- #1

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- TL;DR Summary
- What happens to the existence of n:th x-derivatives of a function f(x,t) in diffusive dynamics?

Suppose I have an initial condition function ##f(x,t_0 )##, which is everywhere twice differentiable w.r.t. the variable ##x##, but the third or some higher derivative doesn't exist at some point ##x\in\mathbb{R}##.

Then, if I evolve that function with the diffusion equation

##\displaystyle\frac{\partial f}{\partial t} = D\frac{\partial^2 f}{\partial x^2}##,

does the dynamics immediately smooth the function ##f## to an arbitrarily many times differentiable one? Such that for any ##t>t_0## the derivatives ##\displaystyle\frac{\partial^n f}{\partial x^n}## exist for any ##n,x##.

This does not happen with just any dynamical equation, because for instance the advection equation

##\displaystyle\frac{\partial f}{\partial t} = v\frac{\partial f}{\partial x}##

only shifts the function ##f## with constant velocity ##v## and the point of non-differentiability moves with the same velocity.

If this problem has a name and has been considered by someone else, I'd be happy to know about it.

Then, if I evolve that function with the diffusion equation

##\displaystyle\frac{\partial f}{\partial t} = D\frac{\partial^2 f}{\partial x^2}##,

does the dynamics immediately smooth the function ##f## to an arbitrarily many times differentiable one? Such that for any ##t>t_0## the derivatives ##\displaystyle\frac{\partial^n f}{\partial x^n}## exist for any ##n,x##.

This does not happen with just any dynamical equation, because for instance the advection equation

##\displaystyle\frac{\partial f}{\partial t} = v\frac{\partial f}{\partial x}##

only shifts the function ##f## with constant velocity ##v## and the point of non-differentiability moves with the same velocity.

If this problem has a name and has been considered by someone else, I'd be happy to know about it.