Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 31, pp. 1–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu INITIAL-BOUNDARY LAYER ASSOCIATED WITH THE 3-D BOUSSINESQ SYSTEM FOR RAYLEIGH-BÉNARD CONVECTION XIAOTING FAN, SHU WANG, WEN-QING XU Abstract. This article concerns the initial-boundary layer effects of the 3- D incompressible Boussinesq system for Rayleigh-Bénard convection with ill- prepared initial data. We consider a non-slip boundary condition for the veloc- ity field and inhomogeneous Dirichlet boundary condition for the temperature. By means of multi-scale analysis and matched asymptotic expansion methods, we establish an accurate approximating solution for the viscous and diffusive Boussinesq system. We also study the convergence of the infinite Prandtl number limit. 1. Introduction In atmospheric fronts and oceanographic circulation, fluid phenomena with heat transfer have received much attentions (see [11, 20, 22, 26]). Here we deal with the Rayleigh-Bénard convection setting of a horizontal layer of fluid confined by two parallel planes a distance h apart and heated at the bottom plane at temper- ature T2 and cooled at the top plane at temperature T1 < T2. In presence of the gravity force, hot fluid at the bottom rises while cool fluid on top sinks. The dy- namic model consists of 3D incompressible Navier-Stokes equation via a buoyancy force proportional to the temperature coupled with the heat advection-diffusion of temperature [1, 4, 32, 33]. We consider the Boussinesq system with rotation for Rayleigh-Bénard convection [2, 9, 17, 31]. ∂tu+ (u · ∇)u+∇p+ 2Ωe3 × u = ν∆u+ gαe3T, ∇ · u = 0, ∂tT + u · ∇T = κ∆T, u|z=0,h = 0, T |z=0 = T2, T |z=h = T1. The unknown functions u = (u1, u2, u3)T , p and T represent the vector velocity field, the scalar pressure and the scalar temperature of the fluid, respectively. ν and κ are the kinematic viscosity and the thermal diffusion coefficient, respectively. Ω is the rotation rate. e3 denotes the unit upward vector. As usual, e3 := (0, 0, 1)T . 2010 Mathematics Subject Classification. 35B25, 35B40, 35K57. Key words and phrases. Boussinesq system; Rayleigh-Bénard convection; infinite Prandtl number limit; initial-boundary layer; asymptotic expansion. c©2020 Texas State University. Submitted October 12, 2019. Published April 7, 2020. 1 2 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 g denotes the gravity acceleration constant. α stands for the thermal expansion coefficient. For simplicity, we impose the periodicity in the horizontal directions. The mathematical analysis of the nondimensional form has attracted much at- tention. Wang [32] studied the infinite Prandtl number limit and derived the conver- gence of Boussinesq system for Rayleigh-Bénard convection to the infinite Prandtl number system (see, also [33]). This singular limit problem was also investigated by Shi et al. [27], in which they considered initial layer problem by an effective approximating expansion and rigorously proved the convergence of the Boussinesq system to the infinite Prandtl number system. In this article, we study a simplification of Boussinesq system by using the Boussi- nesq approximation and non-dimensionalization, ε[∂tu ε + (uε · ∇)uε] +∇pε + 1 Ek e3 × uε = ∆uε +Rae3T ε, (x, y, z, t) ∈ X × (0, S), (1.1) ∇ · uε = 0, (x, y, z, t) ∈ X × (0, S), (1.2) ∂tT ε + uε · ∇T ε = ε∆T ε, (x, y, z, t) ∈ X × (0, S), (1.3) where X := T2 × [0, 1], T2 = (R1/2π)2 is the torus in R2, S > 0. ε = 1/ √ Pr, Pr = ν/κ is the Prandtl number, Ek = ν/(2Ωh2) is the Ekman number and Ra = gα(T2 − T1)h3 νκ is the Rayleigh number. We suplement the above system with no-slip boundary conditions for the vector velocity and inhomogeneous Dirichlet boundary conditions for the temperature: uε|z=0,1 = 0, (x, y, t) ∈ T2 × (0, S), (1.4) T ε|z=0 = a0(x, y, t), T ε|z=1 = a1(x, y, t), (x, y, t) ∈ T2 × (0, S). (1.5) To the above system we impose the initial conditions uε(t = 0) = uε0(x, y, z), T ε(t = 0) = T 0 0 (x, y, z), (x, y, z) ∈ X. (1.6) This nondimensional form (1.1)–(1.6) of Boussinesq system is different from those in [27, 32, 33]. Motivated by results on the existence and the regularities of the