The Existence of Solution for the Nonstationary Two Dimensional Microflow Boundary Layer System

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1 The Existence of Solution for the Nonstationary Two Dimensional Microflow Boundary Layer System XIA YE Jimei University School of Sciences Xiamen, China HUASHUI ZHAN Xiamen University of Technology School of Applied Mathematics Xiamen, China Abstract: The paper concerns with the nonstationary two dimensional microflow boundary layer system. By posing some restrictions on the viscous function, the existence and the uniqueness of local solutions to the system are got. The main technique we used in the paper is Oleinik line method based on a successive approximation, which is used in the study of Prandtl system. However, the corresponding calculations in our paper are much more complicated. Key Words: Two dimensional microflow boundary layer system, Prandtl system, Existence, Local solution. 1 Introduction As we known, the Prandtl system is a simplification of the Navier-Stokes system and describes the motion of a fluid with small viscosity about a solid body in a thin layer which is formed near its surface owing to the adhesion of the viscous fluid to the solid surface. Assume that the motion of a fluid occupying a two dimensional region is characterized by the velocity vector V = (u, v), where u, v are the projections of V onto the coordinate axes x, y, respectively, the Prandtl system for a non-stationary boundary layer arising in an axially symmetric incompressible flow past a solid body has the form u t + uu x + vu y = u yy p x, u x + v y = 0, u(0, x, y) = u 0 (x, y), u(t, 0, y) = u 1 (t, y), u(t, x, 0) = 0, v(t, x, 0) = v 0 (t, x), lim y u(t, x, y) = U(t, x). in a domain D = {0 < t < T, 0 < x < X, 0 < y < }, where = const > 0 is the viscosity coefficient of the incompressible fluid. U t + UU x = p x (t, x), U(t, x) > 0, u 0 > 0, u 1 > 0 for y > 0, u 0y > 0, u 1y > 0 for y 0. U = U(t, x) is the velocity at the outer edge of the boundary layer, p = p(t, x) is the pressure. The density of the fluid ρ is equal to 1. Prandtl boundary theory does not consider both the influence of wall s properties on the characteristic of the boundary layer and the interaction of the actual solid wall with the flow of water. If one considers these influences, the Prandtl system should be modified to the following system { ut + uu x + vu y = ((y)u y ) y p x, u x + v y = 0, with the conditions { u(0, x, y) = u0 (x, y), u(t, 0, y) = u 1 (t, y), u(t, x, 0) = g(t, x), v(t, x, 0) = v 0 (t, x), lim y (1) (2) u(t, x, y) = U(t, x). (3) where t, x, y D, (y) is a boundary function, and g(t, x) satisfies some other restrictions. In recent decades, many scholars have been carrying out research in two dimensional boundary layer, achievements are abundant in literature on theoretical, numerical experimental aspects of the theory [1, 2]. In particular, Oleinik had got the existence and the uniqueness of solutions for the Prandtl system by two different kinds of line methods, one of them is based on Rothe s method [3], another one is based on a successive approximation [3]. If is a sufficiently large positive function, which means that (y) > 0 > 0, 0 is a constant, the system (1)-(3) is called the microflow boundary layer [4], and Li-Zhan [5] had got the local well posedness of the system by a similar method as [3], which is base on Rothe s method. Also, there are many papers to deal with the other related problems in the boundary layer theory, such as the relation between the Navier-Stockes system and the Prandtl system and the long-time behavior of the solutions(see [6-15] and references therein). E-ISSN: Issue 6, Volume 12, June 2013

2 In this paper, we used Oleinik s successive approximation method to study the problem (1)-(3). We use the following change of variables [3], which is known as Crocco transform, τ = t, ξ = x, η = u(t, x, y), w(τ, ξ, η) = u y. (4) By calculation, we can get u y = w, u yy = w η w, u yyy = w ηη w 2 + w 2 ηw, u yt = w η u t + w τ, u yx = w η u x + w ξ, y = η w, yy = ηη w 2 + η w η w. From Eqs.(1)-(3) we obtain the following equation for w L(w) = w ηη w 2 w τ ηw ξ + p x w η + ηη w 3 +2 η w η w 2 = 0, (5) in the domain Ω = {0 < τ < T, 0 < ξ < X, g(τ, ξ) < η < U(τ, ξ)}, with the conditions { w τ=0 = w 0 (ξ, η), w ξ=0 = w 1 (τ, η), w η=u(τ,ξ) = 0, l(w) = ( η w 2 + w η w p x v 0 w g τ (6) gg ξ ) η=g(τ,ξ) = 0, (7) where (y), g(t, x) turn into the corresponding functions of τ, ξ and η, but we still denoted them by (τ, ξ, η), g(τ, ξ). Clearly, if = c, i.e. in the Prandtl boundary system, then (5) has the following simpler form w ηη w 2 w τ ηw ξ + p x w η = 0. Now, due to the nonlinear terms ηη w η w η w 2, the problem becomes more difficult. In order to get the similar results as those of Prandtl boundary layer, some restrictions in, g have to be added. 0 < 0 < (y) < 1, where i, (i = 0, 1), are constants. ηη, η, g τ, g, ηηη and g ξ all are bounded, ηη < 0, η < 0 and g(τ, ξ) < min U(τ,ξ) 4, where U(τ, ξ) is the press function of the flow outside the boundary layer. 2 Some important lemmas Definition 1 A function w(τ, ξ, η) is said to be a weak solution of problem (5)-(7), if w has first order derivatives in equation (5) continuous in Ω, and its derivative w ηη continuous when g(τ, ξ) < η < U(τ, ξ); w satisfies equation (5) almost everywhere in Ω, together with the conditions (6)(7). The solution of problem (5)-(7) will be constructed as the limit of a sequence w n, n, which consists of solutions of the equations L n (w n ) (w n 1 ) 2 w n ηη w n τ ηw n ξ +p x w n η + ηη (w n 1 ) η w n η (w n 1 ) 2 = 0, (8) supplemented by the conditions { w n (0, ξ, η) = w 0 (ξ, η), w n (τ, 0, η) = w 1 (τ, η) w n (τ, ξ, U(τ, ξ)) = 0, (9) l n (w n ) = (w n 1 wη n v 0 w n 1 + η (w n 1 ) 2 p x g τ gg ξ ) η=g(τ,ξ) = 0. (10) As w 0 we take a function which is smooth in Ω, satisfies the conditions (6), and is positive for g(τ, ξ) < η < U(τ, ξ). We assume that there exists φ 0 (τ, ξ, η) with the following properties: φ 0 is smooth in Ω; w 0 φ 0 (0, ξ, η), w 1 φ 0 (τ, 0, η), φ 0 > 0 for g(τ, ξ) < η < U(τ, ξ); moreover, φ 0 m 0 (U(τ, ξ) η) k for some m 0 > 0 and k 1, provided that U(τ, ξ) η < δ 0, where δ 0 is a small positive constant. Assuming that problem (8)(9) admits a solution w n (n = 1, 2...) with continuous third order derivatives in the closed domain Ω, let us show that w n are convergent, as n, to a solution of problem (5)- (7); after that we are going to show that the w n do exist, and we indicate a method for their approximation. A solution will be constructed for problem (1)-(3) in the domain Ω for some T = T 0 and any X, as well as for some X = X 0 and any T. The constant T 0 and X 0 are determined by u 0, u 1, v 0, p x. Lemma 2 Let V be a smooth function such that L n (V ) 0 in Ω, l n (V ) > 0 for η = g(τ, ξ), and V w n for τ = 0 and ξ = 0. Assume that w n 1 > 0 for η = g(τ, ξ). Then V w n everywhere in Ω. Let V 1 be a smooth function such that L n (V 1 ) 0 in Ω, l n (V ) < 0 for η = g(τ, ξ), and V 1 w n for τ = 0 and ξ = 0. Assume that w n 1 > 0 for η = g(τ, ξ). Then V 1 w n everywhere in Ω. Proof: Let us prove the first statement of Lemma 2. The difference z = w n V satisfies the inequalities L n (z) = L n (w n ) L n (V ) 0, l n (z) = l n (w n ) l n (V ) = w n 1 z η < 0. since w n 1 > 0 for η = g(τ, ξ). By assumption V w n for τ = 0, ξ = 0, we have z 0 for τ = 0, and E-ISSN: Issue 6, Volume 12, June 2013

