Boundary conditions
The zeroth Euler flow, denoted as \( u_0^e \), is defined under the no-slip boundary condition at \( y = 0 \), which necessitates that \( u_e + u_0^p(x, 0) = u_b \) and \( v_e^1(x, 0) + v_p^0(x, 0) = 0 \) From the zeroth order of the expansion, we find that \( u_1^p(x, 0) = -u_1^e(x, 0) \) and \( v_p^1(x, 0) = 0 \) are established for the \( \sqrt{\epsilon} \)-order layers Additionally, we assume that as \( y \) approaches infinity, \( \lim_{y \to \infty} u_0^p(x, y) = 0 \) and \( \lim_{y \to \infty} u_1^p(x, y) = 0 \).
In boundary layers, the normal velocities \( v_p^0(x, y) \) and \( v_p^1(x, y) \) are derived from the respective functions \( u_p^0(x, y) \) and \( u_p^1(x, y) \) by applying the divergence-free condition It is important to note that as \( y \) approaches infinity, the limit of \( v_p^j(x, y) \) does not equal zero, necessitating the introduction of cut-off functions to effectively localize \( v_p^j \).
In this section, we examine the boundary conditions at x = 0 and L, noting that the Prandtl layers are governed by parabolic-type equations, which necessitate only initial conditions at x = 0: u₀ₚ(0, y) = ¯u₀(y) and u₁ₚ(0, y) = ¯u₁(y) Additionally, we establish boundary values for the Euler profiles at both x = 0 and L: u₁ₑ(0, Y) = u₁ᵦ(Y), v₁ₑ(0, Y) = Vᵦ₀(Y), and vₑ₁(L, Y) = Vᵦₗ(Y) Compatibility conditions are also required at the domain corners, specifically Vᵦ₀(0) = v₀ₚ(0, 0) and Vᵦₗ(0) = vₚ₀(L, 0).
Finally, we impose the following boundary conditions for the remainder solution [u ε , v ε ]:
[u ε , v ε ] y=0 = 0 (no-slip), [u ε , v ε ] x=0 = 0 (Dirichlet), p ε −2εu ε x = 0, u ε y +εv x ε = 0 at x=L(Neumann or stress-free) (1.13)
When evaluating boundary conditions for [u ε , v ε ] at x=L, it is essential to consider the potential formation of boundary layers near this boundary To mitigate this issue, the Neumann stress-free condition is the most suitable option to apply.
Main result and discussions
We are ready to state our main result:
Theorem 1.1 states that for a smooth Euler flow represented by u₀ in (Y), and given smooth data u₁, V₀, and Vₗ, which exhibit exponential decay at infinity, we consider the case where Y is defined as √εy Additionally, we assume that the difference between Vₗ(Y) and V₀(Y) is bounded by a small constant L, while uₑ remains a positive constant.
There exists a positive number L, determined solely by the given data, such that the boundary layer expansions hold for γ in the range (0, 1/4) Specifically, the solutions [U ε, V ε, P ε] defined in the boundary layer expansions are the unique solutions to the Navier-Stokes equations The remainder solutions [u ε, v ε] satisfy a bounded condition involving L² and L∞ norms, ensuring that the combined terms remain less than or equal to a constant C₀, which is dependent only on the initial data Here, ∇ε represents the gradient with respect to ε, and k·kL p indicates the standard L p norm over the specified domain.
As a direct consequence of our primary theorem, we derive the inviscid limit of steady Navier-Stokes flows, utilizing specified data up to the square root of viscosity order.
Corollary 1.2 Under the same assumption as made in Theorem 1.1, there are exact solutions [U, V]to the original Navier-Stokes equations (1.1)on Ω = [0, L]×R+, withLbeing as in Theorem 1.1, so that sup
ε γ 2 + 1 2 asε→0, for given Euler flowu 0 e , the constructed Euler flows[u 1 e , v e 1 ]and Prandtl layers[u 0 p ,√ εv 0 p ].
In particular, we have the convergence (U, V) → (u 0 e ,0) in the usual L p norm, with a rate of convergence of order ε 1/2p , 1≤p 0, which are closely associated with boundary layers near a moving plate This scenario is commonly observed in wake flows of moving bodies, moving plane jet flows, and shear layers between parallel flows Additionally, it aligns with the established engineering principle that injecting moving fluids at a surface effectively prevents boundary layer separation.
The mathematical study of Prandtl boundary layers and the inviscid limit problem presents significant challenges due to boundary characteristics, such as the condition v = 0 at y = 0, and the inherent instability of boundary layers This article justifies the Prandtl boundary layer theory for steady flows while addressing key issues, including the careful construction of Euler and Prandtl solutions and the derivation of sufficient estimates Complications arise from the need to truncate actual layers to fit our functional framework and the absence of a priori estimates for linearized Prandtl equations The construction of approximate solutions is detailed in Section 2.
After constructing the approximate solutions, it is essential to derive stability estimates for the remaining solutions Given the limited regularity of the Prandtl layers [u 1 p , v 1 p ], we will focus on the linearization around the following approximate solutions: \( u_s(x, y) := u_0 e(\sqrt{\epsilon} y) + u_0 p(x, y) + \sqrt{\epsilon} u_1 e(x, \sqrt{\epsilon} y) \) and \( v_s(x, y) := v_0 p(x, y) + v_1 e(x, \sqrt{\epsilon}) \).
A straightforward calculation provides the equations for the remainder solutions [u ε, v ε, p ε] in (1.9), defined as follows: 1 The equation for u ε involves terms like u s u ε x, v s u ε y, and the remainder R 1 (u ε, v ε), while 2 The equation for v ε includes similar terms with v s v ε y and the remainder R 2 (u ε, v ε) The term ∆ ε is defined as ∂ y 2 + ε∂ x 2, where [u s, v s] represents the leading approximate solutions The standard energy estimate indicates control over k∇εu ε kL 2 and √ εk∇εv ε kL 2, but the analysis remains open due to the significant convective term, such as R u sy u ε v ε, which highlights a common challenge in the stability theory of boundary layers.
The essential aspect of the proof is to establish a bound on \( k∇εv ε kL 2 \) at order one, rather than the order of \( √ε \) derived from the energy estimate To achieve this, it is vital to analyze the vorticity equation.
The equation presented, u s ∆ ǫ v ε + v s ∆ ǫ u ε − u ε ∆ ǫ v s + v ε ∆ ǫ u s = ∆ ǫ ω ε + R 1y − εR 2x, introduces a new multiplier u v ε s Under the assumption stated in (1.14) and the condition that u b > 0, along with the Maximum Principle for the Prandtl equations, it is ensured that u s remains bounded away from zero This analysis is conducted without delving into boundary terms, focusing instead on the integral R.
The term ∆εv ε vanishes, indicating that the dominant factor in the vorticity estimate is related to convection, expressed as −us∆ǫv ε + v ε∆ǫus In the context of boundary layer analysis, this simplifies to −us∂y²v ε + usy v ε A crucial point to note is the positivity of the second-order operator involved.
−∂yy+u syy u s Indeed, a direct calculation yields
Z u syy us v 2 which gives the positivity estimate:
The desired bound onv ε y , and in fact,∇εv ε is derived from this positivity estimate and the weighted estimates from the vorticity equation Precisely,
(1.17) in which the last inequality used the estimate (3.8) on us.
In addition, the Dirichlet boundary condition at x = 0 and the stress-free boundary condition at x = L as imposed in (1.13) are carefully designed to ensure boundary contributions at x = 0 and x=L are controllable.
In Sections 4 and 5, we focus on deriving the L ∞ estimate for the remainder solution [u ε, v ε] to finalize our nonlinear analysis A key challenge is addressing the regularity of solutions to the elliptic problem in domains with corners, particularly the justification of integrability for all terms involved in integration by parts due to the limited regularity near the corners Notably, when u b = 0, our analysis becomes less applicable because of the zero points in the profile solutions u s, which prevents the use of the function v u ε s as a multiplier Consequently, our positivity estimate is compromised in this classical limiting case.
The construction of profiles, essential for establishing error estimates and closing nonlinear iterations, is a delicate process due to the regularity requirement of \( v_1 p_{xx} \) in the remainder \( R^2(u_\epsilon, v_\epsilon) \) To ensure sufficient regularity for the first-order Euler correction \([u_1^e, v_1^e, p_1^e]\), it is necessary to introduce an artificial boundary layer at \( y = 0 \) Additionally, the positivity estimate (1.16) is crucial for constructing both \( v_1^e \) and \( v_1^p \), as discussed in Sections 2.2 and 2.4.
There is a lack of literature addressing the validity of Prandtl boundary layer theory specifically for steady Navier-Stokes flows While significant contributions exist for unsteady flows, particularly within the framework of analyticity and initial vorticity conditions, replicating the same analytical approach for unsteady flows is challenging due to the inherent instability of boundary layers.
Notation Throughout the paper, we shall use hyi = p
In this article, we denote the usual Lp norms, for p ≥ 1, using the notation k·kLp or occasionally k·kp, with integration over the domain Ω = [0, L] × R+ We also specify the Lp norms for integrations over [0, L] and R+ as k·kL p (0,L) and k·kL p (R+), respectively The universal constant C(us, vs) is defined based on the given Euler flow u0e and boundary conditions, and we may simplify this to C or use the notation "≲" in our estimates Uniform estimates refer to those that remain consistent regardless of the smallness of ε and L; specifically, the smallness of L is determined solely by the provided data, while ε can be chosen arbitrarily small once L and the data are established, with the condition that ε is much less than L.
2 Construction of the approximate solutions
To derive approximate solutions, we substitute the Ansatz into the scaled Navier-Stokes equations and align the terms by order of ε to formulate the equations for the profiles We define the approximated velocity and pressure fields as follows: \( u_{app}(x, y) = u_0 e^{(\sqrt{\epsilon}y)} + u_{0p}(x, y) + \sqrt{\epsilon}u_{1e}(x, \sqrt{\epsilon}y) + \sqrt{\epsilon}u_{1p}(x, y) \), \( v_{app}(x, y) = v_{p0}(x, y) + v_{e1}(x, \sqrt{\epsilon}y) + \sqrt{\epsilon}v_{p1}(x, y) \), and \( p_{app}(x, y) = \sqrt{\epsilon}p_{1e}(x, \sqrt{\epsilon}y) + \sqrt{\epsilon}p_{1p}(x, y) + \epsilon p_{2p}(x, y) \).
