Timeline for Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

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May 18, 2019 at 14:58	comment	added	Jay		I've asked a very closely related question on some aspects of your answer on Mathematics StackExchange: math.stackexchange.com/questions/3226098/…
May 11, 2019 at 13:38	comment	added	Carlo Beenakker		yes, the derivative diverges; I'm not sure what you mean by the "test function"; all of these manipulations are in distributional sense
May 11, 2019 at 13:35	comment	added	Jay		Do you mean that in that case the derivative blows up?
May 11, 2019 at 13:24	comment	added	Carlo Beenakker		if the derivative vanishes there is no finite answer; for the same reason that $\int \delta(x^2)dx$ diverges.
May 11, 2019 at 12:43	comment	added	Jay		Also: (1) is there anything we can do if the gradient does vanish on $S(t)$? (2) what is the relationship between the last formula and the distributional derivative of the function, i.e. $\langle \int_{\{u(t,\cdot) >0\}} 1 dx, \partial_t \phi \rangle$? (since $\partial_t u$ is to be intended in the distributional sense).
May 11, 2019 at 12:36	comment	added	Jay		Maybe I got it: do you mean that $S(t)$ is the "interface" (i.e. the boundary of the support)?
May 11, 2019 at 10:29	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 666 characters in body
May 10, 2019 at 20:48	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 367 characters in body
May 10, 2019 at 20:16	history	answered	Carlo Beenakker	CC BY-SA 4.0