Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 33, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS FOR THERMOSOLUTAL CONVECTION XIAOTING FAN, WEI WANG Abstract. This article concerns the behavior of the initial layer appearing at large Prandtl number in Boussinesq equations with the ill initial data. By using the asymptotic expansion methods of singular perturbation theory, we establish an approximate solution and the rate of convergence as the Prandtl number tends to infinity. Our results improve the existing ones concerning thermosolutal convection. 1. Introduction Double diffusive convection has been considered a fundamental fluid dynamical phenomenon that can occur when a layer of fluid with a dissolved solute, is heated from below. This phenomena was first studied in the ocean [14], where heat and salt are relevant properties; so the process is called “thermohaline” or “thermoso- lutal” convection. Many investigations have been studied in the stratification of magma chambers [2] and convection in the sun, in meteorology, geophysics, and astrophysics. In this article, we consider the thermosolutal convection setting of a horizontal layer of fluid confined by two parallel planes a distance h apart. The warm salty fluid tends to rise and salty fluid being heavier than fresh fluid tends to fall. The Boussinesq system with rotation for thermosolutal convection is stated as follows: ∂tu+ (u · ∇)u = 1 ρ0 (∇p+ ρgk) + √ Tak × u+ ν∆u, ∇ · u = 0, ∂tT + u · ∇T − κT∆T = 0, ∂tS + u · ∇S − κS∆S = 0, u|z=0,h = 0, T |z=0 = T0, T |z=h = T1, S|z=0 = S0, S|z=h = S1, where T0 > T1 and S0 > S1. The unknown functions u = (u1, u2, u3)>, p, T and S represent the velocity field, the scalar pressure, the scalar temperature field and the solute concentration of the fluid, respectively. ν, κT and κS are the kinematic 2020 Mathematics Subject Classification. 35Q35, 35B20, 35C20. Key words and phrases. Boussinesq system; thermosolutal convection; Prandtl number; perturbation theory; asymptotic expansion. ©2022. This work is licensed under a CC BY 4.0 license. Submitted November 16, 2021. Published April 21, 2022. 1 2 X. FAN, W. WANG EJDE-2022/33 viscosity, the thermal and solute diffusivity, respectively. ρ is the density, ρ0 > 0 is the fluid density at the lower surface z = 0 and g is the gravity acceleration constant, k := (0, 0, 1)> is the unit upward vector. Ta = 4Ω2h2 ν2 is the Taylor number, Ω is the rotation rate and h is the distance between two plates. The lower (upper) boundary is maintained at a constant temperature T0 (T1) and a constant solute concentration S0 (S1). For simplicity, we consider the periodic boundary conditions in the horizontal directions: (u, T, S)(x, t) = (u, T, S)(x+ k1L1, y + k2L2, z, t), for all k1, k2 ∈ Z. By non-dimensionalization and the Oberbeck-Boussinesq approximation, Park [25] obtained the dimensionless form λ[∂tu λ + (uλ · ∇)uλ] +∇pλ = ∆uλ + √ Tak × uλ + (RTT λ −RSSλ)k, (1.1) ∇ · uλ = 0, (1.2) ∂tT λ + uλ · ∇Tλ = ∆Tλ, (1.3) ∂tS λ + uλ · ∇Sλ = τ∆Sλ, (1.4) in the dimensionless domain D := (0, L1)× (0, L2)× [0, 1] (L1, L2 > 0), t ∈ (0, T∗), T∗ > 0. λ = 1 Pr , Pr = ν κT is the Prandtl number, RT = αg(T0−T1)h3 κT ν is the thermal Rayleigh number, RS = γg(S0−S1)h3 κT ν is the salinity Rayleigh number, τ = κS κT is the Lewis number. α and γ denote the coefficients of thermal expansion and its analogous compositional counterpart, respectively. The Lewis number τ for magmatic substances is less than 10−1, here we consider 0� τ < 1 for simplicity. We complement the above system with the boundary conditions: uλ|z=0,1 = 0, (x, y, t) ∈ X × (0, T∗), (1.5) Tλ, Sλ|z=0 = 1, Tλ, Sλ|z=1 = 0, (x, y, t) ∈ X × (0, T∗), (1.6) for X := (0, L1)× (0, L2), and provide an initial condition (uλ, Tλ, Sλ)(t = 0) = (uλ0 , T λ 0 , S λ 0 )(x, y, z). (1.7) The mathematical analysis of the thermosolutal convection has attracted much attention (see [1, 4, 8, 11, 13, 16, 18, 23, 24, 26, 29, 31]). Hallstrom [10] discussed the global existence and uniqueness of smooth solutions for the infinite Prandtl number limit of the Boussinesq equations. Park [24] considered the thermosolutal convection with or without rotation in the infinite Prandtl number, and investigated that