Long time dynamics and anomalous dissipation of energy in viscous forced active scalar equations

1. Problem Statement

The paper investigates a broad class of active scalar equations in a periodic domain $\Omega = \mathbb{T}^d$ ($d=2,3$) driven by a time-independent forcing term $S$. The system is governed by the equation:
$$\begin{cases} \partial_t \theta + u \cdot \nabla \theta + \lambda D\theta + \kappa \Lambda^\gamma \theta = S, \\ u = u[\theta] = \partial_{x_i} T_{ij}[\theta], \\ \theta(x, 0) = \theta_0(x), \end{cases}$$
where:

• $\theta$ is the scalar unknown (e.g., temperature or magnetic potential).
• $u$ is the velocity field, determined non-locally from $\theta$ via a singular integral operator $T_{ij}$.
• $D = \Lambda + 1$ is a damping operator, with $\Lambda = \sqrt{-\Delta}$.
• $\kappa \Lambda^\gamma$ represents fractional diffusion (thermal diffusion), where $\kappa > 0$ and $\gamma \in (0, 2]$.
• $\lambda \ge 0$ is the damping coefficient.

The study focuses on two specific physical models that fit this abstract framework:

1. Surface Quasi-Geostrophic (SQG) Equation: $\lambda=0, \kappa>0, \gamma=1$.
2. Magnetogeostrophic (MG) Equation: $\lambda=0, \kappa \ge 0, \gamma=2$.

The paper addresses two primary challenges:

1. Anomalous Dissipation: Does the energy dissipation rate remain non-zero as the viscosity/diffusion parameter $\kappa \to 0$ (the inviscid limit), even in the presence of damping $\lambda > 0$?
2. Long-Time Dynamics: Does the system possess a global attractor, and how do these attractors behave as the damping parameter $\lambda \to 0$?

2. Methodology

The authors employ a combination of harmonic analysis, energy estimates, statistical solution theory, and dynamical systems theory.

• Assumptions on Operators: The velocity operator $u[\theta]$ satisfies specific Fourier multiplier bounds (Assumptions A1–A4). Crucially, for convergence results as $\lambda \to 0$, a stronger smoothing assumption (A3') is introduced, requiring the operator to gain two derivatives (satisfied by the MG equation).
• Statistical Solutions: To address anomalous dissipation, the authors construct stationary statistical solutions (Borel probability measures on the phase space) for both the viscous ($\kappa > 0$) and inviscid ($\kappa = 0$) cases. They utilize the Banach limit to relate long-time averages of solutions to these measures.
• De Giorgi Iteration: To establish global well-posedness in $H^s$ spaces for rough initial data, the authors adapt the De Giorgi iteration method to derive $L^\infty$ bounds from $L^2$ data, a technique essential for handling the nonlinearity $u \cdot \nabla \theta$.
• Absorbing Sets and Regularity: The proof of global attractor existence relies on constructing a hierarchy of absorbing sets in $L^\infty$, Hölder spaces ($C^\alpha$), and Sobolev spaces ($H^1, H^{1+\gamma/2}$).
• Fractal Dimension Analysis: To prove finite fractal dimension, the authors demonstrate that the solution map is uniformly differentiable and that the linearized flow contracts volume elements in high dimensions (using trace estimates of the linearized operator).

3. Key Contributions and Results

A. Absence of Anomalous Dissipation (Theorem 2.1)

• Result: For the forced system with damping $\lambda > 0$, the paper proves that there is no anomalous dissipation of energy in the limit $\kappa \to 0$.
• Mathematical Statement:
  $$\lim_{\kappa \to 0} \left( \limsup_{t \to \infty} \frac{1}{t} \int_0^t \|\nabla \theta^{(\kappa)}\|_{L^2}^2 \, ds \right) = 0$$
• Significance: This contrasts with scenarios where forcing is absent or specific eigenfunction forcings are used. The presence of the damping term $\lambda D\theta$ ensures that energy is dissipated through the damping mechanism rather than through a singular cascade at small scales in the inviscid limit. This extends previous results by Constantin, Tarfulea, and Vicol to a broader class of active scalars.

B. Convergence as Damping Vanishes (Theorem 2.2)

• Result: Under the stronger smoothing assumption (A3') (satisfied by the MG equation), the solutions $\theta^{(\lambda)}$ converge to $\theta^{(0)}$ in $H^s$ norms as the damping parameter $\lambda \to 0$.
• Significance: This establishes the continuity of the dynamics with respect to the damping parameter, bridging the gap between the damped and undamped regimes for the MG equation.

C. Existence and Properties of Global Attractors (Theorem 2.3 & 6.14)

• Existence: The solution map $\pi_\lambda(t)$ possesses a unique global attractor $\mathcal{G}_\lambda$ in $H^1$ for all $\lambda \ge 0$ and $\kappa > 0$.
• Regularity: The attractor is bounded in $H^{1+\gamma/2}$ (or $H^{3/2}$ if $\lambda > 0, \gamma \le 1$).
• Finite Fractal Dimension: Under conditions $\lambda \ge 0, \gamma \in [1, 2]$ or $\lambda > 0, \gamma \in (0, 2]$, the attractor $\mathcal{G}_\lambda$ has finite fractal dimension.
• Invariance: The attractor is fully invariant ($\pi_\lambda(t)\mathcal{G}_\lambda = \mathcal{G}_\lambda$) and maximal among bounded invariant sets.

D. Upper Semicontinuity of Attractors (Theorem 7.5)

• Result: The family of global attractors $\{\mathcal{G}_\lambda\}_{\lambda \ge 0}$ is upper semicontinuous at $\lambda = 0$.
• Mathematical Statement:
  $$\sup_{\phi \in \mathcal{G}_\lambda} \inf_{\psi \in \mathcal{G}_0} \|\phi - \psi\|_{H^1} \to 0 \quad \text{as } \lambda \to 0.$$
• Significance: This proves that the long-time dynamics of the damped MG equation converge to the long-time dynamics of the undamped MG equation as the damping vanishes, providing a rigorous link between the two regimes.

4. Applications to Specific Models

The abstract results are explicitly applied to:

1. Fractional SQG Equation: The results confirm the absence of anomalous dissipation for $\lambda > 0$ and establish the existence of finite-dimensional attractors.
2. Magnetogeostrophic (MG) Equation: The paper provides the first rigorous proof of the existence of a global attractor for the forced MG equation with damping and proves its convergence to the attractor of the undamped system.

5. Significance

• Theoretical Rigor: The paper resolves open questions regarding the long-time behavior of active scalar equations with fractional diffusion and damping, specifically addressing the interplay between forcing, damping, and viscosity.
• Anomalous Dissipation: It clarifies the role of damping in preventing anomalous dissipation, a phenomenon often associated with turbulence and the breakdown of energy conservation in inviscid limits.
• Dynamical Systems: By proving the finite fractal dimension of the attractors, the authors show that the infinite-dimensional PDE system effectively behaves like a finite-dimensional dynamical system in the long run, which is crucial for numerical modeling and understanding the complexity of geophysical flows.
• Methodological Advancement: The combination of statistical solution techniques with sharp regularity estimates (De Giorgi iteration) and attractor theory provides a robust framework applicable to other non-local fluid dynamics problems.

In summary, this work provides a comprehensive analysis of the long-time dynamics of viscous forced active scalars, proving that damping suppresses anomalous dissipation and ensuring the existence of finite-dimensional global attractors that behave continuously as damping is removed.