3 z 0 for ξ = 0. Consider the function z 1 = ze τ, clearly, z 1 0 for τ = 0 and ξ = 0; z 1η < 0 for η = U(τ, ξ). It follow that z 1 can t have a negative minimum at η = g(τ, ξ), since at the point of negative minimum z 1η 0. The rest of the proof is similar as in [3].The second statement of Lemma 2 can be proved in a similar fashion. Lemma 3 Suppose that ηη, η,, g τ, g and g ξ all are bounded, g(τ, ξ) < min U(τ,ξ) 4, ηη < 0, η < 0. There is a positive constant T 0 such that for all n and all τ T 0 the inequalities H 1 (τ, ξ, η) w n h 1 (τ, ξ, η), hold in Ω, where H 1 and h 1 are continuous functions in Ω, h 1 0 for g < η < U, τ < T 0. Proof: Let us construct function V and V 1 satisfying the conditions of Lemma 2. To this end, we define a twice continuously differentiable function ψ(τ, ξ, η) such that ψ κ(α 1 (η g)) for g < η < g + δ 1, 0 < δ 1 < min U(τ, ξ)/2 g, κ(s) = e s for 0 s 1, 1 κ(s) 3 for s 1, and ψ = (U(τ, ξ) η) k for U η < δ 0 ; 0 < α 0 ψ 4 for δ 1 < η < U δ 0. Here α 0 is a small constant. We define the functions V and V 1 by V = mψe ατ, V 1 = M(C e β 1η )e βτ, where m, α, α 1, β, β 1, C, M are positive constants. Let us show that T 0 and the constants in the definition of V and V 1 can be chosen independent of n, so that the inequality V w n 1 V 1 for τ T 0 implies that V w n V 1 for τ T 0. Consider l n (V ), l n (V 1 ). For e ατ 1/2, since w n 1 V = mψe ατ and g, g τ, g ξ, p x, η are bounded. If we choose α 1 > 0 and β 1 > 0 large enough, we can get l n (V ) = w n 1 V n η v 0 w n 1 + η (w n 1 ) 2 p x g τ gg ξ me ατ (mα 1 e ατ v 0 ) p x + η (w n 1 ) 2 g τ gg ξ > 0, l n (V 1 ) = w n 1 V n 1η v 0 w n 1 + η (w n 1 ) 2 p x g τ gg ξ me ατ ( β 1 Me βτ v 0 ) p x + η (w n 1 ) 2 g τ gg ξ < 0, due to 1 > > 0, η < 0. The constant m, C and M should be chosen from the conditions φ 0 (τ, ξ, η) mψ(τ, ξ, η), C e β 1η 1, M max{w 0, w 1 }, Let us choose β > 0 such that L n (V 1 ) < 0 in Ω. Taking into account the inequality w n 1 V = mψe ατ, ηη < 0, we find that for large positive β L n (V 1 ) = (w n 1 ) 2 Mβ 2 1e β 1η e βτ M(C e β 1η )βe βτ p x Mβ 1 e β 1η e βτ + ηη (w n 1 ) 3 +2 η Me βτ ( β 1 )e β 1η (w n 1 ) 2 e βτ [(mψe ατ ) 2 Mβ 2 1e β 1η + Mβ For L n (V ), we have +p x Mβ 1 e β 1η ] < 0. L n (V ) = (w n 1 ) 2 mψ ηη e ατ + αmψe ατ mψ τ e ατ ηmψ ξ e ατ + p x mψ η e ατ + ηη (w n 1 ) η mψ η e ατ (w n 1 ) 2. Since 0 w n 1 M(C e β 1η )e βη, and ηη, p x,, η are all bounded. the positive constant α can be chosen independent of n and so large that L n (V ) > 0 in Ω for η < U(τ, ξ) δ 0. because of the inequality ψ min{α 0, 1}. In the region η U(τ, ξ) δ 0 where ψ = (U η) k, we have L n (V ) = me ατ [(w n 1 ) 2 k(k 1)(U η) k 2 k(u η) k 1 U τ + α(u η) k ηk(u η) k 1 U ξ p x k(u η) k 1 2 η (w n 1 ) 2 k(u η) k 1 ] + ηη (w n 1 ) 3. It follows from the Bernoulli relation that U τ + ηu ξ + p x = (U η)u ξ. Due to ηη, η all are bounded, if we choose T 0 such that e αt 0 1/2, then for τ T 0, L n (V ) me ατ [k(u η) k U ξ + α(u η) k 2 η (w n 1 ) 2 ] + ηη (w n 1 ) 3 > 0, for large positive α. Thus, the conditions of Lemma 2 hold for V and V 1 in Ω. The constant α and T 0 depend only on the data of problem (5)-(7). Consequently, if by V w n 1 V 1 for τ T 0, it follow that V 1 w n V for any n and τ T 0. Now, it remains to set h 1 (τ, ξ, η) = V, H 1 (τ, ξ, η) = V 1. E-ISSN: Issue 6, Volume 12, June 2013

4 Lemma 4 Suppose that ηη, η,, g τ, g and g ξ all are bounded, ηη < 0, η < 0, there is a positive constant X 0 such that for all n and ξ X 0 the inequalities H 2 (τ, ξ, η) w n h 2 (τ, ξ, η), hold in Ω, where H 2, h 2 are continuous functions in Ω and h 2 (τ, ξ, η) > 0 for η < U, ξ X 0. Proof: Let us construct functions V and V 1 that satisfy the conditions of Lemma 2. Let ψ(τ, ξ, η) be the function constructed in the proof of Lemma 3, and let φ(s) be a twice differentiable function for s 0 and such that φ(s) = 3 e s for 0 s 1/2, 1 φ(s) 3, φ (s) 3, φ (s) 3 for all s 0. Set V = mψe αξ, V 1 = Mφ(β 1 η)e βξ. Let us assume positive constant m, M, α, α 1, β, β 1 and X 0 can be chosen independent of n, and V 1 w n 1 V for ξ X 0, p x, η, g τ, g, and g ξ are bounded, we have l n (V ) = w n 1 mα 1 e αξ v 0 w n 1 p x + η (w n 1 ) 2 g τ gg ξ me αξ (mα 1 e αξ v 0 ) p x > 0, for large enough α 1, provided that e αξ 1/2. If β 1 is sufficiently large and e αξ 1/2, then l n (V 1 ) me αξ ( Mβ 1 e βξ v 0 ) p x + η (w n 1 ) 2 We have g τ gg ξ < 0. L n (V 1 ) = (w n 1 ) 2 Mβ 2 1φ e βξ ηmφβe βξ +[p x + 2 η (w n 1 ) 2 ]Mβ 1 φ e βξ + ηη (w n 1 ) 3. If β 1 η 1/2, φ 1. By assumption, w n 1 mψe αξ, where the function ψ has already been fixed, and the constant m is determined from the condition: mφ φ 0, e αξ 1/2 for ξ X 0 and sufficiently small X 0. Therefore, β 1 is taken so large that L n (V 1 ) < 0 for β 1 η 1/2. Choosing β > 0 so large that L n (V 1 ) < 0 for β 1 η 1/2, choosing a suitable M, we can ensure the inequality V 1 w n for τ = 0 and for ξ = 0. By Lemma 2, V 1 w n in Ω for ξ X 0. For L n (V ) we have L n (V ) = (w n 1 ) 2 mψ ηη e αξ + αmψe αξ mψ τ e αξ ηmψ ξ e αξ + p x mψ η e αξ + ηη (w n 1 ) η mψ η e αξ (w n 1 ) 2. since mψe αξ w n 1, if α 1 η 1, e αβ 1/2 and α 1 large enough then L n (V ) m 3 α 2 1e 3α 1η e 3αξ +[p x +2 η (w n 1 ) 2 ]α 1 e α 1η e αξ m+ ηη (w n 1 ) 3 > 0, due to the boundedness of η, ηη. If 1/α 1 < η < U δ 0, ψ α 0 > 0,0 w n 1 Mφ(β 1 η)e βξ and α is taken large enough, then L n (V ) > 0. If U(τ, ξ) η < δ 0 from the Bernoulli law, as in the proof of Lemma 3, we take α sufficiently large to assure L n (V ) > 0 for U η < δ 0. Therefore, L n (V ) > 0 in Ω for 0 ξ X 0, if X 0 is chosen such that e αx 0 1/2. Since, owing to our choice of m, we have V w n for τ = 0 and ξ = 0, it follows from Lemma 2 that w n mψe αξ for ξ X 0 and all τ. This completes the proof of Lemma 4, since it may be assumed that V w 0 V 1. In what follows, it is assumed that the constants T 0 and X 0 in the definition of Ω are the same as in Lemma 3 and Lemma 4. In order to estimate the first and the second order derivatives of w n, we pass to new unknown functions W n = w n e αη in (8)(9), where α is a positive constant to be chosen later. Thus, we have Set L n (w n ) = (w n 1 ) 2 W n ηη W n τ ηw n ξ +[p x + 2 η (w n 1 ) 2 2(w n 1 ) 2 α]w n η +[α 2 (w n 1 ) 2 p x α 2 η αe αη (w n 1 ) 2 ]W n + ηη (w n 1 ) 3 e α(η g) = 0. l n (w n ) = W n η α W n W n 1 v 0 + η (W n 1 ) 2 p x g τ gg ξ = 0. L 0 n(w ) (w n 1 ) 2 W ηη W τ ηw ξ + A n W η, A n = p x + 2 η (w n 1 ) 2 2(w n 1 ) 2 α, L 0 n(w n ) + B n W n + ηη (w n 1 ) 3 e αη = 0, B n = α 2 (w n 1 ) p x α 2 η α(w n 1 ) 2. Consider the function Φ n = (W n τ ) 2 + (W n ξ ) 2 + W n η (W n η 2H n +2 g τ + gg ξ 2 ηw n 1 ) + k 0 + k 1 η, E-ISSN: Issue 6, Volume 12, June 2013