there are bifurcating solutions for 2-D and 3-D Boussinesq equations. Park [25] studied the infinite Prandtl number limit and derived the convergence of Boussinesq system for the thermosolutal convection to the infinite Prandtl number system. Our purpose is to study the asymptotic behavior of the thermosolutal convection in the infinite limit of Prandtl number. For one fluid model as a vanishing limit case, there have been large investigations, see for instance [3, 20, 21, 30]. Moti- vated by [28] and the related studies, see [6, 7, 9, 15, 17], we view the Boussinesq system as a small perturbation of the infinite Prandtl number model. Firstly, we derive the appearance of initial layer in detail. Secondly, by using the multi-scale approach [12, 19] and the matched asymptotic expansion methods [5], we construct an approximation solution to (1.1)–(1.7) as the combination of inner and initial expansions. Finally, we study the convergence of (1.1)–(1.7) to the infinite Prandtl EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 3 number system as the Prandtl number tends to∞. Our results improve the existing results in [25]. This article is organized as follows. Section 2 is dedicated to the derivation of initial layer. In Section 3, we construct an approximate solution. The main result and its proof are presented in Section 4. Let C be a positive generic constant, independent on λ, while C may depend on S for fixed T∗ > 0. 2. Derivation of initial layer Let ( uλ, pλ, Tλ, Sλ ) be the global weak solution of the system (1.1)–(1.6) in the Leray’s sense. The initial data has uλ(t = 0) = ( u0 0 + λu1 0 + uλ0E ) (x, y, z), (2.1) Tλ(t = 0) = ( T 0 0 + λT 1 0 + Tλ0E ) (x, y, z), (2.2) Sλ(t = 0) = ( S0 0 + λS1 0 + Sλ0E ) (x, y, z), (2.3) where u0 0, u1 0, T 0 0 , T 1 0 , S0 0 and S1 0 are C∞(D) functions, uλ0E , Tλ0E , Sλ0E denote the remainders for the initial data, uλ0E , Tλ0E , Sλ0E ∈ C∞(D) satisfies ‖(uλ0E , Tλ0E , Sλ0E)(x, y, z)‖L2(D) ≤ Cλ2, (2.4) and u0 0, u1 0, T 0 0 , T 1 0 , S0 0 , S1 0 satisfy the following compatibility conditions: ∇ · u0 0 = ∇ · u1 0 = 0, (2.5) (u0 0, u 1 0) |z=0,1= 0, (2.6) (T 0 0 , S 0 0)|z=0 = (1, 1), (2.7) (T 0 0 , S 0 0)|z=1 = (0, 0), (2.8) (T 1 0 , S 1 0) ∣∣ z=0,1 = (0, 0). (2.9) In this section, by employing the Stokes operator [19] and singular perturbation theory [22, 27], we consider the initial layer behavior of the solution when the Prandtl number tends to infinity, i.e., λ tends to 0. Letting λ = 0 in the system (1.1)–(1.6), we obtain that ∆u0 −∇p0 + √ Tak × u0 + (RTT 0 −RSS0)k = 0, (2.10) ∇ · u0 = 0, (2.11) ∂tT 0 + (u0 · ∇)T 0 = ∆T 0, (2.12) ∂tS 0 + (u0 · ∇)S0 = τ∆S0, (2.13) u0|z=0,1 = 0, (2.14) T 0, S0|z=0 = 1, T 0, S0|z=1 = 0, (2.15) for (x, y, z, t) ∈ D × (0, T∗), T∗ > 0. The initial condition of T 0, S0 is (T 0, S0)(t = 0) = (T 0 0 , S 0 0)(x, y, z), (x, y, z) ∈ D, (2.16) where (T 0 0 , S 0 0)(x, y, z) stands for the limit of (Tλ0 , S λ 0 )(x, y, z) as λ→ 0. Because of the singularity of perturbation, the limit of uλ0 (x, y, z) as λ → 0 can not be satisfied by the velocity u0(t = 0) in the limit system (2.10)–(2.16). 4 X. FAN, W. WANG EJDE-2022/33 Restricting (2.10), (2.11) and (2.14) to t = 0 gives ∆u0(t = 0)−∇p0(t = 0) + √ Tak × u0(t = 0) + (RTT 0(t = 0)−RSS0(t = 0))k = 0, ∇ · u0(t = 0) = 0, u0|z=0,1(t = 0) = 0. By solving the above system, we know that the value of u0(t = 0) is determined by initial value of T 0 and S0, while limλ→0 u λ 0 is given arbitrarily and independently of T 0(t = 0) and S0(t = 0), so limλ→0 u λ 0 6= u0(t = 0). Comparing (1.7) with (2.16) leads to the appearance of an initial layer. 3. Approximate solution In this section, we give a rigorous proof of infinite lint of the Prandtl for system (1.1)–(1.6). Firstly, we establish the approximate solution. We seek an expansion of the form( uλ, pλ, Tλ, Sλ ) (x, y, z, t) ∼ ∞∑ i=0 (λ)i(uIn,i(x, y, z, t) + uI,i(x, y, z, η), pIn,i(x, y, z, t) + pI,i(x, y, z, η), T In,i(x, y, z, t) + T I,i(x, y, z, η), SIn,i(x, y, z, t) + SI,i(x, y, z, η), (3.1) where λ stands for the length of the initial layer and η = t/λ is the fast time variable.