The role of the dimension in uniqueness results for the stationary quasi-geostrophic system

TL;DR

This paper investigates how the spatial dimension affects the uniqueness of solutions to the stationary fractional quasi-geostrophic system, establishing conditions under which only trivial solutions exist in dimensions 2, 3, and 4.

Contribution

It provides a Liouville-type theorem demonstrating the uniqueness of trivial solutions for the stationary fractional quasi-geostrophic system across multiple dimensions under specific integrability conditions.

Findings

Uniqueness of trivial solutions in dimensions 2, 3, and 4.

Dimension and fractional Laplacian power influence solution properties.

Conditions under which non-trivial solutions do not exist.

Abstract

In this paper, we study a Liouville-type theorem for the stationary fractional quasi-geostrophic equation in various dimensions. Indeed, our analysis focuses on dimensions n = 2, 3, 4 and we explore the uniqueness of weak solutions for this fractional system. We demonstrate here that, under some specific Lebesgue integrability information, the only admissible solution to the stationary fractional quasi-geostrophic system is the trivial one and this result provides a comprehensive understanding of how the dimension in connection to the fractional power of the Laplacian influences the uniqueness properties of weak solutions.