suitable global weak solution in [32, 33], and the related models, see [7, 18, 21], the system (1.1)–(1.6) has also suitable weak solution, standard Galerkin approximation procedure implies the existence of weak solution. In this paper, we are interested in the behavior of the system (1.1)–(1.6) with the infinite Prandtl number limit. As the Prandtl number tends to infinity, it is clear that (1.1) and (1.3) become Stokes-hyperbolic coupled equations (instead of parabolic-parabolic coupled equations), the boundary z = 0, 1 become charac- teristic for the temperature due to the non-slip boundary condition uε|z=0,1 = 0. Then, the boundary condition of temperature should be dropped. This leads to the appearance of the boundary layers. The initial layer of velocity arises from ill-prepared initial data. Compared with the studies in [27, 32, 33], the problem in this paper becomes more complicated due to the appearance of boundary layers and initial layer. This perturbed problems have been studied in many other works see for instance, [3, 10, 13, 23, 24, 25, 28, 35, 36, 37] and references therein. The present work is mainly motivated by [8, 14, 16, 27, 30, 34]. Firstly, we derive the appearance of boundary layers and initial layer in detail. Secondly, we construct EJDE-2020/31 3-D BOUSSINESQ SYSTEM 3 an approximation solution to the original equation as the combination of inner, initial and boundary expansions. We mainly use the matched asymptotic expansion methods of singular perturbation theory [5, 12] and the multi-scale approach [12, 20]. Finally, we consider the convergence of (1.1)–(1.6) to the infinite Prandtl number system as the Prandtl number approaches infinity. The rest of this article is organized as follows. The derivation of initial and boundary layers is given in Section 2. The main results are stated in Section 3. In Section 4, we establish the approximating solution. The proof of main results is shown in Section 5. 2. Derivation of initial and boundary layers In this section, by employing the singular perturbation theory [5, 12] and the Stokes operator [15, 17, 19], we consider the behavior of the solution when the Prandtl number tends to ∞, i.e., ε tends to 0. Letting ε = 0 in (1.1)–(1.6), we obtain ∇p0 + 1 Ek e3 × u0 = ∆u0 +Rae3T 0, (2.1) ∇ · u0 = 0, (2.2) ∂tT 0 + ( u0 · ∇ ) T 0 = 0, (2.3) u0|z=0,1 = 0. (2.4) Here we consider the initial data T 0(t = 0) = T 0 0 (x, y, z), where T 0 0 (x, y, z) is the initial value of T ε(x, y, z). We first study the boundary conditions of T 0. Restricting (2.3) to z = 0, 1 and then plugging (2.4) into the resulting equation, we have ∂tT 0|z=0,1 = 0. (2.5) The compatibility conditions for (1.5)–(1.6) yield that T 0 0 (x, y, z)|z=0 = a0(x, y, t = 0), T 0 0 (x, y, z)|z=1 = a1(x, y, t = 0). (2.6) The combination of (2.5) and (2.6) implies T 0|z=0 = a0(x, y, t = 0), T 0|z=1 = a1(x, y, t = 0). (2.7) Moreover, comparing (1.5) and (2.7), we obtain that lim ε→0 a0(x, y, t) 6= a0(x, y, t = 0), lim ε→0 a1(x, y, t) 6= a1(x, y, t = 0), i.e., lim ε→0 (T ε|z=0) 6= T 0|z=0, lim ε→0 (T ε|z=1) 6= T 0|z=1. This leads to the appearance of the boundary layers of the scalar temperature. Then we turn to derive the initial conditions of u0. Because of the singularity of perturbation, generally speaking, the limit of uε0(x, y, z) as ε → 0 can not be satisfied by the velocity u0(t = 0) in the limit system. Restricting (2.1), (2.2) and (2.4) to t = 0 gives ∇p0(t = 0) + 1 Ek e3 × u0(t = 0) = ∆u0(t = 0) +Rae3T 0(t = 0), ∇ · u0(t = 0) = 0, 4 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 u0|z=0,1(t = 0) = 0. By solving above system, we know that the value of u0(t = 0) is determined by the initial data of the temperature, while limε→0 u ε 0 can be given arbitrarily and independently of T 0(t = 0), so limε→0 u ε 0 6= u0(t = 0). This leads to the appearance of an initial layer of the vector velocity. Hence, the infinite Prandtl number limit of the initial and boundary value problem (1.1)–(1.6) is a singular problem involving boundary layers and an initial layer. 