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I already asked the question on MSE, and have tried to figure it out myself. But the problem seems trickier than expected, so I guess MO is a better place to ask.. For the sake of completeness, I ...

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus) be a solution of a Cauchy problem $$\frac{d}{dt} f(t) = A f(t)$$ $$f(0)...