MAGNETO-MICROPOLAR FLUID MOTION: GLOBAL EXISTENCE OF STRONG SOLUTIONS

Here, u(t,x) ∈ R3 denotes the velocity of the fluid at a point x ∈ and time t ∈ [0,T ]; w(t,x) ∈ R3, b(t,x) ∈ R3 and p(t,x) ∈ R denote, respectively, the micro-rotational velocity, the magnetic field and the hydrostatic pressure; the constants μ,χ,α,β,γ,j , and ν are positive numbers associated to properties of the material; f (t,x), g(t,x) ∈ R3 are given external fields. We assume that on the boundary ∂ of the following conditions hold

We assume that on the boundary ∂ of the following conditions hold of local existence (Theorems 2.3, 2.4, and 2.5). In Section 3, we give and prove the global existence theorem analogous to Theorem 2.3 (Theorem 3.1). In Section 4, we give and prove the regularity result (Theorem 4.1). In Section 5, we give the results on pressure, and finally in Section 6, we give results of global existence and regularity when assuming exponential decay in time of the associated external forces.

Statements and notations
Let ⊂ R 3 , be a bounded domain with boundary ∂ of class C 1,1 . Let H s ( ) be the usual Sobolev spaces on with norm · H s (s real); (·, ·) denotes the usual inner product in L 2 ( ) and · the L 2 -norm on . By H 1 0 ( ) we denote the completion of C ∞ 0 ( ) under the norm · H 1 ; the L p -norm on is denoted by · L p , 1 ≤ p ≤ ∞. If B is a Banach space, we denote by L q (0, T ; B) the Banach space of the B-valued functions defined in the interval (0, T ) that are L q -integrable in the sense of Bochner.
The functions, in this paper, are either R or R n -valued and we will not distinguish them in our notations; the situation will be clear from the context.
Throughout the paper, P will denote the orthogonal projection from L 2 ( ) onto H . Then the operator A : D(A) → H → H given by A = −P with domain D(A) = H 2 ( )∩V is called the Stokes operator. It is well known that the operator A is positive definite, self-adjoint operator and is characterized by the relation (2. 3) The operator A −1 is linear continuous from H into D(A), and since the injection of D(A) in H is compact, A −1 is a compact operator in H . As an operator in H , A −1 is also self-adjoint. By a well-known theorem of Hilbert spaces, there exists a sequence of positive numbers µ j > 0, µ j +1 ≤ µ j and an orthonormal basis of H, such that A −1 ϕ j = µ j ϕ j . We denote by λ j = µ −1 j . Since A −1 has range in D(A) we obtain that Aϕ j = λ j ϕ j , ϕ j ∈ D(A), (2.4) 112 Magneto-micropolar fluid motion: global existence of strong solutions form an orthogonal basis in V (with the inner product (∇u, ∇v), u, v ∈ V ) and H 2 ( ) ∩ V (with inner product (Au, Av), u, v ∈ D(A)), respectively. We denote by V k = span{ϕ 1 (x), . . . , ϕ k (x)}.
We observe that for the regularity of the Stokes operator, it is usually assumed that is of class C 3 , this being in order to use Cattabriga's results [3]. We use, instead, the stronger results of Amrouche and Girault [2] which implies, in particular, that when Au ∈ L 2 ( ) then u ∈ H 2 ( ) and u H 2 and Au are equivalent norms when is of class C 1,1 .
Similar considerations are true for the Laplacian operator B ≡ − : D(B) → L 2 ( ) with the Dirichlet boundary conditions with domain D(B) ≡ H 2 ( ) ∩ H 1 0 ( ) and we will denote φ i (x), γ i the eigenfunctions and eigenvalues of B. We denote by By using the properties of P , we can reformulate the problem (1.1), (1.2), and (1.3) as follows: find u, w, b, in suitable spaces, to be exactly defined later on, satisfying Now, we define strong solutions of the problem (2.5).

Remark 2.2.
In what follows, we will prove that if (u, w, b) is a strong solution of (2. To prove existence of strong solutions we will apply the spectral Galerkin method to (2.5). That is, we consider the finite dimensional subspaces V k = span[ϕ 1 , . . . , ϕ k ],

