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Asymptotic behavior of the Navier–Stokes flow in a general 2D domain
Applied Mathematics Letters 18 (2005) 1170–1176 www.elsevier.com/locate/aml
Asymptotic behavior of the Navier–Stokes flow in a general 2D domain Yinnian Hea,∗, Chunshan Zhaob a Department of Mathematics, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR China b Department of Mathematics, The University of Iowa, Iowa city, IA 52242, USA
Received 1 March 2004; received in revised form 20 December 2004; accepted 1 February 2005
Abstract In this work, we consider the convergence of the Navier–Stokes flow to the Euler flow at small viscosity in a general 2D domain and derive some explicit asymptotic formulas for the convergence at small viscosity under some smoothness assumptions on the solution of the Euler flow. © 2005 Elsevier Ltd. All rights reserved. MSC: 35Q30; 35L70; 76D05 Keywords: Navier–Stokes flow; Euler flow; Boundary layer flow; Asymptotic behavior
1. Introduction The asymptotic behavior of wall bounded flow at small viscosity (or large Reynolds number) is one of the most intriguing and largely open problems both in fluid mechanics and in mathematical analysis. On one hand, the nonlinear term tends to mix the modes and propagate energy among them; on the other hand, it is known that important phenomena occur in a thin region near the boundary called the boundary layer, in particular, the generation of vortexes which propagate in the fluid and drive the flow. Many articles were devoted to this problem: for example, Temam and Wang’s [1–3] and Weinan’s [4] for the 2D or 3D Navier–Stokes flow. Temam and Wang [1,2] derived an explicit asymptotic formula for the convergence of the Navier–Stokes flow to the Euler flow at small viscosity. Wang [3] gave a necessary ∗ Corresponding author. Tel.: +86 292665242; fax: +86 293237910.
E-mail addresses: [email protected] (Y. He), [email protected] (C. Zhao). 0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2005.02.005
Y. He, C. Zhao / Applied Mathematics Letters 18 (2005) 1170–1176
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and sufficient condition for the convergence and some applications in several special 3D flows. Later, Weinan gave a review of the current status of the boundary layer theory and the zero-viscosity limit of the 3D Navier–Stokes flow [4]. For the asymptotic behavior of the solutions to the linearized Navier–Stokes flow and the Oseen flow, the reader can refer to [5–7,2]. In this work, we consider the convergence of the Navier–Stokes flow to the Euler flow in a general 2D domain at small viscosity (or large Reynolds number). Under some smoothness assumptions on the solution of the Euler flow as in [1,3], we derive better explicit asymptotic formulas for the convergence at small viscosity. Next, we consider the following Navier–Stokes flow: div u = 0 in Ω × (0, T ]; u t − u + (u · ∇)u + ∇ p = f, (1) u(x, 0) = u 0 in Ω ; ( p, 1) = p(x, t)dx = 0, u = 0 on ∂ Ω × (0, T ]; Ω
2 2 where u t = ∂u ∂t , Ω is an open bounded domain in R with C a smooth boundary, ∂ Ω , u = u(x, t) = (u 1 (x, t), u 2 (x, t)) denotes the velocity of the fluid, p = p(x, t) represents the pressure of the fluid, f = f (x, t) is the prescribed external force, u 0 = u 0 (x) is the initial velocity and > 0 is the viscosity of the fluid. The last condition ( p, 1) = 0 in (1) is introduced for uniqueness of pressure p and clearly (1) is the Euler flow if = 0. Here, we consider the limit of the flow (1) as → 0+ . The corresponding Euler flow is as follows: 0 u t + (u 0 · ∇)u 0 + ∇ p 0 = f, div u 0 = 0 in Ω × (0, T ]; (2) 0 0 u · n = 0 on ∂ Ω × (0, T ]; u (x, 0) = u 0 (x) in Ω ; ( p 0 , 1) = 0,
where n is the unit outer normal of ∂ Ω . In order to consider the convergence of flow (1) to flow (2), we need to introduce the following boundary layer flow which is an extension of the boundary flow provided by Temam and Wang [7,1]: θt − θ + (u · ∇)θ + (θ · ∇)u 0 + ∇η = 0, div θ = 0 in Ω × (0, T ]; (3) u (0) = 0 in Ω ; (η , 1) = 0. θ |∂ Ω = −u 0 on ∂ Ω × (0, T ]; For the discussion of the boundary layer flow, the reader can refer to [7,1]. First, let us introduce some functional spaces as follows: 1 2 2 2 2 2 Y = L (Ω ) , M = L 0 (Ω ) = q ∈ L (Ω ); qdx = 0 . X = H0 (Ω ) , Ω
Spaces L 2 (Ω )m = L 2 (Ω ) × · · · × L 2 (Ω ) (m times), m = 2, 4, are endowed with the L 2 -scalar product and L 2 -norm denoted by (·, ·) and · 0 , respectively. The space X is equipped with its usual scalar product and norm as follows: ((u, v)) = (∇u, ∇v), ∇u 0 = ((u, u))1/2 . The subspaces V and H of X and Y are defined by V = {v ∈ X ; div v = 0 in Ω } and H = {v ∈ Y ; div v = 0 in Ω and v · n = 0 on ∂ Ω }. In this work, we provide the following main results on the asymptotic behaviors of the flow (1).
