[Paper Review] Hölder estimates for nonlocal-diffusion equations with drifts

Zhen-Qing Chen, Xicheng Zhang|arXiv (Cornell University)|Oct 30, 2014

Advanced Mathematical Modeling in Engineering8 references3 citations

TL;DR

This paper establishes a priori Φ-Hölder estimates for solutions to nonlocal diffusion equations with drifts using a probabilistic approach, proving that solutions are regular under intrinsic scaling governed by a function Φ derived from the jump kernel's scaling behavior. The key result is a sharp Hölder continuity estimate in space-time, with the modulus controlled by Φ and dependent on the drift's regularity and the underlying Lévy measure's properties.

ABSTRACT

We study a class of nonlocal-diffusion equations with drifts, and derive a priori $Φ$-Hölder estimate for the solutions by using a purely probabilistic argument, where $Φ$ is an intrinsic scaling function for the equation.

Motivation & Objective

• To establish a priori Hölder regularity estimates for parabolic functions solving nonlocal-diffusion equations with time-dependent drifts.
• To extend existing Hölder estimates beyond the fractional Laplacian case to a general class of nonlocal operators with drifts.
• To introduce and utilize an intrinsic scaling function Φ to measure Hölder continuity in the presence of irregular jump kernels.
• To unify and generalize previous results on Hölder regularity for nonlocal operators with drifts under minimal assumptions on the drift and jump kernel.

Proposed method

• The authors use a purely probabilistic argument based on the associated time-inhomogeneous Lévy process and its Lévy system to analyze the transition density and hitting probabilities.
• They define an intrinsic scaling function Φ(r) := ( ∫_r^2 ds/(sϕ(s)) )^{-1} to capture the scaling behavior of the jump kernel, especially when ϕ is regularly varying with index α ∈ (0,2).
• The proof relies on constructing a time-space parabolic cylinder Q(r) and analyzing the exit time from a larger cylinder Q(φ_a(r)) to control the probability of hitting a compact set K.
• Key estimates are derived using the Lévy system formula and bounds on the jump kernel κ_t(x,z), leveraging the symmetry and integrability conditions (1.2) and (1.3).
• The method involves a change of measure and time-space transformation to reduce the problem to a local regularity estimate in a transformed coordinate system.
• The argument uses a comparison principle and maximal inequality to control the oscillation of solutions, leading to the final Hölder estimate.

Experimental results

Research questions

• RQ1 How can a priori Hölder estimates be established for nonlocal-diffusion equations with drifts when the jump kernel does not satisfy a power-law scaling?
• RQ2 What is the appropriate intrinsic scaling function Φ that governs the Hölder regularity of solutions in the supercritical case (α ∈ (0,1))?
• RQ3 Can the Hölder regularity of solutions be preserved under time-dependent drifts that are only Hölder continuous or bounded measurable?
• RQ4 How does the presence of a drift affect the modulus of continuity of solutions in space-time, especially when the diffusion is of lower order than the drift?
• RQ5 What probabilistic tools are sufficient to derive sharp Hölder estimates for nonlocal equations with general jump kernels and drifts?

Key findings

• The paper establishes a Φ-Hölder estimate for solutions to the nonlocal-diffusion equation ∂_t u = L^b_t u, where Φ is the intrinsic scaling function derived from the jump kernel's scaling behavior.
• For any classical solution u, the estimate |u(t,x) - u(s,y)| ≤ C‖u‖_∞ (|x-y|^β + |t-s|^{β/α}) / t^{β/α} holds, with β ∈ (0,1) depending on d, α, and the drift's regularity.
• When the drift b is bounded measurable (α ∈ [1,2)), the Hölder exponent β depends only on d and α, and the constant C depends on ‖b‖_∞.
• In the supercritical case (α ∈ (0,1)), the drift b must be in the Hölder class C^{1−α} to ensure the same Hölder estimate, reflecting the dominance of the drift at small scales.
• The intrinsic scaling function Φ satisfies lim_{r→0} Φ(r)/ϕ(r) = α, and Φ(r) → 0 as r → 0, which is essential for non-trivial regularity estimates.
• The proof relies on a probabilistic construction of the process and estimates on the exit time from a parabolic cylinder, using the Lévy system and hitting probabilities to control the solution's oscillation.