We then calculate the error caused by the approximation:
R app u := [u app ∂ x +v app ∂ y ]u app +∂ x p app −∆ ε u app (2.2a)
R app v := [u app ∂ x +v app ∂ y ]v app +1 ε∂ y p app −∆ ε v app , (2.2b) or explicitly,
Zeroth-order Prandtl layers
The (leading) zeroth order terms on the right-hand side of (2.2a) consist of
R u,0 :={u 0 e +u 0 p }{u 0 e +u 0 p }x+{v 0 p +v e 1 }{u 0 e +u 0 p }y− {u 0 e +u 0 p }yy in which we note that{u 0 e }x = 0 Since the Euler flows are evaluated at (x, z) = (x,√ εy), we may write
{v p 0 +v e 1 }∂ y u 0 e =√ ε{v 0 p +v e 1 }u 0 ez and, with u e =u 0 e (0), u 0 e u 0 px +v e 1 u 0 py =u e u 0 px +v e 1 (x,0)u 0 py +√ εu ez (√ εy)yu 0 px +√ εv ez 1 (√ εy)yu 0 py +E 0 in which E 0 satisfies
In particular,E 0 is in the high order inε, as to be proved rigorously in the next section; see (2.40).
To leading order, this yields the nonlinear Prandtl problem for u 0 p :
{u e +u 0 p }u 0 px +{v 0 p +v e 1 (x,0)}u 0 py =u 0 pyy , v 0 p (x, y) :Z ∞ y u 0 px dy, u 0 p (x,0) =u b −ue, v p 0 (x,0) +v e 1 (x,0) = 0, u 0 p (0, y) = ¯u0(y).
Having constructed the Prandtl layer [u 0 p , v 0 p ], the zeroth order term R u,0 is reduced to
R u,0 =√ ε{v 0 p +v e 1 }u 0 ez +√ εu ez (√ εy)yu 0 px +√ εv ez 1 (√ εy)yu 0 py −εu 0 ezz +E 0 , (2.5) which will be put into the next order inε.
Lemma 2.2 establishes that for an arbitrary smooth boundary data \( u_0^p(0, y) = \bar{u}_0(y) \) at \( x = 0 \), and under the condition that \( \min_y \{u_e + \bar{u}_0(y)\} > 0 \), there exists a positive number \( L \) such that the problem (2.4) has a unique smooth solution \( u_0^p(x, y) \) in the domain \([0, L] \times \mathbb{R}^+\) Additionally, for all non-negative integers \( n \) and \( k \), there is a constant \( C_0(n, k, \bar{u}_0) \) ensuring the uniform bound given by the inequality \( \sup_{x \in [0, L]} \langle y \rangle^{n/2} \| \partial_x^k u_0^p \|_{L^2(\mathbb{R}^+)} + \langle y \rangle^{n/2} \| \partial_x^k \partial_y u_0^p \|_{L^2(0, L; L^2(\mathbb{R}^+))} \leq C_0(n, k, \bar{u}_0) \).
1 +|y| 2 Here, the constant C(n, k,u¯ 0 ) depends on n, k, and the hyi n -weighted
H 2k (R+) norm of the boundary value u¯ 0 (y).
Corollary 2.3 Let u 0 p be the Prandtl layer constructed as in Lemma 2.2 Then, there holds sup x ∈ [0,L]khyi n/2 ∂ x k ∂ y j [u 0 p , v 0 p ]kL 2 (R + )≤C 0 (n, k, j,u¯ 0 ), (2.7) for arbitrary n, k, j.
Proof Indeed, the proof follows directly from Lemma 2.2 and a use of equations (2.4) for the Prandtl layer to bound∂ y 2 u 0 p by those of lower-order derivative terms.
Proof of Lemma 2.2 Letu e =u 0 e (0) Following Oleinik [12], we use the von Mises transformation: η:Z y 0
The function \( w \) satisfies the equation \( w_x = \{ w w_\eta \} \) on the domain \([0, L] \times \mathbb{R}^+\) According to the Maximum Principle applied to the equation for \( w^2 \), we establish that \( w \geq \min_y \{ u_b, u_e + u_0 p | x=0 \} \geq c_0 > 0 \), indicating that this is a non-degenerate parabolic equation Given that \( w \) does not vanish on the boundary, we redefine \( w \) as \( w = w u_e - [u_b - u_e] e^{-\eta} \) Consequently, \( w \) approaches zero as \( y \) approaches both 0 and infinity The equation then simplifies to \( w_x = [w w_\eta]_\eta - [u_b - u_e][w_e - \eta]_\eta - F_\eta \), where \( F(\eta) = [u_b - u_e][u_e + [u_e - u_b] e^{-\eta}] e^{-\eta} \).
Clearly, hηi n F(ã) ∈W k,p (R+), for arbitraryn, k ≥0 and p∈ [1,∞] We shall solve this equation via the standard contraction mapping.
First, let us derive a priori weighted estimates We introduce the following weighted iterative norm:
By multiplying the equation (2.9) by hηi n w, it follows the standard weighted energy estimate:
, forn≥0, which together with the Young’s inequality yields
The Gronwall inequality then yields sup
Z hηi n |w η | 2 ≤C(L), (2.11) which givesN0(L)≤C(L), for some constantC(L) that depends only on largeLand the give data u e , u b , c 0 in the problem.
Next, takingx−derivatives of (2.9), we get
C j j − α {∂ x j − α w∂ x α w η }η−[u b −u e ][∂ x j we − η ] η forj≥1 Similarly as above, multiplying the equation byhηi n ∂x j wyields the inequality
Let us treat each term on the right For arbitrary positive constant δ, we get
By choosing δ sufficiently small, the first term in the above inequalities can be absorbed into the left-hand side of the inequality (2.12) Next, for 0< α < j, we have
Z hηi n − 1 {∂ x j − α w∂ x α w η }∂ j x w ≤ Ck∂ x j − α wk∞kp hηi n w∂ x α w η kL 2 (R + )khηi n−2 2 ∂ x j wkL 2 (R + )
Whereas in the case α= 0, we instead estimate
Z hηi n − 1 {∂ x j ww η }∂ j x w ≤ kw η k∞kp hηi n ∂ j x wkL 2 (R + )khηi n−2 2 ∂ x j wkL 2 (R + ).
It remains to give bounds on kw η k ∞ and k∂x j − α wk ∞ , for 0 < α≤ j We recall the definition w=w+u e + [u b −u e ]e − η Using the Sobolev embedding, we get k∂ x j − α wk 2 ∞ ≤ C+k∂ x j − α wk 2 ∞ ≤ C+k∂ x j − α wkL 2 (R + )k∂ x j − α w η kL 2 (R + )
≤ C+k∂ x j − α wkL 2 (R + ) hk∂ j x − α w η | x=0kL 2 (R + )+k∂ x j − α w η k 1/2 L 2 k∂ x j − α+1 w η k 1/2 L 2 i in which k ã kL 2 denotes the usual L 2 norm over [0, x]ìR+ In term of the iterative normNj(x), this yields k∂ x j − α wk 2 ∞ ≤ C+N j 1/2 − α (x)h k∂ x j − α w η | x=0kL 2 (R + )+N j 1/2 (x)i
To estimate the supremum norm \( kw η k ∞ \) for \( 0 < α ≤ j \), we utilize the embedding \( kw η k 2 ∞ ≤ C kw η kL 2 (R + ) kw η kH 1 (R + ) \) By applying equation (2.9) and the established lower bound on \( w \), we derive \( kw η η kL 2 (R + ) ≤ C kw x kL 2 (R + ) + kw kH 1 (R + ) + kw 2 η kL 2 (R + ) + kF η kL 2 (R + ) \) The Sobolev embedding for the supremum norm leads to \( kw η 2 kL 2 (R + ) ≤ C kw η k 3/2 L 2 (R + ) kw η k 1/2 H 1 ≤ δ kw η η kL 2 (R + ) + C δ kw η k 3 L 2 (R + ) \) By selecting \( δ \) to be sufficiently small, we conclude that \( kw η k ∞ 1 + kw x kL 2 (R + ) + kw kH 1 (R + ) + kw η k 2 L 2 (R + ) \).
Hence, integrating the above inequality over [0, x], recalling the definition of the iterative norm, and using the uniform bound onN0(L), we obtain
Putting the above estimates altogether into the j th weighted estimates (2.12), integrating the result over [0, x] and rearranging terms, we obtain
The Gronwall inequality then yields
For sufficiently small values of L, the inequality Nj(L) ≤ CNj(0) holds for all j ≥ 0, where both L and the constant C are determined by the specific problem data Utilizing the standard contraction mapping along with a priori bounds demonstrates the existence and uniform boundedness of the solutions to equation (2.9) within the domain [0, L] × R+ By reverting to the original coordinates and acknowledging the relationship y ∼ η due to the established upper and lower bounds of w, the lemma is established.
The iterative scheme discussed allows for the derivation of a global-in-x solution to the Prandtl equation The L2 estimate demonstrates the existence of a bounded weak solution, while applying the standard Nash-Moser iteration to the parabolic equation provides a uniform bound in C1 and H1 spaces The nonlinear nature of the iterative estimate at the initial step for N1 leads to a consistent bound for all Nk, with k≥1, remaining uniform even for small L Consequently, this results in the establishment of a global smooth solution.
