Fractional Laplacian: Singular integral definition

Let $u: \mathbb{R}^{d} \rightarrow \mathbb{R}$, the fractional Laplacian of $u$ is given by \[ \begin{aligned} (-\Delta)^{s}u(x) &= \mathcal{C}(d, s) \; \textup{P.V.} \int_{\mathbb{R}^{d}} \frac{u(x) - u(y)}{|x-y|^{d + 2s}} dy \\ &= \mathcal{C}(d, s) \; \lim_{\epsilon \rightarrow 0} \int_{\mathbb{R}^{d} \setminus B_{\epsilon}} \frac{u(x) - u(y)}{|x-y|^{d + 2s}} dy \end{aligned} \]

Spectral fractional Laplacian

Let $\Omega$ be an open, bounded, Lipschitz domain. Let $\{ \psi_{k} \}_{k \geq 1}$ the eigenfunctions of the Laplace operator with a boundary condition $\mathcal{B}(\psi) = 0$, satisfying the eigenvalue problem \[ \left \{ \begin{array}{ll} -\Delta \psi = \lambda \psi & \textrm{in } \Omega, \\ \mathcal{B}(\psi) = 0 & \textrm{on } \partial \Omega. \end{array} \right. \] The spectral fractional Laplacian with boundary condition $\mathcal{B}$ can then be defined by \[ (-\Delta_{\mathcal{B}})^{s} u := \sum_{k=1}^{\infty} \lambda_{k}^{s} u_{k} \psi_{k} \quad \textrm{with } u_{k} = \int_{\Omega} u \psi_{k} \,\text{d}x. \]

Spectral fractional Sobolev space

Let $\Omega$ be an open, bounded, Lipschitz domain. Let $\{ \psi_{k} \}_{k \geq 1}$ the eigenfunctions of the Laplace operator with a boundary condition $\mathcal{B}(\psi) = 0$, satisfying the eigenvalue problem \[ \mathbb{H}^s_{\mathcal{B}}(\Omega) := \Bigg \{ u(\cdot) = \sum_{k=1}^{\infty} u_{k} \psi_{k}(\cdot) \in L^{2}(\Omega): \|u\|^{2}_{\mathbb{H}^s_{\mathcal{B}}(\Omega)} := \sum_{k=1}^{\infty} \lambda_{k}^{s} u_{k}^{2} < \infty \Bigg \} \] \[ \textrm{with } u_{k} := \int_{\Omega} u(x) \psi_{k}(x) \textrm{d}x. \]

Corollary

Let $\rho$ be be a nonnegative strong solution of the porous medium equation with fractional pressure. The following result holds: \[ \|\rho(t)\|_{L^{\infty}(\Omega)} \leq C \|\rho_{0}\|_{L^{\infty}(\Omega)}. \]

☞ Caffarelli, Soria, Vazquez (2012) proof on $\mathbb{R}^d$ with the integral representation of $(-\Delta)^s$ and no parabolic regularization

☞ Chen, Holzinger, Jüngel, Zamponi (2022) proof on $\mathbb{R}^d$ with the integral representation of $(-\Delta)^s$ with parabolic regularization

Lemma

Let $\rho$ be a nonnegative strong solution of the fractional porous medium equation. Then the following free energy identity holds for all $t\geq 0$ \[ \frac{d}{dt} E(\rho) = - \int_{\Omega} \rho |\nabla (\log \rho - c)|^{2} \, \text{d}x \leq 0 . \]

Theorem

There exists a subsequence of $\{ \rho_{h, \delta, L}^{n} \}_{\delta, h >0}$ and a nonnegative function $\rho_{L}^{n} \in V$ such that, as $\delta, h \to 0_{+}$ \[ \begin{aligned} \rho_{h, \delta, L}^{n} &\to \rho_{L}^{n} &\text{strongly in} \quad &L^{2}(\Omega), \\ \nabla \rho_{h, \delta, L} &\to \nabla \rho_{L}^{n} & \text{weakly in} \quad &L^{2}(\Omega; \mathbb{R}^{d}), \\ \Theta_{\delta}^{L}(\rho_{h, \delta, L}^{n}) &\to \beta^{L}(\rho^{n}_{L}) I & \text{strongly in} \quad &L^{2}(\Omega; \mathbb{R}^{d \times d}), \\ c_{h, \delta, L}^{n} &\to c_{L}^{n} & \text{strongly in} \quad &L^{2}_{\ast}(\Omega), \\ \nabla c_{h, \delta, L}^{n} &\to \nabla c_{L}^{n} & \text{weakly in} \quad &L^{2}_{\ast}(\Omega; \mathbb{R}^{d}). \end{aligned} \] Moreover $\{ \rho^{n}_{L} \}_{n=1, \dots, N}$ solves a discrete-in-time scheme and given $\rho_{L}^{0}$ such that $\frac{1}{|\Omega|} \int_{\Omega} \rho^{0} \, \text{d}x = 1$, one has $\frac{1}{|\Omega|} \int_{\Omega} \rho^{n}_{L} \, \text{d}x = 1$ for all $n=1, \dots, N$ .

