Timeline for Solving $x\partial_x f = 0$ over distributions

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Apr 22, 2012 at 3:56	comment	added	Phil Isett		This answer is correct. The fact that $xu=0$ has only the functions $c_1\delta_0$ as solutions follows from how every smooth test function $\phi(x)$ with $\phi(0)=0$ can be written as $\phi(x)=x\psi(x)$ where $\psi(x)$ is another test function. You can see this by Taylor expanding: $\phi(x)=0+x\int^1_0\phi'(tx)dt$ -- in higher dimensions you should put in a cutoff to make sure this $\psi$ has compact support. Finally, to show $v'(x)=c_1\delta_0$ has only Heaviside solutions, observe that the only solutions to $v'(x)=0$ are constants, which can be proven by mollifying the distribution $v$.
Jun 8, 2011 at 17:33	history	answered	Sylvain Bonnot	CC BY-SA 3.0