Long time dynamics of forced critical SQG

Abstract

We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^{\infty }$ to $C^{\alpha }$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to $L^{\infty }\cap H^1$. Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in $H^1$, obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in $H^1$ along solution trajectories, and use this property to bound the dimension of the global attractor.

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• On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces
• Stability of Solutions to the Quasi-Geostrophic Equations in ${\mathbb{R}}^2$
• On some electroconvection models
• The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
• Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations
• On the generalized Buckley-Leverett equation
• On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
• On the forced surface quasi-geostrophic equation: Existence of steady states and sharp relaxation rates
• Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation