# Heat equation coarsened

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feynman1
how to show or prove that the shape of the solution of a heat equation can only go smoother and smoother but not the opposite as time increases?

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Mentor
how to show or prove that the solution of a heat equation can only go coarsened/smoother as time increases?
Sorry, my Google search of your term coarsened seems to say the opposite of smoother. Could you please expand on your question, and post links to relevant articles?

feynman1
Sorry, my Google search of your term coarsened seems to say the opposite of smoother. Could you please expand on your question, and post links to relevant articles?

graduate school / PhD level is the right category.
maybe coarsen means differently in different fields, so let's forget about coarsen and just consider a solution becoming smoother and smoother.

Mentor
Mentor
how to show or prove that the solution of a heat equation can only go coarsened/smoother as time increases?

feynman1
sorry but if there was such a link there'd be explanations then I wouldn't have posted here. It's just about time irreversibility of heat equations.

hutchphd
Mentor
time irreversibility of heat equations
So you're really going to make us Google search that phrase? Please do that search and tell us what you don't understand. Seriously.

PhDeezNutz, jim mcnamara and hutchphd
Consider the simplest situation: boundary conditions are ##2\pi-##periodic
$$u_t=u_{xx},\quad u\mid_{t=0}=v(x)=\sum_{k\in\mathbb{Z}}v_ke^{ikx}$$ then the solution
is
$$u(t,x)=\sum_{k\in\mathbb{Z}}v_k e^{-|k|^2t}e^{ikx}.$$ It is it.

feynman1 and jasonRF