Interior $$\epsilon$$-regularity theory for the solutions of the magneto-micropolar equations with a perturbation term

A crypto-regularity result for the micropolar fluids equations

“…Let us now explain how the previous crypto-regularity result (Theorem 1.1) can help to deduce a real gain of regularity in terms of usual spaces. The following result heavily relies in our previous work done in [4].…”

“…x which appears in the first equation above in the term ( a • ∇) u + ( u • ∇) a is a given divergence free vector field and the presence of this particular type of perturbation is inspired by some applications (see [2] and [9,Section 12.6]) and it is linked to a previous work [4]. Of course, if ω = a = 0 this system reduces to the classical Navier-Stokes equations.…”

“…Micropolar equations were first introduced in 1966 by Eringen [5] and they have been recently studied by many authors, see e.g. [4], [7], [8], [10] and the references there in. One very interesting feature of the micropolar fluid system is that the variable ω is not a divergence free vector field and this fact makes its study quite different from other systems of PDEs based on the Navier-Stokes equations (such as the Magneto-Hydrodyamic equations for example).…”

“…Finally, we will show how to obtain a classical Hölder regularity result for the couple ( u, ω). However, we will not look for a Serrin type regularity theory and instead we will use the more sophisticated theory of Caffarelli-Kohn-Nirenberg which was displayed in our previous article [4]: doing so we will mix these two theories in order to obtain a different and new regularity framework for the system (1.1).…”

“…3) The external force satisfies 4) The weak solution ( u, p, ω) is suitable in the sense of the Caffarelli-Kohn-Nirenberg partial regularity theory (see the Definition 1.1 of [4], see also the Section 13.8 of the book [9]).…”