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Theorem 1.1. Suppose that Ω , u 0 ∈ H and f ∈ L ∞ (0, T ; Y ) are sufficiently smooth. If the solution (u 0 , p 0 ) of the Euler flow (2) satisfies the following smoothness assumptions:
u 0 (t) 2 + ∇u 0 (t) L ∞ ≤ κ,
∀0 ≤ t ≤ T,
(4)
then the following holds: sup u(t) − u (t) 0 ≤ κ, 0
0≤t≤T
t 0 1/2 sup ( p − p )ds ≤ κ ,
0≤t≤T
0
0
T
1/2
u
− u 0 21 ds
≤ κ 1/2 ,
(5) (6)
0
where · i denotes the norm on the Sobolev space H i (Ω )m for i = 1, 2 and m = 1, 2 and κ is used to denote a general constant depending on the data (u 0 , f, Ω , T ) which may vary from line to line. Remark. The short time existence of the smooth solutions of the Euler flow (2) can be found in Temam [8,9] provided the data (Ω , u 0 , f ) is sufficiently smooth, and the existence and uniqueness of the global solutions for the two-dimensional case can be found in Kato [10]. Here the estimates (5) and (6) are significant improvements of the results given in [1]. 2. Proof of Theorem 1.1 We will give a proof of Theorem 1.1 in this section. First, we introduce the continuous bilinear form d(·, ·) on X × M and trilinear form b(·, ·, ·) on X × X × X : d(v, q) = −(v, ∇q) = (q, div v) ∀v ∈ X, q ∈ M, b(u, v, w) = ((u · ∇)v, w) ∀u, v, w ∈ X. It is well known [11,12] that
q 0 ≤ C sup v∈X
d(v, q)
∇v 0
∀q ∈ M.
(7)
Here and from now on C > 0 is used to denote a general positive constant depending only on Ω , and it may vary from line to line. It is easy to verify that the trilinear form b(·, ·, ·) satisfies the following important properties (see [11,12]): b(u, v, w) = −b(u, w, v) |b(u, v, w)| ≤
∀u ∈ V, v, w ∈ H 1 (Ω )2 ,
1/2 1/2 1/2 1/2 C u 0 ∇u 0 ∇v 0 w 0 ∇w 0
(8) ∀u, v, w ∈ X.
(9)
To prove our main results, we need the following two lemmas: Lemma 2.1. Under the assumptions of Theorem 1.1, (w, r ) = (u − u 0 − θ , p − p 0 − η ) satisfies t
w(s) 21 ds ≤ κ 2 ∀0 ≤ t ≤ T, (10)
w(t) 20 + 0
t 2 r (s)ds ≤ κ 0
0
∀0 ≤ t ≤ T.
