Global well-posedness of slightly supercritical active scalar equations

Abstract

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.

References in corpus (3)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations
• On a 1D transport equation with nonlocal velocity and supercritical dissipation

Cited by in corpus (21)

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• Global solutions for a supercritical drift-diffusion equation
• Remarks on well-posedness of the generalized surface quasi-geostrophic equation
• Critical Keller-Segel meets Burgers on ${\mathbb{S}}^1$: large-time smooth solutions
• Regularity for a Local-Nonlocal Transmission Problem
• Global wellposedness for the 3D Muskat problem with medium size slope
• Finite time singularity formation for the modified SQG patch equation
• On the Differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion
• Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement
• On a 1D transport equation with nonlocal velocity and supercritical dissipation
• Regularity criterion and energy conservation for the supercritical Quasi-Geostrophic equation
• Eventual regularization for the 3D Muskat problem: Lipschitz for finite time implies global existence
• The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
• Global regularity for a 1D Euler-alignment system with misalignment
• On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
• Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime
• Mild criticality breaking for the Navier-Stokes equations
• Regularity results for a class of generalized surface quasi-geostrophic equations