5 where H n v 0 + p x + αw n χ(η) + g τ + gg ξ ηw n 1. We assume that H n is defined in Ω, and v 0, p x have been extended to the region η > g(τ, ξ), so that v 0 = 0, p x = 0 for η > δ 2 = min U(τ, ξ)/2; v 0, p x do not depend on η for η < δ 2 /2 and are sufficiently smooth for all η; χ(η) is smooth function such that χ(η) = 1 for η δ 2 /2, and χ(η) = 0 for η δ 2. Obviously, Wη n = H n for η = g(τ, ξ). Lemma 5 Suppose that ηη, η,, g τ, g and g ξ all are bounded, ηη < 0, η < 0, and as before 0 < 0 < 1, then the constant k 0, k 1, α can be chosen such that Φ n η αφ n α 2 Φ n 1 for η = g(τ, ξ), L 0 n(φ n ) + R n Φ n 0 in Ω, (11) where R n depends on w n 1 and its derivatives up to the second order. Proof: For Φ n η at η = g(τ, ξ), Φ n η = 2W n τ W n τη + 2W ξ W n ξη + W n ηη(w n η 2H n ) +W n η (W n ηη 2H n η ) + 2W n ηη( g τ + gg ξ + ηw n 1 ) + 2Wη n ( g τ + gg ξ ηw n 1 ) η + k 1. Using the boundary condition Wη n = H n at η = g(τ, ξ), we obtain Φ n η = 2W n τ H n τ + 2W n ξ H n ξ 2H n H n η +2W n ηη( g τ + gg ξ +2H n ( g τ + gg ξ + ηw n 1 ) ηw n 1 ) η + k 1. According to Lemma 3 and Lemma 4, Wη n h 0 > 0 for η = g(τ, ξ). For η = g(τ, ξ), we have H n η = v 0 η 2 (p x + g τ + gg ξ )( η W n 1 + η ) ( ) 2 +αw n η χ(η) ( η ) ηw n 1 η W n 1 η. Let us express Wη n and Wη n 1 from the condition Wη n = H n. We find that H n Hη n depends only on, η, ηη, W n, W n 1, W n 2, and therefore, is uniformly bounded with respect to n. Consequently, 2H n Hη n k 2, k 2 being independent of n. Let us estimate Wτ n Hτ n and Wξ nhn ξ. For η = g(τ, ξ), we have χ(η) = 1, Hτ n = v 0τ + p xτ p xwτ n 1 (W n 1 ) 2 + αw τ n + (g τ + gg ξ ) τ (g τ + gg ξ )Wτ n 1 (W n 1 ) 2, Wτ n Hτ n = Wτ n [ v 0τ + p xτ p xwτ n 1 (W n 1 ) 2 +αw n τ + (g τ + gg ξ ) τ (g τ + gg ξ )Wτ n 1 (W n 1 ) 2 ] α(w n τ ) 2 1 α [v 0τ + p xτ + (g τ + gg ξ ) τ ] 2 1 α [p x + g τ + gg ξ (W n 1 ) 2 ] 2 (Wτ n 1 ) 2 α 4 (W τ n ) 2. Due to p x, g τ, g and g ξ all are bounded, 0 < < 1, we can choose a positive α independent of n and such that Then 1 α [p x + g τ + gg ξ (W n 1 ) 2 ] 2 α 4, Wτ n Hτ n 3α 4 (W τ n ) 2 α n 1 (Wτ ) 2 k 3, 4 k 3 > max 1 α [v 0τ + p xτ + (g τ + gg ξ ) τ ] 2, and k 3 does not depend on n. In a similar way, we find that Wξ n Hξ n 3α 4 (W ξ n ) 2 α n 1 (Wξ ) 2 k 4, 4 k 4 max 1 α [v 0ξ + p xξ + (g τ + gg ξ ) ξ ] 2 2Wηη n g τ + gg ξ + η (W n 1 ) 2 = 2{W n τ + ηw n ξ [p x 2(w n 1 ) 2 α]w n η [α 2 (w n 1 ) 2 p x α]w n } g τ + gg ξ + η (W n 1 ) 2 α 4 (W n τ ) 2 1 α (g τ + gg ξ + η (W n 1 ) 2 ) 2 E-ISSN: Issue 6, Volume 12, June 2013

6 α 4 (W n ξ ) 2 1 α (2η(g τ + gg ξ + η (W n 1 ) 2 ) ) 2 2[p x 2(w n 1 ) 2 α]wη n g τ + gg ξ + η (W n 1 ) 2 α 4 (W n ) 2 1 α [(α2 (w n 1 ) 2 p x α)(g τ + gg ξ + η (W n 1 ) 2 ) ] 2 α 4 (W n τ ) 2 α 4 (W n ξ ) 2 α 4 (W n ) 2 k 5. For η = g(τ, ξ), we have Φ n η α[(w n τ ) 2 + (W n ξ ) 2 ] α 2 [(W n τ ) 2 + (W n 1 ξ ) 2 ] k 6 + k 1, where k 6 = k 2 + 2k 3 + 2k 4 + k 5. The function Wη n (Wη n 2H n ) η=g(τ,ξ) is uniformly bounded with respect to n, according to the boundary condition Wη n = H n. Therefore Φ n η αφ n α 2 Φ n 1 k 7 + k 1, where k 7 is a constant that does not depend on n. Let us choose k 1 > k 7. We have Φ n η αφ n 1 2 αφ n 1 for η = g(τ, ξ). Next we consider L 0 n(φ n ). Choosing a suitable k 0, we may assume that Φ n 1 in Ω. Noting that for η δ 2, we have H n = g τ + gg ξ ηw n 1, Φ n = Φ n (W n τ ) 2 + (W n ξ ) 2 + (W n η ) 2 + k 0 + k 1 η. Applying the operator 2W n τ to the equation τ + 2W n ξ ξ + 2W n η L 0 n(w n ) + B n W n = 0, η we can get the conclusion of (11). As the details of the proof, and we will give them in the appendix of the paper. In order to estimate the second derivatives of w n in Ω, consider the function F n = (W n ττ ) 2 +(W n ξξ) 2 +(W n τξ) 2 +W n ξη(w n ξη 2H n ξ ) +W n τη(w n τη 2H n τ ) + f(η)(w n ηη) 2 + N 0 + N 1 η, where N 0, N 1 are constants, and f(η) is a smooth function such that f(g) = 0, f (g) = 0, f(η) > 0 for η > g(τ, ξ), f(η) = 1 for η > δ 2. Lemma 6 The constant N 0 and N 1 can be chosen independent of w n, w n 1, w n 2 or their derivatives, so that F n η αf n α 2 F n 1 for η = g, L 0 n(f n ) + C n F n + N 2 0 in Ω, where the constant N 2 depends only on the first derivatives of w n, w n 1, w n 2 ;the constant C n depends on w n 1 and its derivatives up to the second order. The proof is similar with the way of lemma 5. 3 The solution of the system (5)-(7) Theorem 7 Let w n be solutions of problems(8)(9) (10). Then the derivatives of w n up to the second order are uniformly bounded with respect to n in domain Ω with a positive T depending on the data of problem (1)-(3). Proof: Let us show that there exist constants M 1, M 2 and T > 0 such that the conditions Φ µ M 1, F µ M 2 for τ T, µ n 1, imply that Φ n M 1, F n M 2 for τ T. According to Lemma 5, we have L 0 n(φ n ) + R n Φ n 0, where R n depends on w n 1 and its derivatives up to the second order. Consider the function Φ 1 n = Φ n e γτ with a positive constant γ to be chosen later. We have L 0 n(φ 1 n) + (R n γ)φ 1 n 0 in Ω. Let us choose γ in accordance with M 1 and M 2, so as to have R n γ 1 in Ω, as well as for ξ = X, τ = T, or η = U(τ, ξ). If Φ 1 n attains its largest value at τ = 0 or at ξ = 0, we should have Φ 1 n = Φ n e γτ Φ n < k 11, where the constant k 11 does not depend on n and is determined only by the data of problem (8)(9)(10). If Φ 1 n attains its largest value at some point with η = g, we must have Φ 1 n/ η 0 at that point, and it E-ISSN: Issue 6, Volume 12, June 2013

