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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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- Thread starter feynman1
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- #2

berkeman

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

And your marked your thread start with an "A" prefix, which means you want the discussion to be at the graduate school / PhD level. Is that really what you intended?

- #3

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graduate school / PhD level is the right category.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?

And your marked your thread start with an "A" prefix, which means you want the discussion to be at the graduate school / PhD level. Is that really what you intended?

maybe coarsen means differently in different fields, so let's forget about coarsen and just consider a solution becoming smoother and smoother.

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berkeman

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maybe coarsen means differently in different fields, so let's forget about coarsen

https://www.merriam-webster.com/dictionary/coarsenDefinition of coarsen

transitive verb

: to make coarse

intransitive verb

: to become coarse

- #5

berkeman

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Please post links to your reading about your question. Thank you.how to show or prove that the solution of a heat equation can only go coarsened/smoother as time increases?

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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.Please post links to your reading about your question. Thank you.

- #7

berkeman

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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.time irreversibility of heat equations

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wrobel

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$$u_t=u_{xx},\quad u\mid_{t=0}=v(x)=\sum_{k\in\mathbb{Z}}v_ke^{ikx}$$ then the solution

is

$$u(t,x)=\sum_{k\in\mathbb{Z}}v_k e^{-|k|^2t}e^{ikx}.$$ It is it.

- #9

jasonRF

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- #11

martinbn

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