(11)
Y. He, C. Zhao / Applied Mathematics Letters 18 (2005) 1170–1176
1173
Proof. From (1) and (8) it follows that d ∀0 ≤ t ≤ T.
u(t) 20 + 2 ∇u(t) 20 = 2( f (t), u(t)) dt Integrating (12) from 0 to t and using Hölder’s inequality, we obtain t 2
u(t) 0 + 2
∇u(s) 20 ds ≤ u 0 20 + 2T sup f (s) 0 sup u(s) 0 , 0
0≤t≤T
0≤s≤T
for all 0 ≤ t ≤ T . Moreover, from the above inequality it follows that
1/2
sup u(t) 0 ≤ u 0 0 + 2T sup f (s) 0 0≤t≤T
1/2
sup u(s) 0
0≤t≤T
(12)
,
0≤s≤T
which yields sup u(t) 0 ≤ 2 u 0 0 + 2T sup f (s) 0 . 0≤t≤T
0≤t≤T
Combining these inequalities, we obtain t 2
∇u(s) 20 ds ≤ 2 u 0 20 + 4T (T + 1) sup f (s) 20 ,
u(t) 0 + 2 0
(13)
0≤s≤T
for all 0 ≤ t ≤ T . Now we derive from (1), (2) and (3), wt − w + (u · ∇)w + (w · ∇)u 0 = u 0 − ∇r in Ω × (0, T ); w, u 0 , u ∈ V, r ∈ M, w(0) = 0.
(14)
So (14) leads to the following variational formula: (wt , v) + ((w, v)) + b(u, w, v) + b(w, u 0 , v) = −((u 0 , v)) + d(v, r )
∀v ∈ X,
(15)
with w(0) = 0. We notice that the regularity theory of the Navier–Stokes flow in 2D smooth bounded domain guarantees u ∈ L ∞ (Ω × [0, T ]) if u 0 , Ω and f are sufficiently smooth with r = p − p 0 − η in L 2 (Ω × (0, T )) and ∇u 0 in L ∞ (Ω × [0, T ]) by the assumption (4). Then we obtain w(t) ∈ V for any t ∈ [0, T ] by the regularity theory for parabolic equations (see [13, Theorem 6.1 on page 100]). So taking v = w in (15), noting that w ∈ V or d(w, r ) = 0 and using (8) yields d
w(t) 20 + 2 ∇w(t) 20 ≤ 2 w(t) 20 ∇u 0 (t) L ∞ dt + 2 u 0 (t) 0 ∇w(t) 0 ≤ (1 + 2 ∇u 0 (t) L ∞ ) w(t) 20 + 2 u 0 (t) 22 .
(16)
Noticing (4), we obtain (10) by integrating (16) from 0 to t and using the well-known Grownwall Lemma. Integrating (15) with respect to t, using (7), (9) and applying the Poincaré inequality to v ∈ X , we get t t d v, r (s)ds 0 r (s)ds ≤ C sup ≤ C w(t) 0
∇v 0 0 v∈X 0 t t + C
∇w(s) 0 ds + C
u 0 (s) 2 ds 0
0
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t
1/2
1/2
1/2
1/2
+C
u(s) 0 ∇u(s) 0 w(s) 0 ∇w(s) 0 ds 0 t +
w(s) 0 ∇u 0 (s) L ∞ ds ∀0 ≤ t ≤ T.
(17)
0
Notice that t 1/2 1/2 1/2 1/2
u(s) 0 ∇u(s) 0 w(s) 0 ∇w(s) 0 ds 0
≤ 0
t
u(s) 20 ds
14
t
0
∇u(s) 20 ds
14
t
0
w(s) 20 ds
14 0
t
∇w(s) 20 ds
14
Finally, we arrive at (11) by using (4), (10) and (13). The proof of this lemma is complete.
.
Next, we derive an explicit boundary layer behavior as → 0+ . Lemma 2.2. Under the assumptions of Theorem 1.1 we have t 2
θ (t) 0 +
θ (s) 21 ds ≤ κ 2 , 0 2
t η (s)ds ≤ κ. 0
(18) (19)
0
Proof. From (3) and (8) it follows that for any 0 ≤ t ≤ T , d
θ (t) 20 + 2 ∇θ (t) 20 ≤ 2 θ (t) 20 ∇u 0 (t) L ∞ dt + 2 θ (t) · ∇θ (t) · ndsx + 2 η (t)θ (t) · ndsx ∂Ω ∂Ω 2 0 0 ∞ u (t) · ∇θ (t) · ndsx , = 2 θ (t) 0 ∇u (t) L + 2 ∂Ω
(20)
where we have used the boundary conditions in (2) and (3). Next, we give an estimate of the last term in (20). By the trace theorem for the divergence free functions and the definition of H −1 (Ω )4 , we have 0 2 u (t) · ∇θ (t) · ndsx ≤ 2 u 0 (t) H 3/2 (∂ Ω )2 ∇θ (t) · n H −3/2 (∂ Ω )2 ∂Ω
≤ C u 0 (t) 2 ∇θ (t) H −1 (Ω )4 ,
and
∇θ (t) H −1 (Ω )4 =
sup φ∈H01 (Ω )4
(∇θ (t), φ) −(θ (t), div φ) = sup ≤ C θ (t) 0 .
∇φ 0
∇φ 1 4 0 φ∈H (Ω ) 0
Substituting the above estimates into (20) yields d
θ (t) 20 + 2 ∇θ (t) 20 ≤ θ (t) 20 (1 + 2 ∇u 0 (t) L ∞ ) + C 2 u 0 (t) 22 dt
(21)
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for any 0 ≤ t ≤ T . Integrating (21) from 0 to t and using (4), we obtain (18). The variational formula of (3) is as follows: (θt , v) + ((θ , v)) + b(u, θ , v) + b(θ , u 0 , v) = d(v, η )
∀v ∈ X.
(22)
Finally, by integrating (22) with respect to t, using (7), (9) and applying the Poincaré inequality to v ∈ X , we obtain t t t d v, 0 η (s)ds η (s)ds ≤ C sup ≤ C θ (t) + C
∇θ (s) 0 ds 0
∇v 0 0 0 v∈X 0 t 1/2 1/2 1/2 1/2 +C
u(s) 0 ∇u(s) 0 θ (s) 0 ∇θ (s) 0 ds 0 t +C
θ (s) 0 ∇u 0 (s) L ∞ ds, (23) 0
for any 0 ≤ t ≤ T . Notice that t 1/2 1/2 1/2 1/2
u(s) 0 ∇u(s) 0 θ (s) 0 ∇θ (s) 0 ds 0
≤ 0
t
u(s) 20 ds
14 0
t
∇u(s) 20 ds
14 0
t
θ
(s) 20 ds
14 0
t
∇θ
(s) 20 ds
(19) is obtained from (23) by using (4), (13) and (18). The proof of this lemma is complete.
14
.
Proof of Theorem 1.1. Since u − u 0 = w + θ and p − p 0 = r + η , Theorem 1.1 follows easily from Lemmas 2.1 and 2.2 in this section. Thus Theorem 1.1 is proved.
Acknowledgments Part of this work was completed while the first author visited the Institute for Scientific Computing and Applied Mathematics, Indiana University, USA, in 2002. He is grateful to Professor R. Temam for his valuable suggestions, academic influence and hospitality. The first author was subsidized by the China Scholarship Council, the Research Fund of Indiana University and the NSF of China 10371095. References [1] R. Temam, X. Wang, On the behavior of Navier–Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997) 807–828. [2] R. Temam, X. Wang, Boundary layers associated with incompressible Navier–Stokes equations: The non-characteristic boundary case, J. Differential Equations 179 (2002) 647–686. [3] X. Wang, A Kato type theorem on zero viscosity limit of Navier–Stokes flows, Indiana Univ. Math. J. 50 (50) (2001) 223–241 (special issue). [4] E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000) 207–218. [5] R. Temam, X. Wang, Asymptotic analysis of the linearized Navier–Stokes equations in a channel, Differential Integral Equations 8 (1995) 1591–1618.
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[6] R. Temam, X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana Univ. Math. J. 45 (1996) 863–916. [7] R. Temam, X. Wang, Asymptotic analysis of the linearized Navier–Stokes equations in a general 2D domain, Asymptot. Anal. 14 (1997) 293–321. [8] R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975) 32–43. [9] M. Marion, R. Temam, Navier–Stokes equations, theory and approximations, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. VI, North-Holland, Amsterdam, 1998, pp. 503–688. [10] T. Kato, On Classical solutions of the two-dimensional non-stationary Euler equations, Arch. Ration. Mech. Anal. 25 (1967) 188–200. [11] V. Girault, P.A. Raviart, Finite Element Approximation of the Navier–Stokes Equations, Springer-Verlag, New York, NY, 1981. [12] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, third ed., North-Holland, Amsterdam, 1983. [13] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1998.