7 follows from lemma 5 that Φ 1 n 1 2 Φ1 n 1, i.e, Φ1 n 1 2 M 1. Thus we have Φ 1 n max{ 1 2 M 1, k 11 }, Φ n max{ 1 2 M 1, k 11 }e γτ in Ω. Let us take T 1 T such that e γt 1 = 2, and set M 1 = 2k 11. In this case, Φ n M 1 for τ T 1. We consider F n. By Lemma 6, we have L 0 n(f n ) + C n F n + N 2 0 in Ω, where C n depends on the first and the second derivatives of w n 1, while N 2 depends on the first derivatives of w n, w n 1, w n 2. Set F 1 n = F n e γ 1τ. Then L 0 n(f 1 n)+(c n γ 1 )F 1 n N 2 e γ 1τ N 2 in Ω. Let us choose γ 1 > 0 in accordance with M 1 and M 2, so as to have C n γ 1 1 in Ω 1 = Ω {τ T 1 }. Then, if F 1 n attains its largest value inside Ω 1, or at τ = T 1, or at ξ = X, or at η = U(τ, ξ), we must have F 1 n N 2 (M 1 ). If F 1 n attains its largest value at τ = 0 or at ξ = 0, then F 1 n = F n e γ 1τ F n N 12, where the constant N 12 depends on M 1. If Fn 1 attains its largest value at η = g(τ, ξ), then, according to Lemma 6, at the point of maximum we have and therefore It follows that 0 F 1 n η αf 1 n α 2 F 1 n 1. F 1 n 1 2 F 1 n F n 1e γ 1τ 1 2 M 2. F 1 n max{ 1 2 M 2, N 12, N 2 } in Ω, F n max{ 1 2 M 2, N 12, N 2 }e γ 1τ. Let us take T 2 T such that e γ 1T 2 = 2. Set M 2 = max{2n 12, 2N 2 }. Then F n M 2 for τ T 2 and τ T 1. The constant T 2, like T 1, depends only on M 1 and M 2 chosen above and determined only by the data of problem (1)(2)(3). It may be assumed that w 0 has been chosen such that Φ 0 M 1 and F 0 M 2. The above results show that Φ n and F n are uniformly bounded with respect with respect to n for τ min{t 1, T 2 } = T. The fact that Φ n and F n are bounded with respect to n allows us to conclude that the first and the second derivatives of w n are also bounded, since the boundedness of w n ηη for η δ 2 follows from (8) and the boundedness of the first derivatives of w n. Theorem 7 is proved. By the last theorem, we obtain a solution of problem (8)(9)(10) for any X and a sufficiently small T. The fact that derivatives of w n are bounded for an arbitrary T and a sufficiently small X is established by the following: Theorem 8 Let satisfy the conditions quoted before, and w n be solutions of problems(8)(9) (10). Then w n are uniformly bounded with respect to n in domain Ω with X depending on the data of problem (1)-(3). Proof: Let us show that there exist constants M 1, M 2, and X > 0 such that the conditions Φ µ M 1 and F µ M 2 for ξ X and µ n 1 imply that Φ n M 1 and F n M 2 for ξ X. By Lemma 5, we have L 0 n(φ n ) + R n Φ n 0, where R n depends on w n 1 and its derivatives up to the second order. Let Φ n = Φ 1 ne βξ φ 1 (β 1 η), where φ 1 (s) is a smooth function such that φ 1 (s) = 2 e s /2 for s ln(3/2), 1 φ 1 3/2 for all s; β, β 1 are positive constants that will be chosen later. We have L 0 n(φ 1 n) + 2(w n 1 ) 2 β 1 φ 1 φ 1 Φ 1 nη+ +(R n ηβ + A n φ 1 β 1 + (w n 1 ) 2 β 2 φ 1 1 )Φ 1 n 0. φ 1 φ 1 (12) If β 1 η ln(3/2), then 3/4 φ 1 1/2, φ 1 1/2. By Lemma 4, we have (wn 1 ) 2 γ 0 > 0 for η δ 2 and ξ < X 0. Let η β1 1 ln(3/2) and η δ 2. Due to is bounded, then we can find β 1 such that the coefficient of Φ 1 n in (12), for ξ X, satisfies the inequality R n ηβ + A n φ 1 β 1 + (w n 1 ) 2 β 2 φ φ 1 φ 1 In the region of η > min{δ 2, β1 1 ln(3/2)} this inequality is valid if β > 0 has been chosen sufficiently large. Obviously, β may be assumed independent of M 1, M 2. Then, according to (12), the function Φ 1 n can t attain its largest value inside Ω for ξ < X at any of the points τ = T, ξ = X, or η = U(τ, ξ). If Φ 1 n attains its largest value at τ = 0 or ξ = 0, then Φ 1 n = Φ n e βξ Φ n k 12, φ 1 E-ISSN: Issue 6, Volume 12, June 2013

8 where k 12 does not depend on n, since Φ n τ=0 and Φ n ξ=0 can be expressed through w 0, w 1 and their derivatives. If Φ 1 n attains its largest value at η = g(τ, ξ), then Φ 1 n/ η 0 at the point of maximum, and it follows from Lemma 5 that Φ 1 n 1 2 Φ1 n 1 or Φ 1 n 1 Φ n 1 e βξ 1 2 φ 1 2 M 1. by virtue of our assumption. Thus Φ 1 n max{ 1 2 M 1, k 12 } in Ω for ξ X, Φ n max{ 1 2 M 1, k 12 } max[e βξ φ 1 (β 1 η)]. Since φ 1 (β 1 η) 3/2, we have e βξ φ 1 (β 1 η) 2, if e βξ 4/3. Let us choose X 1 X 0 from the condition e βx 1 4/3. Then Φ n max{m 1, 2k 12 } for ξ X 1. Set M 1 = 2k 12. Then Φ n M 1 for ξ X 1, where X 1 depends on M 1 and M 2. Now, let us consider F n. By Lemma 6 L 0 n(f n ) + C n F n N 2 in Ω for ξ X 1. Let F n = F 1 nφ 1 (β 2 η)e β 3ξ and φ 1 (s) be the function defined above. We have L 0 n(f 1 n) + 2(w n 1 ) 2 β 2 φ 1 φ 1 F 1 nη + [C m ηβ 3 +A n φ 1 β 2 + (w n 1 ) 2 β 2 φ 1 1 ]F 1 e β3ξ n > N 2, φ 1 φ 1 φ 1 (13) If β 2 η ln(3/2), then 3/4 φ 1 1/2, φ 1 1/2, 1 φ 1 3/2. It follows from Lemma 4 that (w n 1 ) 2 γ 0 > 0 for η δ 2. Let η min{δ 2, β2 1 ln(3/2)}. For these values of η, the constant β 2 can be chosen so as to make the coefficient by Fn 1 in (13) satisfy the inequality φ 1 C n ηβ 3 + A n φ 1 β 2 + (w n 1 ) 2 β φ 1 φ 1 This inequality will hold in the region of η > min{δ 2, β2 1 ln(3/2)} if β 3 has been chosen sufficiently large. Clearly, β 3 depends on M 1 and M 2. Just as in the proof of Theorem 7, we find that F 1 n max{ 1 2 M 1, N 2, N 13 } in Ω for ξ X, where N 13 = max{f n } for τ = 0 and for ξ = 0; N 13 depends on M 1. We have F n max{ 1 2 M 2, N 2, N 13 } max[e β3ξ φ 1 (β 2 η)] max{m 2, 2N 2, 2N 13 }, provided that e β3ξ φ 1 (β 2 η) 2 and e β3ξ 4/3. Let us take M 2 = max{2n 2, 2N 13 } and define X 2 X 0 from the inequality e β 3X 2 4/3. Then F n M 2 for ξ X, where X = min{x 1, X 2 }. The fact that Φ n and F n are bounded implies that the derivatives of w n up to the second order are bounded uniformly in n, since wηη, n for η δ 2, can be estimated from equation (8). Theorem 9 The function w n (n )are uniformly convergent in Ω to a solution w n of problem (5)(6)(7) in Ω, where T is defined in Theorem 7 and X may be taken arbitrarily, or X is that of Theorem 8 and T is arbitrary. The function w is continuously differentiable in Ω and its derivative w ηη is continuous for η < U(τ, ξ). Proof: Let v n = w n w n 1. We obtain the following equation from (8) (w n 1 ) 2 v n ηη v n τ ηv n 1 ξ + [p x + 2 η (w n 1 ) 2 ]v n η +[(w n 1 ) 2 (w n 2 ) 2 ]w n 1 ηη +2 η [(w n 1 ) 2 (w n 2 ) 2 ]w n 1 η + ηη [(w n 1 ) 2 (w n 2 ) 3 ] = 0, and also the boundary condition: v n τ=0 = 0, v n ξ=0 = 0, v n η=u(τ,ξ) = 0, w n 1 vη n v 0 v n 1 + wη n 1 v n 1 + η (w n 1 + w n 2 )v n 1 = 0 for η = g. Consider the function v n 1 defined by vn = e ατ+βη v n 1, β < 0. We have (w n 1 ) 2 v n 1ηη v n 1τ ηv n 1ξ + [p x + 2 η (w n 1 ) 2 ]v n 1η +wηη n 1 (w n 1 + w n 2 )v1 n 1 +2 η wη n 1 (w n 1 + w n 2 )v1 n 1 + ηη [(w n 1 ) 2 +w n 1 w n 2 + (w n 2 ) 2 ]v n (w n 1 ) 2 βv n 1η + [(w n 1 ) 2 β 2 + p x β +2 η (w n 1 ) 2 β α]v n 1 = 0, (14) The boundary condition η = g(τ, ξ) w n 1 v1η n + βw n 1 v1 n v 0 v1 n 1 + wη n 1 v1 n η (w n 1 + w n 2 )v n 1 1 = 0. (15) E-ISSN: Issue 6, Volume 12, June 2013