Moreover $\{ \rho^{n}_{L} \}_{n=1, \dots, N}$ solves the discrete-in-time scheme:

For $n = 1, \dots, N$, given $\rho_{L}^{n-1} \in V$ find $\rho_{L}^{n} \in V$ such that \[ \int_{\Omega} \frac{\rho_{L}^{n} - \rho_{L}^{n-1}}{\Delta t} \phi \,\text{d}x = - \int_{\Omega} \nabla \rho_{L}^{n} \cdot \nabla \phi + \int_{\Omega} \beta^{L}(\rho_{L}^{n}) \nabla c_{L}^{n} \cdot \nabla \phi \quad \textrm{for all } \phi \in V, \] where \[ - (-\Delta_{\mathrm{N}})^{s} c_{L}^{n} = (\rho_{L}^{n})^{\ast}, \quad \partial_{n} c_{L}^{n} = 0 \textrm{ on } \partial \Omega \] subject to the initial condition $\rho_{L}^{0} = \rho^{0}$.

Corollary

Let $\rho_{L}^{\Delta t (,\pm)}$ defined as before. The following bound holds: \[ \sup_{t \in [0,T]} \|\rho_{L}^{\Delta t (,\pm)}(t)\|_{L^{\infty}(\Omega)} \leq C \|\rho_{0}\|_{L^{\infty}(\Omega)}. \]

Dubinskiǐ's compactness theorem

Suppose that $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ are Banach spaces, $\mathcal{A}_{0} \hookrightarrow \mathcal{A}_{1}$ (i.e., $\mathcal{A}_{0}$ is continuously embedded in $\mathcal{A}_{1}$), and $\mathcal{C}$ is a seminormed set contained in $\mathcal{A}_{0}$ such that $\mathcal{C}$ is compactly embedded in $\mathcal{A}_{0}$. Consider the set \[ \mathcal{Y} \coloneqq \Bigg \{ \varphi: [0, T] \to \mathcal{C}: [\varphi]_{L^{p}(0, T; \mathcal{C})} + \bigg\| \frac{\mathrm{d} \varphi}{\text{d}t}\bigg\|_{L^{p_{1}}(0, T; \mathcal{A}_{1})} < \infty \Bigg \}, \] where $1\leq p \leq \infty$, $1\leq p_{1} \leq \infty$, $\| \cdot \|_{\mathcal{A}_{1}}$ is the norm of $\mathcal{A}_{1}$ and $\frac{\text{d}\varphi}{\text{d}t}$ is understood in the sense of $\mathcal{A}_{1}$-valued distributions on the open interval $(0, T)$. Then $\mathcal{Y}$, with \[ [\varphi]_{\mathcal{Y}} \coloneqq [\varphi]_{L^{p}(0, T; \mathcal{C})} + \bigg\| \frac{\mathrm{d} \varphi}{\text{d}t} \bigg\|_{L^{p_{1}}(0, T; \mathcal{A}_{1})}, \] is a seminormed set in $L^{p}(0, T; \mathcal{A}_{0}) \cap W^{1, p_{1}}(0, T; \mathcal{A}_{1})$, and $\mathcal{Y}$ is compactly embedded in $L^{p}(0, T; \mathcal{A}_{0})$ if either $1 \leq p \leq \infty$ and $1< p_{1} < \infty$, or if $1 \leq p < \infty$ and $p_{1}=1$.

Theorem

There exists a subsequence $\{ \rho^{\Delta t(,\pm)} \}_{\Delta t>0}$ and a function $\widehat{\rho}$ such that \[ \widehat{\rho} \in L^{\infty}(0, T, L^{\infty}(\Omega)) \cap H^{1}(0, T, H^{-\beta}(\Omega)), \quad \beta = d+1, \] with $\widehat{\rho} \geq 0$ almost everywhere on $\Omega \times (0, T)$ and $\frac{1}{|\Omega|} \int_{\Omega} \widehat{\rho}(x, t) \, \text{d}x = 1$ for a.e. $t \in [0, T]$, and a function $\widehat{c}(\cdot, t) \in \mathbb{H}^{s}(\Omega) \cap H^{1}_{\ast}(\Omega)$, defined as $-(-\Delta_{\mathrm{N}})^{s} \widehat{c} = \widehat{\rho}^{\ast}$, such that, for all $p \in [1, \infty)$, as $\Delta t \to 0$, \[ \begin{aligned} \rho^{\Delta t(,\pm)} & \to \widehat{\rho} &\text{strongly in} \quad & L^{p}(0,T;L^{1}(\Omega)), \\ \nabla \sqrt{\rho^{\Delta t(,\pm)}} & \to \nabla \sqrt{\widehat{\rho}} &\text{weakly in} \quad & L^{2}(0,T;L^{2}(\Omega; \mathbb{R}^{d})), \\ \frac{\partial \rho^{\Delta t}}{\partial t} & \to \frac{\partial \widehat{\rho}}{\partial t} &\text{weakly in} \quad & L^{2}(0,T; H^{-\beta}(\Omega)), \\ \rho^{\Delta t(,\pm)} & \to \widehat{\rho} &\text{weakly-$*$ in} \quad & L^{\infty}(0,T;L^{\infty}(\Omega)), \\ \rho^{\Delta t(,\pm)} & \to \widehat{\rho} &\text{strongly in} \quad & L^{p}(0,T;L^{p}(\Omega)), \end{aligned} \]