( uIn,i, pIn,i, T In,i, SIn,i ) (x, y, z, t) denotes the inner functions for the velocity field, the scalar pressure, the temperature field, and the solute concentration, respectively, independent of λ. (uI,i, pI,i, T I,i, SI,i)(x, y, z, η) stands for the initial layer functions and the initial layer functions satisfying (uI,i, pI,i, T I,i, SI,i)(η → +∞) = 0. (3.2) We assume that the asymptotic expansion of the system (1.1)–(1.7) including the initial corrections is of the form( uλa , p λ a , T λ a , S λ a ) (x, y, z, t) = 1∑ i=0 λi(uIn,i(x, y, z, t) + uI,i(x, y, z, η), pIn,i(x, y, z, t) + pI,i(x, y, z, η), T In,i(x, y, z, t) + T I,i(x, y, z, η), SIn,i(x, y, z, t) + SI,i(x, y, z, η)). (3.3) Moreover, to match the boundary and initial conditions (1.5)–(1.7), we shall impose the following restrictions uλa |z=0,1 = (uλIn + uλI )|z=0,1 = uλI |z=0,1 = 1∑ i=0 λiuI,i|z=0,1, Tλa |z=0 = (TλIn + TλI )|z=0 = 1 + 1∑ i=0 λiT I,i|z=0, Tλa |z=1 = (TλIn + TλI )|z=1 = 1∑ i=0 λiT I,i|z=1, EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 5 Sλa |z=0 = (SλIn + SλI )|z=0 = 1 + 1∑ i=0 λiSI,i|z=0, Sλa |z=1 = (SλIn + SλI )|z=1 = 1∑ i=0 λiSI,i|z=1, uλa(t = 0) = 1∑ i=0 λi(uIn,i(t = 0) + uI,i(η = 0)), Tλa (t = 0) = 1∑ i=0 λi(T In,i(t = 0) + T I,i(η = 0)), Sλa (t = 0) = 1∑ i=0 λi(SIn,i(t = 0) + SI,i(η = 0)). i.e., (uIn,0 + uI,0)|z=0,1 = 0, (3.4) (uIn,1 + uI,1)|z=0,1 = 0, (3.5) (T In,0 + T I,0)|z=0 = 1, (T In,1 + T I,1)|z=0 = 0, (3.6) (T In,0 + T I,0)|z=1 = 0, (T In,1 + T I,1)|z=1 = 0, (3.7) (SIn,0 + SI,0)|z=0 = 1, (SIn,1 + SI,1)|z=0 = 0, (3.8) (SIn,0 + SI,0)|z=1 = 0, (SIn,1 + SI,1)|z=1 = 0, (3.9) uIn,0(t = 0) + uI,0(η = 0) = u0 0, uIn,1(t = 0) + uI,1(η = 0) = u1 0, (3.10) T In,0(t = 0) + T I,0(η = 0) = T 0 0 , T In,1(t = 0) + T I,1(η = 0) = T 1 0 , (3.11) SIn,0(t = 0) + SI,0(η = 0) = S0 0 , SIn,1(t = 0) + SI,1(η = 0) = S1 0 . (3.12) We discuss the construction of the inner functions and initial layer functions( uλa , p λ a , T λ a , S λ a ) := ( uλIn, p λ In, T λ In, S λ In ) (x, y, z, t) + ( uλI , p λ I , T λ I , S λ I ) (x, y, z, η), η = t λ , (3.13) where ( uλIn, p λ In, T λ In, S λ In ) = 1∑ i=0 λi ( uIn,i, pIn,i, T In,i, SIn,i ) , (3.14) ( uλI , p λ I , T λ I , S λ I ) = 1∑ i=0 λi ( uI,i, pI,i, T I,i, SI,i ) . (3.15) First, we study inner expansion away from t = 0 in Section 3.1. Then, we study the initial layer expansion near t = 0 in Section 3.2. Finally, we consider the approximate solution in Section 3.3. 3.1. Inner expansion. Away from t = 0 in (3.1), the solution to system (1.1)– (1.6) has the expansion( uλ, pλ, Tλ, Sλ ) (x, y, z, t) ∼ ∞∑ i=0 λi(uIn,i, pIn,i, T In,i, SIn,i)(x, y, z, t). 6 X. FAN, W. WANG EJDE-2022/33 First, inserting the above expansion into (1.1)–(1.6) and using direct calculations, we obtain that ∞∑ i=0 λi(λ[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i −∆uIn,i − √ Tak × uIn,i − (RTT In,i −RSSIn,i)k) = 0, ∞∑ i=0 λi∇ · uIn,i = 0, ∞∑ i=0 λi ( ∂tT In,i + i∑ j=0 uIn,j · ∇T In,i−j −∆T In,i ) = 0, ∞∑ i=0 λi ( ∂tT In,i + i∑ j=0 uIn,j · ∇SIn,i−j − τ∆SIn,i ) = 0, ∞∑ i=0 λiuIn,i|z=0,1 = 0, ∞∑ i=0 λi(T In,i, SIn,i)|z=0 = (1, 1), ∞∑ i=0 λi(T In,i, SIn,i)|z=1 = (0, 0). Then ( uλIn, p λ In, T λ In, S λ In ) satisfies λ[∂tu λ In + (uλIn · ∇)uλIn] +∇pλIn = ∆uλIn + √ Tak × uλIn + (RTT λ In −RSSλIn)k +RλIn,u, (3.16) ∇ · uλIn = 0, (3.17) ∂tT λ In + (uλIn · ∇)TλIn = ∆TλIn +RλIn,T , (3.18) ∂tS λ In + (uλIn · ∇)SλIn = τ∆SλIn +RλIn,S , (3.19) uλIn|z=0,1 = 0, (3.20) (TλIn, S λ In)|z=0 = (1, 1), (TλIn, S λ In)|z=1 = (0, 0), (3.21) where the remainders RλIn,u, RλIn,T and RλIn,S are RλIn,u = − ∞∑ i=2 λi(λ[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i −∆uIn,i − √ Tak × uIn,i − (RTT In,i −RSSIn,i)k) + λ3uIn,1 · ∇uIn,1, RλIn,T = − ∞∑ i=2 λi ( ∂tT In,i + i∑ j=0 uIn,j · ∇T In,i−j −∆T In,i ) + λ2uIn,1 · ∇T In,1, RλIn,S = − ∞∑ i=2 λi ( ∂tS In,i + i∑ j=0 uIn,j · ∇SIn,i−j − τ∆SIn,i ) + λ2uIn,1 · ∇SIn,1. EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 7 Thus, RλIn,u, RλIn,T and RλIn,S satisfy ‖(RλIn,u, RλIn,T , RλIn,S)‖L∞(0,T∗;Hs(D)) ≤ Cλ2, (3.22) for T∗ > 0 and s ≥ 1. Now, we set the coefficient of O(λ0) in system (3.16)-(3.19) as zero and use (3.20)-(3.21), (2.16). Then for ( uIn,0, pIn,0, T In,0, SIn,0 ) , we have ∆uIn,0 + √ Tak × uIn,0 + (RTT In,0 −RSSIn,0)k −∇pIn,0 = 0, (3.23) ∇ · uIn,0 = 0, (3.24) ∂tT In,0 + (uIn,0 · ∇)T In,0 = ∆T In,0, (3.25) ∂tS In,0 + (uIn,0 · ∇)SIn,0 = τ∆SIn,0, (3.26) uIn,0|z=0,1 = 0, (3.27) (T In,0, SIn,0)|z=0 = (1, 1), (T In,0, SIn,0)|z=1 = (0, 0), (3.28) (T In,0, SIn,0)(t = 0) = (T 0 0 , S 0 0). (3.29) Similarly, we consider the O(λ1) terms. We set the coefficient of O(λ1) in the system (3.16)-(3.19) as zero and use (3.20)-(3.21), the initial conditions (3.11)- (3.12). At first, (uIn,1, pIn,1, T In,1, SIn,1) satisfy ∆uIn,1 + √ Tak × uIn,1 + (RTT In,1 −RSSIn,1)k −∇pIn,1 − ∂tuIn,0 − (uIn,0 · ∇)uIn,0 = 0, (3.30) ∇ · uIn,1 = 0, (3.31) ∂tT In,1 + uIn,0 · ∇T In,1 + uIn,1 · ∇T In,0 = ∆T In,1, (3.32) ∂tS In,1 + uIn,0 · ∇SIn,1 + uIn,1 · ∇SIn,0 = τ∆SIn,1, (3.33) uIn,1|z=0,1 = 0, (3.34) (T In,1, SIn,1)|z=0,1 = (0, 0), (3.35) T In,1(t = 0) = T 1 0 − T I,1(η = 0), SIn,1(t = 0) = S1 0 − SI,1(η = 0). (3.36) We have the compatibility condition (T 1 0 − T I,1(η = 0), S1 0 − SI,1(η = 0)) ∣∣ z=0,1 = (0, 0). (3.37) The rotating system (3.23)-(3.29) has stationary Stokes equations by a buoyancy force proportional to the temperature and the solute concentration coupled with heat advection-diffusion equations of the temperature and the solute concentration. The system (3.30)-(3.36) can be considered as one linear system of Stokes equations coupled with a linearized heat advection-diffusion equations. So, the existence of the smooth solutions is the same to those of the incompressible Stokes equations. Since the proof is basic, we omit the details. Proposition 3.1. Assume that T 0 0 , T 1 0 , S 0 0 , S 1 0 , T In,1(t = 0), SIn,1(t = 0) ∈ C∞(D) satisfy the suitable compatibility conditions like (2.7)–(2.9), (3.37) etc. There is a unique and global C∞(D × [0,+∞)) smooth solution of system (3.23)– (3.29) and (3.30)–(3.36), respectively. 8 X. FAN, W. WANG EJDE-2022/33 3.2. Initial layer expansion. We now derive the systems satisfying the initial layer functions. Near t = 0, inserting (3.13) into system (1.1)–(1.4), one obtains λ[∂tu λ a + (uλa · ∇)uλa ] +∇pλa −∆uλa − √ Tak × uλa − (RTT λ a −RSSλa )k = λ[∂t(u λ In + uλI ) + ((uλIn + uλI ) · ∇)(uλIn + uλI )] +∇(pλIn + PλI ) −∆(uλIn + uλI )− √ Tak × (uλIn + uλI )− (RT (TλIn + TλI )−RS(SλIn + SλI ))k = RλIn,u + λ[∂tu λ I + (uλIn · ∇)uλI + uλI · ∇(uλIn + uλI )] +∇pλI −∆uλI − √ Tak × uλI − (RTT λ I −RSSλI )k, (3.38) ∇ · uλa = ∇ · (uλIn + uλI ) = ∇ · uλI = 1∑ i=0 λi∇ · uI,i, (3.39) ∂tT λ a + (uλa · ∇)Tλa −∆Tλa = ∂t(T λ In + TλI ) + ((uλIn + uλI ) · ∇)(TλIn + TλI )−∆(TλIn + TλI ) = RλIn,T + ∂tT λ I + uλIn · ∇TλI + (uλI · ∇)(TλIn + TλI )−∆TλI , (3.40) ∂tS λ a + (uλa · ∇)Sλa − τ∆Sλa = ∂t(S λ In + SλI ) + ((uλIn + uλI ) · ∇)(SλIn + SλI )− τ∆(SλIn + SλI ) = RλIn,S + ∂tS λ I + uλIn · ∇SλI + (uλI · ∇)(SλIn + SλI )− τ∆SλI . (3.41) We consider uIn,i(x, y, z, t) = uIn,i(x, y, z, λη) = uIn,i(x, y, z, 0) + λ∂tu In,i(t = 0)η + · · · , T In,i(x, y, z, t) = T In,i(x, y, z, λη) = T In,i(x, y, z, 0) + λ∂tT In,i(t = 0)η + · · · , SIn,i(x, y, z, t) = SIn,i(x, y, z, λη) = SIn,i(x, y, z, 0) + λ∂tS In,i(t = 0)η + · · · . Next we compare the coefficients of O(λi), i ≥ 0 in the resulting system and derive the systems satisfying the initial layer functions. Taking the coefficient of O(λ−1) in (3.40)–(3.41) as zero, it follows that ∂ηT I,0 = 0, ∂ηS I,0 = 0, By this and (3.2), we obtain T I,0(x, y, z, τ) = 0, (3.42) SI,0(x, y, z, τ) = 0, (3.43) which indicates that the temperature and the solute concentration have no zero order initial layer. Setting the coefficients of O(λ0) in (3.38)−(3.41) as zero and using (3.42)-(3.43), we have ∂ηu I,0 +∇pI,0 −∆uI,0 − √ Tak × uI,0 = 0, (3.44) ∇ · uI,0 = 0, (3.45) ∂ηT I,1 + uI,0 · ∇(T In,0(t = 0)) = 0, (3.46) ∂ηS I,1 + uI,0 · ∇(SIn,0(t = 0)) = 0. (3.47) EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 9 and using the boundary conditions (3.4), (3.27), and the initial condition (3.10), one obtains uI,0|z=0,1 = 0, (3.48) uI,0(η = 0) = u0 0 − uIn,0(t = 0). (3.49) Then (uI,0, pI,0, T I,0, SI,0) satisfy the system (3.42)-(3.45), (3.48), and (3.49) as T I,0(x, y, z, τ) = 0, SI,0(x, y, z, τ) = 0, ∂ηu I,0 +∇pI,0 −∆uI,0 − √ Tak × uI,0 = 0, ∇ · uI,0 = 0, uI,0|z=0,1 = 0, uI,0(η = 0) = u0 0 − uIn,0(t = 0). Now, we investigate the initial and boundary conditions of T I,1, SI,1. Using (3.46)-(3.47) and the decay condition (3.2), we obtain T I,1 = − ∫ ∞ η [uI,0 · ∇(T In,0(t = 0))](s)ds, (3.50) TS,1 = − ∫ ∞ η [uI,0 · ∇(SIn,0(t = 0))](s)ds. (3.51) Then, we restrict (3.50)-(3.51) to η = 0, and replace the right term of result by T I,1 , S I,1 . Namely, T I,1(η = 0) = T I,1 , (3.52) SI,1(η = 0) = S I,1 . (3.53) By restricting (3.50)-(3.51) to z = 0, 1 and using (3.48), one obtains T I,1|z=0,1 = 0, (3.54) SI,1|z=0,1 = 0. (3.55) Moreover, from (3.11)-(3.12) it follows that T I,1(τ = 0) = T 1 0 − T In,1(t = 0), (3.56) TS,1(τ = 0) = S1 0 − SIn,1(t = 0). (3.57) Setting the coefficients of O(λ1) in (3.38), (3.39) as zero, we have ∂ηu I,1 +∇pI,1 −∆uI,1 − √ Tak × uI,1 − (RTT I,1 −RSSI,1)k = −(uI,0 · ∇)uIn,0(t = 0)− (uIn,0(t = 0) · ∇)uI,0 − (uI,0 · ∇)uI,0, (3.58) ∇ · uI,1 = 0. (3.59) From the boundary conditions and the initial condition from (3.5), (3.34), and (3.10), we deduce that uI,1 ∣∣ z=0,1 = 0, (3.60) uI,1(η = 0) = u1 0 − uIn,1(t = 0). (3.61) 10 X. FAN, W. WANG EJDE-2022/33 Hence, we obtain the system ∂ηT I,1 + uI,0 · ∇(T In,0(t = 0)) = 0, ∂ηS I,1 + uI,0 · ∇(SIn,0(t = 0)) = 0, ∂ηu I,1 +∇pI,1 −∆uI,1 − √ Tak × uI,1 − (RTT I,1 −RSSI,1)k = −(uI,0 · ∇)uIn,0(t = 0)− (uIn,0(t = 0) · ∇)uI,0 − (uI,0 · ∇)uI,0, ∇ · uI,1 = 0, uI,1 ∣∣ z=0,1 = 0, uI,1(η = 0) = u1 0 − uIn,1(t = 0). In view of [28], we obtain the following result, whose proof is easy. Proposition 3.2. Assume that (2.1)-(2.3) hold, u0 0, u 1 0, T 0 0 , T 1 0 , S 0 0 , S 1 0 ∈ C∞(D) satisfy the compatibility conditions (2.5)-(2.9). There exist a unique and smooth solution (uI,0, pI,0) to the system (3.42)-(3.45), (3.48) and (3.49) and a unique and smooth solution (uI,1, pI,1, T I,1, SI,1) to the system (3.46)-(3.47) and (3.58)-(3.61) satisfying the exponential decay to zero as η →∞, in the sense that ‖(uI,0, uI,1, T I,1, SI,1)(·, η)‖Hs(D) ≤ Ce−βη, for some constant β > 0 and any s ≥ 1. 3.3. Approximate solution. Simple computations yield λ[∂tu λ a + (uλa · ∇)uλa ] +∇pλa −∆uλa − √ Tak × uλa − (RTT λ a −RSSλa )k = RλIn,u + (∂ηu I,0 +∇pI,0 −∆uI,0 − √ Tak × uI,0) + λ[∂ηu I,1 +∇pI,1 −∆uI,1 − √ Tak × uI,1 − (RTT I,1 −RSSI,1)k + (uI,0 · ∇)uIn,0(t = 0) + (uIn,0(t = 0) · ∇)uI,0 + (uI,0 · ∇)uI,0] +RλI,u, (3.62) ∂tT λ a + (uλa · ∇)Tλa −∆Tλa = RλIn,T + λ−1∂ηT I,0 + (∂ηT I,1 + uI,0 · ∇(T In,0(t = 0)) + (uIn,0 + uI,0) · ∇T I,0 +RλI,T , (3.63) ∂tS λ a + (uλa · ∇)Sλa − τ∆Sλa = RλIn,S + λ−1∂ηS I,0 + (∂ηS I,1 + uI,0 · ∇(SIn,0(t = 0)) + (uIn,0 + uI,0) · ∇SI,0 +RλI,S , (3.64) where the remainders RλI,u, RλI,T and RλI,S , caused by the initial layer, are shown by RλI,u = λ2[η(uI,0 · ∇)∂tu In,0(θ1t) + η∂tu In,0(θ2t) · ∇uI,0 + (uIn,0(t = 0) · ∇)uI,1 + (uIn,1 · ∇)uI,0 + uI,0 · ∇(uIn,1 + uI,1) + uI,1 · ∇(uIn,0(t = 0) + uI,0)] + λ3[η∂tu In,0(θ3t) · ∇uI,1 + η(uI,1 · ∇)∂tu In,0(θ4t) + (uIn,1 · ∇)uI,1 + uI,1 · ∇(uIn,1 + uI,1)], 0 < θi < 1, i = 1, 2, 3, 4, (3.65) EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 11 RλI,T = λ[η(uI,0 · ∇)∂tT In,0(t = 0) + uIn,0(t = 0) · ∇T I,1 + uI,0 · ∇(T In,1(t = 0) + T I,1) + uI,1 · ∇T In,0(t = 0) + (uIn,1 + uI,1) · ∇T I,0 −∆T I,1] + λ2[ 1 2 η2(uI,0 · ∇)∂ttT In,0(θ5t) + η(uI,1 · ∇)∂tT In,0(θ6t) + η(uI,0 · ∇)∂tT In,1(θ7t) + η∂tu In,0(θ8t) · ∇T I,1 + (uIn,1 · ∇)T I,1 + uI,1 · ∇(T In,1 + T I,1)], (3.66) 0 < θi < 1, i = 5, 6, 7, 8, RλI,S = λ[η(uI,0 · ∇)∂tS In,0(t = 0) + uIn,0(t = 0) · ∇SI,1 + uI,0 · ∇(SIn,1(t = 0) + SI,1) + uI,1 · ∇SIn,0(t = 0) + (uIn,1 + uI,1) · ∇SI,0 − τ∆SI,1] + λ2[ 1 2 η2(uI,0 · ∇)∂ttS In,0(θ9t) + η(uI,1 · ∇)∂tS In,0(θ10t) + η(uI,0 · ∇)∂tS In,1(θ11t) + η∂tu In,0(θ12t) · ∇SI,1 + (uIn,1 · ∇)SI,1 + uI,1 · ∇(SIn,1 + SI,1)], (3.67) 0 < θi < 1, i = 9, 10, 11, 12. Hence, the previous computations show that (uλa , p λ a , T λ a , S λ a ) solves the initial- boundary problem λ[∂tu λ a + (uλa · ∇)uλa ] +∇pλa = ∆uλa + √ Tak × uλa + (RTT λ a −RSSλa )k +RλIn,u +RλI,u, (3.68) ∇ · uλa = 0, (3.69) ∂tT λ a + (uλa · ∇)Tλa = ∆Tλa +RλIn,T +RλI,T , (3.70) ∂tS λ a + (uλa · ∇)Sλa = τ∆Sλa +RλIn,S +RλI,S , (3.71) uλa |z=0,1 = 0, (3.72) (Tλa , S λ a )|z=0 = (1, 1), (3.73) (Tλa , S λ a )|z=1 = (0, 0), (3.74) (uλa , T λ a , S λ a )(t = 0) = (u0 0 + λu1 0, T 0 0 + λT 1 0 , S 0 0 + λS1 0), (3.75) where the remainders RλIn,u, RλIn,T , RλIn,S satisfy (3.22) and RλI,u, RλI,T , RλI,S de- fined by (3.65)-(3.67) respectively satisfy ‖RλI,u(., η)‖L∞(D) ≤ Cλ2(η + 1)e−βη, ‖(RλI,T , RλI,S)(·, η)‖L∞(D) ≤ Cλ(η2 + η + 1)e−βη, (3.76) for some constant β > 0 and for any t ∈ [0, S] and any fixed S > 0. The estimate (3.76) can be easily derived by the definitions of RλI,u, RλI,T , RλI,S and Proposition 3.2. 