Consider heat equation on torus: $$\partial_tu(x,t) + (- \Delta)^{\alpha/2} u(x,t)=0, u (x, 0)=u_0(x)$$ where $(x, t) \in \mathbb T^d \times \mathbb R, \alpha>0$

Formally, we may write the solution of heat equation as follows:

$$u(x,t) = \sum_{n\in \mathbb Z} e^{ -tn^{\alpha}} \widehat{u_0}(n) e^{inx}$$

(Is this form of formal solution is correct?)

My question is following: (1) Is there a well-posedness results for the heat equation on torus in the literature? Any reference? (2) Is Fourier multiplier with symbol $e^{-tn^{\alpha}}$ bounded on $L^p(\mathbb T^d)$? What are the corresponding results on $\mathbb R^d$?


EDIT: The following is (mostly) for $\alpha < 2$; scroll to the bottom for more on general $\alpha$. Kudos to Abdelmalek Abdesselam (again).

As for $\mathbb{R}^d$, this is classical: the solution is given by the "fractional heat kernel": $$u(t,x) = u_0 * p_t(x),$$ where $p_t(x)$ is the inverse Fourier transform of $\exp(-t |\xi|^\alpha)$. Since $p_t$ is a probability density function, convolution with $p_t$ is a contraction on every $L^p(\mathbb{R}^d)$. The kernel $p_t$ has several alternative representations; in particular, we have Bochner's subordination formula, which asserts that $p_t(x)$ is a mixture of Gaussians: $$p_t(x) = \int_0^\infty q_s(x) \eta_t(s) ds,$$ where $$q_s(x) = (4 \pi s)^{-d/2} \exp(-|x|^2 / (4t))$$ is the Gaussian and $\eta_t(s)$ is a probability density distribution with Laplace transform $\exp(-t \xi^{\alpha/2})$.

For the torus, all you need to do is to periodize the heat kernel: if we write $$\tilde p_t(x) = \sum_{n \in \mathbb{Z}^d} p_t(x + n),$$ then the solution on the torus is given by $$u(t, x) = u_0 * \tilde p_t(x),$$ where the convolution on the torus is defined as $$ \int_{[0,1)^d} u_0(y) \tilde p_t(x - y) dy .$$

You can find a number of references for the $\mathbb{R}^d$ case in my two survey papers: a more probabilistic view in Section 4 of

M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019

or an analytical perspective in Section 2.6 of

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal. 20(1) (2017): 7–51

I do not know any reference specifically for the torus. You may search for papers on the fractional heat equation on manifolds (or even "fractals"/"$d$-sets"/"metric measure spaces"), but this will likely be too general and abstract for your needs. (I only know of a paper Fractional Laplacian on the torus by Luz Roncal and Pablo Raúl Stinga, but this one is about the extension technique, not very useful for the heat equation.)

EDIT: what about general $\alpha > 0$?

For general $\alpha > 0$, in $\mathbb{R}^d$, a solution is again given by the convolution with $p_t$ given as an inverse Fourier transform of $\exp(-t |\xi|^\alpha)$. This is no longer a positive function if $\alpha > 2$, but it is anyway an integrable function. Here is a short (but perhaps not the most elementary) proof of this fact.

If $\alpha$ is an even integer, then the Fourier transform of $p_t$ is a Schwartz class function, and hence $p_t$ is Schwartz class. If $\alpha$ is not an even integer, then $p_t$ is still smooth, but it no longer decays rapidly. Now the result of:

K. Soni, R.P. Soni, Slowly Varying Functions and Asymptotic Behavior of a Class of Integral Transforms I, II, III. J. Anal. Appl. 49 (1975): 166--179; 477--495; 612--628

applied to the $d$-dimensional Hankel transform provides an asymptotic expansion of $p_t(x)$ at infinity, which in particular implies that $p_t$ is of constant sign in a neighbourhood of infinity. This easily leads to the conclusion that $p_t$ is integrable: otherwise, its Fourier transform would necessarily diverge at zero.

The convolution with $p_t$ is therefore again a bounded operator on $L^p(\mathbb{R}^d)$ for every $p \in [1, \infty]$. With no doubt it is written somewhere, but I do not have a reference at hand.

Of course, periodization leads to similar results on the torus. However, for general $\alpha > 0$ it is way easier to simply note that $\tilde{p}_t$ is given by a Fourier series with rapidly decreasing coefficients, and hence it is infinitely smooth. For this reason, the convolution with $\tilde p_t$ is a bounded operator on $L^p(\mathbb{T}^d)$.

| cite | improve this answer | |
  • 2
    $\begingroup$ +1 but what you said about $p_t$ being a probability density (so its $L^1$ norm is 1 allowing for the use of Young's convolution inequality etc.) is only true if $\alpha\le 1$. See my answer to mathoverflow.net/questions/256526/… $\endgroup$ – Abdelmalek Abdesselam Jun 18 '19 at 20:38
  • 1
    $\begingroup$ @AbdelmalekAbdesselam: Ah, actually all of my answer is about $\alpha \le 1$! Too sloppy again... Thanks! $\endgroup$ – Mateusz Kwaśnicki Jun 18 '19 at 20:49
  • $\begingroup$ I figured you knew this but just typed too fast...I did too, I meant α≤2... $\endgroup$ – Abdelmalek Abdesselam Jun 18 '19 at 21:23
  • 1
    $\begingroup$ BTW Thm 2.1 of the ref by Blumenthal and Getoor I gave in my other MO answer does prove the asymtotic expansiion and constant sign at infinity you mentioned. $\endgroup$ – Abdelmalek Abdesselam Jun 18 '19 at 21:29
  • $\begingroup$ @AbdelmalekAbdesselam: That's interesting! I did not know this, I thought they only dealt with the $\alpha \le 2$ case. (This sort of confirms what one of my masters told me long ago: whatever you prove, there is a related result of one of "The Martians" – the result for $d = 1$ is due to Pólya...) $\endgroup$ – Mateusz Kwaśnicki Jun 18 '19 at 22:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.