9 Due to, ηη and η are bounded, the constant β < 0 should be choose such that in the boundary condition for v1 n at η = g(τ, ξ) the coefficients by vn 1 and vn 1 1 satisfy the inequality max w n 1 η v 0 + η (w n 1 + w n 2 ) q min w n 1 β, for q < 1. Having fixed β, let us choose α > 0 such that max w n 1 ηη (w n 1 + w n 2 ) +2 η w n 1 η (w n 1 + w n 2 ) + ηη [(w n 1 ) 2 +w n 1 w n 2 + (w n 2 ) 2 ] q(α max (w n 1 ) 2 β 2 +p x β+2 η (w n 1 ) 2 β ). Now, if v1 n attains its largest value at an interior point of Ω or on its boundary, it follow equations (14), (15) that max v n 1 q max v n 1 1, where means that the series v v vn 1 +, whose partial sums have the form w n e ατ βη, is majorized by a geometrical progression, and therefore, is uniformly convergent. The fact w n and its derivatives up to the second order are bounded implies that the first derivatives of w n are uniformly convergent as n. It follows from equation (8) that w ηη are also uniformly convergent as n for η < U(τ, ξ) δ 3, where δ 3 < min U(τ, ξ) max g(τ, ξ) and which is an arbitrary positive constant. 4 The existence of the solution of the system (8)-(9) and the main result Now, let us establish the existence of the solution w n (τ, ξ, η) for problem (8)(9). the way is the similar as [3]. Consider the operator L ε (w) ε(w ττ +w ξξ +w ηη )+a 1 w ττ +a 2 w ξξ +a 3 w ηη +[(w n 1 ) 2 ] ε w ηη w τ ηw ξ +[p x +2 η (w n 1 ) 2 ] ε w η +[ ηη (w n 1 ) 3 ] ε 2(a 1 + ε)w. Consider the following elliptic boundary value problem: L ε (w) = (f) ε in Q, (16) w n = (F ) ε on S. (17) where n is unit inward normal to S. The function f in (14) is defined in Q by f = L(w ) + a 1 w ττ + a 2 w ξξ + a 3 w ηη 2a 1 w, in Q\Ω 1, f = 0 in Ω, f coincides with an arbitrary smooth extension of this function on the rest of Q. The function F is given by F = v 0 + p x + g τ + gg ξ ηw n 1 on S 0, F = w on γ, n where γ is the intersection of S with the boundary of Q Ω 1 ; on the rest of S, the function F in (19) coincides with any smooth extension of F just defined on S 0 and γ. Clearly, owing to the properties of w, we may assume that f has bounded derivatives up to the fourth order in Q and is infinitely differentiable outside a δ-neighborhood of Ω; F has bounded derivatives up to the fourth order in a neighborhood of S 0 and is infinitely differentiable on the rest of S. The boundary value problem (16)-(17) has a uniqueness solution wε n in Q, one can see the fact in [3]. Let us show that the functions wε n and their derivatives up to the fourth order are bounded uniformly with respect to ε. Lemma 10 The solutions w n ε of problem (16)(17) in Q are bounded uniformly in ε. Proof: Set wε n = v ε ψ 1, where ψ 1 (τ) = 1 for τ 1, ψ 1 (τ) = 1 + b(1 + τ) 3 for 1 τ T + 2, choose suitable b > 0 such that ψττ 1 ψ 1 in Q. Let 6b(T + 3) < 1. Then v 3 satisfies the following equation in Q: (f) ε ψ 1 = ε( v ε) + a 1 v εττ + a 2 v εξξ + a 3 v εηη +[(w n 1 ) 2 ] ε v εηη v ετ ηw εξ +[p x + 2 η (w n 1 ) 2 ] ε v εη + 2(a 1 + ε) ψ1 τ ψ 1 v ετ +[(a 1 + ε) ψ1 τ ψ 1 ψ1 τ ψ 1 2(a 1 + ε)]v ε + [ ηη(w n 1 ) 3 ] ε ψ 1, (18) as well as the boundary conditions on S v ε n = (F ) ε ψ 1 for 2 τ T + 1, (19) E-ISSN: Issue 6, Volume 12, June 2013

10 v ε n + 1 ψ 1 ψ 1 n v ε = (F ) ε ψ 1 for τ T + 1. (20) Since ψ 1 n = τ ψ1 τ 0 for τ T + 1 on S, n the coefficient of v ε in the boundary condition (20) is non-positive. (The domain Q may be assumed convex for τ T + 1.) The coefficient of v ε in equation (18) is negative. Indeed, we have (a 1 + ε) + (a 1 + ε) ψ1 ττ ψ 1 0. since ψττ 1 /ψ 1 1, and ψτ 1 > 0 for τ > 1, while a 1 > 0 for τ < 1/2. Applying to the solution of problem (18)(19)(20) the a priori estimate established by Theorem 4 of [4], we find that v ε are bounded in Q by a constant independent of ε but depending only on the maximum moduli of the coefficients in equation (18), as well as on max (f) ε ψ 1, max (F ) ε ψ 1, min[(a 1 + ε) ψ1 ττ ψ 1 ψ1 τ ψ 1 2(a 1 + ε)]. Lemma 11 The solution w n ε of problem (16)(17) have their derivatives up to the fourth order in Q bounded by a constant independent of ε. Proof: First of all, we note that equation (16) is uniformly (with respect to ε) elliptic in Q for τ > T + δ + r 1, and for τ < 1/2 r 1, where r 1 is any positive constant. Therefore, according to the well-known estimates of Schauder type (see [3]), the m-th order derivatives of w n ε have their absolute values bounded by a constant independent of ε, for τ > T +δ +r 1 and for τ < 1/2 r 1, provided that w n 1 have bounded (m 1)-th order derivatives (with m 2) in the same region. Let P (ξ, η) be a point of σ δ such that ξ 2δ, and let A δ be the intersection of its δ neighborhood on the plane ξ, η with the domain G. Consider the cylinder B δ = [ 1 2 r 1, T + δ + r 1 ] A δ. Let us show that in this domain the functions w n ε have their derivatives up to the fourth order bounded by a constant independent of ε. We may assume that in B δ the coefficient a 1 depends merely on τ, whereas a 2 and a 3 depend only on ξ and η. In the domain A δ, let us introduce new coordinates ξ and η, so that the part of the boundary σ belonging to A δ would turn into a subset of the straight line η = 0, and the direction of the normal n to σ would coincide with that of the η -axis. In the new coordinates, again denoted by ξ, η, the boundary condition (17) takes the form w n ε / η = F ε. Let Y (τ, ξ, η) be a function defined on B δ which satisfies the condition Y η η=0= F ε. The function z w n ε Y satisfies the equation M(z) (ε + a 1 )z ττ z τ + a 11 z ξξ + 2a 12 z ξη +a 22 z ηη + b 1 z ξ + b 2 z η 2(ε + a 1 )z = f ε, (21) in B δ and also the condition z η = 0 on S, where a 11 α a 12 α 1 α 2 + a 22 α 2 2 λ 0 (α α 2 2), the constant λ 0 being positive and independent of ε. In order to estimate the first order derivatives of z with respect to ξ and η, consider the function Λ 1 = ρ 2 δ(ξ, η)[z 2 ξ + z 2 η] + c 1 z 2 + c 2 η, c 2>0 Here the constant c 1 is assumed sufficiently large and will be chosen later; ρ δ (ξ, η) = 1 in A δ/2, and ρ δ (ξ, η) = 0 in a small neighborhood of the boundary of A δ that does not belong to σ; ρ δη = 0 on σ. It is easy to see that Λ 1 / η = c 2 > 0 on S and, therefore, Λ 1 cannot attain its largest value on S. If the maximum of Λ 1 is reached on the boundary of B δ at a point where ρ δ = 0, then Λ 1 max(c 1 z 2 + c 2 η) c 3, where c 3 is a constant independent of ε. It is easy to verify that for large enough c 1 we have M(Λ 1 ) Λ 1 c 4 in B δ, provided that c 4 is sufficiently large. Therefore, if Λ 1 takes its largest value inside B δ, then Λ 1 c 4. As shown above, for τ = T + δ + r 1 and τ = 1/2 r 1, the function Λ 1 is uniformly bounded in ε. Thus, Λ 1 in B δ is bounded by a constant independent of ε and, therefore, z ξ, z η are bounded in B δ1,δ 1 < δ. Let us rewrite equation (21) in the firm M(z) Γ(z)+M 1 (z) = f ε, Γ(z) (ε+a 1 )z ττ z τ. The coefficients of the operator M 1 may be assumed independent of τ. Hence, it is easy to see that Γ satisfies the following equation and the boundary condition: M(Γ) Γ(Γ) + M 1 (Γ) = Γ(f ε ) in B δ, E-ISSN: Issue 6, Volume 12, June 2013

11 Γ η η=0 = 0 on S. (22) In B δ1 consider the function Λ 2 ρ 2 δ 1 [z 2 ξξ + z 2 ξη + Γ 2 (z)] + c 5 (z 2 ξ + z 2 η) + c 6 η. Making use of equations (21) and (22), we easily find that M(Λ 2 ) Λ 2 c 7 in B δ1, Λ 2 η = c 6 > 0 on S. provided that c 5 > 0 is sufficiently large. Hence we see that Λ 2 is uniformly bounded in B δ1, and Γ(z), z ξξ,z ξη are uniformly bounded in B δ2, δ 2 < δ 1. It follows from (21) that z ηη is also bounded uniformly in ε. If we consider an equation of the form (a 1 + ε)z ττ z τ = Γ for z τ and use the fact that Γ is bounded in B δ2 and z τ is bounded for τ = 1/2 r 1 and τ = T + δ + r 1, we easily find that z τ is bounded in B δ2 uniformly with respect to ε. Since Γ(z) is bounded in B δ2 and satisfies equation (22), as well as the boundary condition Γ η η=0 = 0 on S, we can consider functions similar to Λ 1 and Λ 2 for Γ in B δ2 (as we have done for z) and thus obtain uniform estimates in B δ3, δ 3 < δ 2, for the derivatives Γ ξ, Γ η, Γ ξξ, Γ ξη, Γ ηη, Γ τ. Differentiating equation (22) in τ, we obtain the following equation for Γ τ : (a 1 + ε)γ τττ (1 a 1)Γ ττ + M 1 (Γ τ ) = (Γ(f ε )) τ, as well as the boundary condition Γ τη η=0 = 0 on S. By assumption, a 1 (τ) is small in B δ. Therefore, the equation for Γ τ is similar to (22). Thus, for the derivatives of Γ of the form Γ τξ, Γ τη, Γ τξξ, Γ τηξ, (a 1 + ε)γ τττ (1 a 1)Γ ττ, Γ τηη, Γ ττ. in B δ4, δ 4 < δ 3, we obtain estimates uniform in ε, just as we have done for z. Similar arguments applied to Γ ττ allow us to show that the derivatives Γ ττξ, Γ ττη, Γ ττξξ, Γ ττηξ, (a 1 + ε)γ ττττ (1 a 1)Γ τττ, Γ ττηη, Γ τττ. are bounded in B δ5, δ 5 < δ 4, uniformly with respect to ε. These estimates show that in B δ5 the third and the fourth order derivatives of z involving more than one differentiation in τ are bounded uniformly in ε, while the first order derivatives of Γ(Γ) in ξ and η satisfy the Lipschitz condition in ξ, η with constants independent of ε, τ. From the Schauder estimates for an elliptic equation of the form M 1 (Γ) = Γ(Γ) + Γ(f ε ). it follows that the derivatives of Γ in ξ and η up to the third order are bounded and satisfy the Hölder condition in B δ6, δ 6 < δ 5, uniformly with respect to ε and τ. Schauder estimates for the solution z of equation (21) written in the form M 1 (z) = Γ(z) + f ε, allow us to claim that z has its derivatives in ξ and η up to the forth order bounded in B δ7, δ 7 < δ 6, uniformly with respect to ε and τ. Thus, we have obtained estimate for the derivatives of w n ε in τ, ξ, η up to the fourth order in a neighborhood of the entire boundary S, except for a neighborhood of S 0 and a neighborhood w of the intersection S {ξ = 0} which is a subset of the plane {η = η 1 }. In equation (16) and the boundary condition (17), let us pass to another unknown function W given by w = W e φ 2(η), φ 2 (η) = αη η 1 η η 1, α = const > 0. For W we obtain the following equations W η αw = (F ) ε for η = 0, W η αw = (F ) ε for η = η 1. In order to estimate the first order derivatives of w n ε in Q, consider the following function in Q r1 { 1 2 r 1 < τ < T + δ + r 1 }: X 1 = W 2 ξ + W 2 τ + W η (W η 2Y ) + k(η), where Y = (αw + (F ) ε )κ 1 (η), we define function κ 1 (η) such that κ 1 (η) = 1 for η < δ; κ 1 (η) = 1 for η η 1 < δ; κ 1 (η) = 0 for 2δ < η < η 1 2δ, k(η) is a positive function to chosen later. Clearly, W/ η Y = 0 on the parts of the boundary S belonging to the planes η = 0 or η = η 1. We have X 1 η η=0= 2W ξ W ξη + 2W τ W τη 2W η Y η + k (0) = 2α[W 2 ξ +W 2 τ ] 2Y Y η +2W ξ (F ) εξ +2W τ (F ) ετ + E-ISSN: Issue 6, Volume 12, June 2013