4. Main results Theorem 4.1. Assume that (2.1)-(2.3) hold. And assume that u0 0, u 1 0, T 0 0 , T 1 0 , S 0 0 , S1 0 ∈ C∞(D) satisfy the compatibility conditions (2.5)-(2.9). Then, as λ → 0, for 12 X. FAN, W. WANG EJDE-2022/33 any 0 < T∗ <∞, we have ‖(uλ − uλa , Tλ − Tλa , Sλ − Sλa )‖L∞(0,T∗;L2(D)) ≤ Cλ3/2, (4.1) and we obtain ‖(uλ − uλa , Tλ − Tλa , Sλ − Sλa )‖L2(0,T∗;H1(D)) ≤ Cλ3/2, (4.2) where H1(D)=W 1,2(D), for some positive constants C independent of λ. Proof. We use the classical L2-energy method, and separate the proof into five steps. Step 1. We define the error functions as uλe = uλ − uλa , pλe = pλ − pλa , Tλe = Tλ − Tλa , Sλe = Sλ − Sλa , which satisfies λ[∂tu λ e + (uλa · ∇)uλe + (uλe · ∇)(uλa + uλe )] +∇pλe = ∆uλe + √ Tak × uλe + (RTT λ e −RSSλe )k −RλIn,u −RλI,u, (4.3) ∇ · uλe = 0, (4.4) ∂tT λ e + (uλa · ∇)Tλe + (uλe · ∇)(Tλa + Tλe ) = ∆Tλe −RλIn,T −RλI,T , (4.5) ∂tS λ e + (uλa · ∇)Sλe + (uλe · ∇)(Sλa + Sλe ) = τ∆Sλe −RλIn,S −RλI,S , (4.6) (uλe , T λ e , S λ e )|z=0,1 = (0, 0, 0) (4.7) (uλe , T λ e , S λ e )(t = 0) = (uλ0E , T λ 0E , S λ 0E)(x, y, z). (4.8) where RλIn,u, RλIn,T , RλIn,S , RλI,u, RλI,T and RλI,S are the remainders, uλ0E , Tλ0E and Sλ0E are defined in Section 2. Step 2. Taking the L2-inner product of (4.5) with Tλe and integrating over D, one obtains 1 2 d dt ‖Tλe ‖2L2(D) = ∫ D ∆Tλe T λ e dx dy dz − ∫ D (RλIn,T +RλI,T )Tλe dx dy dz − ∫ D ( uλa · ∇ ) Tλe T λ e dx dy dz − ∫ D ( uλe · ∇ ) (Tλa + Tλe )Tλe dx dy dz =: I1 + I2 + I3 + I4. (4.9) Applying Green’s first formula to I1 and taking into account (4.7) yield I1 = ∮ ∮ Γ Tλe ∂Tλe ∂n dS − ∫ D |∇Tλe |2 dx dy dz = − ∫ D |∇Tλe |2 dx dy dz, (4.10) where Γ represents the boundary surface. Next, we estimate I2 by the estimates (3.22), (3.76). We obtain |I2| ≤ ξ1‖Tλe ‖2L2(D) + C(ξ1)‖RλIn,T +RλI,T ‖2L2(D) ≤ ξ1‖Tλe ‖2L2(D) + C(ξ1)(Cλ4 + Cλ2(η2 + η + 1)2e−2βη), (4.11) where the constant C(ξ1) > 0 is independent of λ, and ξ1 is a small constant. EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 13 According to (3.69), (3.72), and (4.7), I3 = − ∫ D uλa · ∇ ( (Tλe )2 2 ) dx dy dz = − ∫ D ∇ · ( uλa (Tλe )2 2 ) dx dy dz + ∫ D ∇ · uλa (Tλe )2 2 dx dy dz = 0. (4.12) The last integral term I4 can be established similarly as in estimating I3. We obtain I4 = − ∫ D ( uλe · ∇ ) Tλa T λ e dx dy dz − ∫ D (uλe · ∇)Tλe T λ e dx dy dz = − ∫ D (uλe · ∇)Tλa T λ e dx dy dz ≤ | − ∫ D (uλe · ∇)Tλa T λ e dx dy dz| ≤ C(ξ2)‖∇Tλa ‖2L∞(D)‖u λ e‖2L2(D) + ξ2‖Tλe ‖2L2(D), (4.13) where the constant C(ξ2) > 0 is independent of λ, and ξ2 is a small constant. Putting estimates (4.10)-(4.13) into (4.9) yields 1 2 d dt ‖Tλe ‖2L2(D) + ‖∇Tλe ‖2L2(D) ≤ C(ξ1)(Cλ4 + Cλ2(η2 + η + 1)2e−2βη) + (ξ1 + ξ2)‖Tλe ‖2L2(D) + C(ξ2)‖∇Tλa ‖2L∞(D)‖u λ e‖2L2(D). Assuming that ξ1 is small enough but independent of λ, it follows that d dt ‖Tλe ‖2L2(D) + ‖∇Tλe ‖2L2(D) ≤ 2‖∇Tλa ‖2L∞(D)C(ξ2)‖uλe‖2L2(D) + 2C(ξ1)(Cλ4 + Cλ2(η2 + η + 1)2e−2βη). (4.14) Integrating (4.14) over [0, t] for t ∈ [0, T∗] and any fixed T∗ > 0, we have ‖Tλe (t)‖2L2(D) + ∫ t 0 ‖∇Tλe ‖2L2(D)dt ≤ ‖Tλe (t = 0)‖2L2(D) + C ∫ t 0 ‖uλe‖2L2(D)dt+ Cλ3. (4.15) Step 3. Similarly, taking the L2-inner product of (4.6) with Sλe and integrating over D yield 1 2 d dt ‖Sλe ‖2L2(D) = ∫ D τ∆Sλe S λ e dx dy dz − ∫ D (RλIn,S +RλI,S)Sλe dx dy dz − ∫ D ( uλa · ∇ ) Sλe S λ e dx dy dz − ∫ D ( uλe · ∇ ) (Sλa + Sλe )Sλe dx dy dz. Similarly, we derive d dt ‖Sλe ‖2L2(D) + 2τ‖∇Sλe ‖2L2(D) ≤ 2‖∇Sλa‖2L∞(D)C(ξ4)‖uλe‖2L2(D) + 2C(ξ3)(Cλ4 + Cλ2(η2 + η + 1)2e−2βη). (4.16) 14 X. FAN, W. WANG EJDE-2022/33 Integrating (4.16) over [0, t] for t ∈ [0, T∗] and any fixed T∗ > 0, we obtain ‖Sλe (t)‖2L2(D) + 2τ ∫ t 0 ‖∇Sλe ‖2L2(D)dt ≤ ‖Sλe (t = 0)‖2L2(D) + C ∫ t 0 ‖uλe‖2L2(D)dt+ Cλ3. (4.17) Step 4. Similarly, testing (4.3) by uλe and integrating over D, it follows that∫ D (λ[∂tu λ e + (uλa · ∇)uλe + (uλe · ∇)(uλa + uλe )] +∇pλe )uλe dx dy dz = ∫ D (∆uλe + √ Tak × uλe + (RTT λ e −RSSλe )k −RλIn,u −RλI,u)uλe dx dy dz. (4.18) First, taking into account divergence formula, Proposition 3.2, the approximate solution’s property (3.69), (4.4) and