Function differentiability and diffusion

[hilbert2]

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.

 

[S.G. Janssens]

There is a lot to say about this. By comparing the diffusion equation (with appropriate BCs) and the advection equation (with constant velocity), you have touched upon the difference between semigroups that are analytic or merely strongly continuous.

For your specific question about the diffusion equation (and much more), I would recommend A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995. In a nutshell, the answer is: yes.

 

[hilbert2]

Thanks. I'll try to find that book. I guess if I make a fractional order PDE where the x-derivative is of order between 1 and 2, there's some limiting value after which the dynamics smooths any non-analytic function.

 

[S.G. Janssens]

I'll try to find that book.

It may be that your institute has it as part of some Springer package of e-books. (That tends to happen here.)

I guess if I make a fractional order PDE where the x-derivative is of order between 1 and 2, there's some limiting value after which the dynamics smooths any non-analytic function.

Nice question. I don't know the answer (presently don't have expertise in evolution equations with fractional derivatives) but it is natural to wonder and I would expect there to be literature on it. (Alternatively, you could go back to the foundations (Engel and Nagel or Pazy, for example) and try to see how far you can stretch the proof of analyticity of the semigroup for the integer case.)

 

[hilbert2]

An interesting implication of this is that a backward diffusion with $D<0$ can suddenly make a function non-analytic at some instant of time. But it shouldn't be able to add new non-differentiable points at more than one instant.

Edit: this is a bit confusing because any twice differentiable function $f(x,t)$ obtainable as a product of backward diffusion should also be obtainable by forward diffusion. This seems to contradict the claim that any forward diffusion immediately makes the function arbitrarily many times differentiable.

There's probably some version of the advection equation that has a non-constant coefficient $v$ and acts to scale the initial function:

$\displaystyle f(x,t) = f\left(\frac{x}{t-t_0 +1},t_0 \right)$.

This wouldn't remove non-differentiabilities either. A multiplier $v$ that makes the non-analytic point escape to infinity in finite time, like $\displaystyle v=\frac{1}{t' -t}$ with $t'$ a constant, wouldn't apparently be a proper advection equation in this sense.

 

[hilbert2]

I was thinking about something like forming a piecewise defined function

$f(x)=\left\{\begin{array}{c}\exp (-kx^2 ),\hspace{10pt}when\hspace{5pt} |x|>1 \\ ax^3 + bx^2 + cx + d,\hspace{10pt}when\hspace{5pt} |x|\leq 1\end{array}\right.$

and setting the coefficients so that the first two derivatives of $f$ exist also at the points $x=\pm 1$. Then one could attempt to form an exact solution for the diffusion equation with this initial condition and see how the existence of higher derivatives changes in the time evolution. Or what does antidiffusion to negative time direction do to that property. However, this is probably not an easy initial value problem to solve.

The Schrödinger equation for a free particle is also a diffusion equation, but with a complex diffusion coefficient $D\in\mathbb{C}$, which makes the time evolution qualitatively similar in both positive and negative time directions.

Numerical solution methods can't be applied to the problem of the existence of derivatives at some point $x$, but from that kind of solution you could at least see how quickly the sudden "jump" in the values of higher derivatives in the neighborhood of some point smooths out in the time evolution.