ε 1/2 -order corrections
Next, we collect all terms with a factor √ ε from (2.2a), together with the new √ ε-order terms arising from R u,0 (see (2.5)), to get
R u,1 := [u 1 e +u 1 p ]∂ x [u 0 e +u 0 p ] + [u 0 e +u 0 p ]∂ x [u 1 e +u 1 p ] +v p 1 ∂ y [u 0 e +u 0 p ] + [v 0 p +v e 1 ]∂ y [u 1 e +u 1 p ] +p 1 ex +p 1 px −∂ y 2 [u 1 e +u 1 p ] + [yu 0 px +v 0 p +v 1 e ]u 0 ez +yv 1 ez u 0 py
In our analysis, we establish the Euler and Prandtl layers to ensure that the term R u,1 is proportional to √ ε We organize the terms according to the interior variables (x, √ εy) and the boundary-layer variables (x, y) It is important to note that when the partial derivative ∂y interacts with an interior term scaled by √ εy, that term can be elevated to the next order For example, the expression [v 0 p + v 1 e ]∂ y u 1 e simplifies to √ ε[v p 0 + v e 1 ]u 1 ez (√ εy) Consequently, the principal interior terms are represented by the equation u 0 e u 1 ex + v 1 e u 0 ez + p 1 ex = 0, while the boundary-layer terms are also defined accordingly.
The equation presented, \( [u_1 e + u_1 p] u_0 px + u_0 p u_1 ex + [u_0 e + u_0 p] u_1 px + v_1 p [u_0 py + \sqrt{\epsilon} u_0 ez] + [v_0 p + v_1 e] u_1 py + p_1 px - u_1 pyy + [yu_0 px + v_0 p] u_0 ez + yv_1 ez u_0 py = 0 \), is formulated to eliminate leading terms effectively By constructing these layers, the resulting error is significantly minimized.
Next, let us consider the normal component (2.2b) Clearly, the leading term is √ 1 εp 1 py , which leads to the fact that Prandtl’s pressure is independent of y: p 1 p =p 1 p (x) (2.20)
The next (zeroth) order in (2.2b) consists of
We will again enforce the condition R v,0 = 0, allowing for a possible error of order √ ε It is important to note that v p 1 has been established through the divergence-free condition and the construction of u 1 p We will consider the interior layer [u 1 e, v 1 e, p 1 e] to satisfy the equation u 0 e v ex 1 + p 1 ez = 0.
Whereas, the next layer pressurep 2 p is taken to be of the form p 2 p (x, y) Z ∞ y h[u 0 e +u 0 p ]v 0 px +u 0 p v ex 1 + [v 0 p +v e 1 ][v 0 py +√ εv ez 1 ]−v 0 pyy −εv 1 ezz i
With this choice ofp 2 p and (2.21), the error termR v,0 in this leading order is reduced to
The set of equations (2.17), (2.21), together with the divergence-free condition, constitutes the profile equations for the Euler correction [u 1 e , v e 1 , p 1 e ], whereas equations (2.18) and (2.20) are for the divergence-free Prandtl layers [u 1 p , v p 1 , p 1 p ].
Euler correctors
Euler profiles
We now construct the Euler corrector [u 1 e , v 1 e , p 1 e ] that is used in the boundary-layer expansion (1.9).
We takev 1 e =ve, whereve solves the modified elliptic problem (2.26), with an extra sourceE b By a view of (2.21) and the divergence-free condition, we take u 1 e :=u 1 b (z)−
(2.34) for any constant p b Without loss of generality, we take p b = 0 Clearly, by definition and the uniformH 2 estimates on v 1 e (Lemma 2.5), we have ku 1 e k ∞ +khzi n u 1 e kH 1 ≤C 0 , n≥0 (2.35)
By construction, [u 1 e , v 1 e , p 1 e ] solves (2.21), the divergence-free condition, and instead of (2.17), the equation u 0 e u 1 ex +v e 1 u 0 ez +p 1 ex =−
As compared to (2.17), this contributes a new error term into (2.19), which is now defined as
To estimate the error term, we recognize that our analysis uses the coordinates (x, y), while the Euler flows are assessed at (x, z) = (x, √εy) Utilizing Corollary 2.3 and the boundedness of v_e, along with equation (2.35), we derive that the L2 norm of k√ε[v_p0 + v_1e]u_1ezk is bounded by √εkv_p0 + v_e1k∞ku_1ez(√εã)kL2, which simplifies to Cε^(1/4)ku_1ez(ã)kL2.
Similarly, by definition and the estimates from Lemma 2.5, we have εku 1 ezz (√ εã)kL 2 ≤ε 3/4 ku 1 bz kL 2 +Cε 3/4 khzi n v 1 ezzz kL 2 ≤Cε 1/4 , and by the estimate (2.31),
L 2 ≤Cε − 1/4 khzi n E b (ã)kL 2 ≤Cε 1/4 Hence, we obtain the uniform error estimate: kR u,1 kL 2 ≤Cε 1/4 (2.38)
We provide an estimate for \( R_{v,0} \) as defined in equation (2.23) Utilizing the boundedness of \( v_{p,0} \) and \( v_{1,e} \), along with the estimates from Lemma 2.5 on \( v_{e,1} \), we derive that \( \| R_{v,0} \|_{L^2} \leq \sqrt{\epsilon} \| v_{0,p} + v_{e,1} \|_{L^\infty} \| v_{e,1} \|_{L^2} + \epsilon \| v_{1,e} z \|_{L^2} \leq C \epsilon^{1/4} \).
We estimate \( E_0 \) as defined in equation (2.3) Given that \( u_0^p \) rapidly decays at infinity and \( v_1 e_{zz} \) is in \( L^2 \), we find that \( \|E_0\|_{L^2} \leq \epsilon \|hy_i u_0^p\|_{L^2} \|u_0 e_{zz}\|_{L^\infty} + \epsilon \|hy_i u_0^y\|_{L^2} \|v_1 e_{zz}\|_{L^2} \leq C \epsilon^{3/4} \).
Prandtl correctors
Construction of Prandtl layers
The linearized Prandtl equation (2.41) provides a natural energy estimate that bounds \( p_y \) in terms of \( v_p \) However, the challenge lies in managing the unknown \( v_p \) To address this, we begin with the essential positivity estimate (1.16) and introduce a specific inner product to facilitate our analysis.
The quantity [[v, v]] provides a bound on the norm kv y k 2 L 2 (R + ) Consequently, we will reformulate the equation in terms of v p, incorporating u p into the source term Specifically, by differentiating equation (2.41) with respect to y and applying the divergence-free condition, we obtain the desired results.
−u 0 v pyy +v p u 0 yy +u p u 0 xy + [v p 0 +v 1 e ]u pyy +√ εv ez 1 u py −u pyyy =F py , or equivalently, in a view of the inner product,
F py −u p u 0 xy −[v 0 p +v e 1 ]u pyy −√ εv 1 ez u py
Furthermore, taking x-derivative of (2.42) yields
−v pxyy +u 0 yy u 0 v px +v pyy u 0 yy =G px −u 0 yy u 0 xv p + u py 1 u 0 x yy (2.43) with G p defined as in (2.42) We shall solve the problem (2.42)-(2.43) for v p , with u px +v py = 0, and the boundary conditions: v p (x,0) = 0, v py (x,0) =u 1 ex (x,0) (2.44)
Lemma 2.6 There exists a unique smooth divergence-free solution [u p , v p ] solving (2.41) with initial condition u p (0, y) = ¯u 1 (y) and the boundary conditions (2.44) Furthermore, there hold k[u p , v p ]kL ∞ + sup
0 ≤ x ≤ Lkhyi n v pyy kL 2 (R + )+khyi n v pxy kL 2 C(L, κ)ε − κ , (2.45) for arbitrary small κ, and high regularity estimates sup
0 ≤ x ≤ Lkhyi n v pxyy kL 2 (R + )+khyi n v pxxy kL 2 C(L)ε − 1 , (2.46) uniformly in small ε, L, in which the bounds depend only on the constructed profiles [u 0 , v p 0 ], the given boundary data, and small L.
The proof consists of several steps First, we express the boundary conditions of v p in term of the given dataup(0, y) = ¯u1(z).
Lemma 2.7 For smooth solutions[u p , v p ]solving (2.41)with u p (0, y) = ¯u 1 (y) Then, there hold khyi n vp(0,ã)kH ˙ k+1 (R + )≤C0
1 +khyi n u¯1kH k+3 (R + ) khyi n v px (0,ã)kH ˙ k+1 (R + )≤C 0
1 +khyi n u¯ 1 kH k+3 (R + )+khyi − m u 1 exx (0,√ εã)kH k (R + ) forn, m, k≥0, for some constantC 0 =C 0 (u 0 , v p 0 ,[u 1 e , v e 1 ](0,ã))depending only on the profile [u 0 , v 0 p ] and given data [u 1 e , v 1 e ](0,ã).
Proof Define the stream function ψ=R y
In this article, we define the quantities \( u_p \) and \( v_p \) as \( u_p = \psi_y \) and \( v_p = -\psi_x \) We introduce \( w = u_0 \psi_y - u_0 y \psi \) and derive the equation \( w_x = -u_0 xy \psi - [v_{p0} + v_1 e] u_{py} + u_{pyy} + F_p \) The boundary values of \( w \) and \( w_x \) can be calculated directly from the boundary data \( \bar{u}_1(y) \), \( [u_{0p}, v_{0p}] \), and \( F_p(0, y) \) Specifically, for \( k \geq 0 \), we find that \( k \partial_y k w(0, \tilde{a}) k_{L^2} \leq C(u_0, v_{0p}) k \bar{u}_1 k_{H^k}(R^+) \) and \( k \partial_y k w_x(0, \tilde{a}) k_{L^2} \leq C(u_0, v_{p0}) k \bar{u}_1 k_{H^{k+2}}(R^+) + k \partial_y k F_p(0, \tilde{a}) k_{L^2}(R^+) \), where \( C(u_0, v_{0p}) \) depends on the high regularity norms of \( u_0 \) and \( v_{0p} \), as detailed in Corollary 2.3.
The boundary conditions at x = 0 for the variables ∂z k [u 1 e, v e 1, u 1 ex] imply that the decay factor arises directly from the decay properties of u 0 p Furthermore, the term C(u 0, v p 0, [u 1 e, v 1 e](0, ã)) is influenced by the high regularity norms of u 0, v p 0, and [u 1 e, v e 1](0, ã), leading to the inequality k∂y k F p (0, ã)kL 2 (R +) ≤ C(u 0, v p 0, [u 1 e, v 1 e](0, ã)).