Asymptotic behavior of the Navier–Stokes flow in a general 2D domain Yinnian Hea,∗, Chunshan Zhaob a Department of Mathematics, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, PR China b Department of Mathematics, The University of Iowa, Iowa city, IA 52242, USA
Received 1 March 2004; received in revised form 20 December 2004; accepted 1 February 2005
Abstract In this work, we consider the convergence of the Navier–Stokes flow to the Euler flow at small viscosity in a general 2D domain and derive some explicit asymptotic formulas for the convergence at small viscosity under some smoothness assumptions on the solution of the Euler flow. © 2005 Elsevier Ltd. All rights reserved. MSC: 35Q30; 35L70; 76D05 Keywords: Navier–Stokes flow; Euler flow; Boundary layer flow; Asymptotic behavior
1. Introduction The asymptotic behavior of wall bounded flow at small viscosity (or large Reynolds number) is one of the most intriguing and largely open problems both in fluid mechanics and in mathematical analysis. On one hand, the nonlinear term tends to mix the modes and propagate energy among them; on the other hand, it is known that important phenomena occur in a thin region near the boundary called the boundary layer, in particular, the generation of vortexes which propagate in the fluid and drive the flow. Many articles were devoted to this problem: for example, Temam and Wang’s [1–3] and Weinan’s [4] for the 2D or 3D Navier–Stokes flow. Temam and Wang [1,2] derived an explicit asymptotic formula for the convergence of the Navier–Stokes flow to the Euler flow at small viscosity. Wang [3] gave a necessary ∗ Corresponding author. Tel.: +86 292665242; fax: +86 293237910.
E-mail addresses: [email protected] (Y. He), [email protected] (C. Zhao). 0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2005.02.005
Y. He, C. Zhao / Applied Mathematics Letters 18 (2005) 1170–1176
1171
and sufficient condition for the convergence and some applications in several special 3D flows. Later, Weinan gave a review of the current status of the boundary layer theory and the zero-viscosity limit of the 3D Navier–Stokes flow [4]. For the asymptotic behavior of the solutions to the linearized Navier–Stokes flow and the Oseen flow, the reader can refer to [5–7,2]. In this work, we consider the convergence of the Navier–Stokes flow to the Euler flow in a general 2D domain at small viscosity (or large Reynolds number). Under some smoothness assumptions on the solution of the Euler flow as in [1,3], we derive better explicit asymptotic formulas for the convergence at small viscosity. Next, we consider the following Navier–Stokes flow: div u = 0 in Ω × (0, T ]; u t − u + (u · ∇)u + ∇ p = f, (1) u(x, 0) = u 0 in Ω ; ( p, 1) = p(x, t)dx = 0, u = 0 on ∂ Ω × (0, T ]; Ω
2 2 where u t = ∂u ∂t , Ω is an open bounded domain in R with C a smooth boundary, ∂ Ω , u = u(x, t) = (u 1 (x, t), u 2 (x, t)) denotes the velocity of the fluid, p = p(x, t) represents the pressure of the fluid, f = f (x, t) is the prescribed external force, u 0 = u 0 (x) is the initial velocity and > 0 is the viscosity of the fluid. The last condition ( p, 1) = 0 in (1) is introduced for uniqueness of pressure p and clearly (1) is the Euler flow if = 0. Here, we consider the limit of the flow (1) as → 0+ . The corresponding Euler flow is as follows: 0 u t + (u 0 · ∇)u 0 + ∇ p 0 = f, div u 0 = 0 in Ω × (0, T ]; (2) 0 0 u · n = 0 on ∂ Ω × (0, T ]; u (x, 0) = u 0 (x) in Ω ; ( p 0 , 1) = 0,
where n is the unit outer normal of ∂ Ω . In order to consider the convergence of flow (1) to flow (2), we need to introduce the following boundary layer flow which is an extension of the boundary flow provided by Temam and Wang [7,1]: θt − θ + (u · ∇)θ + (θ · ∇)u 0 + ∇η = 0, div θ = 0 in Ω × (0, T ]; (3) u (0) = 0 in Ω ; (η , 1) = 0. θ |∂ Ω = −u 0 on ∂ Ω × (0, T ]; For the discussion of the boundary layer flow, the reader can refer to [7,1]. First, let us introduce some functional spaces as follows: 1 2 2 2 2 2 Y = L (Ω ) , M = L 0 (Ω ) = q ∈ L (Ω ); qdx = 0 . X = H0 (Ω ) , Ω
Spaces L 2 (Ω )m = L 2 (Ω ) × · · · × L 2 (Ω ) (m times), m = 2, 4, are endowed with the L 2 -scalar product and L 2 -norm denoted by (·, ·) and · 0 , respectively. The space X is equipped with its usual scalar product and norm as follows: ((u, v)) = (∇u, ∇v), ∇u 0 = ((u, u))1/2 . The subspaces V and H of X and Y are defined by V = {v ∈ X ; div v = 0 in Ω } and H = {v ∈ Y ; div v = 0 in Ω and v · n = 0 on ∂ Ω }. In this work, we provide the following main results on the asymptotic behaviors of the flow (1).
