[Paper Review] Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation

Michael Dabkowski|arXiv (Cornell University)|Jul 18, 2010

Navier-Stokes equation solutions6 references5 citations

TL;DR

This paper establishes that solutions to the supercritical quasi-geostrophic (SQG) equation, which are smooth up to a finite time, become globally smooth for all time—eventual regularity—by introducing a modified class of dual test functions inspired by Kiselev and Nazarov. Using a refined duality argument and energy estimates, it proves that Hölder continuity with exponent arbitrarily close to 1 emerges after a finite time, implying global $ C^∞ $ regularity beyond that point.

ABSTRACT

Recently, Silvestre proved that certain weak solutions of the slightly supercritical surface quasi-geostrophic equation eventually become smooth. To prove this, he employed a De Giorgi type argument originated in the work of Caffarelli and Vasseur. Kiselev and Nazarov proved a variation of the result of Caffarelli and Vasseur by introducing a class of test functions. Motivated by the results of Silvestre, we will modify the class of test functions from the work of Kiselev and Nazarov and use this modified class to show that a solution to the supercritical SQG that is smooth up to a certain time must remain smooth forever.

Motivation & Objective

• To establish eventual regularity for solutions to the supercritical quasi-geostrophic equation when initial data is smooth up to a finite time.
• To extend the duality method of Kiselev and Nazarov by introducing a modified class of test functions to control Hölder norms.
• To show that uniform control of the solution's pairing with these test functions implies Hölder continuity and thus global smoothness.
• To provide an alternative proof of Silvestre's result on eventual regularity for slightly supercritical SQG, now extended to the full supercritical regime.
• To demonstrate that viscosity solutions to the perturbed SQG equation converge to classical solutions with global smoothness after a time depending only on $ \alpha $ and $ \|\theta_0\|_{\infty} $.

Proposed method

• Introduces a new class of test functions $ \mathcal{U}(r) $ that are dual to Hölder continuous functions, with $ L^p $ and duality bounds depending on $ r $.
• Uses a modified version of the Class Evolution Lemma to propagate control of solution pairing with $ \mathcal{U}(r) $ over time.
• Applies energy estimates and fractional Laplacian properties on the torus to show exponential decay of $ L^q $ norms for mean-zero initial data.
• Employs a time-translation argument and iterative estimates to show that solution pairing with $ \mathcal{U}(r) $ remains bounded by $ r^\beta $ for $ \beta > 1 - \alpha $, after a time $ T $.
• Relies on duality estimates to derive $ C^\beta $ regularity for $ \beta $ arbitrarily close to 1, using the fact that $ \|\theta(\cdot,t)\|_{C^\beta} \leq C $ implies $ C^\infty $ regularity.
• Applies results from Constantin, Cordoba, and Wu to conclude that uniform $ C^{1-\alpha+\delta} $ bounds imply classical global smoothness.

Experimental results

Research questions

• RQ1 Can solutions to the supercritical SQG equation that are smooth up to a finite time be shown to remain smooth for all future times?
• RQ2 Does a modified duality framework with a new class of test functions allow for improved control of Hölder regularity in the supercritical regime?
• RQ3 Can the eventual regularity result of Silvestre for slightly supercritical SQG be extended to the full supercritical range $ \alpha < 1 $?
• RQ4 Is it possible to derive global $ C^\infty $ regularity from uniform bounds on the solution’s pairing with a dual class of functions?
• RQ5 Does the viscosity solution approach yield classical solutions with global smoothness after a time depending only on $ \alpha $ and $ \|\theta_0\|_{\infty} $?

Key findings

• There exists a time $ T = T(\alpha, \|\theta_0\|_{\infty}) $ such that any solution smooth on $ \mathbb{T}^d \times [0,T] $ extends smoothly to $ \mathbb{T}^d \times [0,\infty) $.
• After time $ T $, the solution exhibits Hölder continuity with exponent $ \beta $ arbitrarily close to 1, uniformly in time.
• The solution’s pairing with the dual class $ \mathcal{U}(r) $ is bounded by $ r^\beta $ for all $ r \in (0,1] $, implying $ C^\beta $ regularity for $ \beta > 1 - \alpha $.
• The time $ T $ depends only on $ \alpha $ and the $ L^\infty $ norm of the initial data, not on higher Sobolev norms.
• The method applies to viscosity solutions of the perturbed SQG equation, showing they become classical solutions after time $ T $, with regularity uniform in the viscosity parameter $ \epsilon $.
• The result provides an alternative proof of Silvestre’s eventual regularity result and extends it to the full supercritical regime.