\[ \begin{aligned} c^{\Delta t(,\pm)} & \to \widehat{c} & \text{strongly in} \quad & L^{p}(0, T; \mathbb{H}^{s}(\Omega)), \\ \nabla c^{\Delta t, +} & \to \nabla \widehat{c} & \text{weakly in} \quad & L^{2}(0, T; L^{2}(\Omega; \mathbb{R}^{d})). \end{aligned} \] Moreover the function $\widehat{\rho}$ is a global weak solution to the problem

\[ \begin{gathered} -\int_{0}^{T} \int_{\Omega} \widehat{\rho} \frac{\partial \phi}{\partial t} \, \text{d}x \, \text{d}t + \int_{0}^{T} \int_{\Omega} \nabla \widehat{\rho} \cdot \nabla \phi \, \text{d}x \, \text{d}t + \int_{0}^{T} \int_{\Omega} \widehat{\rho} \, \nabla \widehat{c} \cdot \nabla \phi \, \text{d}x \, \text{d}t= \int_{\Omega} \rho_{0} \;\phi|_{t=0} \, \text{d}x \\ \text{for all } \phi \in W^{1,1}(0, T; H^{\beta}(\Omega)) \text{ such that } \phi(., T) = 0. \end{gathered} \]

In addition, the function $\widehat{\rho}$ is weak-$*$ continuous as a mapping from $[0,T]$ to $L^{\infty}(\Omega)$ and it is weakly continuous as a mapping from $[0, T]$ to $L^{1}(\Omega)$. The energy functional $E(\cdot)$ satisfies the inequality \[ E(\widehat{\rho}(t)) + \int_{0}^{t} \int_{\Omega} \bigg| 2 \nabla \sqrt{\widehat{\rho}} - \sqrt{\widehat{\rho}} \nabla \widehat{c} \bigg|^{2} \text{d}x \text{d}\tau \leq E(\rho_{0}), \] for a.e. $t \in [0, T]$.

\[\text{Recall} \quad (-\Delta_{h})^{s} u_{h} := \sum_{k=1}^{N_{h}} (\lambda_{k}^{h})^{s} u_{k}^{h} \varphi_{k}^{h} \]

Problem : solve efficiently the fractional Poisson equation \[ (-\Delta)^{s} u = f \quad \text{in } \Omega. \]

For the standard Poisson equation, using finite elements, we need to solve a linear system, \[ -\Delta u = f \quad \leadsto \quad SU = F, \] where $S$ is the stiffness matrix.
For the fractional Poisson equation we should compute the $s$-th power of a matrix \[ (-\Delta)^{s} u = f \quad \leadsto \quad A^{s}U = F. \]

It turns out that $A = M^{-1}S$, where $M$ is the mass matrix.

Compute the $s$-th power using a rational approximation $r(x) \sim x^{-s}$, where \[ r \in \mathcal{R}_{n,n} = \bigg \{ \frac{p}{q}: p \in \mathbb{R}_{m}[x], q \in \mathbb{R}_{n}[x]\bigg \}, \] \[ r (x) = R_{0} + \frac{R_{1}}{ x- t_{1}} + \dots + \frac{R_{n}}{x - t_{n}}. \]

Work with matrix \[ A^{-s} \sim r(A) = R_{0} I + R_{1} (A - t_{1}I)^{-1} + \dots + R_{n} (A - t_{n}I)^{-1}. \]

$\star$ To compute the rational approximation $r$ we use the minimax algorithm
☞ Filip-Nakatsukasa-Trefethen-Bernhard (2018)
Computational benefits: no need to know all the (discrete) spectrum (only min and max eigenvalues ), no storage of dense matrices, at each time step just solve $n$ sparse linear systems, available error estimates

We can get simulations for the fractional porous medium equation for different fractional orders on a large variety of domains, in an efficient and fast way. We can compare different self similar profiles to see the effect of having fractional Laplacian in the potential

central section of 2D simulations

Property: the comparison principle does not hold.

☞ Caffarelli, Vazquez (2011) proof on $\mathbb{R}^d$ with the integral representation of $(-\Delta)^s$ and no parabolic regularization
☞ del Teso, Jakobsen (2025) numerical experiment in 1D, finite difference scheme

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

Property: the comparison principle does not hold.

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

diagonal section $x_1=x_2$

central section $x_2=0$

Property: the comparison principle does not hold.

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

Property: the comparison principle does not hold.

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

Property: the comparison principle does not hold.

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

Property: the comparison principle does not hold.

Consider different initial data $u_{1}$ and $u_2$ on $\Omega = (-5,5)^{2}$, $s=0.75$
initially ordered $u_1 \leq u_2$: for some $c>0$

\[ u_1(x) = \exp(-|x-2|^2/c), \qquad u_2(x) = \exp(-|x-2|^2/c) + 2 \exp(-|x+2|^2/c). \]

diagonal section $x_1=x_2$

central section $x_2=0$