the boundary condition (3.72), the left-hand side terms of (4.18) can be expressed as∫ D λ∂tu λ eu λ e dx dy dz = λ 2 d dt ‖uλe‖2L2(D),∫ D λ(uλa · ∇)uλeu λ e dx dy dz = ∫ D λ∇ · ( uλa (uλe )2 2 ) dx dy dz − ∫ D λ∇ · uλa ( uλe )2 2 dx dy dz = 0,∫ D λ(uλe · ∇)(uλa + uλe )uλe dx dy dz = ∫ D λ(uλe · ∇)uλau λ e dx dy dz + ∫ D λ∇ · ( uλe (uλe )2 2 ) dx dy dz − ∫ D λ∇ · uλe (uλe )2 2 dx dy dz = ∫ D λ ( uλe · ∇ ) uλau λ e dx dy dz ≤ ∣∣ ∫ D λ ( uλe · ∇ ) uλau λ e dx dy dz ∣∣ ≤ λ‖∇uλa‖L∞(D)‖uλe‖2L2(D),∫ D ∇pλeuλe dx dy dz = ∫ D ∇ · (pλeuλe ) dx dy dz − ∫ D ∇ · uλepλe dx dy dz = 0. For the right-hand side terms of (4.18), we have∫ D ∆uλeu λ e dx dy dz = ∫ D 3∑ i=1 (∂xx + ∂yy + ∂zz)u λ ieu λ ie dx dy dz EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 15 = ∫ 1 0 ∫ L2 0 3∑ i=1 ( ∂xu λ ieu λ ie ∣∣x=L1 x=0 − ∫ L1 0 (∂xu λ ie) 2dx ) dy dz + ∫ 1 0 ∫ L1 0 3∑ i=1 ( ∂yu λ ieu λ ie ∣∣y=L2 y=0 − ∫ L2 0 (∂yu λ ie) 2dy ) dx dz + ∫ L2 0 ∫ L1 0 3∑ i=1 ( ∂zu λ ieu λ ie|z=1 z=0 − ∫ 1 0 (∂zu λ ie) 2dz ) dx dy = − ∫ D (∇uλe )2 dx dy dz, where we use uλe = (uλ1e, u λ 2e, u λ 3e).∫ D √ Tak × uλeuλe dx dy dz = ∫ D √ Ta(−uλ2e, uλ1e, 0)(uλ1e, u λ 2e, u λ 3e) > dx dy dz = 0,∫ D (RTT λ e −RSSλe )kuλe dx dy dz ≤ | ∫ D (RTT λ e −RSSλe )kuλe dx dy dz| ≤ ξ5‖uλe‖2L2(D) + C(ξ5)(R2 T ‖Tλe ‖2L2(D) +R2 S‖Sλe ‖2L2(D)), − ∫ D (RλIn,u +RλI,u)uλe dx dy dz ≤ | ∫ D (RλIn,u +RλI,u)uλe dx dy dz| ≤ ξ6‖uλe‖2L2(D) + C(ξ6)‖RλIn,u +RλI,u‖2L2(D) ≤ ξ6‖uλe‖2L2(D) + C(ξ6)(Cλ4 + Cλ4(η + 1)2e−2βη), where the constants C(ξi) > 0, (i = 5, 6) are independent on λ, and ξi > 0, (i = 5, 6) are small constants. Substituting the above derivation equations into (4.18) yields λ 2 d dt ‖uλe‖2L2(D) + ‖∇uλe‖2L2(D) ≤ λ‖∇uλa‖L∞(D)‖uλe‖2L2(D) + (ξ5 + ξ6)‖uλe‖2L2(D) + C(ξ5)(R2 T ‖Tλe ‖2L2(D) +R2 S‖Sλe ‖2L2(D)) + C(ξ6)(Cλ4 + Cλ4(η + 1)2e−2βη). By restricting λ to be small enough satisfying λ‖∇uλa‖L∞(D) ≤ Cλ ≤ 1/4 and taking ξ5, ξ6 to be small enough (ξ5 + ξ6 = 1/4) but independent of λ, one obtains λ d dt ‖uλe‖2L2(D) + ‖∇uλe‖2L2(D) ≤ 2C(ξ5)(R2 T ‖Tλe ‖2L2(D) +R2 S‖Sλe ‖2L2(D)) + 2C(ξ6)(Cλ4 + Cλ4(η + 1)2e−2βη), (4.19) that is, λ d dt ‖uλe‖2L2(D) + ‖uλe‖2L2(D) ≤ 2C(ξ5)(R2 T ‖Tλe ‖2L2(D) +R2 S‖Sλe ‖2L2(D)) + 2C(ξ6)(Cλ4 + Cλ4(η + 1)2e−2βη), 16 X. FAN, W. WANG EJDE-2022/33 i.e., d dt (e t λ ‖uλe‖2L2(D)) ≤ [2C(ξ5)(R2 T ‖Tλe ‖2L2(D) +R2 S‖Sλe ‖2L2(D)) + 2C(ξ6)(Cλ4 + Cλ4(η + 1)2e−2βη)]λ−1e t λ . (4.20) Integrating (4.20) over [0, t] for t ∈ [0, T∗] and any fixed T∗ > 0, we have ‖uλe (t)‖2L2(D) ≤ ‖u λ e (t = 0)‖2L2(D) + 2C(ξ5)(R2 T ‖Tλe (t)‖2L∞(0,t;L2(D)) +R2 S‖Sλe (t)‖2L∞(0,t;L2(D))) + 2C(ξ6)Cλ4. (4.21) Step 5. Combining (4.15), (4.17) and (4.21), and restricting ξ2 to be small enough independent of λ yields ‖uλe (t)‖2L2(D) ≤ C‖(u λ e , T λ e , S λ e )(t = 0)‖2L2(D) + C‖uλe (t)‖2L2(0,t;L2(D)) + Cλ3. Using Gronwall’s lemma and (2.4) yields ‖uλe (t)‖2L2(0,t;L2(D)) ≤ Cλ 3, (4.22) ‖uλe (t)‖2L∞(0,T∗;L2(D)) ≤ Cλ 3. (4.23) Inserting (4.22) into (4.15) and (4.17) yields ‖Tλe (t)‖2L2(D) + ∫ t 0 ‖∇Tλe ‖2L2(D)dt ≤ Cλ 3, (4.24) ‖Sλe (t)‖2L2(D) + 2τ ∫ t 0 ‖∇Sλe ‖2L2(D)dt ≤ Cλ 3. (4.25) From the two inequalities above, we have ‖Tλe (t)‖2L∞(0,T∗;L2(D)) ≤ Cλ 3, (4.26) ‖Sλe (t)‖2L∞(0,T∗;L2(D)) ≤ Cλ 3, (4.27) ‖Tλe (t)‖2L2(0,T∗;H1(D)) ≤ Cλ 3, (4.28) ‖Sλe (t)‖2L2(0,T∗;H1(D)) ≤ Cλ 3. (4.29) Simple computations yield λ‖uλe (t)‖2L2(D) + ∫ t 0 ‖∇uλe‖2L2(D)dt ≤ λ‖uλe (t = 0)‖2L2(D) + 2C(ξ5)(R2 T ‖Tλe (t)‖2L2(0,t;L2(D)) +R2 S‖Sλe (t)‖2L2(0,t;L2(D))) + Cλ4. (4.30) Inserting (4.26)-(4.29) into (4.30) yields λ‖uλe (t)‖2L2(D) + ∫ t 0 ‖∇uλe‖2L2(D)dt ≤ Cλ 3. Hence, ‖uλe (t)‖2L2(0,T∗;H1(D)) ≤ Cλ 3. (4.31) Estimates (4.23), (4.26)-(4.29) and (4.31) lead to (4.1)-(4.2) in Theorem 4.1. � Acknowledgments. This work is supported by NNSF of China (No. 11901360). EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 17 References [1] F. H. Busse, Fundamentals of thermal convection, In: Mantle Convection: Plate Tectonics and Global Dynamics, Gordon and Breach, 1989. [2] C. F. Chen, J. S. Turner; Crystallization in a double-diffusive system, J. Geophys. Res., 85 (1980), 2573-2593. [3] W. Chen, D. Han, X. Wang, Y. Zhang; Uniquely Solvable and Energy Stable Decoupled Numerical Schemes for the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq System, J. Sci. Comput., 85 (45) (2020), 1-28. [4] P. Constantin, C. R. Doering; Infinite Prandtl number convection, J. Stat. Phys. 94 (1999), 159-172. [5] J. Cousteix, J. Mauss; Asymptotic analysis and boundary layers, Berlin, Springer, 2007. [6] X. T. Fan, S. Wang, W.