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A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

Jae-Hwan Choi, Jaehoon Kang, Daehan Park, Jinsol Seo

TL;DR

The paper develops a comprehensive regularity theory for time-fractional evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces. By leveraging a probabilistic heat-kernel representation for the time-changed independent array of subordinate Brownian motions, it overcomes Fourier-analytic obstacles and derives sharp heat-kernel bounds, BMO–$L^{\infty}$ estimates, and maximal regularity in $L_q((0,T);L_p(\mathbb{R}^d))$ spaces. It introduces adapted Littlewood–Paley projections to handle anisotropic symbols and identifies optimal initial data spaces via generalized real interpolation, including trace and extension results. The results yield existence, uniqueness, and quantitative a priori estimates for solutions, with precise control of spatial and temporal regularity through the parameters $\alpha$, $p$, $q$, and the Bernstein functions $\phi_i$. This framework advances the analysis of space-time non-local PDEs with anisotropic diffusion, providing sharp, implementable estimates for applications in fractional kinetics and related fields.

Abstract

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is $(Δ_{x})^{β_{1}/2}+(Δ_{y})^{β_{2}/2}$. We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.

A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

TL;DR

The paper develops a comprehensive regularity theory for time-fractional evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces. By leveraging a probabilistic heat-kernel representation for the time-changed independent array of subordinate Brownian motions, it overcomes Fourier-analytic obstacles and derives sharp heat-kernel bounds, BMO– estimates, and maximal regularity in spaces. It introduces adapted Littlewood–Paley projections to handle anisotropic symbols and identifies optimal initial data spaces via generalized real interpolation, including trace and extension results. The results yield existence, uniqueness, and quantitative a priori estimates for solutions, with precise control of spatial and temporal regularity through the parameters , , , and the Bernstein functions . This framework advances the analysis of space-time non-local PDEs with anisotropic diffusion, providing sharp, implementable estimates for applications in fractional kinetics and related fields.

Abstract

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is . We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.
Paper Structure (14 sections, 18 theorems, 282 equations)

This paper contains 14 sections, 18 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Motivations and goals
  3. Historical Results
  4. Description of Approaches
  5. Notations
  6. Main Results
  7. Definition of Non-Local Operators
  8. Solution spaces
  9. Statement of Main result
  10. Heat Kernel Estimates for Space-Time Anisotropic Non-Local Operators
  11. Maximal Regularity Estimates of Solutions in Mixed-Norm Lebesgue Space
  12. Trace and Extension Theorem for Solution Spaces
  13. Proof of Theorem \ref{['main theorem']}
  14. Proofs of Proposition \ref{['lem 01.13.15:58']} and Proposition \ref{['prop 03.21.16:12']}

Key Result

Proposition 2.4

Let $1<p<\infty$ and $\gamma\in\mathbb{R}$. (i) The space $H_p^{\vec{\phi},\gamma}$ is a Banach space. (ii) For any $\mu\in\mathbb{R}$, the map $(1-\vec{\phi}\cdot\Delta_{\vec{d}})^{\mu/2}$ is an isometry from $H^{\vec{\phi},\gamma}_{p}$ to $H^{\vec{\phi},\gamma-\mu}_{p}$. (iii) If $\mu>0$, then we where the constant $C$ is independent of $u$. (iv) For any $u\in H^{\vec{\phi},\gamma+2}_{p}$, we h

Theorems & Definitions (42)

  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • Remark 2.11
  • ...and 32 more