbb{R}^{n})} \right) \end{matrix} \right\}? $$ Note: Here, $ \dot{u} $ denotes the time-derivative of $ u: (0,\infty) \to {L^{2}}(\Bbb{R}^{n}) $.
This is formally true, but when one tries to formulate a rigorous proof, obstacles arise. One obstacle is the fact that the time-derivative $ \dot{u} $ of $ u $ is not taken ‘pointwise’ at points in $ \Bbb{R}^{n} $ but rather as a map from $ (0,\infty) $ to $ {L^{2}}(\Bbb{R}^{n}) $.
I could show that the map is locally Lipschitz continuous, so it is differentiable almost everywhere. However, this is far from answering the question posed.
Thank you!