3. Main results Let (uε, pε, T ε) be the global weak solution of (1.1)–(1.6) in the Leray’s sense. We assume that the initial data has an expansion up to the 0 order as follows (uε, T ε)(t = 0) = (u0 0 + uε0E , T 0 0 )(x, y, z), (3.1) where u0 0 and T 0 0 are all C∞(X) functions, uε0E denotes the remainders except for 0 order term for the initial data of uε, uε0E ∈ C∞(X) satisfies ‖uε0E(x, y, z)‖L2(X) ≤ C √ ε. (3.2) Theorem 3.1. Assume that (3.1) holds. Also, assume that u0 0 and T 0 0 ∈ C∞(X), a0, a1 ∈ C∞(T2 × [0,+∞)) satisfy some compatibility conditions as (2.6) and (uεa, p ε a, T ε a) is an approximate solution of the system (1.1)–(1.6). Then, as ε → 0, for any 0 < S <∞, we have ‖(uε − uεa, T ε − T εa)‖L∞(0,S;L2(X)) ≤ Cε1/4, (3.3) ‖∇(uε − uεa)‖2L2(0,S;L2(X)) + ε‖∇(T ε − T εa)‖2L2(0,S;L2(X)) ≤ C √ ε, ‖uε − uεa‖L2(0,S;H1(X)) ≤ Cε1/4, (3.4) where H1(X) = W 1,2(X), for some positive constants C independent of ε. The functions uεa, T εa , and pεa are defined in Section 4. By a standard method [6, 19, 29], we formulate any mth, m = 0, 1, 2, . . . , order compatibility conditions. 4. Approximate solution In this section, we construct the approximate solution including the inner ex- pansion away from z = 0, 1, initial layer expansion near t = 0 and the boundary layers expansion near z = 0, 1. Some useful properties of approximating solution are also derived. It is easy to see that (uε, pε, T ε)(x, y, z, t) ∼ ∞∑ i=0 ( √ ε)i(uIn,i(x, y, z, t) + uI,i(x, y, z, τ), pIn,i(x, y, z, t) + pI,i(x, y, z, τ), T In,i(x, y, z, t) + f(z)TB,i+ (x, y, Z, t) + h(z)TB,i− (x, y, Z, t)), where τ = t/ε is the fast time variable, Z = z√ ε and Z = 1−z√ ε are the fast space variables. ε and √ ε are the lengths of the initial layer and boundary layers, respec- tively. (uIn,i, pIn,i, T In,i)(x, y, z, t) are the inner functions for the velocity field, pressure and temperature field, respectively, independent of ε. (uI,i, pI,i)(x, y, z, τ) are the initial layer functions near t = 0 for the velocity field and pressure, respec- tively. The initial layer functions satisfy that uI,i, pI,i decay to zero exponentially, EJDE-2020/31 3-D BOUSSINESQ SYSTEM 5 as τ → ∞. TB,i+ (x, y, Z, t) represents the lower boundary layer function of temper- ature field near z = 0. TB,i− (x, y, Z, t) stands for the upper boundary layer function of temperature field near z = 1. The boundary layer functions satisfy that TB,i+ decays to zero exponentially, as Z → ∞, and TB,i− decays to zero exponentially, as Z → ∞. Here f(z) and h(z) are smooth C2 cut-off functions [23], near z = 0, we define f(z) = { 1, z ∈ [0, 1 4 ], 0, z ∈ [ 1 2 , 1]. Similarly, near z = 1, we define h(z) = { 0, z ∈ [0, 1 4 ], 1, z ∈ [ 1 2 , 1], which, in turn, imply that f(0) = h(1) = 1 and f(1) = f ′(1) = f ′′(1) = f ′(0) = f ′′(0) = h(0) = h′(0) = h′′(0) = h′(1) = h′′(1) = 0. We assume that the asymptotic expansion of the system (1.1)–(1.6) including initial and boundary corrections is of the form (uεa, p ε a, T ε a)(x, y, z, t) = ( uIn,0(x, y, z, t) + uI,0(x, y, z, τ), pIn,0(x, y, z, t) + pI,0(x, y, z, τ), T In,0(x, y, z, t) + f(z)TB,0+ (x, y, Z, t) + h(z)TB,0− (x, y, Z, t) ) . (4.1) Moreover, to match the boundary and initial conditions (1.4)–(1.6), we impose the following restrictions (uIn,0 + uI,0)|z=0,1 = 0, (4.2) T In,0|z=0 + TB,0+ |Z=0 = a0(x, y, t), T In,0|z=1 + TB,0− |Z=0 = a1(x, y, t), (4.3) uIn,0(t = 0) + uI,0(τ = 0) = u0 0, (T In,0 + f(z)TB,0+ + h(z)TB,0− )(t = 0) = T 0 0 , (4.4) We discuss the construction of the inner, initial layer and boundary layers functions (uεa, p ε a, T ε a) := (uεIn, p ε In, T ε In) + (uεI , p ε I , T ε B), (4.5) where (uεIn, p ε In, T ε In) = (uIn,0, pIn,0, T In,0), (4.6) (uεI , p ε I) = (uI,0, pI,0), (4.7) T εB = TB,0 = f(z)TB,0+ + h(z)TB,0− . (4.8) First, we study inner expansion away from the boundary z = 0 and z = 1 in Section 4.1. Then, we study the initial layer expansion near t = 0 and lower boundary layer expansion near z = 0 in Section 4.2 and the upper boundary layer expansion near z = 1 can be used by the similar method in Section 4.3. Finally, we consider the approximating solution in Section 4.4. 