Question about the regularity of fractional Heat equation

Question

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation:

‎$‎‎$‎ ‎\begin{cases}‎ ‎u_t = (-\Delta)^s u + ‎f(x,t) & \quad \mathrm{in} ‎\Omega \times (0,T),\\ u(x,0)=u_0 & \quad \mathrm{in} ‎\mathbb{R}^N‎, ‎\\ u(x,t)=0 & \quad \mathrm{in} ‎(\mathbb{R}^N \setminus \Omega )\times (0,T)‎. ‎\end{cases}‎ ‎$‎‎$‎ I want to know that, does $(-\Delta)^s$ generate a semigroup and is it analytic? Is the regularity results for this problem, depending on the regularities of $f(x,t)$ and $u_0$, well-known?

Can someone give a survey of reference for these questions?

Answer 1

It depends on $s$, if $1/2\leq s \leq 1$, then you get analyticity. If $0<s<1/2$, then you get Gevrey class only. See Section 8.3, arXiv:1606.00873

Answer 2

Pablo Raúl Stinga's User’s guide to the fractional Laplacian and the method of semigroups (2018) may provide a helpful entry point to the literature. The semigroup connection is expressed by:

The fractional Laplacian $L^s=(-\Delta)^s$, $0<s<1$ can be expressed in terms of the heat diffusion semigroup $v=e^{-tL}u$ generated by $L$ acting on $u$ through the integral formula $$L^s > u=\frac{1}{\Gamma(-s)}\int_0^\infty\left(e^{-tL}u-u\right)\frac{dt}{t^{1+s}}.$$ The solution to $L^s u=f$ can then be written as $$u=\frac{1}{\Gamma(s)}\int_0^\infty e^{-tL}f\frac{dt}{t^{1-s}}.$$

This connection forms the starting point of the regularity study reviewed by Stinga, see in particular theorems 13-15 (Schauder–Hölder–Zygmund estimates).

Answer 3

Regarding analyticity, there is the following nice result by Gomilko and Tomilov (On Subordination of Holomorphic Semigroups, Theorem 1.1):

Gomilko, Alexander; Tomilov, Yuri, On subordination of holomorphic semigroups, Adv. Math. 283, 155-194 (2015). ZBL1319.47034.

A smooth function $\psi\colon (0,\infty)\to (0,\infty)$ is called Bernstein function if $(-1)^n f^{(n+1)}\geq 0$ for all $n\in\mathbb{N}$. If $\psi$ is a Bernstein function and $A$ generates a bounded holomorphic semigroup of angle $\theta$ on the Banach space $X$, then $\psi(A)$ also generates a bounded holomorphic semigroup of angle $\theta$.

Clearly, $\lambda\mapsto \lambda^s$ is a Bernstein function for $s\in (0,1)$. Thus the fractional Laplacian generates a bounded holomorphic semigroup whenever the Laplacian does. In particular, this is the case on $L^p(\Omega)$ for $p\in[1,\infty)$. As a consequence you get $W^{\alpha,p}$ regularity of $u(t,\cdot)$ in terms of $u_0$ and $f$ by the usual semigroup methods. Of course, if you are only interested in the case $p=2$, all of this can also be established via the spectral theorem.