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Theorem 1.1. Suppose that Ω , u 0 ∈ H and f ∈ L ∞ (0, T ; Y ) are sufficiently smooth. If the solution (u 0 , p 0 ) of the Euler flow (2) satisfies the following smoothness assumptions:
u 0 (t) 2 + ∇u 0 (t) L ∞ ≤ κ,
∀0 ≤ t ≤ T,
(4)
then the following holds: sup u(t) − u (t) 0 ≤ κ, 0
0≤t≤T
t 0 1/2 sup ( p − p )ds ≤ κ ,
0≤t≤T
0
0
T
1/2
u
− u 0 21 ds
≤ κ 1/2 ,
(5) (6)
0
where · i denotes the norm on the Sobolev space H i (Ω )m for i = 1, 2 and m = 1, 2 and κ is used to denote a general constant depending on the data (u 0 , f, Ω , T ) which may vary from line to line. Remark. The short time existence of the smooth solutions of the Euler flow (2) can be found in Temam [8,9] provided the data (Ω , u 0 , f ) is sufficiently smooth, and the existence and uniqueness of the global solutions for the two-dimensional case can be found in Kato [10]. Here the estimates (5) and (6) are significant improvements of the results given in [1]. 2. Proof of Theorem 1.1 We will give a proof of Theorem 1.1 in this section. First, we introduce the continuous bilinear form d(·, ·) on X × M and trilinear form b(·, ·, ·) on X × X × X : d(v, q) = −(v, ∇q) = (q, div v) ∀v ∈ X, q ∈ M, b(u, v, w) = ((u · ∇)v, w) ∀u, v, w ∈ X. It is well known [11,12] that
q 0 ≤ C sup v∈X
d(v, q)
∇v 0
∀q ∈ M.
(7)
Here and from now on C > 0 is used to denote a general positive constant depending only on Ω , and it may vary from line to line. It is easy to verify that the trilinear form b(·, ·, ·) satisfies the following important properties (see [11,12]): b(u, v, w) = −b(u, w, v) |b(u, v, w)| ≤
∀u ∈ V, v, w ∈ H 1 (Ω )2 ,
1/2 1/2 1/2 1/2 C u 0 ∇u 0 ∇v 0 w 0 ∇w 0
(8) ∀u, v, w ∈ X.
(9)
To prove our main results, we need the following two lemmas: Lemma 2.1. Under the assumptions of Theorem 1.1, (w, r ) = (u − u 0 − θ , p − p 0 − η ) satisfies t
w(s) 21 ds ≤ κ 2 ∀0 ≤ t ≤ T, (10)
w(t) 20 + 0
t 2 r (s)ds ≤ κ 0
0
∀0 ≤ t ≤ T.
