Global well-posedness of space-time fractional diffusion equation with Rockland operator on graded Lie group
Aparajita Dasgupta, Michael Ruzhansky, Abhilash Tushir
TL;DR
The paper addresses global well-posedness for a general space-time fractional diffusion equation on graded Lie groups with a Rockland operator $\mathcal{R}$ of homogeneous degree $\nu$, using a Caputo-type derivative $\mathbb{D}_{(g)}$. By applying the group Fourier transform, the PDE reduces to a family of scalar equations in each representation, enabling a rigorous existence, uniqueness, and regularity theory in $L^{2}(\mathbb{G})$ and $L^{2}_{\gamma}(\mathbb{G})$ Sobolev spaces. The main contributions include complete well-posedness results for both homogeneous and non-homogeneous problems with time-dependent coefficients $a(t)$ and $b(t)$, along with detailed regularity estimates for $\mathbb{D}_{(g)}u$ and $\mathcal{R}^{s}u$. The work extends fractional diffusion models to hypoelliptic operators on graded Lie groups, providing a unified framework that encompasses examples like the Heisenberg group and broad classes of Rockland operators, with potential applications to diffusion processes in non-Euclidean settings and variable external fields.
Abstract
In this article, we examine the general space-time fractional diffusion equation for left-invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time-fractional diffusion equation, the space-time fractional diffusion equation when diffusion is under the influence of sub-Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well-posedness of the Cauchy problem for the general space-time fractional diffusion equation for the Rockland operator on a graded Lie group in the associated Sobolev spaces. More precisely, we establish the existence and uniqueness results for both homogeneous and inhomogeneous fractional diffusion equations. In addition, we also develop some regularity estimates.