Next, from the definition of w, we can write ψ=u 0
0 w(x, θ) {u 0 } 2 dθ and so we have khyi n ∂ y k+1 v p (0,ã)kL 2 (R + )=khyi n ∂ y k+1 ψ x (0,ã)kL 2 (R + )≤C(kw(0,ã)kH k (R + )+kw x (0,ã)kH k (R + )).
This proves the claimed estimate forv p (0,ã) in ˙H k+1 (R+) Next, as forv x estimate, we differentiate (2.47) with respect to xand get kw xx (0,ã)kH k (R + ) ≤C(u 0 , v 0 p ) khyi − 2 ψ x (0,ã)kH k (R + )+kv p (0,ã)kH k+3 (R + )
+kF px (0,ã)kH k (R + ). Again by definition of F p in (2.41), we have kFpx(0, y)kH k (R + )≤C(u 0 , v 0 p ,[u 1 e , v e 1 ](0,ã))
1 +khyi − m u 1 exx (0,ã)kH k (R + ) in which C(u 0 , v 0 p , v 1 e (0,ã)) depends on high regularity norms of u 0 , v 0 p , and v 1 e (0,ã) Similarly, v pyy (0, y) can be written in term ofu p (0, y), v p (0, y), F py (0, y), according to (2.42) This gives kv p (0,ã)kH k+3 (R + )≤C(u 0 , v p 0 ,[u 1 e , v e 1 ](0,ã))
This yields estimates onvpx on the boundary as claimed.
Lemma 2.8 There exists a positive number L > 0 so that for each N large, the fourth order elliptic equation:
−v yyx +u 0 yy u 0 v x +nv yy u 0 o yy =f y +g (2.48) on [0, L]×[0, N] has a unique solution satisfying initial condition v(0, y) = ¯v 0 (y) and boundary conditions: [v, vy] = 0 at y = 0, N, with sources f, g in weighted L 2 spaces Furthermore, there holds sup
0 ≤ x ≤ Lkhyi n ∂ x j v yy kL 2 ([0,N])+khyi n ∂ x j v xy kL 2
X k=0 khyi n ∂ x k vyykL 2 ( { x=0 } )+khyi n ∂ x k fkL 2 +khyi n+ 3 2 ∂ x k gkL 2
(2.49) for j= 0,1, for any n≥0, as long as the right-hand side is finite.
Proof Let us choose an orthogonal basis{e i (y)} ∞ i=1 inH 2 ([0, N]) with [e i , e i y ] = 0 aty = 0, N, for all i≥1 The orthogonality is obtained with respect to the [[ã]] inner product:
Such an orthogonal basis exists, since [[ã]] is equivalent to the usual inner product in H 1 ([0, N]). Then, we introduce the weak formulation of (2.48):
Z v yy e i yy u 0 dy Z (−f e i y +ge i )dy for alle i (y),i≥1 Next, for each fixedk, we construct an approximate solution in Span{e i (y)} k i=1 defined as v k (x, y) : k
Z v yy k e i yy u 0 Z (−f e i y +ge i )dy (2.50) which by orthogonality yields a system of ODE equations: a i x + k
Since f, g∈L 2 (R+), the ODE system has the unique smooth solution a k and hence, v k is defined uniquely and smooth Multiplying (2.50) by a i x and taking the sum over i, we get
Z v k yy v k yyx u 0 Z (−f v k xy +gv k x )dy which is equivalent to
By the positivity estimate (1.16) and (1.17) withu 0 , we note that [[v k x , v x k ]]≥θ 0 kv k xy k 2 L 2 (0,N) The standard Gronwall inequality then yields sup x ∈ [0,L]
|v yy k | 2 +kv xy k k 2 L 2 ≤C kv yy k k 2 L 2 ( { x=0 } ) +kfk 2 L 2 +khyi 3/2 gk 2 L 2
In the analysis of the inequality kgv x k kL 1 ≤Ckv k xy kL 2 khyi 3/2 gkL 2, we take the limit as k approaches infinity, which allows us to derive the solution to equation (2.48) directly This process also validates the assertion made in (2.49) when both j and n are equal to zero.
Next, we shall derive high regularity estimates We take x-derivative of (2.50) to get
Recall now thatv xx i =Pk i=1a i xx e i , which is a smooth function, sincea i , e i are both smooth Using v i xx as a test function, we then get
The Gronwall inequality, combined with equation (2.51), supports the assertion in (2.49) for unweighted estimates by integrating the third term on the right by parts with respect to y Similarly, we can incorporate the weight function w(y) = hyi^2n and apply inner products against w(y)v_xk and w(y)v_kxx in the energy estimates to derive the claimed weighted estimates, without reiterating the detailed process.
To prove Lemma 2.6, we begin by addressing the non-zero boundary conditions outlined in equation (2.44) We define a cutoff function χ(ã) that is close to zero, ensuring that χ(0) equals 1 Subsequently, we introduce the function ¯v, which is defined as v p (x, y) minus yχ(y)u 1 ex (x,0).
Hence, ¯v= ¯v y = 0 at both y= 0 and y=N.In addition, from (2.43), ¯v solves
−¯v xyy +u 0 yy u 0 v¯ x +v¯ yy u 0 yy =G px −u 0 yy u 0 xv p + u py 1 u 0 x yy−(yχ) yy u 1 exx (x,0)
+ u 0 yy u 0 yχu 1 exx (x,0) +(yχ) yy u 0 yyu 1 ex (x,0)
(2.52) in which G p is defined as in (2.42) Explicitly, we have defined f :=u pyy 1 u 0 x+u py 1 u 0 xy+v p 0 +v 1 e u 0 v pyy + 2v pyy 1 u 0 y+v 1 ex u 0 u py g:= 1 u 0 x
Fpy−upu 0 xy −[v p 0 +v 1 e ]upyy−√ εv ez 1 upy
F pxy −u px u 0 xy −u p u 0 xxy −v px 0 u pyy −√ εv 1 exz u py −√ εv ez 1 u pxy
−u 0 yy u 0 xvp−(yχ)yyu 1 exx (x,0) + u 0 yy u 0 yχu 1 exx (x,0) +(yχ) yy u 0 yyu 1 ex (x,0) with
F p =−u 0 ez [yu 0 px +v 0 p ]−yv 1 ez u 0 py −u 1 e u 0 px −u 0 p u 1 ex
Here, we note that the divergence-free condition is imposed: u px = −v py =−¯v y + (yχ) y u 1 ex (x,0).
We construct the unique solution ¯v to the above problem, and hence the solution vp to (2.43) via a contraction mapping theorem We shall work with the norm:
Lemma 2.8 (or precisely, the estimate (2.51)) yields
|||v¯||| ≤C kv¯ yy k 2 L 2 ( { x=0 } ) +kfk 2 L 2 +khyi 3/2 gk 2 L 2
(2.53) with (f, g) defined as in (2.52) Recall that ¯v = vp(x, y) +yχ(y)v 1 ez (x,0) with v e 1 given on the boundaryx= 0 Hence, ¯v yy (0, y) can be estimates as follows, thanks to Lemma 2.7, kv pyy (0,ã)kL 2 (R + )≤C(u 0 , v 0 p ,[u 1 e , v e 1 ](0,ã))
The uniform bound onkv¯ yy k 2 L 2 ( { x=0 } ) follows.
Next, let us give bounds on f, g For instance, |v¯| ≤ y 3/2 kv yy kL 2 (R + ), |v¯ y | ≤y 1/2 kv yy kL 2 (R + ), and thus Z Z hyi − n |v¯| 2 ≤Csup x kv yy k 2 L 2 (R + )
Z Z hyi − n+1 ≤CL|||v¯|||, for some large n Such a spatial decay hyi − n is produced by the rapid decay property of u 0 p Similarly, we have
, using the fact that u px = −v py = −¯v y + (yχ) y u 1 ex (x,0) As for u py , we use |u py | ≤ |u¯ 1y |+
In this study, we analyze the behavior of the term hyi − n upyy, noting that the decaying factor hyi − n consistently appears for large values of n, attributed to the decay properties of [u 0 p , v p 0 ] To establish a bound, we utilize equation (2.41) for estimation purposes.
2 which is again bounded byC+CL|||v||| Finally, we note that ku 1 exx (x,0)k 2 L 2 ≤
≤ kv exz 1 k L q ′ kv exzz 1 kL q ≤C(L, q)ε − 1+1/q , for arbitrary pair (q, q ′ ) so that 1/q+ 1/q ′ = 1; here, we take q→1.
TakingLsufficiently small in the above estimates and in (2.53) yields a uniform bound on|||v¯|||:
The equation \( ||| \bar{v} ||| \leq C(L, \kappa) \epsilon - \kappa \) holds for any arbitrarily small \( \kappa > 0 \) This linearity in \( \bar{v} \ guarantees the existence of a unique solution to equations (2.52) and (2.43) Furthermore, this method can be applied repeatedly to achieve a global solution in \( x \) for any specified \( L \), ensuring the existence of the Euler data.
N → ∞, we obtain the solution to (2.41) over [0,∞]×R+ The claimed weighted estimates follow similarly, using the rapid decay property of u 0 p
In addition, the boundedness of v p follows by the calculation:
(2.54) which is bounded thanks to the previous bound on |||v||| and the uniform estimate of v p on the boundaryx= 0 Similarly, boundedness ofu p follows from the definition up(x, y) = ¯u1(y) +
Z Z hyi n |v pyy | 2 which is again bounded by|||v|||.