(11)
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Proof. From (1) and (8) it follows that d ∀0 ≤ t ≤ T.
u(t) 20 + 2 ∇u(t) 20 = 2( f (t), u(t)) dt Integrating (12) from 0 to t and using Hölder’s inequality, we obtain t 2
u(t) 0 + 2
∇u(s) 20 ds ≤ u 0 20 + 2T sup f (s) 0 sup u(s) 0 , 0
0≤t≤T
0≤s≤T
for all 0 ≤ t ≤ T . Moreover, from the above inequality it follows that
1/2
sup u(t) 0 ≤ u 0 0 + 2T sup f (s) 0 0≤t≤T
1/2
sup u(s) 0
0≤t≤T
(12)
,
0≤s≤T
which yields sup u(t) 0 ≤ 2 u 0 0 + 2T sup f (s) 0 . 0≤t≤T
0≤t≤T
Combining these inequalities, we obtain t 2
∇u(s) 20 ds ≤ 2 u 0 20 + 4T (T + 1) sup f (s) 20 ,
u(t) 0 + 2 0
(13)
0≤s≤T
for all 0 ≤ t ≤ T . Now we derive from (1), (2) and (3), wt − w + (u · ∇)w + (w · ∇)u 0 = u 0 − ∇r in Ω × (0, T ); w, u 0 , u ∈ V, r ∈ M, w(0) = 0.
(14)
So (14) leads to the following variational formula: (wt , v) + ((w, v)) + b(u, w, v) + b(w, u 0 , v) = −((u 0 , v)) + d(v, r )
∀v ∈ X,
(15)
with w(0) = 0. We notice that the regularity theory of the Navier–Stokes flow in 2D smooth bounded domain guarantees u ∈ L ∞ (Ω × [0, T ]) if u 0 , Ω and f are sufficiently smooth with r = p − p 0 − η in L 2 (Ω × (0, T )) and ∇u 0 in L ∞ (Ω × [0, T ]) by the assumption (4). Then we obtain w(t) ∈ V for any t ∈ [0, T ] by the regularity theory for parabolic equations (see [13, Theorem 6.1 on page 100]). So taking v = w in (15), noting that w ∈ V or d(w, r ) = 0 and using (8) yields d
w(t) 20 + 2 ∇w(t) 20 ≤ 2 w(t) 20 ∇u 0 (t) L ∞ dt + 2 u 0 (t) 0 ∇w(t) 0 ≤ (1 + 2 ∇u 0 (t) L ∞ ) w(t) 20 + 2 u 0 (t) 22 .
(16)
Noticing (4), we obtain (10) by integrating (16) from 0 to t and using the well-known Grownwall Lemma. Integrating (15) with respect to t, using (7), (9) and applying the Poincaré inequality to v ∈ X , we get t t d v, r (s)ds 0 r (s)ds ≤ C sup ≤ C w(t) 0
∇v 0 0 v∈X 0 t t + C
∇w(s) 0 ds + C
u 0 (s) 2 ds 0
0
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t
1/2
1/2
1/2
1/2
+C
u(s) 0 ∇u(s) 0 w(s) 0 ∇w(s) 0 ds 0 t +
w(s) 0 ∇u 0 (s) L ∞ ds ∀0 ≤ t ≤ T.
(17)
0
Notice that t 1/2 1/2 1/2 1/2
u(s) 0 ∇u(s) 0 w(s) 0 ∇w(s) 0 ds 0
≤ 0
t
u(s) 20 ds
14
t
0
∇u(s) 20 ds
14
t
0
w(s) 20 ds
14 0
t
∇w(s) 20 ds
14
Finally, we arrive at (11) by using (4), (10) and (13). The proof of this lemma is complete.
.
Next, we derive an explicit boundary layer behavior as → 0+ . Lemma 2.2. Under the assumptions of Theorem 1.1 we have t 2
θ (t) 0 +
θ (s) 21 ds ≤ κ 2 , 0 2
t η (s)ds ≤ κ. 0
(18) (19)
0
Proof. From (3) and (8) it follows that for any 0 ≤ t ≤ T , d
θ (t) 20 + 2 ∇θ (t) 20 ≤ 2 θ (t) 20 ∇u 0 (t) L ∞ dt + 2 θ (t) · ∇θ (t) · ndsx + 2 η (t)θ (t) · ndsx ∂Ω ∂Ω 2 0 0 ∞ u (t) · ∇θ (t) · ndsx , = 2 θ (t) 0 ∇u (t) L + 2 ∂Ω
(20)
where we have used the boundary conditions in (2) and (3). Next, we give an estimate of the last term in (20). By the trace theorem for the divergence free functions and the definition of H −1 (Ω )4 , we have 0 2 u (t) · ∇θ (t) · ndsx ≤ 2 u 0 (t) H 3/2 (∂ Ω )2 ∇θ (t) · n H −3/2 (∂ Ω )2 ∂Ω
≤ C u 0 (t) 2 ∇θ (t) H −1 (Ω )4 ,
and
∇θ (t) H −1 (Ω )4 =
sup φ∈H01 (Ω )4
(∇θ (t), φ) −(θ (t), div φ) = sup ≤ C θ (t) 0 .
