6

[Pardon my lack of rigor; I am an engineer by training. Also, for convenience, allow me to make this question as concrete as possible.]

Assume the simplest linear diffusion equation: $\alpha \dfrac{\partial^2u}{\partial x^2} = \dfrac{\partial u}{\partial t}$, where $u$ is the temperature and $\alpha$ is the thermal diffusivity.

The domain is finite, say, $[-100, 100]$. (If the assumption of an infinite domain makes it possible (or more convenient) to answer this question, then please assume so. However, the question of interest primarily pertains to a finite domain.)

Assume that the initial temperature profile has a compact support, say over $[-1, 1]$.

After the passage of an arbitrarily small but finite duration of time:

(i) would the temperature profile necessarily have support everywhere over the entire domain?

(ii) or, is it possible that a solution may still have some compact support over some finite interval that is smaller than the whole domain?

Can it be proved either way? Given the sum totality of today's mathematics (i.e. all its known principles put together), is it possible to pick between the above two alternatives in general?

A subsidiary question only if the alternative (ii) is possible: please supply an example, better so, it is of a kind wherein the initial profile is infinitely differentiable, e.g. the bump function $e^\frac1{x^2-1}$.

Thanks in advance.

--Ajit [E&OE]

Ajit R. Jadhav
  • 61
  • Alternative (i) holds. This can be proven easily by means of the formula: $$u(x, t)=\int_{-\infty}^\infty \frac{e^{-\frac{\left\lvert x-y\rught\rvert^2}{2t}}}{\sqrt{4\pi t}}u_0(y), dy.$$ – Giuseppe Negro Jan 01 '13 at 16:58

1 Answers1

7

The classical heat equation exhibits infinite propagation speed, unlike the wave equation. For every domain $\Omega$, such as $[-100,100]$, there is an associated heat kernel $\Phi_\Omega(x,y,t)$ from which the solution (with zero boundary data) is obtained as $$u(x,t)=\int_\Omega u(y,0)\,\Phi(x,y,t)\,dy$$ An explicit form of $\Phi$ is available for certain domains, e.g., Giuseppe Negro gave it for the line. But no matter what the domain, for any fixed $t>0$ all values of the kernel $\Phi$ are strictly positive. This implies that if $u(x,0)$ is positive on some part of the domain and zero elsewhere, the solution $u(x,t)$ will be positive on all of the domain.

Infinite propagation is not physically realistic, but is a consequence of mathematical process of abstraction in which a finite number of molecules of positive mass are replaced by infinite number of massless points on a line. In other words, it is built into the heat equation. If one wishes to model finite propagation speed, there are two options:

  • change the equation. The recent article Some diffusion equations with finite propagation speed by F. Andreu, V. Caselles, J.M.Mazon, and S. Moll gives an overview of such efforts. The key term here is flux-limited diffusion. Unfortunately, there is a steep price to pay: such modified equations are a lot harder to deal with (they are nonlinear).
  • keep the equation but replace the notion of "support". Namely, we may decide that the increase of temperature, say, by $10^{-3}$ of a Celsius degree is not physically important. (This threshold should be picked based on the details of the problem.) Then define the "substantial" support of the solution at time $t$ as the region where $u(x,t)>10^{-3}$. This set will not immediately expand to the entire domain.
  • Thanks Pavel and Giuseppe. Referring to Einstein's 1905 analysis of the Brownian movement, as I noted in my 2006 ISTAM conference paper, the stochastic theory of diffusion does not necessarily carry instantaneous action at a distance (IAD for short), but still refers to the same (above) diffusion equation. Here, I was wondering if a parallel was already available in the classical theory. Even if not, the assumption that the kernel must have strictly positive values (or have support over the entire domain) seems questionable. Has anyone ever proved that it must be strictly positive? Regards. – Ajit R. Jadhav Jan 02 '13 at 08:00
  • @AjitR.Jadhav Indeed, heat equation (and heat kernel) are related to a stochastic process known as the Wiener process or as the standard Brownian motion. The increments of the Wiener process are normally distributed, and therefore can attain arbitrary large values in arbitrarily small time. Hence, this model of Brownian motion does not exhibit limited propagation speed. –  Jan 02 '13 at 08:20
  • @AjitR.Jadhav Concerning proofs, I can now give a precise reference: the book by Evans, Lawrence C., Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9. In section 2.3.3 Evans proves that if the solution is nonnegative on all of the domain and is positive somewhere in the domain at $t=0$, then it is positive everywhere on the domain for all times $t>0$. –  Jan 02 '13 at 08:24
  • But: (i) As Einstein noted in his 1905 paper (section 4, "On the irregular movement of particles suspended..."), the PDF (prob. distr. function) for the displacements of a single particle "only differs from zero for very small values of [the displacement]." He still got to the same diffusion equation. (ii) As K. Huang (book: "Intro to Stat. Phys." Taylor & Francis, 2001) notes, the diffusion law is independent of the form of the PDF. Putting the two together, the normal distribution is only a convenience of practical computations, not a basic requirement. Support of PDF can be compact. – Ajit R. Jadhav Jan 02 '13 at 09:11
  • Further, since the practical computer is finite, the actual MC/random walk simulations have to use compact support for the PDF. In fact, they usually use just a constant step size, not even a PDF, let alone a PDF going to infinity. They still end up modeling the diffusion equation right---rigorously right (so long as the boundary conditions are handled right.) The whole research program of diffusion-limited aggregation (DLA) simply is a variation on that basic theme. DLA assumes that a finite step size + particles not sticking, will rigorously model the simple diffusion equation. – Ajit R. Jadhav Jan 02 '13 at 09:28
  • The premises of Evans' proof would be interesting to check. ...Usually, all such discussions are somewhere circular: they first assume an infinite domain, and assume infinite support for the PDF, then use the normal PDF, and then reach conclusions of the kind you quote. Simpler is to directly use the Fourier theory: Every frequency is spread all over the domain---that is an assumption. Ergo, the Dirac's delta spreads to the entire domain in infinitesimal time---becomes strictly nonzero. Obvious. But, is it necessary to assume that the Fourier theory applies? Or that PDF is infinite? No! – Ajit R. Jadhav Jan 02 '13 at 09:35
  • An aside: Is it OK if I expand on my question by adding an answer myself, below? I would like to clarify the question a bit more, as in my comments here, in reference to the Brownian movement. (I am new to this forum; don't know if I could or should give an "answer" myself for considerably expanding the question. But the main answer, it seems, would allow me more space than these comments.) Could you please clarify this part of the rules of this forum? Thanks. – Ajit R. Jadhav Jan 02 '13 at 09:40
  • @AjitRJadhav If you would like to clarify the question or add more to it, the best way to do so is to edit the question. The edit link is under the text of the question. The answer box should be used for actual answers to the question. –  Jan 02 '13 at 15:51