A global well-posedness result for the Rosensweig system of ferrofluids

TL;DR

This paper establishes the global well-posedness of the two-dimensional Rosensweig system for ferrofluids by employing a fractional time derivative variation of the Aubin-Lions lemma, and analyzes the long-term behavior and regularity propagation of solutions.

Contribution

It provides the first well-posedness result for the 2D Rosensweig system using a novel fractional derivative approach and studies solution regularity and long-term dynamics.

Findings

Existence and uniqueness of solutions in 2D

Long-time behavior of weak solutions analyzed

Propagation of Sobolev regularities demonstrated

Abstract

In this Paper we study a Bloch-Torrey regularization of the Rosensweig system for ferrofluids. The scope of this paper is twofold. First of all, we investigate the existence and uniqueness of solutions \`a la Leray of this model in the whole bidimensional space. Interesting enough, the well-posedness relies on a variation of the Aubin-Lions lemma for fractional time derivatives. In the second part of this paper we investigate both the long- time behavior of weak solutions and the propagation of Sobolev regularities in dimension two