12 +k (0) > 0, provided that k (0) is positive and sufficiently large. Likewise, taking k(η) in X 1 such that k (η 1 ) is negative and sufficiently large in absolute value, we find that X 1 η η=η 1 < 0. By the methods already used in the proof of Lemma 5 we obtain where L 0ε (X 1 ) + c 8 X 1 c 9, (23) L 0ε (W ) L ε (W ) + 2[(ε + a 3 ) + (w n 1 ) 2 ε]φ 2η W η +{((w n 1 ) 2 ε + ε + a 3 )[φ 2ηη + (φ 2η ) 2 ] +(p x + 2 η (w n 1 ) 2 ) ε φ 2η }W, the constant c 8 and c 9 do not depend on ε. In Q r1, consider the function X 1 = X 1 e βt, β = const > 0. For sufficiently large β, the coefficient of X1 in (23) is less than -1. It follows from (23) that if X1 takes its largest value inside Q r1, then X1 is bounded by a constant independent of ε. Neither for η = 0 nor for η = η 1 can X1 attain its largest value. It follows from the above estimates that on the remaining part of the boundary of Q r1 the function X1 is bounded uniformly in ε. Likewise, we can estimate the second and the third order derivatives of wε n by considering the functions X 2 = W 2 ττ + W 2 ξξ + W 2 τξ + W ηξ (W ηξ 2Y ξ ) +W ητ (W ητ 2Y τ ) + g 2 1(η)W 2 ηη + k(η), X 3 = (X 3 ) + g 2 1(η)[W 2 ηηη + W 2 ηηξ + W 2 ηητ ] +W ηξξ (W ηξξ 2Y ξξ ) + W ηττ (W ηττ 2Y ττ ) +W ηξτ (W ηξτ 2Yξτ ) + k(η), where (X 3 ) stands for the sum of third order derivatives of W in ξ and τ { 0 for η < δ g 1 (η) = 2 or η > η 1 δ 2, 1 for η 1 δ > η > δ. The required estimates for X 2 and X 3 can be obtained by the method used above in relation to X 1 ; in order to establish the inequality of type (23) for X 2 and X 3, we can use the fact that in (16) the coefficient of W ηη is positive for η < δ and η 1 η < δ, as we have done in the proof of Lemma 6. While estimating the fourth order derivatives of W, the following observations are useful. Consider the function X 4 = (X 4 ) + g 2 1(η)(X 4 ) + W ηξξξ (W ηξξξ 2Y ξξξ ) +W ητττ (W ητττ 2Y τττ ) + W ηξξτ (W ηξξτ 2Y ξξτ ) +W ηττξ (W ηττξ 2Y ττξ ) + k(η), where (X 4 ) is the sum of squared fourth order derivatives of W except those involving a differentiation in η, and (X 4 ) is the sum of squared fourth order derivatives of W involving more than one differentiation in η. The expression for X 4 contains third order derivatives of Y and, therefore, of (F ) ε. The operator L 0ε (X 4 ) can be estimated through the expressions L 0ε (Y τττ ), L 0ε (Y ξξξ ), L 0ε (Y ττξ ), L 0ε (Y τξξ ). which contain fifth order derivatives of (F ) ε. By construction, F is infinitely differentiable outside the δ- neighborhood of S 0 and has its fourth order derivatives bounded in ε on S. In the intersection of the domain Q with the δ-neighborhood of S 0, the operator L 0ε involves second order derivatives in ξ and τ with the coefficient ε, namely, ε 2 τ 2, ε 2 ξ 2. Since F has its fourth order derivatives bounded in ε, the fifth order derivatives of its regularization (F ) ε can be written as O(ε 1 ). Therefore, the operator L 0ε applied to the third order derivatives of (F ) ε results in a quantity uniformly bounded in ε. For the rest, the proof of the estimate for X 4 literally follows the case of X 1, X 2, and X 3. Thus, we finally see that the derivatives of w n ε up to the fourth order are bounded uniformly in ε. Theorem 12 The solutions w n ε of problem (16)(17) in Q converge, as ε 0, to the function w n which is a solution of problem (8)(9) in Ω and has its derivatives up to the fourth order bounded in Ω. Proof: By Lemma 11, the derivatives of w n ε up to the fourth order are uniformly bounded in ε. Therefore, there is a subsequence w n εk such that wn εk, together with their derivatives up to the third order, are uniformly convergent to w n in Q as ε k 0. The limit function w n (τ, ξ, η) satisfies equation (8) in Ω, as well as the boundary condition (10). Let us show that the condition in (9) hold for w n. To this end, we prove E-ISSN: Issue 6, Volume 12, June 2013

13 that w n = w in Q \ Ω 1. Set z = w n w. By construction, we have a 1 z ττ + a 2 z ξξ + a 3 z ηη + (w ) 2 z ηη z τ ηz ξ +[p x + 2 η (w ) 2 ]w η 2a 1 z = 0, in Q \ Ω 1, and z/ n = 0 on the part of the boundary of Q \ Ω 1 that belongs to S. In Q \ Ω 1, consider the function z defined by z = z ψ 1 (τ), where ψ 1 (τ) is the function constructed in the proof of Lemma 10. For z we obtain an equation in Q \ Ω 1 with the coefficient of z being strictly negative in the closure of Q \ Ω 1. Let E(τ, ξ, η) be a smooth function in Q such that E/ n < 0 on S, and E > 1. Set z 1 = z (E + C), where C is a positive constant. It is easy to see that in the equation for z 1 the coefficient of z 1 is negative if C is sufficiently large. The boundary condition on S for z 1 will have the form z 1 n α 1z 1 = 0, where α 1 = E n > 0. Clearly, z 1 cannot attain its largest value on S, for at the point of maximum of z 1 on S we must have z 1 z 1 n α 1(z 1 ) 2 < 0, which is incompatible with the boundary condition on S. The largest value of z 1 cannot be attained inside Q \ Ω 1, for at the point of its maximum we must have z 1τ = 0, z 1ξ = 0, z 1η = 0, z 1 z 1ηη 0, z 1 z 1ξξ 0, and z 1 z 1ττ 0, which is in contradiction with the equation obtain for z 1 at that point. In a similar way, it can be shown that the maximum of z 1 can be attained neither for τ = 0 nor for ξ = 0 on the boundary of Q \ Ω 1. It follows that z 1 0 in Q \ Ω 1 and, therefore, w n w in Q \ Ω 1. Hence, we see that w n (0, ξ, η) = w 0, w n (τ, 0, η) = w 1. Let us show that w n = 0 on the surface η = U(τ, ξ). It follows from the above results that w n = 0 for τ = 0 and η = U(0, ξ), as well as for ξ = 0 and η = U(τ, 0). Since w n 1 = 0 on the surface η = U(τ, ξ), the equation w n τ + ηw n ξ p x w n η = 0, holds for w n on that surface. As indicated above, the vectors (1, η, p x ) belong to planes tangential to the surface η = U(τ, ξ) and form a vector field on that surface. Integral curves of that field, being extended for smaller values of τ, will cross the border of the surface either at ξ = 0 or at τ = 0, where w n = 0. Since w n is constant on these integral curves, w n = 0 on the entire surface η = U(τ, ξ). Note that the function w n constructed above has its third order derivatives in Ω satisfying the Lipschitz condition. Theorem 13 Assume that p(t, x), v 0 (t, x), u 0 (x, y), u 1 (t, y), w 0 (ξ, η), w 1 (τ, η), (y), g(t, x) are sufficiently smooth and satisfy the compatibility conditions which amount to the existence of the function w mentioned earlier. Then there is one and only one solution of problem (1)-(3) in the domain D, with X being arbitrary and T depending on the data of problem (1)-(9), or T being arbitrary and X depending on the data. This solution has the following properties: u > 0 for y > 0, u y > 0 for y 0; the derivatives u t, u x, u y, u yy, v y are continuous and bounded in D; moreover, the ratios are bounded in D. u yy, u yyyu y u 2 yy, u y Proof: Let w be the solution of problem (5)-(7) constructed in the proof of Theorem 12. Let u be defined by the condition w = u y, or y = u 0 u 3 y ds w(t, x, s). (24) Since w(t, x, s) > 0 for s < U(t, x), and w = 0 for s = U(t, x), we have u U(t, x) as y, and 0 < u < U(t, x) for 0 < y <, u(t, x, 0) = 0. The condition u(0, x, y) = u 0 and u(t, 0, y) = u 1 are also valid, since w 0 = u 0y and w 1 = u 1y. The function defined by (24) possesses the derivatives u y = w, u yy = w η w, u yyy = w ηη u 2 y + w η u yy, y = η w, yy = ηη w 2 + η w η w. The derivatives u t and u x are given by Set u w t (t, x, s) u t = w 0 w 2 (t, x, s) ds, u w x (t, x, s) u x = w 0 w 2 (t, x, s) ds. v = u t uu x p x + (u y ) y u y. (25) Let us show that u and v defined by (24) and (25) satisfy system (1). Differentiating the relation u y = w, we find that there exist the derivatives u yx = w ξ + u x w η, u yt = w τ + u t w η. E-ISSN: Issue 6, Volume 12, June 2013

