A PRIORI BOUNDS AND WELL-POSEDNESS OF A SYSTEM ASSOCIATED WITH UNSTEADY BOUNDARY LAYER FLOWS

An integral equation with singularities is introduced to characterize unsteady laminar boundary layer flows and some properties of solutions of this integral equation are investigated. Utilizing these properties, a priori bounds are obtained for the skin friction function and the similarity stream function and the well-posedness of solutions is proved.


Introduction
The following nonautonomous system has been used to study unsteady laminar boundary layer flows [5,9] and is reduced from the governing unsteady two-dimensional Navier-Stokes equations and energy equation via the similarity transformations [5,18] ∂u ∂x + ∂v ∂y = 0, subject to the boundary conditions For more details, one may refer to [5,18].
In the problem (1.1)- (1.4), f is the similarity stream function, f ′′ is the skin friction function, A > 0 is the unsteady parameter, f 0 > 0 (respectively, f 0 < 0) corresponds to suction (respectively, injection) of fluid at the surface, and P r is the Prandtl number. Ishak et al. [5] investigated the problem numerically and the nature of the solutions as the physical parameters are varied. Recently, Paullet [9] studied the existence and uniqueness of the solutions for some (but not all) values of the parameters and obtained a priori bounds for the skin friction coefficient and local Nusselt number.
This paper extends and strengthens the study [5,9] in two aspects: (i) A priori bounds are presented for the skin friction function f ′′ and the similarity stream function f .
(ii) The solutions obtained in [9] are well-posed related to the parameters involved in the system.
It is well-known that numerical and analytical study of similarity solutions is very important in many fields and can provide a standard of comparison without introducing the complication of non-similar solutions. Much attention is always focused on this subject. One may refer to some recent research achievements such as boundary layer flows [3,8], magnetohydrodynamic(MHD) [2,4,8], heat transfer [5,6,9,11,18], manufacturing polymer sheets, processing paper products [5], designing heat exchangers and chemical processing equipment [1] and the references therein. Also, one may refer to the review and extension of similarity solutions [12,13]. This paper is organized as follows: in section 2, we establish the relation between the BVP (1.1), (1.3) and an integral equation with singularities and study some properties of solutions of this integral equation. Utilizing these properties, a priori bounds are obtained. In section 3, the well-posedness of solutions is proved.
has a solution z ∈ Q satisfying where . Then z(t) is continuous and z(t) < 0 on (0, 1]. It follows from Lemma 2.1 that z(t) is strictly decreasing on [0, 1] and Integrating the last equality from t to 1, we have ds.
Integrating the previous equality from 0 to t, noticing that z(0) = 0 and we have that z satisfies (2.5). Since (2.5) contains the improper integrals Az(t) and Bz(t), the following results provide some properties of z, which will be applied to prove the main results of this paper.

Lemma 2.3.
Let z ∈ Q be a solution of (2.5). Then the following facts hold: where Since both integrands in the right hand in (2.5) are continuous in [0, 1]. This implies so we have a contradiction. Since < 0 for s ∈ (0, 1), we know that the Lebesgue integral Az(1) converges and Az(t) exists and is finite for t ∈ [0, 1]. Hence, (i) holds.
) ≥ 0. This implies that the right side of (2.10) holds.

Lemma 2.4.
Let z ∈ Q be a solution of (2.5). Then the BVP (1.1), (1.3) has a solution f ∈ P satisfying and and by s = f ′ (σ) Hence By the differentiation of f ′ (η) = t with respect to η, we have Substituting η, f (η), f ′ (η), f ′′ (η), f ′′′ (η) into (1.1) and utilizing (2.8), we obtain It follows from Lemma 2.3 (P 3 ) and the decrease in  The following Lemma can be found in [9](see, Theorems 1, 2 and 5). In order to prove the well-posedness, we need to prove that the solutions in P × Θ of (1.1)-(1.4) depend on parameters A, f 0 and P r continuously. For this, we need the following Helly selection principle (see [ and P r = P (n) r , and (f, θ) ∈ P × Θ denote the solutions of (1.1)-(1.4). We prove the following theorem. Proof. It follows from Theorem 2.1 (i) that there exists z n ∈ Q which is a solution of (2.5) satisfying (2.6), that is, By Lemma 2.3 (P 3 ), there exist constants a > 0, c < d < 0 and N > 0 such that We first prove the following fact: If it is false, then there exist ε > 0 and {η n k } such that Lemma 2.3 (P 3 ), together with the decrease in z, implies that . From this, we obtain η n k → ∞(n k → ∞).
Without loss of generality, we may assume f ′ n k (η n k ) − f ′ (η n k ) ≥ ε for all k. By the decrease in f ′ n k , we know when η n k ≥ η Taking limit when k → ∞, we have f ′ (η) ≥ ε.
Next, we start to prove the well-posedness.
In this paper, utilizing integral methods, we treat some nonautonomous boundary layer problems analytically. Integral methods are used to treat autonomous boundary layer problems, one may refer to [7,[14][15][16][17].