∇φ 0
∇φ 1 4 0 φ∈H (Ω ) 0
Substituting the above estimates into (20) yields d
θ (t) 20 + 2 ∇θ (t) 20 ≤ θ (t) 20 (1 + 2 ∇u 0 (t) L ∞ ) + C 2 u 0 (t) 22 dt
(21)
Y. He, C. Zhao / Applied Mathematics Letters 18 (2005) 1170–1176
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for any 0 ≤ t ≤ T . Integrating (21) from 0 to t and using (4), we obtain (18). The variational formula of (3) is as follows: (θt , v) + ((θ , v)) + b(u, θ , v) + b(θ , u 0 , v) = d(v, η )
∀v ∈ X.
(22)
Finally, by integrating (22) with respect to t, using (7), (9) and applying the Poincaré inequality to v ∈ X , we obtain t t t d v, 0 η (s)ds η (s)ds ≤ C sup ≤ C θ (t) + C
∇θ (s) 0 ds 0
∇v 0 0 0 v∈X 0 t 1/2 1/2 1/2 1/2 +C
u(s) 0 ∇u(s) 0 θ (s) 0 ∇θ (s) 0 ds 0 t +C
θ (s) 0 ∇u 0 (s) L ∞ ds, (23) 0
for any 0 ≤ t ≤ T . Notice that t 1/2 1/2 1/2 1/2
u(s) 0 ∇u(s) 0 θ (s) 0 ∇θ (s) 0 ds 0
≤ 0
t
u(s) 20 ds
14 0
t
∇u(s) 20 ds
14 0
t
θ
(s) 20 ds
14 0
t
∇θ
(s) 20 ds
(19) is obtained from (23) by using (4), (13) and (18). The proof of this lemma is complete.
14
.
Proof of Theorem 1.1. Since u − u 0 = w + θ and p − p 0 = r + η , Theorem 1.1 follows easily from Lemmas 2.1 and 2.2 in this section. Thus Theorem 1.1 is proved.
Acknowledgments Part of this work was completed while the first author visited the Institute for Scientific Computing and Applied Mathematics, Indiana University, USA, in 2002. He is grateful to Professor R. Temam for his valuable suggestions, academic influence and hospitality. The first author was subsidized by the China Scholarship Council, the Research Fund of Indiana University and the NSF of China 10371095. References [1] R. Temam, X. Wang, On the behavior of Navier–Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997) 807–828. [2] R. Temam, X. Wang, Boundary layers associated with incompressible Navier–Stokes equations: The non-characteristic boundary case, J. Differential Equations 179 (2002) 647–686. [3] X. Wang, A Kato type theorem on zero viscosity limit of Navier–Stokes flows, Indiana Univ. Math. J. 50 (50) (2001) 223–241 (special issue). [4] E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier–Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000) 207–218. [5] R. Temam, X. Wang, Asymptotic analysis of the linearized Navier–Stokes equations in a channel, Differential Integral Equations 8 (1995) 1591–1618.
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[6] R. Temam, X. Wang, Asymptotic analysis of Oseen type equations in a channel at small viscosity, Indiana Univ. Math. J. 45 (1996) 863–916. [7] R. Temam, X. Wang, Asymptotic analysis of the linearized Navier–Stokes equations in a general 2D domain, Asymptot. Anal. 14 (1997) 293–321. [8] R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975) 32–43. [9] M. Marion, R. Temam, Navier–Stokes equations, theory and approximations, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. VI, North-Holland, Amsterdam, 1998, pp. 503–688. [10] T. Kato, On Classical solutions of the two-dimensional non-stationary Euler equations, Arch. Ration. Mech. Anal. 25 (1967) 188–200. [11] V. Girault, P.A. Raviart, Finite Element Approximation of the Navier–Stokes Equations, Springer-Verlag, New York, NY, 1981. [12] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, third ed., North-Holland, Amsterdam, 1983. [13] G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1998.