To complete the proof of the lemma, we are now concerned with the higher regularity estimate. Again, applying Lemma 2.8 to the equation (2.52) yields sup
0 ≤ x ≤ Lkhyi n v xyy kL 2 (R + )+khyi n v xxy kL 2
X k=0 khyi n ∂ x k v yy kL 2 ( { x=0 } )+khyi n ∂ k x fkL 2 +khyi n+ 3 2 ∂ x k gkL 2
Let us give bounds on the boundary term on x = 0 Recall that v = v p (x, y)−yχ(y)u 1 ex (x,0). Lemma 2.7 gives khyi n v pxyy (0,ã)kL 2 (R + )≤C 0
Using the inequality |Lf(0)| ≤RL
LkfkL 2 +L 3/2 kf x kL 2 , we have ku 1 exx (0,√ εã)kL 2 (R + )≤Cε − 1/4 kv exz 1 (0,ã)kL 2 (R + )
≤Cε − 1/4 L − 1/2 kv 1 exz (ã)kL 2 +kv exz 1 (ã)k 1/2 L 2 kv 1 exxz (ã)k 1/2 L 2
≤Cε − 1/2 L − 1/2 and ku 1 exxy (0,√ εã)kL 2 (R + ) ≤Cε 1/4 kv exzz 1 (0,ã)kL 2 (R + )
≤Cε 1/4 L − 1/2 kv 1 exzz (ã)kL 2 +kv 1 exzz (ã)k 1/2 L 2 kv exxzz 1 (ã)k 1/2 L 2
≤Cε − 3/4 L − 1/2 in which the estimates on v 1 e from Lemma 2.5 were used Also, we have
|u 1 exx (0,0)| 2 ≤ ku 1 exx (0,ã)kL 2 (R + )kv exzz 1 (0,ã)kL 2 (R + )
≤CL − 1/2 ku 1 exx (0,ã)kL 2 (R + ) kv 1 exzz (ã)kL 2 +kv 1 exzz (ã)k 1/2 L 2 kv 1 exzzz (ã)k 1/2 L 2
This proves that khyi n v pxyy (0,ã)kL 2 (R + )≤Cε − 3/4 L − 1/2 , uniformly in small ε and L Next, estimates for f and g are treated similarly as done above In particular, we note ku 1 exxx (x,0)k 2 L 2 ≤
≤ kv exxz 1 kL 2 kv 1 exxzz kL 2 ≤C(L)ε − 2 , which together with the previous estimates yield the estimate (2.46).
This completes the proof of the lemma.
Cut-off Prandtl layers
We are now prepared to present the Prandtl layers for our boundary layer expansion We define a cutoff function χ(ã) supported on the interval [0,1], and we construct the velocities [u p, v p] as previously described Additionally, we introduce the term u 1 p = χ(√εy)up + √εχ′(√εy).
Clearly, [u 1 p , v 1 p ] is a divergence-free vector field By the estimates from Lemma 2.6 on [u p , v p ], we get
≤√ εy|χ ′ (√ εy)|ku p k∞≤C(L, κ)ε − κ Hence, Lemma 2.6 now reads we have k[u 1 p , v 1 p ]k ∞ + sup
0 ≤ x ≤ Lkhyi n v 1 pyy kL 2 (R + )+khyi n v pxy 1 kL 2 ≤C(L, κ)ε − κ sup
0 ≤ x ≤ Lkhyi n v 1 pxyy kL 2 (R + )+khyi n v 1 pxxy kL 2 ≤C(L)ε − 1 (2.56) uniformly in smallε, L, and for arbitrarily small κ In addition, thanks to the cut-off function, we also have kv 1 px k 2 L 2 ≤
Z Z hyi n |v pxy 1 | 2 dxdy≤C(L, κ)ε − 1/2 − 2κ kv py 1 k 2 L 2 ≤
Z Z hyi n |v 1 pyy | 2 dxdy ≤C(L, κ)ε − 1/2 − 2κ kv 1 pxx k 2 L 2 ≤
We now plug [u 1 p , v p 1 ] into (2.41), or equivalently, (2.18) It does not solve it completely, yielding a new error due to the above cut-off:
The expression involves several components, including terms related to rapidly decaying functions, specifically u 0 p and its derivatives These terms contribute to the error estimate in the context of R u,1 and R u app Notably, the decay properties of u 0 p imply that both u 0 x and u 0 px diminish quickly at infinity, leading to a manageable integral for u 0 x Ry This analysis underscores the significance of decay rates in evaluating the overall error term.
0 u p ds is uniformly bounded by ε − κ Together with boundedness of the constructed Euler and Prandtl layers, we have L 2 norm of the first three big terms involving u p in R u,1 p is bounded by
Cε − κ √ εkχ(√ εã)kL 2 ≤Cε 1/4 − κ Now, as for the term
In the analysis of the Prandtl layers, we observe that the terms [u₀p, vp₀] decay rapidly as y approaches infinity, leading to an error term bounded by Ce⁻ʸ, which behaves as εⁿ in the region where √(εy) ≥ 1 for any large n ≥ 0 This observation, combined with previous estimates, demonstrates that the norm kRu₁p kL² is bounded by C(L, κ)ε¹/₄⁻κ Furthermore, utilizing the definition of p²p from (2.22), we can estimate p²px as follows: p²px = ∫₀^∞ y.
(x, θ)dθ, the last identity was obtained with a use of the divergence-free condition on the vector field [u 0 p , v p 0 ].
We estimate each term on the right Thanks to the boundedness of u 0 e ,[u 0 p , v 0 p ] and v 1 e , and the rapid decay property of the Prandtl layers [u 0 p , v 0 p ], we note that
[u 0 e +u 0 p ]v 0 pxx ≤Chyi − n ku 0 e +u 0 p kL ∞ khyi n v 0 pxx kL 2 (R + )
Z ∞ y u 0 p v exx 1 ≤Chyi − n khyi n u 0 p kL 2 (R + )kv exx 1 (x,√ εã)kL 2 (R + )
[v p 0 +v 1 e ]v pxy 0 ≤Chyi − n kv p 0 +v e 1 kL ∞ khyi n v 0 pxy kL 2 (R + )
Z ∞ y v pxyy 0 ≤Chyi − n khyi n v pxyy 0 kL 2 (R + ).
Hence, taking n≥2 and using the known bounds on the profile solutions, we immediately get kp 2 px kL 2 ε − 1/4 (2.59)
Proof of Proposition 2.1
Having constructed the Euler and Prandtl layers, we now calculate the remaining errors in R u app and R v app from (2.2a) and (2.2b), respectively, and hence complete the proof of Proposition 2.1.
To do so, collecting errors from R u,0 in (2.5), R u,1 in (2.38), the new error R u,1 p in (2.58), and the remainingε-order terms inR app u , we get
[u 1 e +u 1 p ]∂x+v 1 p ∂y i [u 1 e +u 1 p ] +εp 2 px −ε∂ x 2 {u 0 p +√ ε[u 1 e +u 1 p ]} in which reading the estimates (2.38), (2.40), (2.58), (2.59) and using the fact thatu 0 ezz are bounded inL 2 in the original coordinates, we immediately have kE 0 −εu 0 ezz +√ εR u,1 +√ εR p u,1 +εp 2 px kL 2 ≤C(L, κ)ε 3/4 − κ
Using the boundedness of \( u_1^e \), \( u_1^p \), and \( v_1^p \), along with the \( L^2 \) bounds on the derivatives of \( u_0^p \), \( u_1^e \), and \( u_1^p \), we analyze the Euler flows evaluated at \( (x, \sqrt{\epsilon}y) \) This leads to the inequality \( \epsilon k [u_1^e + u_1^p] \partial_x [u_1^e + u_1^p] k_{L^2} \leq \epsilon [k u_1^e k_{L^\infty} + k u_1^p k_{L^\infty}][k u_1^e_x k_{L^2} + k u_1^p_x k_{L^2}] \leq C(L, \kappa) \epsilon^{3/4 - \kappa} \) Furthermore, we derive that \( \epsilon k v_1^p \partial_y [u_1^e + u_1^p] k_{L^2} \leq \epsilon k v_1^p k_{L^\infty} [k \sqrt{\epsilon} u_1^e_z k_{L^2} + k u_1^p_y k_{L^2}] \leq C(L, \kappa) \epsilon^{1 - \kappa} \) Additionally, we establish that \( \epsilon k \partial_x^2 \{u_0^p + \sqrt{\epsilon}[u_1^e + u_1^p]\} k_{L^2} \leq \epsilon k u_0^{p_{xx}} k_{L^2} + \epsilon^{3/2} [k u_1^{e_{xx}} k_{L^2} + k u_1^{p_{xx}} k_{L^2}] \leq C \epsilon \) This ultimately demonstrates that \( k R_{app} u k_{L^2} \leq C(L, \kappa) \epsilon^{3/4 - \kappa} \).
Next, we calculate the error R v app from (2.2b) Simply collecting the remaining terms in R v,0 (see (2.23)) and all terms with a factor√ εor small, we get
By (2.39), we have kR v,0 kL 2 ≤Cε 1/4 We now estimate the remaining terms one by one in R v app Similarly as above, using the boundedness of all profile solutions, we get
≤C(L, κ)ε 1/4 − κ upon recalling the bound kv 1 px kL 2 ≤C(L)ε − 1/4 − κ Next, we have
L 2 ≤C√ εk[u 1 e , u 1 p , v 1 p ]kL ∞ kv 0 px +v ex 1 kL 2 +kv 0 py +√ εv 1 ez kL 2
The analysis reveals that the norm \( kC(L, κ)ε^{1/4 - κ} \) is bounded, indicating that \( kv^{1}_{ex}(ã, \sqrt{εã})k_{L^2} \leq Cε^{-1/4} \) Additionally, it is evident that \( k\sqrt{ε}v_{pyy}(1 - ε)∂x^2(v_{p0} + v_{1}e + \sqrt{ε}v_{1p})k_{L^2} \leq C(L)ε^{1/4 - κ} \) The uniform bounds \( k[v_{pyy1}, v_{pxx0}]k_{L^2} \) are maintained at \( C(L, κ)ε^{-κ} \), while \( kv_{exx1}k_{L^2} \) and \( kv_{pxx1}k_{L^2} \) are constrained by \( Cε^{-1/4} \) and \( C(L)ε^{-5/4} \), respectively, as detailed in Section 2.4.2 By aggregating these findings into \( R_{v_{app}} \) and considering that \( ε \ll L \), we conclude that \( kR_{app} v k_{L^2} \leq C(L, κ)ε^{1/4 - κ} \) This finalizes the proof of Proposition 2.1.