6 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 4.1. Inner expansion. Away from the boundary z = 0 and z = 1, from (4.1), the solution to (1.1)–(1.5) has the expansion (uε, pε, T ε) (x, y, z, t) ∼ ∞∑ i=0 ( √ ε)i(uIn,i, pIn,i, T In,i)(x, y, z, t). First, inserting above expansion into (1.1)–(1.5) and using direct calculations, we obtain ∞∑ i=0 ( √ ε)i ( ε[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i + 1 Ek e3 × uIn,i −∆uIn,i −Rae3T In,i ) = 0, ∞∑ i=0 ( √ ε)i∇ · uIn,i = 0, ∞∑ i=0 ( √ ε)i ( ∂tT In,i + i∑ j=0 uIn,j · ∇T In,i−j − ε∆T In,i ) = 0, ∞∑ i=0 ( √ ε)iuIn,i|z=0,1 = 0, ∞∑ i=0 ( √ ε)iT In,i(t = 0) = T 0 0 . Then (uεIn, p ε In, T ε In) satisfies ε[∂tu ε In + (uεIn · ∇)uεIn] +∇pεIn + 1 Ek e3 × uεIn = ∆uεIn +Rae3T ε In +RεIn,u, (4.9) ∇ · uεIn = 0, (4.10) ∂tT ε In + (uεIn · ∇)T εIn = ε∆T εIn +RεIn,T , (4.11) uεIn|z=0,1 = 0, (4.12) T εIn(t = 0) = T 0 0 , (4.13) where the remainders are RεIn,u = − ∞∑ i=1 ( √ ε)i(ε[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i + 1 Ek e3 × uIn,i −∆uIn,i −Rae3T In,i), and RεIn,T = − ∞∑ i=1 ( √ ε)i ( ∂tT In,i + i∑ j=0 uIn,j · ∇T In,i−j − ε∆T In,i ) . So we know that RεIn,u and RεIn,T satisfy the estimates ‖(RεIn,u, RεIn,T )‖L∞(0,S;Hs(X)) ≤ C √ ε, (4.14) for S > 0 and s ≥ 1. Denote C by a positive constant, independent of ε. EJDE-2020/31 3-D BOUSSINESQ SYSTEM 7 We now set the coefficient of O ( ( √ ε)0 ) in (4.9)–(4.11) as zero and use the initial and boundary conditions (4.12)–(4.13). We obtain a system for (uIn,0, pIn,0, T In,0), ∇pIn,0 + 1 Ek e3 × uIn,0 = ∆uIn,0 +Rae3T In,0, (4.15) ∇ · uIn,0 = 0, (4.16) ∂tT In,0 + (uIn,0 · ∇)T In,0 = 0, (4.17) uIn,0|z=0,1 = 0, (4.18) T In,0(t = 0) = T 0 0 (x, y, z). (4.19) The rotating system (4.15)–(4.19) has stationary Stokes equations via a buoy- ancy force proportional to temperature coupled with heat advection of the tem- perature. Hence, 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 4.1. Assume that T 0 0 ∈ C∞(X) satisfies some compatibility condi- tions like (2.6). There is a unique and global C∞(X × [0,+∞)) smooth solution to the system (4.15)–(4.19). Now we turn to the construction of the initial layer and lower boundary layer function. 4.2. Initial layer and lower boundary layer expansion. We now derive the systems satisfying the initial layer and lower boundary layer function, which is divided into six steps. Step 1. Near t = 0, z = 0, f(z) = 1 and h(z) = 0, equation (4.8) turns into T εB = TB,0 = TB,0+ (x, y, Z, t). (4.20) Step 2. Inserting (4.5) into (1.1)–(1.3), then using direct calculation yields ε[∂tu ε a + (uεa · ∇)uεa] +∇pεa + 1 Ek e3 × uεa −∆uεa −Rae3T ε a = ε[∂t(u ε In + uεI) + ((uεIn + uεI) · ∇)(uεIn + uεI)] +∇(pεIn + P εI ) + 1 Ek e3 × (uεIn + uεI)−∆(uεIn + uεI)−Rae3(T εIn + T εB) = RεIn,u + ε[∂tu ε I + (uεIn · ∇)uεI + uεI · ∇(uεIn + uεI)] +∇pεI + 1 Ek e3 × uεI −∆uεI −Rae3T ε B , (4.21) ∇ · uεa = ∇ · (uεIn + uεI) = 0, (4.22) ∂tT ε a + (uεa · ∇)T εa − ε∆T εa = ∂t(T ε In + T εB) + ((uεIn + uεI) · ∇)(T εIn + T εB)− ε∆(T εIn + T εB) = RεIn,T + uεI · ∇T εIn + ∂tT ε B + (uεIn · ∇)T εB − ε∆T εB + uεI · ∇T εB . (4.23) 8 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 Step 3. We deduce after plugging (4.7), (4.20) into (4.21) that ε[∂tu ε a + (uεa · ∇)uεa] +∇pεa + 1 Ek e3 × uεa −∆uεa −Rae3T ε a = RεIn,u + ε[∂tu ε I + (uεIn · ∇)uεI + uεI · ∇(uεIn + uεI)] +∇pεI + 1 Ek e3 × uεI −∆uεI −Rae3T ε B = RεIn,u + (∂τu I,0 +∇pI,0 + 1 Ek e3 × uI,0 −∆uI,0) + ε[(uIn,0 · ∇)uI,0 + (uI,0 · ∇)(uI,0 + uIn,0)]−Rae3T B,0 + . (4.24) Step 4. Now we set the coefficient of order O ( ( √ ε)0 ) in (4.24) as zero and use (4.6), (4.7), (4.16), the boundary conditions (4.2), (4.18) and initial condition (4.4). Then we have the following system for the initial layer function (uI,0, pI,0)(x, y, z, τ) ∂τu I,0 + 1 Ek e3 × uI,0 +∇pI,0 = ∆uI,0, (4.25) ∇ · uI,0 = 0, (4.26) uI,0|z=0,1 = 0, (4.27) uI,0|τ=0 = u0 0 − uO,0(t = 0), (4.28) uI,0 → 0, as τ → +∞. (4.29) As in [27], we obtain the property of the initial layer function. Proposition 4.2. Let the assumptions of Theorem 3.1 hold. There is a unique and smooth solution ( uI,0, pI,0 ) of (4.25)–(4.29) satisfying the exponential decay to zero as τ →∞, ‖uI,0(·, τ)‖Hs(X) ≤ Ce−λτ , for some positive constants C, λ and any s ≥ 1. Step 5. We deduce after plugging (4.7), (4.20) into (4.23) that ∂tT ε a + (uεa · ∇)T εa − ε∆T εa = RεIn,T + uεI · ∇T εIn + ∂tT ε B + (uεIn · ∇)T εB − ε∆T εB + uεI · ∇T εB = RεIn,T + (uI,0 · ∇)T In,0 + ∂tT B,0 + + (uIn,0 · ∇)TB,0+ − ε∆TB,0+ + (uI,0 · ∇)TB,0+ , where ∂tT B,0 + + (uIn,0 · ∇)TB,0+ − ε∆TB,0+ = ∂tT B,0 + + ( uIn,01 ∂x + uIn,02 ∂y ) TB,0+ + 1√ ε uIn,03 ∂ZT B,0 + − ε(∂xx + ∂yy)TB,0+ − ∂ZZTB,0+ = ∂tT B,0 + + [ (uIn,01 (x, y, 0, t) + √ ε∂zu In,0 1 (z = 0)Z + · · · )∂x + (uIn,02 (x, y, 0, t) + √ ε∂zu In,0 2 (z = 0)Z + · · · )∂y ] TB,0+ + 1√ ε ( uIn,03 (x, y, 0, t) + √ ε∂zu In,0 3 (z = 0)Z + · · · ) ∂ZT B,0 + − ε(∂xx + ∂yy)TB,0+ − ∂ZZTB,0+ , (4.30) EJDE-2020/31 3-D BOUSSINESQ SYSTEM 9 where uIn,0 = ( uIn,01 , uIn,02 , uIn,03 ) . Here the Taylor series expansion is used uIn,0(x, y, z, t) = uIn,0(x, y, √ εZ, t) = uIn,0(x, y, 0, t) + √ ε∂zu In,0(z = 0)Z + · · · . Step 6. Now we set the coefficients of order O ( ( √ ε)k ) (k = −1, 0) in (4.30) equal to zero. First, collecting O ( ( √ ε)−1 ) terms we have uIn,03 (z = 0)∂ZT B,0 + = 0, from which, together with (2.4), we cannot obtain more properties of ∂ZT B,0 + . Next, collecting O(( √ ε)0) terms we obtain ∂tT B,0 + + Z∂zu In,0 3 (z = 0)∂ZT B,0 + = ∂ZZT B,0 + . (4.31) Restricting (2.2) to z = 0, one gets ∇ · uIn,0|z=0 = 0, from which, with (2.4), we have ∂zu In,0 3 (z = 0) = 0. (4.32) Plugging (4.32) into (4.31), we obtain ∂tT B,0 + = ∂ZZT B,0 + . (4.33) Finally, by using the boundary condition (2.7) (4.3) and the initial condition (4.4), equation (4.19) yields TB,0+ |Z=0 = a0(x, y, t)− a0(x, y, t = 0), (4.34) TB,0+ (t = 0) = 0. (4.35) Thus, the lower boundary layer function TB,0+ (x, y, Z, t) satisfies the system ∂tT B,0 + = ∂ZZT B,0 + , TB,0+ |Z=0 = a0(x, y, t)− a0(x, y, t = 0), TB,0+ (t = 0) = 0, TB,0+ → 0, as Z → +∞. Now we state the property of the lower boundary layer function. Proposition 4.3. Let the assumptions of Theorem 3.1 hold. Then there exists a unique and smooth solution TB,0+ (x, y, Z, t) of the above system satisfying ‖ ( TB,0+ , Z∂ZT B,0 + ) ‖L∞(0,S;L2(X)) ≤ Cε1/4, for any t ∈ [0, S], positive constant C independent of ε. Proof. The proof in a straightforward and consists of two steps. Step 1. First we estimate ‖TB,0+ (x, y, Z, t)‖L∞(0,S;L2(X)). Multiplying (4.33) by TB,0+ (x, y, Z, t) and integrating over X, we obtain 1 2 d dt ‖TB,0+ ‖2L2(X) = ∫ X ∂ZZT B,0 + TB,0+ dx dy dz =: J1. (4.36) 10 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 For J1 we have J1 = ∫ T2 ∫ 1 0 ∂ZZT B,0 + TB,0+ dz dy dx = ∫ T2 ∫ σ 0 ∂ZZT B,0 + TB,0+ dz dy dx = √ ε ∫ T2 ∫ σ√ ε 0 ∂ZZT B,0 + TB,0+ dZ dy dx = √ ε ∫ T2 ( ∂ZT B,0 + TB,0+ )∣∣Z= σ√ ε Z=0 dy dx− √ ε ∫ T2 ∫ σ√ ε 0 ( ∂ZT B,0 + )2 dZ dy dx = √ ε ∫ T2 ( ∂ZT B,0 + TB,0+ )∣∣Z= σ√ ε Z=0 dy dx− ∫ T2 ∫ σ 0 ( ∂ZT B,0 + )2 dz dy dx = √ ε ∫ T2 ( ∂ZT B,0 + TB,0+ )∣∣Z= σ√ ε Z=0 dy dx− ∫ X ( ∂ZT B,0 + )2 dx dy dz, (4.37) where we have used integration by parts. Here σ is a sufficiently small positive constant. Then, inserting the estimates derived in (4.37) into (4.36) leads to the inequality 1 2 d dt ‖TB,0+ ‖2L2(X) + ∫ X ( ∂ZT B,0 + )2 dx dy dz ≤ ∣∣√ε∫ 2π 0 ∫ 2π 0 ( ∂ZT B,0 + TB,0+ )∣∣Z= σ√ ε Z=0 dy dx ∣∣ ≤ C √ ε. (4.38) Finally, integrating (4.38) with respect to t over [0, t], for any t ∈ [0, S] and any fixed S > 0, together with (4.35), yields ‖TB,0+ ‖2L2(X) + 2 ∫ t 0 ∫ X ( ∂ZT B,0 + )2 dx dy dzdξ ≤ C √ ε, which implies ‖TB,0+ ‖L∞(0,S;L2(X)) ≤ Cε1/4. Step 2. We now prove an estimate for ‖Z∂ZTB,0+ ‖L∞(0,S;L2(X)). First, multiplying (4.33) by ∂tT B,0 + and integrating it over X, by integrating by parts, yields 0 = ∫ X ( ∂tT B,0 + )2 dx dy dz − ∫ X ∂tT B,0 + ∂ZZT B,0 + dx dy dz = ∫ X ( ∂tT B,0 + )2 dx dy dz + 1 2 d dt ∫ X ( ∂ZT B,0 + )2 dx dy dz − ∫ X ∂Z(∂tT B,0 + ∂ZT B,0 + ) dx dy dz. (4.39) EJDE-2020/31 3-D BOUSSINESQ SYSTEM 11 Let us consider the last term on the right-hand side of (4.39), one has − ∫ X ∂Z(∂tT B,0 + ∂ZT B,0 + ) dx dy dz = − ∫ T2 ∫ σ 0 ∂Z(∂tT B,0 + ∂ZT B,0 + ) dz dy dx = − √ ε ∫ T2 ∫ σ√ ε 0 ∂Z(∂tT B,0 + ∂ZT B,0 + ) dZ dy dx = − √ ε ∫ T2 ( ∂tT B,0 + ∂ZT B,0 + )∣∣Z= σ√ ε Z=0 dy dx. (4.40) Substituting the above estimate into (4.39), we obtain∫ X ( ∂tT B,0 + )2 dx dy dz + 1 2 d dt ∫ X ( ∂ZT B,0 + )2 dx dy dz = √ ε ∫ T2 ( ∂tT B,0 + ∂ZT B,0 + )∣∣Z= σ√ ε Z=0 dy dx ≤ ∣∣√ε∫ T2 ( ∂tT B,0 + ∂ZT B,0 + )∣∣Z= σ√ ε Z=0 dy dx ∣∣ ≤ C √ ε, which leads