THE STOKES PROBLEM IN FRACTAL DOMAINS: ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS

. We study a Stokes problem in a three dimensional fractal domain of Koch type and in the corresponding prefractal approximating domains. We prove that the prefractal solutions do converge to the limit fractal one in a suitable sense. Namely the approximating velocity vector ﬁelds as well as the approximating associated pressures converge to the limit fractal ones respectively. the associated abstract Cauchy problems ¯ P and ¯ P h (see Section 3). In both cases, we prove existence and uniqueness of a mild solution via a standard semigroup approach (see Theorems 3.7 and 3.8), which turns out to be a weak solution of problem ( P ) and ( P h ) respectively. The construction of the associated pressure p ∈ L 2 (0 , T ) , L 2 loc ( Q )) and p h ∈ L 2 (0 , T ) , L 2 ( Q )) respectively follows straightforward (see [13]

1. Introduction. The study of viscous flow in rough microchannels is object of many papers due to the quick development of Micro Electro-Mechanical Systems such as micro-motors micro-turbines. Flow characteristics have remarkable effect on the design and process control of MEMS and heat transfer processes [16], [14] and [12] and they are modeled by Stokes equations. Fractal geometry are a good tool to model such irregular geometries. The numerical approximation of BVPs in fractal domains is a crucial issue ( see e.g. [2] and [3]).
In order to achieve this ambitious goal in the framework of Stokes problems in fractal domains a first step is to study how to approximate the solution of an unsteady Stokes problem in a cylindrical domain with a fractal boundary in terms of the smoother solutions of the associated Stokes problems in the corresponding prefractal domains. A second step will be to consider the numerical approximation of Stokes problems in prefractal domains which will be object of a further investigation.
More precisely, in the present paper, we consider unsteady Stokes problem (P ) and (P h ) in a cylindrical domain Q with a Koch-type cross section with no-slip boundary conditions and in the corresponding approximating prefractal domains in Q h , Our aim is to prove that the velocity vector fields u h converge to u , that the pressure gradients ∇p h converge to the pressure gradient of p as well as the pressures p h converge to p. Here f , f h are given source terms belonging to suitable functional spaces as well as the initial data u 0 and u h 0 . The main difficulty in this asymptotic analysis is due to the irregular geometry of the boundary of the domain Q and to the fact that the approximating polyhedral domains Q h change accordingly to h, namely they are an increasing sequence of domains invading Q.
To prove the existence and uniqueness of a weak solution u and u h of problem (P ) and (P h ) respectively in the sense of Definition 3.9, we consider the associated abstract Cauchy problemsP andP h (see Section 3). In both cases, we prove existence and uniqueness of a mild solution via a standard semigroup approach (see Theorems 3.7 and 3.8), which turns out to be a weak solution of problem (P ) and (P h ) respectively. The construction of the associated pressure p ∈ L 2 (0, T ), L 2 loc (Q)) and p h ∈ L 2 (0, T ), L 2 (Q)) respectively follows straightforward (see [13] and [9]).
Our main result is to prove that the velocity vector fields u h converge to u in L 2 ([0, T ] : H 1 0 (Q)) see Theorem 4.6. This is achieved via the M-convergence of the approximating prefractal energy forms a h associated to (P h ) to the limit fractal energy form a associated to (P ), see Theorem 4.4 which in turn implies the convergence of the associated semigroups see Theorem 4.5.
In Theorem 4.8 we prove that the gradients of the pressures p h weakly converge in L 2 ((0, T ), H −1 (Q)) to the gradient of the pressure p and in Theorem 4.9 we prove that p h weakly converge to p in L 2 ((0, T ) × Q). The proof of theorem 4.8 follows from the weak convergence of the approximating Stokes operators A h to A and of the time derivatives given in Theorem 4.6. The convergence of the pressures is a delicate issue, it deeply relies on some recent results on the solutions of the divergence operator on John domains such as our domain Q.
