Can the Solution of a Heat Equation Only Get Smoother as Time Increases?

[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?

 

[berkeman]

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?

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?

 

[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?

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?

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.

 

[berkeman]

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

Definition of coarsen

transitive verb
: to make coarse

intransitive verb
: to become coarse

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

 

[berkeman]

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

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

 

[feynman1]

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

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.

 

[berkeman]

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.

 

[wrobel]

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.

 

[jasonRF]

A similar way to look at this is to consider the same problem as Wrobel did, $u_t = u_{xx}$ with $u(x,t=0) = v(x)$, but over the entire real line. You should be able to derive the kernel $g$ such that the solution is $u(x,t) = \int g(x-y, t) v(y) \, dy$.

 

[feynman1]

thanks for the suggested explanation, however is there any approach of not using a kernel/Green's function/fourier expansion, since an explanation via a summation of functions isn't the most convincing because one needs to assess the shape along the whole curve?

 

[martinbn]

https://www.math.ucdavis.edu/~hunter/pdes/ch6.pdf