to ‖∂tTB,0+ ‖L∞(0,S;L2(X)) ≤ Cε1/4. (4.41) Then, one multiplies (4.33) by Z2∂tT B,0 + and integrates it over X, integrating by parts we obtain 0 = ∫ X Z2 ( ∂tT B,0 + )2 dx dy dz − ∫ X Z2∂tT B,0 + ∂ZZT B,0 + dx dy dz = ∫ X Z2(∂tT B,0 + )2 dx dy dz − ∫ X ∂Z(Z2∂tT B,0 + ∂ZT B,0 + ) dx dy dz + ∫ X ∂Z(Z2∂tT B,0 + )∂ZT B,0 + dx dy dz = ∫ X Z2(∂tT B,0 + )2 dx dy dz − ∫ X ∂Z(Z2∂tT B,0 + ∂ZT B,0 + ) dx dy dz + 1 2 d dt ∫ X Z2(∂ZT B,0 + )2 dx dy dz + 2 ∫ X Z∂ZT B,0 + ∂tT B,0 + dx dy dz. his can be reduced to∫ X Z2 ( ∂tT B,0 + )2 dx dy dz + 1 2 d dt ∫ X Z2 ( ∂ZT B,0 + )2 dx dy dz = ∫ T2 ∫ σ 0 ∂Z(Z2∂tT B,0 + ∂ZT B,0 + ) dz dy dx− 2 ∫ X Z∂ZT B,0 + ∂tT B,0 + dx dy dz = √ ε ∫ T2 ∫ σ√ ε 0 ∂Z(Z2∂tT B,0 + ∂ZT B,0 + ) dZ dy dx− 2 ∫ X Z∂ZT B,0 + ∂tT B,0 + dx dy dz = √ ε ∫ T2 (Z2∂tT B,0 + ∂ZT B,0 + )| Z= σ√ ε Z=0 dy dx− 2 ∫ X Z∂ZT B,0 + ∂tT B,0 + dx dy dz ≤ | √ ε ∫ T2 (Z2∂tT B,0 + ∂ZT B,0 + )(Z = σ√ ε ) dy dx| 12 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 + |2 ∫ X Z∂ZT B,0 + ∂tT B,0 + dx dy dz| ≤ C √ ε+ η1‖Z∂ZTB,0+ ‖2L2(X) + C(η1)‖∂tTB,0+ ‖2L2(X), (4.42) where we apply Hölder inequality and Young inequality. η1 is a small constant, C(η1) > 0 is a constant independent of ε. Finally, plugging (4.41) into (4.42) and using Gronwall’s inequality that ‖Z∂tTB,0+ ‖L2(0,S;L2(X)) + ‖Z∂ZTB,0+ ‖L∞(0,S;L2(X)) ≤ Cε1/4. This completes the proof of ‖Z∂ZTB,0+ ‖L∞(0,S;L2(X)) ≤ Cε1/4. � 4.3. Initial layer and upper boundary layer expansion. Near z = 1, f(z) = 0 and h(z) = 1, instead of T εB = TB,0 = TB,0+ (x, y, Z, t) in (4.8), we obtain T εB = TB,0 = TB,0− (x, y, Z, t). The upper boundary layer function TB,0− ( x, y, Z, t ) has corresponding results with minor difference in some equations. ε[∂tu ε a + (uεa · ∇)uεa] +∇pεa + 1 Ek e3 × uεa −∆uεa −Rae3T ε a −∆(uεIn + uεI)−Rae3(T εIn + T εB) = RεIn,u + ε[∂tu ε I + (uεIn · ∇)uεI + uεI · ∇(uεIn + uεI)] +∇pεI + 1 Ek e3 × uεI −∆uεI −Rae3T ε B , ∇ · uεa = ∇ · (uεIn + uεI) = 0, ∂tT ε a + (uεa · ∇)T εa − ε∆T εa = RεIn,T + uεI · ∇T εIn + ∂tT ε B + (uεIn · ∇)T εB − ε∆T εB + uεI · ∇T εB . By a similar method, we have ∂tT ε a + (uεa · ∇)T εa − ε∆T εa = RεIn,T + (uI,0 · ∇)T In,0 + ∂tT B,0 − + (uIn,0 · ∇)TB,0− − ε∆TB,0− + (uI,0 · ∇)TB,0− , where ∂tT B,0 − + (uIn,0 · ∇)TB,0− − ε∆TB,0− = ∂tT B,0 − + (uIn,01 ∂x + uIn,02 ∂y)TB,0− − 1√ ε uIn,03 ∂ZT B,0 − − ε(∂xx + ∂yy)TB,0− − ∂ZZT B,0 − = ∂tT B,0 − + [(uIn,01 (x, y, 1, t)− √ ε∂zu In,0 1 (z = 1)Z + · · · )∂x + (uIn,02 (x, y, 1, t)− √ ε∂zu In,0 2 (z = 1)Z + · · · )∂y]TB,0− − 1√ ε (uIn,03 (x, y, 1, t)− √ ε∂zu In,0 3 (z = 1)Z + · · · )∂ZT B,0 − − ε(∂xx + ∂yy)TB,0− − ∂ZZT B,0 − . Here, we also use the Taylor series expansion uIn,0(x, y, z, t) = uIn,0(x, y, 1− √ ε Z, t) = uIn,0(x, y, 1, t)− √ ε∂zu In,0(z = 1)Z+· · · . Applying a similar approach to the one in Section 4.2, we find that the upper boundary layer function TB,0− (x, y, Z, t). It satisfies ∂tT B,0 − = ∂ZZT B,0 − , EJDE-2020/31 3-D BOUSSINESQ SYSTEM 13 TB,0− |Z=0 = a1(x, y, t)− a1(x, y, t = 0), TB,0− (t = 0) = 0, TB,0− → 0, as Z → +∞. Now we state the property of the upper boundary layer function. Proposition 4.4. Let the assumptions of Theorem 3.1 hold. Then there exists a unique and smooth solution TB,0− (x, y, Z, t) to the above system satisfying ‖ ( TB,0− , Z∂ZT B,0 − ) ‖L∞(0,S;L2(X)) ≤ Cε1/4, for any t ∈ [0, S], C is a positive constant independent of ε. 4.4. Approximate solution. Now we study approximate solution (4.1). The com- bination of inner function, initial and boundary layers expansions in Sections 4.1–4.3 implies ε[∂tu ε a + (uεa · ∇)uεa] +∇pεa + 1 Ek e3 × uεa −∆uεa −Rae3T ε a = RεIn,u + (∂τu I,0 +∇pI,0 + 1 Ek e3 × uI,0 −∆uI,0) + ε[(uIn,0 · ∇)uI,0 + (uI,0 · ∇)(uI,0 + uIn,0)]−Rae3T B,0 = RεIn,u + ε[(uIn,0 · ∇)uI,0 + (uI,0 · ∇)(uI,0 + uIn,0)]−Rae3T B,0 =: RεIn,u +RεC,u, and ∂tT ε a + (uεa · ∇)T εa − ε∆T εa = RεIn,T + (uI,0 · ∇)T In,0 + ∂tT B,0 + (uIn,0 · ∇)TB,0 − ε∆TB,0 + (uI,0 · ∇)TB,0 = RεIn,T + (uI,0 · ∇)T In,0 + (uIn,0 · ∇)TB,0 − ε(∂xx + ∂yy)TB,0 + (uI,0 · ∇)TB,0 =: RεIn,T +RεC,T , where TB,0 = f(z)TB,0+ + h(z)TB,0− , the remainders RεC,u and RεC,T , caused by the initial layer and the boundary layer, are given exactly by RεC,u = ε[(uI,0 · ∇)uIn,0 + (uIn,0 · ∇)uI,0 + (uI,0 · ∇)uI,0]−Rae3T