This section is devoted to prove the following crucial linear stability estimates for the linearized equations around the constructed approximate solutions [u app , v app ] Recall (1.15).
Proposition 3.1 states that for the approximate solution [u s, v s] defined in (1.15), there exists a positive constant L for any functions f and g in L² This leads to a linear problem described by equations (3.1) and (3.2), which includes the terms involving u and v, along with their respective derivatives Additionally, the divergence-free condition, expressed as ux + vy = 0, and the relevant boundary conditions must be satisfied.
[u, v]y=0 = 0 (no-slip), [u, v]x=0 = 0 (Dirichlet), p−2εu x = 0, u y +εv x = 0 atx=L (Neumann or stress-free) (3.3) has a unique solution [u, v, p] on [0, L]×R+ Furthermore, there holds k∇εukL 2 +k∇εvkL 2 kfkL 2 +√ εkgkL 2 (3.4)
The proof of Proposition 3.1 consists of several steps First, we construct the solution in the artificial cut-off domain:
Ω N :={0≤x≤L, 0≤y≤N}, with the no-slip boundary conditions [u, v] = 0 prescribed at y=N We shall apply the standard Schaefer’s fixed theorem (see, for instance, [4, Section 9.2.2]) for the space X ={k[u, v]kH 1 (Ω N )≤
Assuming the source terms f and g are smooth, the standard regularity theory for the Stokes problem indicates that the velocity components [u, v] possess H^3 regularity, except at the four corners: [0,0], [0,N], [L,0], and [L,N] Moreover, it has been established that [u, v] belongs to H^(3/2)+ and the pressure p is in H^(1/2)+, even at the corners, ensuring that the H^1 norm of [u, v] is well-defined According to the trace theorem, the relationship k∇[u, v]kL^(2+)(Γ) + kpkL^(2+)(Γ) ≤ k[u, v]kH^(3/2+) + kpkH^(1/2+) holds for any finite piecewise C^1 curve Γ For our analysis, we define Γ = Γδ as the curve formed by the intersection of ΩN and a circle of radius δ centered at the four corners.
.o(1)h k[u, v]kH 3/2+ +kpkH 1/2+ i whereo(1)→0 asδ→0.This justifies the meaning ofH 1 norm of the solution [u, v] in the presence of corners.
We will derive uniform a priori estimates for equations (3.1)-(3.3) By taking the limit as N approaches infinity, we obtain the uniform a priori bound (3.4) Consequently, the existence of the solution, as stated in Proposition 3.1, can be established through a direct application of Schaefer’s fixed point theorem, as referenced in [4, Theorem 4, p.].
504] and Section 3.3, below As will be seen shortly, the positivity estimate (1.16) plays a crucial role.
Energy estimates
Lemma 3.2 Let [u, v]be the solution to the problem (3.1)-(3.3) Assume that ε≪L There holds k∇εuk 2 2+
Proof We multiply (3.1) with u and (3.2) withεv (or equivalently, take [u, v] as the test function in the weak formulation) get
By writing ∆ ε u = 2εu xx + (u y +εv x ) y ,∆ ε v= (u y +εv x ) x + 2v yy and performing the integration by parts multiple times, the left-hand side of the above is reduced to
By using the boundary conditionsp= 2εu x atx=Land [u, v] = 0 atx= 0,and the divergence-free condition u sx +v sy = 0, the energy estimate now becomes
2k∇εuk 2 L 2 −2εk∇εvk 2 L 2 , in which we have used the Young inequality, giving the estimate 2εRR u y v x ≤ 1 2 ku y k 2 L 2 +2εk∇εvk 2 L 2
In addition, since [u, v] = 0 at x = 0, we have the embedding inequalities kukL 2 ≤ Lku x kL 2 Lkv y kL 2 ,kvkL 2 ≤Lkv x kL 2 Hence,
Z Z vg ≤ kukL 2 kfkL 2 +εkvkL 2 kgkL 2
Similarly, since v= 0 at y= 0, we can estimate v=Ry
This proves the claimed inequality in the lemma, with
|yv sy |dy+Lsup x s Z y{u sy } 2 dy+L 2 + 2ε
To establish the bound on the constant C(ε, L, u_s, v_s), we note that u_s is defined as u_0^e + u_0^p + √εu_1^e and v_s as v_0^p + v_1^e The zeroth-order Prandtl layers, represented by [u_0^p, v_0^p], exhibit smoothness with high Sobolev regularity and decay rapidly in the y-direction Consequently, we find that the infinity norm of u_s is bounded by the sum of the infinity norm of u_0^p and the term √ε multiplied by the infinity norm of u_1^e, leading to the conclusion that ku_sxk_∞ ≤ C(u_0^p) + √εkv_1^ek_∞ ≤ C, as detailed in Section 2.3.1 regarding the bounds on the Euler flows Similarly, we can assert the supremum over x.
|zv exz 1 |dxdz 1 +Cε − 1/2 L − 1 khzi n v e 1 k 2 H 2 , (3.7) which is bounded by C(L)ε − 1/2 , thanks to the estimates from Lemma 2.5 Also, we have sup x
Z yh ε|u 0 ez | 2 +|u 0 py | 2 +ε 2 |u 1 ez | 2 i dy
Z Z z|u 1 exz | 2 dxdz 1 +εL − 1 khzi n v e 1 k 2 H 2 , which is again bounded by C, for ε ≪ L Putting this together into the above definition ofC(ε, L, us, vs) and the fact that ε≪L yield the lemma at once.
Positivity estimates
In this section, we establish the following crucial positivity estimate:
Lemma 3.3 Let [u, v]be the solution to the problem (3.1)-(3.3) Assume that ε≪L There holds k∇εvk 2 L 2 +ε 2
Z x=L v 2 y k∇εuk 2 L 2 +Lk∇εvk 2 L 2 +kfk 2 L 2 +εkgk 2 L 2 (3.9)
Proof We start from the identity: ∂ y n v u s o ×(3.1)−ε∂ x n v u s o ×(3.2) Formally, this is the vorticity equation multiplied by the test function u v s This yields
Again, we use the inequality|v| ≤√yR y
0 v 2 y 1/2 , together with the estimates (3.6)-(3.8) on [u s , v s ] to estimate the right-hand side of (3.10) We have
{kfkL 2 +√ εkgkL 2 }k∇εvkL 2 , in which the Young inequality can be applied to absorb the L 2 norm of ∇εv to the left hand side of (3.9).
Next, we treat each term on the left-hand side of (3.10) First, integrating by parts multiple times, we have
Z Z ε∂ y u s vv x u 2 s , in which the last equality is precisely due to the positivity estimate (1.16) From (1.17), we obtain a lower bound
Z Z ε∂yusvvx u 2 s , (3.11) which crucially yields a bound on theL 2 norm of∇εv; or precisely, the L 2 norm of∇εvappearing on the left-hand side of (3.9).
Next, we treat the pressure term Integrating by parts, with recalling that p= 2εu x at x=L, we have
1 u s vv y in which we can estimate ε
≤ {C(u 0 ey , u 0 py ) +ε 1/2 ku 1 ez k∞}Lεkv x kL 2
Here, thanks to the bounds on the Euler flows, summarized in Section 2.3.1, we in particular have ε 1/2 ku 1 ez k∞≤Cε 1/2 ku 1 e kH 3 1 Together with the Young inequality, we thus obtain
Z x=L v 2 y u s +C(u s , v s )L 2 k∇εvk 2 L 2 , (3.12) in which we stress that the boundary term is favorable.
In this section, we will discuss terms that involve the Laplacian It is important to note that, as established in previous studies, the Stokes problem results in the conditions [u, v] ∈ H 3/2+ and p ∈ H 1/2+ Consequently, their traces, [∇u, ∇v] ∈ L 2+ (Γ) and p ∈ L 2+ (Γ), are valid on any smooth curve Γ Additionally, away from the four corners of the domain [0, L]×[0, N], we have [u, v] ∈ H 2 and p ∈ H 1, allowing for further evaluation.
[u yy +εu xx ] + v x us − v∂ x u s u 2 s ε[v yy +εv xx ]i
Z Z hu yy u x u s +εu xx u x u s +εv x v yy u s +ε 2 v x v xx u s i
.Now taking integration by parts respectively in each integration above, with a special attention on the boundary contributions, we get
To address the boundary contributions, we observe that there is a single favorable boundary term at x=0, represented as -ε/2 R x=0 v(x)² u(s) At the boundary x=L, we can simplify the contributions using the relationships u(y) + εv(x) = 0 and u(x) + v(y) = 0, allowing us to streamline the calculations for the boundary terms at this location.
∂ y v∂ y u s u 2 s u, which can be estimated by
We estimate the norms of \( u_s \) similarly to the methods outlined in equations (3.6)-(3.8) Recall that \( u_s = u_0^e + u_0^p + \sqrt{\epsilon} u_1^e \), and note that \( u_s \) remains bounded away from zero due to the assumption stated in (1.14) Following the approach used in (3.8), we find that the supremum over \( x \) applies.
Z yh ε|u 0 ezz |+|u 0 pyy |+ε 3/2 |u 1 ezz |i dy
1 +C(L)ε − 1/2+1/q , for anyq >1 Here, Lemma 2.5 was used Takingq→1 in the above estimates so that−1/2+1/q >
The expression is uniformly bounded by a constant C(us, vs), which remains constant regardless of small ε and L The integral ∫ y |usy|² dy has already been estimated in (3.8) Additionally, we find that ku sy k∞ is less than or equal to √ε ku 0 ez k∞ + ku 0 py k∞ + ε ku 1 ez k∞, which simplifies to C(u 0 e, u 0 p) + ε kv e 1 (ã) kH 3, ultimately leading to a bounded result C This, combined with Young's inequality, reinforces the established bounds.
Z x=L v y 2 u s +C(u s , v s )Lk∇εvk 2 2, (3.16) upon using the divergence-free condition u x =−v y Here,∇ε= (√ ε∂ x , ∂ y ) The boundary term in (3.16) can be absorbed into the good boundary term in (3.12).