The plan of the paper is the following. In section 2 we introduce the geometry and the relevant functional spaces. In Section 3 we state the problems (P ) and (P h ). We consider the abstract problemsP andP h , we prove the existence of a unique mild solution which in turn is a weak solution (see Theorem 3.11). Finally we construct the associated pressures. In Section 4 we adapt the results in [13] to the present case. We prove the M-convergence of the energy forms in Theorem 4.4 and we deduce the convergence of the associated semigroups. In Theorems 4.6,4.8 and 4.9 we prove the convergence of the velocity vector fields u h , of the associated gradient pressures and of the pressures.
2. Preliminaries. We denote by |P − P 0 | the Euclidean distance in R 3 . By the snowflake F we denote the union of three coplanar Koch curves K i (see [5]).
We assume that the junction points A 1 , A 3 , A 5 are the vertices of a regular triangle with unit side length, that is |A 1 − A 3 | = |A 1 − A 5 | = |A 3 − A 5 | = 1. One can define, in a natural way, a finite Borel measure µ F supported on F by where µ i denotes the normalized d f -dimensional Hausdorff measure, restricted to K i , i = 1, 2, 3.   We denote by F h the closed polygonal curve approximating F at the h−th step. By S h we denote F h × I, where F h is the prefractal approximation of F at the step h, I = [0, 1]. S h is a surface of polyhedral type. We give a point P ∈ S h the Cartesian coordinates P = (x, x 3 ), where x = (x 1 , x 2 ) are the coordinates of the orthogonal projection of P on the plane containing F h and x 3 is the coordinate of the orthogonal projection of P on the x 3 -line containing the interval I.
By Ω h we denote the open bounded two-dimensional domain with boundary F h . By Q h we denote the domain with S h as lateral surface andΩ h : where dl is the arc-length measure on F h and dx 3 is the one-dimensional Lebesgue measure on I. We introduce the fractal surface S = F × I given by the Cartesian product between F and I. It can be defined on S the finite Borel measure supported on S. By Ω we denote the two-dimensional domain whose boundary is  We note that the sequence {Q h } is an increasing sequence invading Q, that is the Let K be a compact set of R N , by C(K) we denote the space of continuous functions on K and by C ∞ 0 (K) the space of continuous infinitely differentiable functions with compact support in K. Let M be an open set of R 3 . By L 2 (M ) we denote the Lebesgue space with respect to the Lebesgue measure L.
and by H −1 (M ) its dual.
In the following we identify by we denote the closed subspace of H 1 0 (M ) 3 moreover the following result holds.
. We denote byP the adjoint ofJ sinceJ is the restriction of J to H 1 0,σ (M ),P is an extension of P to (H 1 0,σ (M )) , the dual of H 1 0,σ (M ). 3.1. The abstract problem. We introduce the bilinear symmetric form a(u, v) : We note that it is coercive in H 1 0,σ (Q) thanks to Poincaré inequality and closed in L 2 σ (Q). By Kato's theorem [7] there exists a unique non positive self-adjoint operator A 0 : The following proposition holds (see [9]): and generates an analytic contraction semigroup T (t) = e +tA : From Lemma 2.2.1 in [13] we have that there exists a unique positive self-adjoint operator (−A) We note that it is coercive in H 1 0,σ (Q h ) thanks to Poincaré inequality and closed in L 2 σ (Q h ). By Kato's theorem there exists a unique non positive self-adjoint operator We can now consider the abstract problems associated to Problems (P ) and (P h ) respectively.P where −A and −A h are the Stokes operators in Q and Q h respectively. The following existence results hold.
where T (t) is the analytic semigroup generated byÃ. Then u is the unique mild solution of (P ) , moreover u ∈ H 1 ((0, T )]; L 2 σ (Q)) L 2 ((0, T ); D(A)), u t (t) = Au(t) + P f (t), for almost every t ∈ [0, T ], u(0) = u 0 . and there exists c such that the following inequality holds:  . (4) where T h (t) is the analytic semigroup generated by A h . Then u h is the unique mild solution of (P h ), moreover for almost every t ∈ [0, T ], u h (0) = u h 0 , and there exists C, independent from h, such that the following inequality holds: .
The proofs of Theorems 3.7 and 3.8 follow from Theorem 1.5.2 and Lemma 1.6.1 -1.6.2 in Ch. IV (for s=2) [13]. We remark that in Theorem 3.8 the independence of C from h in (5) follows from estimates 1.6.20-1.6.22 in Ch. IV of [13].

