[Paper Review] The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$

Alexey Cheskidov, Mimi Dai|arXiv (Cornell University)|Feb 19, 2014

Navier-Stokes equation solutions17 references13 citations

TL;DR

This paper establishes the existence of a compact global attractor in $L^2(\mathbb{T}^2)$ for the forced critical surface quasi-geostrophic equation when the forcing term $f$ belongs to $L^p(\mathbb{T}^2)$ for some $p>2$. By applying the De Giorgi iteration method to obtain uniform $L^\infty$ bounds on viscosity solutions and using Littlewood-Paley decomposition to control energy flux, the authors prove strong continuity of solutions and the existence of a compact global attractor in the $L^2$ topology, extending previous results that required stronger regularity on the forcing term.

ABSTRACT

We prove that the critical surface quasi-geostrophic equation driven by a force $f$ possesses a compact global attractor in $L^2(\mathbb T^2)$ provided $f\in L^p(\mathbb T^2)$ for some $p>2$. First, the De Giorgi method is used to obtain uniform $L^\infty$ estimates on viscosity solutions. Even though this does not provide a compact absorbing set, the existence of a compact global attractor follows from the continuity of solutions, which is obtained by estimating the energy flux using the Littlewood-Paley decomposition.

Motivation & Objective

• To establish the existence of a compact global attractor in $L^2(\mathbb{T}^2)$ for the forced critical surface quasi-geostrophic equation.
• To extend previous results on global attractors, which required $f \in L^\infty \cap H^1$, to the case where $f \in L^p$ for some $p>2$.
• To prove uniform $L^\infty$ bounds on viscosity solutions using the De Giorgi iteration method despite the absence of a compact absorbing set.
• To establish strong continuity of solutions via energy flux estimates using Littlewood-Paley decomposition, enabling the construction of a global attractor in $L^2$.

Proposed method

• Application of the De Giorgi iteration method to derive uniform $L^\infty$ bounds on viscosity solutions of the forced critical SQG equation for $t > 0$.
• Use of the Littlewood-Paley decomposition to estimate the energy flux and establish strong continuity of solutions in $L^2$.
• Construction of an evolutionary system $\mathcal{E}$ consisting of viscosity solutions that remain in a bounded absorbing set $X \subset L^2$.
• Proof of the existence of a weak global attractor $\mathcal{A}_w$ via compactness and diagonalization arguments in the weak topology of $L^2$.
• Establishment of strong compactness of the attractor by verifying conditions A1–A3 (sequential compactness, continuity, and convergence in energy norm).
• Use of the energy equality to control the $L^2$-norm of solutions and to prove asymptotic compactness in $L^2$.

Experimental results

Research questions

• RQ1 Does the forced critical surface quasi-geostrophic equation possess a compact global attractor in $L^2(\mathbb{T}^2)$ when the forcing term $f$ is only in $L^p$ for $p>2$?
• RQ2 Can uniform $L^\infty$ bounds be established for viscosity solutions of the forced critical SQG equation using the De Giorgi method under minimal integrability assumptions on $f$?
• RQ3 Is the solution map strongly continuous in $L^2$ under the given forcing conditions, enabling the construction of a strong global attractor?
• RQ4 Can the global attractor be constructed in $L^2$ without requiring $f$ to be in $L^\infty$ or $H^1$?

Key findings

• The critical SQG equation with forcing $f \in L^p(\mathbb{T}^2)$ for some $p>2$ admits a compact global attractor in $L^2(\mathbb{T}^2)$.
• Uniform $L^\infty$ bounds on viscosity solutions are established for all $t > 0$ via the De Giorgi iteration method, even without a compact absorbing set.
• Strong continuity of solutions in $L^2$ is proven using energy flux estimates derived from the Littlewood-Paley decomposition.
• The global attractor $\mathcal{A}$ is compact and strongly attracting in $L^2$, satisfying $\mathrm{d}_s(\theta(t), v(t)) < \epsilon$ for all $t \in [t^*, t^* + T]$ and some complete trajectory $v \in \mathcal{E}((-∞,\infty))$.
• The attractor is constructed as $\mathcal{A} = \{ \theta_0 : \theta_0 = \theta(0) \text{ for some } \theta \in \mathcal{E}((-∞,\infty)) \}$, confirming its strong compactness.
• The result extends prior work requiring $f \in L^\infty \cap H^1$ to the strictly weaker assumption $f \in L^p$ for $p>2$, broadening the applicability of global attractor theory to this class of equations.