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

Susan Friedlander, Anthony Suen

Abstract

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and the other as the coefficient of the fractional Laplacian. We establish existence and convergence results in specific parameter regimes and limits. In particular, we demonstrate the absence of anomalous energy dissipation for long-time averaged solutions. Moreover, we investigate the long time dynamics and prove the existence of a unique global attractor. These results are then applied to two specific classes of active scalar equations in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

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

Abstract

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and the other as the coefficient of the fractional Laplacian. We establish existence and convergence results in specific parameter regimes and limits. In particular, we demonstrate the absence of anomalous energy dissipation for long-time averaged solutions. Moreover, we investigate the long time dynamics and prove the existence of a unique global attractor. These results are then applied to two specific classes of active scalar equations in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.
Paper Structure (13 sections, 35 theorems, 300 equations)

This paper contains 13 sections, 35 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Stationary statistical solution and absence of anomalous dissipation when $\lambda>0$
  5. Existence and convergence of $H^s$-solutions when $\kappa>0$
  6. Existence of $H^s$-solutions
  7. Convergence of $H^s$-solutions as $\lambda\to0$
  8. Long time dynamics of solutions when $\kappa>0$
  9. Existence of global attractors in $H^1$-space
  10. Further properties for the global attractors
  11. Applications to magneto-geostrophic equations
  12. The MG equations in the class of drift-diffusion equations
  13. Behaviour of global attractors for varying $\lambda\ge0$

Key Result

Theorem 2.1

Let $\theta_0,S\in L^\infty$, and assume that $\lambda>0$ and $\gamma\in(0,2]$ be fixed. Under the assumptions A1, A2, A3 and A4, for each $\kappa\ge0$, there exists a unique solution $\theta=\theta^{(\kappa)}(x,t)$ to abstract active scalar eqn satisfying

Theorems & Definitions (80)

  • Remark 1.1
  • Theorem 2.1: Absence of anomalous dissipation of energy as $\kappa\to0$ when $\lambda>0$
  • Theorem 2.2: $H^s$-convergence as $\lambda\rightarrow0$ when $\kappa>0$
  • Theorem 2.3: Existence of global attractors
  • Remark 2.4
  • Proposition 4.1
  • proof
  • Definition 4.2
  • Proposition 4.3
  • proof
  • ...and 70 more