A Semi Discrete Dynamical System for a 2D Dissipative Quasi Geostrophic Equation

TL;DR

This paper develops and analyzes a semi-discrete dynamical system for the 2D dissipative quasi-geostrophic equation, establishing existence, uniqueness, regularity, and the existence of a global attractor in the subcritical case.

Contribution

It introduces a semi-discretization scheme for the 2D dissipative quasi-geostrophic equation and proves key properties of the resulting dynamical system and its attractor.

Findings

Existence and uniqueness of solutions for the discretized system.

Regularity results for solutions in the subcritical case.

Existence of a global attractor for the semi-discrete dynamical system.

Abstract

A semi-discretization in time, according to a full implicit Euler scheme, for a 2D dissipative quasi geostrophic equation, is studied. We prove existence, uniqueness and regularity results of the solution to the predicted discretization, in the subcritical case for any initial data in $\dot{L}^2$. Hence, we define an infinite semi-discrete dynamical system, then we prove the existence and the regularity of the corresponding global attractor, for a source term $f$ in $\dot{L}^{p_{\alpha}}$, for a fixed $p_{\alpha} = \frac{2}{1-\alpha} $.