-Q. Xu; Initial-boundary layer associated with the 3-D Boussinesq system for Rayleigh-Bénard convection, Electron. J. Differential Equations, 2020 (31) (2020), 1-20. [7] X. T. Fan, W.-Q. Xu, S. Wang, W. Wang; Boundary layers associated with the 3-D Boussi- nesq system for Rayleigh-Bénard convection, Appl. Anal., 99 (12) (2020), 2026-2044. [8] G. P. Galdi, B. Straughan; A nonlinear analysis of the stabilizing effect of rotation in the Bénard problem, Proc. Royal. Soc. London A, 402 (1985), 257-283. [9] E. Grenier, N. Masmoudi; Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations, 22 (1997), 953-975. [10] C. Hallstrom; Heat transfer in rotating infinite Prandtl number convection, Ph.D. thesis, University of Chicago, 2000. [11] U. Hansen, D. A. Yuen; Subcritical double-dfiffusive convection at infinite Prandtl number, Geophys. Astrophys. Fluid Dynam., 47 (1989), 199-224. [12] M. H. Holmes; Introduction to perturbation methods, Springer, New York, 1995. [13] C. H. Hsia, T. Ma, S. H. Wang; Bifurcation and stability of two-dimensional double-diffusive convection, Commun. Pure. Appl. Anal., 7 (1) (2008), 23-48. [14] W. S. Jevons; On the cirrous form of cloud, Philos. Mag. J. Sci., 14 (1857), 22-35. [15] S. Jiang, J. W. Zhang, J. N. Zhao, Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, J. Differential Equations 250 (2011) 3907- 3936. [16] F. F. Jin, M. Ghil; Intraseasonal oscillations in the extratropics: Hopf bifurcation and topo- graphic instabilities, J. Atmos. Sci., 47 (1990), 3007-3022. [17] J. P. Kelliher, R. Temam, X. Wang; Boundary layer associated with the Darcy-Brinkman- Boussinesq model for convection in porous media, Phys. D, 240 (2011), 619-628. [18] J. L. Lions, R. Temam, S. Wang; On the equations of large scale ocean, Nonlinearity 5 (1992), 1007-1053. [19] J. L. Lions; Perturbations Singuliéres Dans les Problèmes aux Limites et en Contrôle Opti- mal, Lecture Notes in Math., Springer-Verlag, New York, 323, 1973. [20] H. V. Ly, E. S. Titi; Global Gevrey regularity for the Bénard convection in a porous medium with zero Darcy-Prandtl number, J. Nonlinear Sci., 9 (1999), 333-362. [21] A. Mazzucato, M. Taylor; Vanishing viscosity plane parallel channel flow and related singular perturbation problems, Anal. PDE, 1 (1) (2008), 35-93. [22] O. A. Oleinik, V. N. Samokhin; Mathematical Models in Boundary Layer, Chapman and Hall, London, 1999. [23] J. Park; Bifurcation of infinite prandtl number rotating convection, Nonlinear Anal., 73 (7) (2010), 2010-2021. [24] J. Park; Thermosolutal convection at infinite prandtl number with or without rotation: bi- furcation and stability in physical space, J. Math. Phys., 52 (2011). [25] J. Park; Thermosolutal convection at infinite Prandtl number: initial layer and infinite Prandtl number limit, Appl. Anal., 92 (9) (2013), 1829-1847. [26] Y. Y. Renardy; Pattern selection for the oscillatory onset in thermosolutal convection. Phys. Fluids A, 5 (6) (1993), 1376-1389. [27] H. Schlichting, K. Gersten; Boundary-layer theory, Springer Press, Berlin, New York, 2000. [28] J. G. Shi, K. Wang, S. Wang; The initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection. Commun. Math. Sci. 5 (1) (2007), 53-66. [29] M. E. Stern; The ‘salt fingers’ and thermohaline convection, Tellus 12 (1960), 172-175. 18 X. FAN, W. WANG EJDE-2022/33 [30] X. Wang; Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure Appl. Math., 57 (2004), 1265-1282. [31] X. Yan; On the limits to convective heat transport at infinite Prandtl number with or without rotation, J. Math. Phys., 45 (7) (2004), 2718-2743. Xiaoting Fan College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Email address: xiao ting fan@163.com Wei Wang (corresponding author) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Email address: weiw10437@163.com 1. Introduction 2. Derivation of initial layer 3. Approximate solution 3.1. Inner expansion 3.2. Initial layer expansion 3.3. Approximate solution 4. Main results Acknowledgments References