Fractional heat equation

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The fractional heat equation refers to the parabolic equation \[ u_t + (-\Delta)^s u = 0,\] where $(-\Delta)^s$ stands for the fractional Laplacian.

In principle one could study the equation for any value of $s$. The values in the range $s \in (0,1]$ are particularly interesting because in that range the equation has a maximum principle.

Heat kernel

The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation \begin{align*} p(0,x) &= \delta_{\{x\}} \\ p_t(t,x) + (-\Delta)^s p &= 0 \end{align*}

The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables for general values of $s$, but the following inequalities are known to hold for some constant $C$ \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]

Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]

For the special case $s=1/2$, the heat kernel coincides with the Cauchy kernel for the Laplace equation in the upper half space \[ p(t,x) = \frac 1 {\omega_{n+1}} \frac t {(x^2+t^2)^{\frac{n+1}2}}. \]

More generally, the heat kernel can be shown to exists for certain nonlocal regular Dirichlet forms $(\mathcal{E}, D(\mathcal{E}))$. Assume \[ \mathcal{E}(u,v) = \int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} \big( u(y)-u(x) \big) \big( v(y)-v(x) \big) J(x,y) \, dx dy \] and $D(\mathcal{E})$ is the closure of smooth, compactly supported functions with respect to $\mathcal{E}(u,u) + \|u\|^2_{L^2}$.

Then the corresponding transition semigroup has a heat kernel $p(t,x,y)$ under quite general assumptions on $J(x,y)$[1].

If $J(x,y)$ is comparable to $|x-y|^{-d-\alpha}$, $p(t,x,y)$ satisfies a bound like above [2][3]. One can relax the assumptions significantly and still prove sharp bounds for small time as well as for large time [4].


References

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