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Chris
Chris
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We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;H)$$u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the $'$ means the weak derivative.

Is there a formula involving weak derivatives for $$\frac{d}{dt}f(t)(u(t))$$ where $f(t) \in V^*$ and $u(t) \in V$?

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;H)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the $'$ means the weak derivative.

Is there a formula involving weak derivatives for $$\frac{d}{dt}f(t)(u(t))$$ where $f(t) \in V^*$ and $u(t) \in V$?

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the $'$ means the weak derivative.

Is there a formula involving weak derivatives for $$\frac{d}{dt}f(t)(u(t))$$ where $f(t) \in V^*$ and $u(t) \in V$?

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Chris
Chris
  • 29
  • 2

"Integration by parts" formula for functionals

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;H)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the $'$ means the weak derivative.

Is there a formula involving weak derivatives for $$\frac{d}{dt}f(t)(u(t))$$ where $f(t) \in V^*$ and $u(t) \in V$?