We integrate the untreated terms from the left side of equation (3.10) with all interior terms from equation (3.13) and the final term from equation (3.11) To do this, we will apply standard embedding inequalities, specifically kukL² ≤ Lku x kL² and |v| ≤ √yR y.
{u∂ x v s +v s v y +v∂ y v s }+ ε∂yusvvx u 2 s Let us give estimates on each term on the right We claim that
R 0 k∇εuk 2 L 2 +k∇εukL 2 k∇εvkL 2 +C(L)√ εk∇εvk 2 L 2 , (3.17) Proof of (3.17) First, we have
1 u s u y u x ≤Ck∇u s k ∞ ku y kL 2 (ku y kL 2 +kv y kL 2 ) in which the bounds (3.5) and (3.15) gives k∇u s k∞ 1 Next, upon recalling the definition
≤Ck∇u s k∞ k∇εuk 2 L 2 +k∇εukL 2 k∇εvkL 2 +√ εk∇εvk 2 L 2 which again gives the bound as claimed, sincek∇u s k∞.1 We now estimate the third line on the right ofR 0 We first have
0 y(|u syy | 2 +|u sy | 4 )1/2 ku y kL 2 kv y kL 2 , and
The final two terms on the third line of R 0 can be estimated in a comparable manner We provide bounds for the norms of [u s, v s] Following the approach outlined in (3.14), we determine the supremum of x.
Z yh ε 2 |u 0 ezz | 2 +|u 0 pyy | 2 +ε 3 |u 1 ezz | 2 i dy
Z yh ε 2 |u 0 ez | 4 +|u 0 py | 4 +ε 4 |u 1 ez | 4 i dy
The estimates from Lemma 2.5 ensure that both C(u₀e, u₀p) + ε³L⁻¹khzinve₁k²W²,₄ and C(L)ε³ are bounded Additionally, similar bounds can be established for the weighted integrals of ε|usxy|² and ε|usx|⁴, incorporating the extra factor of ε in these calculations.
In addition, we have sup x
≤ C(u 0 e , u 0 p ) +ε 1/2 L − 1 khzi n v e 1 k 2 W 3,q 1 +C(L)ε − 3/2+2/q , which are again bounded, thanks to the Euler bounds from Lemma 2.5, with q being arbitrarily close to 1.
We now give bounds on the last line on the right of R 0 We have
+kv s k ∞ ku y kL 2 kv y kL 2 in which we note that v s is uniformly bounded The estimate (3.8) gives the weighted bound on u sy Next, we estimate sup x
This gives the desired bound on the first term on the last line inR 0 Next, we have ε
0 y(|usxvsx|+|usxvs| 2 +|usxvsy|)1/2 kuykL 2 kvykL 2
In the analysis, we utilize the inequality kukL2 ≤ Lku x kL2 along with the divergence-free condition v y = −u x This approach allows us to derive a uniform estimate on the given terms.
||vsx||∞ ≤ ||v 0 px ||∞+||v 1 ex ||∞.1 +kv e 1 k 2 W 2,q ≤C(L) thanks to the estimates onv e 1 , withq >2 Finally, we estimate ε
0 y|u sy | 2 1/2 kv y kL 2 kv x kL 2 in which the integralR y|u sy | 2 dyis already estimated in (3.8) Putting all above estimates together, we have completed the proof for the claim (3.17).
Finally, using the Young inequality and the smallness ofε, theL 2 norm of∇εvcan be absorbed into the left-hand side of (3.9).
This completes the proof of the positivity estimate and the lemma.
Proof of Proposition 3.1
The proof of Proposition 3.1 is derived directly from the energy estimate in Lemma 3.2 and the positivity estimate in Lemma 3.3, utilizing Schaefer’s fixed point theorem as outlined in [4, Theorem 4, p 504] By integrating these estimates, we can effectively establish the proposition.
L sufficiently small, we get k∇εukL 2 +k∇εvkL 2 ≤ C(u s , v s )h kfkL 2 +√ εkgkL 2 i uniformly in N TakingN → ∞ yields the stability estimate (3.4).
To apply Schaefer's fixed point theorem, we analyze the system defined by the equations involving variables u and v, where the operator S[u, v] is expressed as S[u, v] = λT[u, v] for the parameter λ within the interval [0, 1] The existence of a solution to this system is directly linked to finding a fixed point of the operator S - 1 T Additionally, the compactness of the operator plays a crucial role in establishing the conditions necessary for the application of the theorem.
S − 1 T follows directly from that of the Stokes operator To derive uniform bounds on the set of solutions [u λ , v λ ], we may rewrite the above system as
The equation (1−λ)v ε+λ[u s v x +uv sx +v s v y +vv sy ] +p y ε −∆ ε v=λg is subject to a divergence-free condition and specific boundary conditions The uniform estimates can be derived similarly from the previously established energy and positivity estimates.
We omit to repeat the details This completes the proof of Proposition 3.1.
In order to perform nonlinear iteration, we shall need to derive bounds inL ∞ for the solution We prove the following:
Lemma 4.1 Consider the scaled Stokes system p x −∆ ε u=f, p y ε −∆ ε v=g together with the divergence-free condition ux+vy = 0 and the (same) boundary conditions
[u, v]y=0 = 0 (no-slip), [u, v]x=0 = 0 (Dirichlet), p−2εu x = 0, u y +εv x = 0 atx=L (Neumann or stress-free) (4.1) Then, there holds ε γ 4 kuk∞+ε γ 4 + 1 2 kvk∞.Cγ,L nkukH 1 +√ εkvkH 1 +kfkL 2 +√ εkgkL 2 o , for some constant C γ,L
In our analysis, we leverage the standard extension theorem to demonstrate that our rectangle domain can be covered by two \( C^{0,1} \) charts, resulting in the existence of functions \( \bar{u} \in H^{1+\beta}(\mathbb{R}^2) \) and \( \bar{v} \in H^{1+\beta}(\mathbb{R}^2) \) for \( \beta \in (0,1) \) These functions satisfy the condition \( [\bar{u}, \bar{v}] = [u, v] \) in the region \( [0, L] \times \mathbb{R}^+ \), with bounds on their norms given by \( \| \bar{u} \|_{H^{1+\beta}} \leq C_{\beta, L} \| u \|_{H^{1+\beta}} \) and \( \| \bar{v} \|_{H^{1+\beta}} \leq C_{\beta, L} \| v \|_{H^{1+\beta}} \), where \( C_{\beta, L} \) is a constant dependent on \( \beta \) and \( L \) Furthermore, applying Sobolev's embedding theorem in \( \mathbb{R}^2 \) along with an interpolation inequality for \( \bar{u} \) and \( \bar{v} \), we establish that for any \( 0 < \tau < \alpha \), it holds that \( \| u \|_{\infty} \leq \| \bar{u} \|_{\infty} \leq \| \bar{u} \|_{H^{1+\tau}} \).
≤ C τ,α,L [kukH 1 ] α−τ α [kukH 1+α ] α τ , and similarly, ε 1/2 kvk∞ ≤C τ,α,L [ε 1/2 kvkH 1 ] α−τ α [ε 1/2 kvkH 1+α ] τ α for some constant C τ,α,L Here, we have used the standard interpolation between Sobolev spaces
H 1 , H 1+τ , and H 1+α , with 0< τ < α We note that thanks to our uniform estimates for kukH 1 and ε 1/2 kvkH 1 ,it suffices to give estimates on the H 1+α norm of [u,√ εv].
We establish that for a fixed α approximately equal to 1/2 and a small τ, there exists a significant number mα > 0 such that the inequality kukH^(1+α) + √εkvkH^(1+α) ≤ mα(kfkL^2 + √εkgkL^2 + kukH^1 + √εkvkH^1) holds Consequently, this leads to the conclusion that kuk∞ + √εkvk∞ is bounded above by a constant C depending on τ, α, and L, multiplied by ε^(-mατ^α) times the sum of the norms kfkL^2, √εkgkL^2, kukH^1, and √εkvkH^1.
The lemma would then be proved at once by choosingτ ≪α so that m α α τ ≤ γ 4
We shall now prove the claim for α = 1/2 and m α = 7/4 To do so, let us introduce the (original) scaling: uε(x, y)≡u(Lx, L y
Clearly, direct calculations yieldu εx +v εy = 0 and
Plugging these in the Stokes problem, we yield a normalized Stokes system: p εx −∆u ε =ε − 1 L 2 f(Lx, L y
√ε) (4.4) in a fixed domain 0≤x≤1 and 0≤y≤ ∞ with boundary conditions
[u ε , v ε ] = 0 on both boundaries: {y= 0} and {x= 0} p ε + 2u εx = 0, v εx +u εy = 0 at x= 1.
In the context of the Stokes problem within a fixed domain, we apply the standard elliptic estimate, which is supported by Poincaré's inequality, stating that the norms of the velocity fields \(u_\varepsilon\) and \(v_\varepsilon\) are bounded by their spatial derivatives Furthermore, standard energy estimates indicate that the combined gradient norms of \(u_\varepsilon\) and \(v_\varepsilon\) are constrained by a constant multiplied by the inverse of \(\varepsilon\) and the \(L^2\) norm of the forcing function \(f\).
We provide an L² estimate for the pressure pε by demonstrating that for any function h ∈ L², there exists a vector-valued function φ ∈ H¹ such that φ(0, y) = 0 and φ(x, 0) = 0, with the gradient ∇φ equal to h and the norm kφkH¹ bounded by khkL² Specifically, we can express h as a sum of characteristic functions, allowing us to define qn = 1_{n ≤ y < n+1}(y)h, which is supported within the unit square [0,1] × [n, n + 1] for each n.
By [13, page 27], we can then find φ n ∈ H 1 , such that ∇ ãφ n = h n on the unit square, with φ n (0, y) ≡φ n (x, n)≡φ n (x, n+ 1)≡0 Furthermore, we have kφ n kH 1 ≤Ckh n kL 2 ,uniformly in n.
If we now define φ ≡P ∞ n=1φ n , it then follows that kφkH 1 ≤ CkhkL 2 ,∇ ãφ =h and φ(0, y) ≡0, φ(x,0)≡0.