14 Therefore, v admits the derivatives in y. Differentiating (25) in y, we obtain v y u y + vu yy = u ty u y u x uu xy + yy u y + 2 y u yy + u yyy. (26) The function w satisfies equation (5). Replacing in (5) the derivatives of w by their expressions in terms of derivatives of u, we find that u 2 u y u yyy u 2 yy u yy y u 3 u yt +u t u(u yx u xu yy )+ y u y u y +p x u yy u y + yy u y + y u yy = 0. (27) It follows from (26) and (27) that u x + v y = 0. (28) Let us show that v(t, x, 0) = v 0 (t, x). It follows from (7) that v 0 = ( ww η p x + η w 2 g τ gg ξ ) η=g. w From (25) we find that v y=0 = ( u t uu x p x + (u y ) y u y ) y=0 = ( ww η p x + η w 2 g τ gg ξ ) η=g = v 0. w Thus we have proved the existence of a solution for problem (1)-(3) in the class of smooth functions. Its uniqueness is able to been established by a similar way as Theorem of [3], we omit the details here. we find that 0 = (w n 1 ) 2 Φ nηη Φ nτ ηφ nξ + An Φ nξ + 2Bn Φ n 2(w n 1 ) 2 {(W n τη) 2 + (W n ξη )2 + (W n ηη) 2 } +[2((w n 1 ) 2 ) τ W n ηηw n τ + 2((w n 1 ) 2 ) ξ W n ηηw n ξ +2((w n 1 ) 2 ) η W n ηηw n η ] + [ 2W n ξ W n η + 2A n η (W n η ) 2 +2A n ξ W n η W n ξ + 2An τ W n η W n τ +2W (B n η W n η + B n ξ W n ξ + Bn τ W n τ )] B n (k 1 η + k 0 ) A n k 1 + 2W n τ [ ηη e αη (w n 1 ) 3 ] τ +2W n ξ [ ηηe αη (w n 1 ) 3 ] ξ + 2W n η [ ηη e αη (w n 1 ) 3 ] η. (29) for the last three terms, we have 2W n τ [ ηη e αη (w n 1 ) 3 ] τ + 2W n ξ [ ηη e αη (w n 1 ) 3 ] ξ +2W n η [ ηη e αη (w n 1 ) 3 ] η = 2αW n τ ηη e αη (w n 1 ) 3 +2W n τ ηη e αη [(w n 1 ) 3 ] τ + 2αW n ξ ηη e αη (w n 1 ) 3 +2W n ξ ηη e αη [(w n 1 ) 3 ] ξ + 2W n η ηηη e αη (w n 1 ) 3 +2αW n η ηη e αη (w n 1 ) 3 + 2W n η ηη e αη [(w n 1 ) 3 ] η. according to ab a2 +b 2 2,, ηη, ηηη are bounded, we have 2αW n τ ηη e αη (w n 1 ) 3 + 2αW n ξ ηη e αη (w n 1 ) 3 +(2α ηη + 2 ηηη )W n η e αη (w n 1 ) 3 h 1 [(W n τ ) 2 + (w n 1 ) 6 ] + h 2 [(W n ξ ) 2 + (w n 1 ) 6 ] +h 3 [(W n η ) 2 + (w n 1 ) 6 ] h[(w n τ ) 2 + (W n ξ ) 2 + (W n τ ) 2 ] + h(w n 1 ) 6, where h 1, h 2, h 3 are taken large enough, h = h 1 + h 2 + h 3 2W n τ ηη e αη [(w n 1 ) 3 ] τ + 2W n ξ ηη e αη [(w n 1 ) 3 ] ξ +2W n η [ ηη e αη (w n 1 ) 3 ] η h[(w n τ ) 2 + (W n ξ ) 2 + (W n τ ) 2 ] + h(w n 1 ) 6 + 2W n τ ηη e αη [(w n 1 ) 3 ] τ +2W n ξ ηη e αη [(w n 1 ) 3 ] ξ 5 Appendix At the last of the paper, we give the details of the proof to the inequality (11). As in the section 2, we have Φ n = Φ n (W n τ ) 2 + (W n ξ ) 2 + (W n η ) 2 + k 0 + k 1 η. Applying the operator 2W n τ to the equation τ + 2W n ξ ξ + 2W n η L 0 n(w n ) + B n W n = 0, η +2W n η ηη e αη [(w n 1 ) 3 ] η h[(w n τ ) 2 + (W n ξ ) 2 + (W n τ ) 2 ] + h(w n 1 ) 6 +h 4 [(W n τ ) 2 + (W n ξ ) 2 + (W n η ) 2 ] + ηηe αη h 4 {[((w n 1 ) 3 ) τ ] 2 + [((w n 1 ) 3 ) ξ ] 2 +[((w n 1 ) 3 ) η ] 2 }. Denote by I 1 the terms in the first square brackets in (29), we obtain the following estimate from above I 1 R 1 [(W n τ ) 2 + (W n ξ ) 2 + (W n η ) 2 ] + R 1 {[((w n 1 ) 2 ) τ ] 2 + [((w n 1 ) 2 ) ξ ] 2 +[((w n 1 ) 2 ) η ] 2 }(W n ηη) 2, E-ISSN: Issue 6, Volume 12, June 2013

15 where R 1 is a constant. It is well known that (see for instance [19]) any non-negative function q(x) defined on interval < x < + and having bounded second derivatives on that interval satisfies the inequality (q x ) 2 2{max q xx }q(x). The function (w n 1 ) 3 or (w n 1 ) 2 can be extended to the entire real axis with respect to any of its independent variable, so that its extension is a non-negative bounded function whose second derivative has its absolute value less than or equal to the maximum modulus of the second derivative of (w n 1 ) 3 or (w n 1 ) 2. Therefore ηη e α(η g) {[((w n 1 ) 3 ) τ ] 2 + [((w n 1 ) 3 ) ξ ] 2 h 4 +[((w n 1 ) 3 ) η ] 2 } ηη (w n 1 ) 3, 2 R 1 {[((w n 1 ) 2 ) τ ] 2 + [((w n 1 ) 2 ) ξ ] 2 +[((w n 1 ) 2 ) η ] 2 }(W n ηη) 2 (w n 1 ) 2 (W n ηη) 2, where R 1, h 4 is chosen sufficiently large. The constant R 1 depends on the second derivative of the functions (w n 1 ) 2 and h 4 depends on the second derivative of the functions (w n 1 ) 3. Denote by I 2 the terms enclosed by the second square brackets in (29). By virtue of the inequality 2ab a 2 + b 2, these terms can be estimated from above by the expression R 2 Φ n + k 8, where the constant R 2 depends on the first order derivative of the functions w n 1, k 8 is independent of n. Therefore, in the region η δ 2, we have L 0 n(φ n )+R 3 Φ n +k 9 0 or L 0 n(φ n )+R n Φ n 0, where the constant k 9 is independent of n, and the function R n depends on the first and the second derivatives of w n 1. In order to estimate L 0 n(φ n ) in Ω for η δ 2, we should also calculate L 0 n(2wη n H n ) L 0 n(2w n η H n ) = 2H n L 0 n(w n η ) + 2W n η L 0 n(h n ) +4(w n 1 ) 2 W n ηηh n η = 2H n { (w n 1 ) 2 ηw n ηη + W n ξ A n η W n η B n η W n B n W n η [ ηη (w n 1 ) 3 e α(η g) ] η } +2Wη n [L 0 n( v 0 ) + L0 n( ) αχ(η)bn W n αχ(η) ηη (w n 1 ) 3 e α(η g) + αw n L 0 (χ) p x +2α(w n 1 ) 2 W n η χ + L 0 n( g τ gg ξ +4(W n 1 ) 2 W n ηηh n η. ηw n 1 )] According to Lemma 3 and Lemma 4, (w n 1 ) 2 γ 0 > 0 for η δ 2. There, the terms I 1 in (11) together with 2H n (w n 1 ) 2 ηw n ηη, can be estimated with the help of the inequality 2ab a 2 /h + hb 2 as follows: I 1 +2H n (w n 1 ) 2 ηw n ηη 1 2 γ 0(W n ηη) 2 +R 4 Φ n +k 10, where the constant R 4 does not depend on n, It follows from (11) and L 0 n(2w n η H n ) that L 0 n(φ n ) + R 5 Φ n + R 6 0 for η δ 2, where R 5 and R 6 are constant that depend neither on w n 1 nor on its derivatives up to the second order. Since Φ n 1, we have R 6 Φ n R 6, Therefore L 0 n(φ n ) + R n Φ n 0. Acknowledgements: The research was supported by the NSF of Fujian Province of China (grant No. 2012J01011) and supported by SF of Xiamen University of Technology, China. References: [1] I. Anderson and H. Toften, Numerical solutions of the laminar boundary layer equations for power-law fluids, Non-Newton, Fluid Mech., 32(1989), pp [2] H. Schliching, K. Gersten and E. Krause, Boundary Layer Theory, Springer, [3] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, CHAP- MAN and HALL, 1999, pp [4] Shumin Wen, The theory of microflow boundary layer and its application, Metallurgical Industry Press, Bejing, [5] Long Li, Huashui Zhan, The study of microfluid boundary layer theory, WSEAS Transaction on Mathematics, 8(12)(2009), pp [6] J. W. Zhang and J. N. Zhao, On the Global Existence and Uniqueness of Solutions to Nonstationary Boundary Layer System, Science in China, Ser. A, 36(2006), pp [7] E. Weinan, Blow up of solutions of the unsteadly Pradtl s equations, Comm. pure Appl. Math., 1(1997), pp [8] Z. Xin and L. Zhang, On the global existence of solutions to the Pradtl s system, Adv. in Math., 191(2004), pp [9] Y. Amirat, O. Bodart, G. A. Chechkin, A. L. Piatnitski, Boundary homogenization in domains with randomly oscillating boundary, Stoch. Proc. Appl., 121(2011), pp E-ISSN: Issue 6, Volume 12, June 2013