E. E. Ortega-Torres and M. A. Rojas-Medar 113
H k = span[φ 1 , . . . , φ k ], k ∈ N, the corresponding orthogonal projections P k : L 2 → V k and R k : L 2 → H k and the approximate solutions (2.6) developed in terms of eigenfunctions of the Stokes and Laplace operators. Then, the coefficients c ik (t), d ik (t), and e ik (t) are found by requiring that u k , w k , and b k satisfy the following equations: This is equivalent to the weak form By using these approximations, Rojas-Medar [9] proved a local in time existence theorem for (2.5). Their results are the following.
Theorem 2.3. Let the initial values u 0 , b 0 ∈ V , w 0 ∈ H 1 0 ( ) and the external forces f, g ∈ L 2 (0, T ; L 2 ( )). Then, on a (possibly small) time interval [0, T 1 ], 0 < T 1 ≤ T , problem (2.5) has a unique strong solution (u, w, b). This solution belongs to 114 Magneto-micropolar fluid motion: global existence of strong solutions (2.9) Also, the same kind of estimates hold uniformly in k ∈ N for the Galerkin approximations (u k , w k , b k ).
With stronger assumptions on the initial values and external fields, we are able to prove the following.

Theorem 2.4. In addition to the assumptions of Theorem 2.3, assume that
(2.11) Moreover, the same kind of estimates hold uniformly in k for the Galerkin approximations (u k , w k , b k ).
As a consequence of Theorems 2.3 and 2.4, by using the results of Amrouche and Girault [2], we conclude the following theorem.
Finally, we would like to say that as it is usual in this context, we will denote by c, C generic constants depending at most on and the fixed parameters in the problem (µ, χ, r, j, ν, α, β, γ ) and the initial conditions. When for any reason we want to emphasize the dependence of a certain constant on a given parameter we will denote this constant with a subscript.

Global existence
The analogue to Theorem 2.3 is the following result.
. Moreover, for any θ > 0 there exists some finite positive constants M and C such that Also, the same kind of estimates hold uniformly in k for the Galerkin approximations.
Proof. We will combine arguments used by Rojas-Medar [9] and Heywood and Rannacher [5]. We start by proving the boundness in time of ∇u(t) 2 +j ∇w(t) 2 + ∇b(t) 2 . From Rojas-Medar [9, page 11], we have the following differential equality: where   (3.7) are also uniformly bounded. Now, we proceed to prove the other stated estimates. They should be proved first for the approximations (u k , w k , b k ) and then carried to (u, w, b) in the limit. Since that is a standard procedure and the computations are exactly the same, to ease the notation, we will work directly with (u, w, b) in the rest of the paper.
Also, we would like to mention that the technique of using exponentials as weight functions in time was inspired by Heywood and Rannacher [5]. Now, by taking ϕ = u t in (2.5a), φ = w t in (2.5b), and ψ = b t in (2.5c), we obtain  Thus, in a finite interval [0, T ], we can take the last estimates with θ = 0.

Remark 3.2.
As in the end of the previous proof, we observe that all these estimates hold true for θ = 0 on the time interval [0, ∞) if we also include in the hypothesis f, g ∈ L 2 ([0, ∞); L 2 ( )).

More regular solution
The following result is the analogue of Theorem 2.4.
Also, the same kind of estimates hold uniformly in k for the Galerkin approximations.

Results on the pressure
In a standard way we can obtain information on the pressure. In fact, we have the following proposition. Proof. We observe that (1.1) is equivalent to where F = f + χ rot w + rb · ∇b − u · ∇u − u t . Now, we observe that under the hypotheses of Theorem 3.1 (respectively, of Theorem 4.1), we have F ∈ L 2 Loc ([0, ∞); L 2 ( )) (respectively, F ∈ L ∞ ([0, ∞); L 2 ( ))). Therefore, Amrouche and Girault's results [2] imply that there exists a unique Now, it is enough to take p = p * − (r/2)b · b and the proposition is proved. Estimates (5.1) (respectively (5.3)) follows easily from the previous estimates and the estimates given in the above section. This completes the proof of the proposition.
124 Magneto-micropolar fluid motion: global existence of strong solutions

Global existence with exponential decay in time of the external forces
The analogue to Theorem 3.1 is the following theorem. where C is a generic constant independent of t. Also, the same kind of estimates hold uniformly in k for the Galerkin approximations.
The following result is the analogue of Theorem 4.1.
Theorem 6.2. In addition to the assumptions of Theorems 4.1 and 6.1, assume that e γ t f t , e γ t g t ∈ L ∞ ([0, ∞); L 2 ( )). Then, the unique global strong solution (u, w, b) given by Theorem 4.1, for the same γ * and θ of Theorem 6.1, satisfies the following estimates: sup t≥0 e θt u t (t) 2 + w t (t) 2 + b t (t) 2 ≤ C, where σ (t) = min{1, t} e θt and C is a generic constant independent of t. Also, the same kind of estimates hold uniformly in k for the Galerkin approximations.