[Paper Review] Eventual regularization of the slightly supercritical fractional Burgers equation

Chi Hin Chan, Magdalena Czubak+1 more|ArXiv.org|Nov 26, 2009

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes that weak solutions to the slightly supercritical fractional Burgers equation become Hölder continuous for large times, even when the initial data may lead to singularities. By combining parabolic De Giorgi-type oscillation lemmas with non-local scaling arguments, the authors prove eventual regularization for $ s \in \left(\frac{1-\alpha}{2}, \frac{1}{2}\right] $, with $ \alpha \in (0, \frac{1}{2}) $, ensuring $ C^\alpha $ regularity after a time $ T^* $ depending only on the $ L^2 $-norm of the initial data.

ABSTRACT

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Holder continuous for large time.

Motivation & Objective

• To establish that solutions to the slightly supercritical fractional Burgers equation become Hölder continuous for large times despite potential initial singularities.
• To extend the De Giorgi-type regularity method to the supercritical regime by compensating for poor scaling using improved oscillation lemmas.
• To provide a self-contained proof of the oscillation lemma that avoids the extension method, enabling generalization to other non-local operators.
• To show that the regularization time $ T^* $ depends only on the $ L^2 $-norm of the initial data, vanishing as $ s \to \frac{1}{2}^- $.

Proposed method

• Uses a vanishing viscosity approximation to construct weak solutions as limits of smooth solutions to a regularized equation.
• Applies a new, self-contained parabolic oscillation lemma adapted to non-local operators, avoiding the extension method used in prior works.
• Employs a scaling argument in parabolic cylinders $ Q_r = [-r,r] \times [-r^{2s}, 0] $ to balance oscillation improvement against equation deterioration.
• Implements a recursive iteration scheme in dyadic cylinders to propagate Hölder regularity from large to small scales.
• Uses a cutoff function $ \psi(x) $ to control the oscillation of rescaled solutions and ensure sub-solution behavior.
• Relies on energy estimates and decay of the $ L^\infty $-norm of $ \theta $ to eventually bound the solution below one, enabling application of the oscillation lemma.

Experimental results

Research questions

• RQ1 Can weak solutions of the slightly supercritical fractional Burgers equation become Hölder continuous for large times, even if they develop singularities initially?
• RQ2 How can the De Giorgi method be adapted to the supercritical regime where the non-local dissipation is too weak for immediate regularization?
• RQ3 What is the role of the oscillation lemma in achieving eventual regularity, and can it be proven independently of the extension technique?
• RQ4 How does the regularization time $ T^* $ depend on the initial data and the parameter $ s $, particularly as $ s \to \frac{1}{2} $?
• RQ5 Can the method be generalized to other non-local operators beyond the fractional Laplacian?

Key findings

• For any initial data $ \theta_0 \in L^2 $, there exists a time $ T^* > 0 $, depending only on $ \|\theta_0\|_{L^2} $, such that $ \theta(t) $ becomes $ C^\alpha $-Hölder continuous for all $ t > T^* $.
• The regularity exponent $ \alpha \in (0, \frac{1}{2}) $ is universal and independent of the initial data, though it depends on the fractional order $ s $.
• The solution's $ L^\infty $-norm decays over time, ensuring that after a finite time, it becomes bounded by 1, which triggers the oscillation lemma argument.
• The proof provides a self-contained derivation of the oscillation lemma, avoiding the use of the extension method, and thus is applicable to general non-local operators.
• The regularization time $ T^* \to 0 $ as $ s \to \frac{1}{2}^- $, indicating that the critical case is the threshold for eventual regularity.
• The result extends to initial data in $ L^p $ for $ 1 \leq p < \infty $, though the paper focuses on $ L^2 $ for simplicity.