Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

TL;DR

This paper investigates the effects of fast rotation and inviscid limits on the 2-D dissipative surface quasi-geostrophic equation with dispersive forcing, revealing linear limit dynamics and transported initial data under combined limits.

Contribution

It provides new results on the behavior of the SQG equation under simultaneous fast rotation and inviscid limits with ill-prepared initial data, including convergence proofs.

Findings

Limit dynamics are linear under fast rotation with fixed viscosity.

Initial data is transported along the flow in combined limits.

Convergence established using Aubin-Lions lemma.

Abstract

In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term $A \mathcal{R}_{1} \vartheta$, in the domain $\Omega =\mathbb{T}^{1} \times \mathbb{R}$. In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described by a linear equation. On the other hand, performing the combined fast rotation and inviscid limits, we show that the initial data $\overline{\vartheta}_{0}$ is transported along the motion. The proof of the convergence is based on an application of the Aubin-Lions lemma.