  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