Weak formulation.
In this section we prove that the solution of the abstract Cauchy problems (P ) and (P h ) are weak solutions of the Stokes system (P ) and (P h ) respectively. Adapting [13] (see chapter IV 2) we now give the definition of weak solution for problem (P ), The definition of weak solution for problem (P h ) can be obtained from 3.9 with the obvious changes.

Remark 2.
We note that if u is the solution of (P ) then the following identity holds: An analogous statement holds for the solution u h of (P h ).
Proof. In order to prove our thesis we use Lemma 3.10 by setting Γ = C ∞ 0,σ (Q). We choose as test function in (6) Since u is a solution and P is self-adjoint we get: One can proceed analogously for u h .
3.3. Associated pressure. We construct the pressure p and p h associated to the weak solution of the Stokes system (P ) and (P h ) respectively. That is a distribution p in (0, T ) × Q satisfying u t − ∆u + ∇p = f and p h in (0, T ) × Q h satisfying u h t − ∆u h + ∇p h = f h . We will carry out the construction of p , the one of p h is analogous with the obvious changes.
Under the hypothesis of Theorem 3.7, let u be the solution (3) that is u is the solution with data f and u 0 , by Remark 2 the following functional is well defined (see [13] page 252): In the following we denote by We now prove that G ∈ L 2 ((0, T ), H −1 (Q)).
where the last inequality follows from the a priori estimate (4).
Moreover it turns out that [G, v] T,Q = 0 for every v ∈ C ∞ 0 ((0, T ) × Q) 3 hence by Remark 1 we have there exists a unique p in L 2 ((0, T ), L 2 loc (Q)) with Q p(t)dL = 0 for almost every t such that G = ∇p in the sense of distributions.
As to the problem P h we introduce the functional by proceeding as above, we can prove that G h ∈ L 2 ((0, T ), 4. The convergence of forms and semigroups. In this Section we study the convergence of the approximating energy forms a (h) to the fractal energy a. The convergence of functional is here intended in the sense of the M-convergence which we define below. In the following for any given function v h defined in Q h , we denote byṽ h its trivial extension to (0,

4.1.
The M-convergence of forms. We recall, for the sake of completeness, the definition of M-convergence of forms introduced by Mosco in [10].
We extend the forms a defined in H 1 0,σ (Q) and a h defined in . We note that taking into account the definition of the space H 1 0,σ (Q h ) the forms a h turn out to be well defined in the whole L 2 σ (Q) because the functions u h in the domain of a h can be trivially extended to H 1 0,σ (Q) that isũ h (P ) = 0 in Q \ Q h .
has a subsequence strongly convergent in L 2 σ (Q). Proposition 4.3. The sequence of forms a h is asymptotically compact in L 2 σ (Q). Remark 3. We point out that, as the sequence of forms a h is asymptotically compact in L 2 σ (Q), M -convergence is equivalent to the Γ-convergence (see Lemma 2.3.2 in [10]), thus we can take in (a) v h strongly converging to u in L 2 σ (Q). We can now state the main theorem of this section. Proof. We start by proving condition a). We can suppose that v h ∈ H 1 0,σ (Q h ) and that lim a h [v h ] < ∞ then there exists a subsequence still denoted by v h such that a h (v h ) converges to its liminf; therefore there exists a c independent from h such that v h H 1 0 (Q h ) ≤ c . We consider the trivial extension to Q of v h . Therefore ṽ h H 1 0 (Q) = v h H 1 0 (Q h ) from this we can deduce that ṽ h H 1 0 (Q) ≤ c with c independent from h . There exists a subsequence still denoted byṽ h weakly converging to w in H 1 0 (Q) and strongly in L 2 σ (Q) from the uniqueness of the limit we deduce that u = w a.e. in L 2 σ (Q) and in particular ∇ṽ h weakly converges to ∇u in L 2 (Q), the thesis follows from the lower semicontinuity of the L 2 − norm of ∇u.
We now prove condition b). Let us assume that u ∈ H 1 0,σ (Q), from Riesz Theorem we have: for any given u ∈ H 1 0,σ (Q) there exists a ψ ∈ (H 1 0,σ (Q)) such that Since for every h ∈ N, ψ belongs also to (H 1 0,σ (Q h )) the problem From Corollary 2, Section 3 in [11], u h strongly converges to u in H 1 0 (Q) and hence lim h a h (u h ) = a(u).
The proof can be carried out as in Theorem 5.3 in [8]. We only remark that from Proof. Since u, p is a solution of P, for every v ∈ L 2 ((0, T ), C ∞ 0 (Q)) 3 we have We note that from i) of Theorem (4.6) the first term in the right-hand side converges to T 0 From iv) of Theorem 4.6 we obtain that 3 ,H 1 0 (Q) 3 dt, hence the thesis.
In order to prove the convergence of the pressures a key tool is that the domain Q is a John domain (for its definition we refer the reader to [6]). Proof. In order to prove our statement we preliminary prove that the sequence {p h } is equi-bounded in L 2 ((0, T ) × Q). for almost every t ∈ (0, T ), {p h } ⊂ L 2 (Q) and Qp h dL = 0. From Theorem 3.2 we have that there exists a function w h ∈ H 1 0 (Q) 3 such that divw h (t) =p h (t) for almost every t ∈ (0, T ) and w h (t) (H 1 0 (Q)) 3 ≤ c p h (t) L 2 (Q) . We multiply equation i) in problem (P h ) by w h and we obtain for almost every t ∈ (0, T ). We have that Since the distributional gradient ofp h weakly converges to the distributional gradient ofp in L 2 ((0, T ), H −1 (Q)) 3 , from Theorem 4.8 and the uniqueness of the limit we have that ∇p = ∇p in the distributional sense. Hence for almost every t ∈ [0, T ] p(t) =p(t) almost everywhere in Q.