B,0, (4.43) RεC,T = (uI,0 · ∇)T In,0 + (uIn,0 · ∇)TB,0 − ε(∂xx + ∂yy)TB,0 + (uI,0 · ∇)TB,0. (4.44) Therefore, (uεa, p ε a, T ε a) solves the initial-boundary problem ε[∂tu ε a + (uεa · ∇)uεa] +∇pεa + 1 Ek e3 × uεa = ∆uεa +Rae3T ε a +RεIn,u +RεC,u, (4.45) ∇ · uεa = 0, (4.46) ∂tT ε a + (uεa · ∇)T εa = ε∆T εa +RεIn,T +RεC,T , (4.47) uεa|z=0,1 = 0, (4.48) T εa |z=0 = a0(x, y, t), T εa |z=1 = a1(x, y, t), (4.49) (uεa, T ε a)(t = 0) = (u0 0, T 0 0 ), (4.50) 14 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 where the remainders RεIn,u and RεIn,T satisfy the estimate (4.14), and RεC,u, R ε C,T defined by (4.43) and (4.44) respectively satisfy ‖RεC,u(t)‖L2(X) ≤ Cεe−λτ + Cε1/4, (4.51) ‖RεC,T (t)‖L2(X) ≤ Ce−λτ + Cε1/4, (4.52) for some positive constants C and λ. The estimates (4.51), (4.52) can be obtained by (4.43)–(4.44) and Propositions 4.2–4.4. 5. Proof of main results In this section, we use the classical L2-energy method to prove Theorem 3.1. Without loss of generality, in the following, we denote C by a positive generic constant independent of ε. Noting that C may depend upon S for any fixed S > 0. We divide the proof into six steps. Step 1. We define the error functions uεe = uε − uεa, pεe = pε − pεa, T εe = T ε − T εa , which satisfies ε[∂tu ε e + (uεa · ∇)uεe + (uεe · ∇)(uεa + uεe)] +∇pεe + 1 Ek e3 × uεe = ∆uεe +Rae3T ε e −RεIn,u −RεC,u, (5.1) ∇ · uεe = 0, (5.2) ∂tT ε e + (uεa · ∇)T εe + (uεe · ∇)(T εa + T εe ) = ε∆T εe −RεIn,T −RεC,T , (5.3) uεe|z=0,1 = 0, (5.4) T εe |z=0,1 = 0, (5.5) (uεe, T ε e )(t = 0) = (uε0E , 0), (5.6) where RεIn,u, RεIn,u, RεC,u and RεC,T are the remainders, uε0E is defined in Section 3. Step 2. Testing the velocity equation (5.1) by uεe and integrating over X with respect to (x, y, z), we obtain∫ X ( ε[∂tu ε e + (uεa · ∇)uεe + (uεe · ∇)(uεa + uεe)] +∇pεe + 1 Ek e3 × uεe ) uεe dx dy dz = ∫ X ( ∆uεe +Rae3T ε e −RεIn,u −RεC,u ) uεe dx dy dz. (5.7) First, by the divergence formula, divergence theorem, (4.46), (5.2) and the boundary condition (4.48), (5.4), we deal with the left-hand side terms of (5.7).∫ X ε∂tu ε eu ε e dx dy dz = ε 2 d dt ‖uεe‖2L2(X),∫ X ε(uεa · ∇)uεeu ε e dx dy dz = ∫ X ε∇ · ( uεa (uεe) 2 2 ) dx dy dz − ∫ X ε∇ · uεa (uεe) 2 2 dx dy dz = 0, EJDE-2020/31 3-D BOUSSINESQ SYSTEM 15∫ X ε(uεe · ∇)(uεa + uεe)u ε e dx dy dz = ∫ X ε(uεe · ∇)uεau ε e dx dy dz + ∫ X ε∇ · ( uεe (uεe) 2 2 ) dx dy dz − ∫ X ε∇ · uεe (uεe) 2 2 dx dy dz = ∫ X ε (uεe · ∇)uεau ε e dx dy dz ≤ ∣∣ ∫ X ε (uεe · ∇)uεau ε e dx dy dz ∣∣ ≤ ε‖∇uεa‖L∞(X)‖uεe‖2L2(X),∫ X ∇pεeuεe dx dy dz = ∫ X ∇ · (pεeuεe) dx dy dz − ∫ X ∇ · uεepεe dx dy dz = 0,∫ X 1 Ek e3 × uεeuεe dx dy dz = ∫ X 1 Ek (−uε2e, uε1e, 0)(uε1e, u ε 2e, u ε 3e) T dx dy dz = 0. Next, we deal with the right-hand side terms of (5.7):∫ X ∆uεeu ε e dx dy dz = ∫ X 3∑ i=1 (∂xx + ∂yy + ∂zz)u ε ieu ε ie dx dy dz = ∫ 1 0 ∫ 2π 0 3∑ i=1 ( ∂xu ε ieu ε ie|x=2π x=0 − ∫ 2π 0 (∂xu ε ie) 2dx ) dy dz + ∫ 1 0 ∫ 2π 0 3∑ i=1 ( ∂yu ε ieu ε ie| y=2π y=0 − ∫ 2π 0 (∂yu ε ie) 2dy ) dx dz + ∫ T2 3∑ i=1 ( ∂zu ε ieu ε ie|z=1 z=0 − ∫ 1 0 (∂zu ε ie) 2dz ) dx dy = − ∫ X (∇uεe)2 dx dy dz, ∫ X Rae3T ε eu ε e dx dy dz ≤ ∣∣ ∫ X Rae3T ε eu ε e dx dy dz ∣∣ ≤ η2‖uεe‖2L2(X) + C(η2)Ra2‖T εe‖2L2(X), and − ∫ X (RεIn,u +RεI,u)uεe dx dy dz ≤ ∣∣ ∫ X (RεIn,u +RεC,u)uεe dx dy dz ∣∣ ≤ η3‖uεe‖2L2(X) + C(η3)‖RεIn,u +RεC,u‖2L2(X) ≤ η3‖uεe‖2L2(X) + C(η3) ( Cε2e−2λτ + Cε1/2 ) , where we have used Hölder inequality, Young inequality and estimates (4.14), (4.51). Here ηi > 0, (i = 2, 3) are small constants, C(ηi) > 0 is a constant which is independent of ε. 16 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 Then, putting the above equations into (5.7), we obtain that ε 2 d dt ‖uεe‖2L2(X) + ‖∇uεe‖2L2(X) ≤ ε‖∇uεa‖L∞(X)‖uεe‖2L2(X) + η2‖uεe‖2L2(X) + C(η2)Ra2‖T εe‖2L2(X) + η3‖uεe‖2L2(X) + C(η3) ( Cε2e−2λτ + Cε1/2 ) . With the help of the Poincaré inequality, restricting ε to be sufficiently small such that ε‖∇uεa‖L∞(X) ≤ Cε ≤ 1 4 and taking η2, η3 to be sufficiently small (η2 + η3 = 1/4) but independent of ε, one obtains ε 2 d dt ‖uεe‖2L2(X) + 1 2 ‖∇uεe‖2L2(X) ≤ C(η2)Ra2‖T εe‖2L2(X) + C(η3) ( Cε2e−2λτ + Cε1/2 ) , (5.8) which implies ε d dt ‖uεe‖2L2(X) + ‖uεe‖2L2(X) ≤ 2C(η2)Ra2‖T εe‖2L2(X) + 2C(η3) ( Cε2e−2λτ + Cε1/2 ) , i.e., d dt (e t ε ‖uεe‖2L2(X)) ≤ [ 2C(η2)Ra2‖T εe‖2L2(X) + 2C(η3) ( Cε2e−2λτ + Cε1/2 )] ε−1e t ε . (5.9) Integrating (5.9) with respect to t over [0, t] for t ∈ [0, S] and any fixed S > 0, we have ‖uεe(t)‖2L2(X) ≤ ‖u ε e(t = 0)‖2L2(X) + 2C(η2)Ra2‖T εe (t)‖2L∞(0,t;L2(X)) + 2C(η3)Cε1/2. (5.10) Step 3. By performing the L2-inner product of temperature error equation (5.3) with T εe and integrating over X with respect to (x, y, z), we obtain 1 2 d dt ‖T εe‖2L2(X) = ∫ X ε∆T εeT ε e dx dy dz − ∫ X (RεIn,T +RεC,T )T εe dx dy dz − ∫ X (uεa · ∇)T εeT ε e dx dy dz − ∫ X (uεe · ∇) (T εa + T εe )T εe dx dy dz =: J3 + J4 + J5 + J6. (5.11) Using Green’s first formula and the boundary condition (5.5), we have J3 = ∮ ∮ Γ εT εe ∂T εe ∂n dS − ε ∫ X |∇T εe |2 dx dy dz = −ε ∫ X |∇T εe |2 dx dy dz, (5.12) where Γ denotes the boundary surface. EJDE-2020/31 3-D BOUSSINESQ SYSTEM 17 In view of Hölder inequality, Young inequality and (4.14), (4.52), we have the estimate |J4| ≤ η4‖T εe‖2L2(X) + C(η4)‖(RεIn,T +RεC,T )‖2L2(X) ≤ η4‖T εe‖2L2(X) + C(η4) ( Ce−2λτ + Cε1/2 ) , (5.13) where η4 > 0 is a small constant, and C(η4) > 0 is a constant which is independent of ε. It follows from the divergence formula, divergence theorem, (4.46) and the bound- ary condition (4.48) that J5 = − ∫ X uεa · ∇ ( (T εe )2 2 ) dx dy dz = − ∫ X ∇ · ( uεa (T εe )2 2 ) dx dy dz + ∫ X ∇ · uεa (T εe )2 2 dx dy dz = 0. (5.14) Similarly, J6 = − ∫ X (uεe · ∇)T εaT ε e dx dy dz − ∫ X (uεe · ∇)T εeT ε e dx dy dz = − ∫ X (uεe · ∇)T εaT ε e dx dy dz ≤ ∣∣− ∫ X (uεe · ∇)T εaT ε e dx dy dz ∣∣ ≤ η5‖uεe‖2L2(X) + C(η5)‖∇T εa‖2L∞(X)‖T ε e‖2L2(X), (5.15) where we used Hölder inequality and Young inequality. η5 > 0 is a small constant, and C(η5) > 0 is a constant which is independent of ε. Finally, using (5.12)–(5.15) in (5.11) yields 1 2 d dt ‖T εe‖2L2(X) + ε ∫ X |∇T εe |2 dx dy dz ≤ η4‖T εe‖2L2(X) + C(η4) ( Ce−2λτ + Cε1/2 ) + η5‖uεe‖2L2(X) + C(η5)‖∇T εa‖2L∞(X)‖T ε e‖2L2(X). (5.16) Step 4. Combining (5.8) and (5.16) yields ε 2 d dt ‖uεe‖2L2(X) + 1 2 ∫ X (∇uεe)2 dx dy dz + 1 2 d dt ‖T εe‖2L2(X) + ε ∫ X |∇T εe |2 dx dy dz ≤ η5‖uεe‖2L2(X) + η4‖T εe‖2L2(X) + [ C(η2)Ra2 + C(η5)‖∇T εa‖2L∞(X) ] ‖T εe‖2L2(X) + C(η3) ( Cε2e−2λτ + Cε1/2 ) + C(η4) ( Ce−2λτ + Cε1/2 ) ≤ η5‖uεe‖2L2(X) + C1‖T εe‖2L2(X) + C2 [ e−2λτ (ε2 + 1) + ε1/2 ] , where C1 = η4 + C(η2)Ra2 + C(η5)‖∇T εa‖2L∞(X) and C2 = (C(η3) + C(η4))C. 18 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 Using the Poincaré inequality and restricting η5 to be sufficiently small indepen- dent of ε we have d dt ( ε‖uεerr‖2L2(X) + ‖T εerr‖2L2(X) ) + ( ‖∇uεerr‖2L2(X) + ε‖∇T εerr‖2L2(X) ) ≤ C ( ε‖uεerr‖2L2(X) + ‖T εerr‖2L2(X) ) + C [ e−2λτ (ε2 + 1) + ε1/2 ] . (5.17) Step 5. It follows from (3.2), (5.6) and Gronwall’s inequality that ε‖uεe‖2L2(X) + ‖T εe‖2L2(X) ≤ e ∫ t 0 Cdξ [ ε‖uεe(t = 0)‖2L2(X) + ‖T εe (t = 0)‖2L2(X) + ∫ t 0 C(e−2λτ (ε2 + 1) + ε1/2)dξ ] ≤ C √ ε, (5.18) which leads to ε‖uεe‖2L∞(0,S;L2(X)) ≤ Cε 1/2, and ‖T εe‖L∞(0,S;L2(X)) ≤ Cε1/4. 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Ser.), 16 (2000)207-218. [36] C. Y. Qian; Blow-up criterion for the 2D Euler-Boussinesq system in terms of temperature, Electron. J. Differential Equations, 2016 (2016) no. 73, 1-11. [37] Y. Wang; Existence and asymptotic behavior of solutions to the generalized damped Boussi- nesq equation, Electron. J. Differential Equations, 2012 (2012) no. 96, 1-11. 20 X. FAN, S. WANG, W.-Q. XU EJDE-2020/31 Xiaoting Fan (corresponding author) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China Email address: xiao ting fan@163.com Shu Wang College of Applied Sciences, Beijing University of Technology, Ping Le Yuan100, Chao Yang District, Beijing 100124, China Email address: wangshu@bjut.edu.cn Wen-Qing Xu Department of Mathematics and Statistics, California State University, Long Beach, California 90840, USA Email address: wxu@csulb.edu 1. Introduction 2. Derivation of initial and boundary layers 3. Main results 4. Approximate solution 4.1. Inner expansion 4.2. Initial layer and lower boundary layer expansion 4.3. Initial layer and upper boundary layer expansion 4.4. Approximate solution 5. Proof of main results Acknowledgments References