Timeline for Weak derivative under the integral sign
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 15, 2020 at 21:55 | comment | added | leo monsaingeon | This will work as soon as $u$ belongs to some Bochner space, and more precisely as soon as $\partial_t u\in L^p(0,T;X')$ for some reasonable dual space $X'$ such that the constant function belongs to $X$. Then by calssical composition $Lip\circ $Sobolev the function $w$ belongs to the same space (the function $\min(0,.)$ being 1-Lipschitz), and therefore your equality amounts to checking that $\frac{d}{dt}<w(t),1>_{X',X}=<w',1>_{X',X}$ | |
| Aug 12, 2019 at 22:26 | comment | added | Christian Remling | I think you need to assume a bit more for the classical result (about $u$), otherwise examples such as $u(t,x)=x\sin (t/x)$ on $x\in\Omega=(0,1)$ become problematic. | |
| Aug 12, 2019 at 22:20 | comment | added | Nate Eldredge | Hint: try integrating both sides from $0$ to $t$ and applying Fubini's theorem. | |
| Aug 12, 2019 at 22:03 | history | asked | David Lingard | CC BY-SA 4.0 |