The pressure estimate is derived from the vector field φ, allowing us to approximate p ε with smooth functions of the form q = ∇ ãφ, leading to the relationship kp ε kL 2 kqkL 2 = k∇φkL 2 By utilizing the vector field φ as a test function in the Stokes problem and applying Young's inequality, we obtain the inequality kp ε kL 2 ≤ Cn (k∇u ε kL 2 + k∇v ε kL 2 + ε − 1 L 2 kf(Lx, Ly)).
≤ Cε − 3/4 Lh kfkL 2 +√ εkgkL 2 i thanks to the estimate (4.5).
Now, it remains to derive estimates in the higher regularity norms We multiply the Stokes system by an arbitrary cut-off function χ(x,√ y ε) to obtain:
If we choose χ =χ 1 (x,√ y ε) which has a compact support away from the corners, then the Stokes problem has an H 2 estimate so that kχ 1 p ε kH 1 +kχ 1 u ε kH 2 +kχ 1 v ε kH 2
≤ ε − 1 {ku ε kL 2 +kv ε kL 2 }+ε − 1 2 {k∇u ε kL 2 +k∇v ε kL 2 +kp ε kL 2 }+ε − 3/4 LkfkL 2 +ε − 1/4 LkgkL 2
Using the estimate (4.5), we derive a uniform estimate for the unscaled solution [u, v, p] through the variable transformation (4.3) The estimate for the function χ₁u in the H² norm is approximately ε⁻¹ + (1/4)k∇²(χ₁u)kL² + ε⁻¹/₂ + (1/4)k∇(χ₁u)kL² + ε¹/₄kχ₁ukL² Similarly, for the function χ₁v, the estimate in the H² norm is ε¹/₂ + ε⁻¹ + (1/4)k∇²(χ₁v)kL² + (1/4)k∇(χ₁v)kL² + ε¹/₂ + (1/4)kχ₁vkL² These results collectively yield a significant relationship between the gradients of the functions u and v, highlighting their dependence on the parameter ε.
≤ε − 5/4 Ln kfkL 2 +√ εkgkL 2 o+ε − 1/4 n kukH 1 +√ εkvkH 1 o.
We select the cut-off function χ = χ²(x, √y ε) with support concentrated near the corners, where we lack an H² estimate of the solution However, as referenced in [14], we possess a weaker estimate that states: the sum of the norms kχ²p εkH³/², kχ²u εkH¹+³/², and kχ²v εkH¹+³/² is less than or equal to the sum of kχ²x p ε, Δχ²u ε, 2∇χ²ã ∇u ε, and χ²fkL².
The inequality ε − 1 {ε − 3/4 LkfkL 2 + ε − 1/4 LkgkL 2 } establishes a crucial relationship in the context of scaling through (4.3) Specifically, we find that k∇{χ 2 u ε }k H 1/2 is equal to ε − 3/4 + 1/4 k∇{χ 2 u}k H 1/2, and k∇{χ 2 v ε }k H 1/2 equals ε 1/2 ε − 3/4 + 1/4 k∇{χ 2 v}k H 1/2 This leads to the conclusion that ε − 3/2 + 1/4 k∇{χ 2 u}k H 1/2 + ε 1/2 ε − 3/4 + 1/4 k∇{χ 2 v}k H 1/2 is bounded by ε − 1 {ε − 3/4 LkfkL 2 + ε − 1/4 LkgkL 2 } Ultimately, by integrating the estimates on χ 1 [u, v] and χ 2 [u, v], we successfully derive the bound (4.2) for α = 1/2 and m α = 7/4, thereby completing the proof of the lemma.
5 Proof of the main theorem
We are now ready to give the proof of our main theorem Consider the nonlinear scaled Navier- Stokes equations (1.4) and write the solutions [U ε , V ε , P ε ] in the asymptotic expansion (1.9):
The equations for the remainder solutions [u ε , v ε , p ε ] are defined as follows: [U ε , V ε , P ε ](x, y) = [u app , v app , p app ](x, y) + ε γ + 1/2 [u ε , v ε , p ε ](x, y) The leading approximate solutions, excluding the Prandtl layers, are given by u s (x, y) = u 0 e + u 0 p + √ε u 1 e and v s (x, y) = v 0 p + v e 1 The remainder solutions must satisfy the equations: u s u ε x + u ε u sx + v s u ε y + v ε u sy + p ε x − ∆ ε u ε = R 1 (u ε , v ε ), u s v x ε + u ε v sx + v s v y ε + v ε v sy + p ε y ε − ∆ ε v ε = R 2 (u ε , v ε ), and u ε x + v y ε = 0, where ∆ ε = ∂ y 2 + ε ∂ x 2, and the remainders R 1 (u ε , v ε ) and R 2 (u ε , v ε ) are also defined.
Here, the errorsR app u , R v app from the approximation are estimated in Proposition 2.1.
We will utilize the standard contraction mapping theorem to establish the existence of solutions for the nonlinear problem To do this, we define the function space X with the norm given by \( k[u_\epsilon, v_\epsilon]k_X \equiv k\nabla_\epsilon u_\epsilon k_{L^2} + k\nabla_\epsilon v_\epsilon k_{L^2} + \epsilon \gamma^2 k u_\epsilon k_\infty + \epsilon^{1/2} + \gamma^2 k v_\epsilon k_\infty \) For each pair \([u_\epsilon, v_\epsilon] \in X\), we address the corresponding linear problem for \([\bar{u}_\epsilon, \bar{v}_\epsilon]\), which is formulated as: \[u_s \bar{u}_\epsilon x + \bar{u}_\epsilon u_s x + v_s \bar{u}_\epsilon y + \bar{v}_\epsilon u_s y + \bar{p}_\epsilon x - \Delta_\epsilon \bar{u}_\epsilon = R_1(u_\epsilon, v_\epsilon),\]\[u_s v_{\bar{v}} x_\epsilon + \bar{u}_\epsilon v_s x + v_s \bar{v}_\epsilon y_\epsilon + \bar{v}_\epsilon v_s y + p_{\bar{v}} y_\epsilon - \Delta_\epsilon \bar{v}_\epsilon = R_2(u_\epsilon, v_\epsilon).\]
We are in the position to use the linear stability estimates obtained in Proposition 3.1, yielding k∇εu¯ ε kL 2 +k∇εv¯ ε kL 2 ≤ kR 1 (u ε , v ε )kL 2 +√ εkR 2 (u ε , v ε )kL 2 (5.6)
We provide estimates for the remainders \( R_1 \) and \( R_2 \) as defined in (5.4) According to Proposition 2.1, the inequality \( \epsilon - \gamma - \frac{1}{2} h k R u_{\text{app}} k_{L^2} + \sqrt{\epsilon} k R_{\text{app}} v k_{L^2} \leq C(L, u_s, v_s) \epsilon - \gamma - \kappa + \frac{1}{4} \) holds for any arbitrary \( \kappa > 0 \) We select any \( \gamma < 1 \) and choose \( \kappa \) such that \( \gamma + \kappa < \frac{1}{4} \) Furthermore, by examining the definition of \( k \cdot k_X \), the divergence-free condition \( u_x = -v_y \), and the nearly bounded nature of \( [u_1^p, v_1^p] \), we derive our estimates.
√εk(u 1 p +ε γ u ε )u ε x kL 2 ≤√ ε(ku 1 p k ∞ +ε γ ku ε k ∞ )ku ε x kL 2
√εk(v p 1 +ε γ v ε )u ε y kL 2 ≤√ ε(kv p 1 k∞+ε γ kv ε k∞)ku ε y kL 2
Also, using the inequality |[u ε , v ε ]| ≤ √yk[u ε y , v y ε ]kL 2 (R + ) and the uniform H 1 bounds obtained in (2.56) onu 1 p , we get
√εku ε u 1 px +v ε u 1 py kL 2 ≤√ εsup x khyi n u 1 px kL 2 (R + )+khyi n ku 1 py kL 2 (R + ) k[u ε y , v ε y ]kL 2
Similarly for terms in R 2 , we have
√εk(u 1 p +ε γ u ε )v ε x kL 2 ≤√ ε(ku 1 p k∞+ε γ ku ε k∞)kv x ε kL 2
√εk(v p 1 +ε γ v ε )v y ε kL 2 ≤√ ε(kv 1 p k∞+ε γ kv ε k∞)kv y ε kL 2
C(v 1 p ) +ε γ 2 k[u ε , v ε ]kX k[u ε , v ε ]kX and √ εku ε v px 1 +v ε v 1 py kL 2 ≤√ ε ku ε k∞kv 1 px kL 2 + sup x khyi n kv py 1 kL 2 (R + )kv ε y kL 2
≤C(v p 1 )ε 1/4 − γ 2 − κ k[u ε , v ε ]kX, in which we have used the bound (2.57): kv px 1 kL 2 ε − 1/4 − κ
Combining the above estimates into (5.6) yields kR 1 (u ε , v ε )kL 2 +√ εkR 2 (u ε , v ε )kL 2
In analyzing the bounds of the gradient for the functions \( \bar{u}_\epsilon \) and \( \bar{v}_\epsilon \), we note that the parameters satisfy \( \gamma + \kappa \leq \frac{1}{4} \) and \( \epsilon \ll 1 \) This leads to a significant estimate based on the Stokes problem, which involves the functions \( f \) and \( g \) defined in terms of the variables \( u_s \), \( v_s \), and their respective gradients Utilizing the Stokes estimates alongside previous results, we establish that the sup norm is bounded by \( \epsilon \gamma^2 \| \bar{u}_\epsilon \|_\infty + \epsilon^{1/2} + \gamma^2 \| \bar{v}_\epsilon \|_\infty \leq C \gamma \epsilon \gamma^4 \left\{ \|\nabla_\epsilon \bar{u}_\epsilon\|_{L^2} + \|\nabla_\epsilon \bar{v}_\epsilon\|_{L^2} \right\} \).
+ε γ 4 k −u s u ε x −u sx u ε −v s u ε y −v ε u sy +R 1 kL 2 (5.8)+ε γ 4 + 1 2 k −usv ε x −u ε vsx−vsv ε y −v ε vsy+R2kL 2