Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and uniqueness for a class of fractional drift-diffusion equations

Thomas Hudson, Matthaeus Ragg

TL;DR

The paper analyzes a fractional drift-diffusion equation on the torus with Caputo time derivative of order $\alpha$ and a spatial fractional Laplacian of order $\beta$, establishing existence and uniqueness of a weak solution under $\beta>\tfrac{1}{2}$ and regularity assumptions on the data. It develops a Fourier-Galerkin framework, derives a coercivity bound for the associated bilinear form, proves mass conservation when the forcing has zero mean, and proves Sobolev regularity up to $H^{2\beta}$, enabling a rigorous convergence analysis. A spectral numerical scheme based on Fourier truncation is shown to converge to the true weak solution, with the finite-dimensional system reducible to an $\alpha$-order ODE system solved via Mittag-Leffler functions. The results provide a solid analytical foundation and a practical algorithm for simulating fractional drift-diffusion processes with memory and anomalous diffusion, relevant to systems driven by $2\beta$-stable Lévy dynamics on periodic domains.

Abstract

This work establishes the existence and uniqueness of solutions to the fractional diffusion equation $$\frac{\partial^αu}{\partial t^α} + K(-Δ)^β u - \nabla \cdot (\nabla V u) = f$$ on a $d$-dimensional torus, subject to sufficient conditions on the input parameters. The focus is on fractional orders $α$ and $β$ less than 1. The strategy uses a Galerkin method and focuses on the additional complexity that comes from the fractional-order derivatives. Additional Sobolev regularity of the solution is shown. The spectral approach to the existence proof suggests an algorithm to compute explicit solutions numerically, and the regularity results are used to support a rigorous convergence analysis of the proposed numerical scheme.

Existence and uniqueness for a class of fractional drift-diffusion equations

TL;DR

The paper analyzes a fractional drift-diffusion equation on the torus with Caputo time derivative of order and a spatial fractional Laplacian of order , establishing existence and uniqueness of a weak solution under and regularity assumptions on the data. It develops a Fourier-Galerkin framework, derives a coercivity bound for the associated bilinear form, proves mass conservation when the forcing has zero mean, and proves Sobolev regularity up to , enabling a rigorous convergence analysis. A spectral numerical scheme based on Fourier truncation is shown to converge to the true weak solution, with the finite-dimensional system reducible to an -order ODE system solved via Mittag-Leffler functions. The results provide a solid analytical foundation and a practical algorithm for simulating fractional drift-diffusion processes with memory and anomalous diffusion, relevant to systems driven by -stable Lévy dynamics on periodic domains.

Abstract

This work establishes the existence and uniqueness of solutions to the fractional diffusion equation on a -dimensional torus, subject to sufficient conditions on the input parameters. The focus is on fractional orders and less than 1. The strategy uses a Galerkin method and focuses on the additional complexity that comes from the fractional-order derivatives. Additional Sobolev regularity of the solution is shown. The spectral approach to the existence proof suggests an algorithm to compute explicit solutions numerically, and the regularity results are used to support a rigorous convergence analysis of the proposed numerical scheme.
Paper Structure (15 sections, 24 theorems, 144 equations, 4 figures)

This paper contains 15 sections, 24 theorems, 144 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Fourier series
  5. Fractional Sobolev Spaces
  6. The fractional time derivative
  7. Weak formulation
  8. A coercivity bound
  9. Initial results
  10. Existence and uniqueness of weak solutions
  11. Existence results
  12. Uniqueness
  13. Regularity
  14. Numerical scheme
  15. Conclusion

Key Result

Theorem 1.1

Suppose $\beta > \frac{1}{2}$, and $V$, $f$ and $h$ are sufficiently regular. Then there is a unique solution $u:\mathbb{T}^d \times [0,T] \to \mathbb{R}$ satisfying eq:weak_FPDE_0. Furthermore, the solution $u$ is sufficiently regular to make sense of $\frac{\partial^{\alpha} u}{\partial t^{\alpha}

Figures (4)

  • Figure 1: Distributions underlying the $2\beta$-stable Lévy process. The case where $\beta=1$ corresponds to a normal distribution.
  • Figure 2: An example of a smooth potential $V$. The example was used to compute the numerical solutions shown in the figures which follow.
  • Figure 3: The approximate solution at final time as $\beta$ varies. To compute the solution we used $\alpha=1$ and a final time of $T=10000$. Smaller $\beta$ entails fatter tails of the noise distribution, and we see the solution is more peaked in this case, but also has more mass around the metastable state close to $\tfrac{\pi}{2}$.
  • Figure 4: A numerical approximation of the solution for the case $\alpha = \beta = 0.8$, where the initial condition is a characteristic function centred at $\tfrac{\pi}{2}$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 2.1: Parseval's identity
  • Definition 2.2: Fractional Laplacian
  • Proposition 2.3: Poincaré's inequality
  • proof
  • Remark 2.4
  • Definition 2.5: Caputo fractional derivative
  • Remark 2.6
  • Theorem 2.7: A version of the Lebesgue differentiation theorem
  • Lemma 2.8: Divergence Theorem for the Fractional Laplacian
  • ...and 41 more