Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit

Xiaoting Fan; Shu Wang; Wen-Qing Xu; Mingshuo Liu

doi:10.1515/math-2018-0094

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Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit

Xiaoting Fan , Shu Wang , Wen-Qing Xu and Mingshuo Liu

Published/Copyright: October 29, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

The main purpose of this paper is to study the initial layer problem and the infinite Prandtl number limit of Rayleigh-Bénard convection with an ill prepared initial data. We use the asymptotic expansion methods of singular perturbation theory and the two-time-scale approach to obtain an exact approximating solution and the convergence rates O(ε32) and O(ε₂).

Keywords: Boussinesq system; Rayleigh-Bénard convection; Infinite Prandtl number limit; Initial layers; Asymptotic expansion; Two-time-scale approach

MSC 2010: 35B25; 35B40; 35K57

1 Introduction

In atmospheric and oceanographic sciences, fluid phenomena with heat transfer has been extensively studied in a large variety of contexts, see, for instance, [1, 4]. The thermal convection of a fluid powered by the difference of temperature between two horizontal parallel plates, known as Rayleigh-Bénard convection see [2, 4, 5, 6, 7, 8, 9, 10], obeys the rotating Boussinesq system:

∂tu+(u⋅∇)u+∇p+2Ωe3×u=νΔu+gαe3T,∇⋅u=0,∂tT+u⋅∇T=κΔT,u|z=0,h=0,T|z=0=T2,T|z=h=T1,

where T₂>T₁, u is the vector velocity field of the fluid, p represents the scalar pressure, Ω is the rotation rate, and e₃ is the unit upward vector. As usual, e₃:=(0,0,1), ν is the kinematic viscosity, g is the gravity acceleration constant, α is the thermal expansion coefficient, T is the scalar temperature field of the fluid, and κ is the thermal diffusion coefficient. Here we also impose the periodic boundary conditions in the horizontal directions for simplicity.

This system with rotation is a dynamic model having 3D incompressible Navier-Stokes equations via a buoyancy force proportional to temperature coupled with the heat advection-diffusion of the temperature [5,10, 11, 12, 13].

We can use the Boussinesq approximation and non-dimensionalization to obtain the simplification of Boussinesq system, namely,

ε[∂tuε+(uε⋅∇)uε]+∇pε+1Eke3×uε=Δuε+Rae3Tε,(x,y,z,t)∈Q×(0,S),(1.1)

∇⋅uε=0,(x,y,z,t)∈Q×(0,S),(1.2)

∂tTε+uε⋅∇Tε=εΔTε,(x,y,z,t)∈Q×(0,S),(1.3)

with the boundary and initial conditions:

uε|z=0,1=0,(x,y,t)∈T×(0,S),(1.4)

Tε|z=0=1,Tε|z=1=0,(x,y,t)∈T×(0,S),(1.5)

uε(t=0)=u0ε(x,y,z),Tε(t=0)=T0ε(x,y,z),(x,y,z)∈Q,(1.6)

where Q=T×[0,1], T=(R1/2π)2 is the torus in R2, S>0, ε=(1Pr)12, Pr=νk is the Prandtl number, Ek=ν2Ωh2 is the Ekman number and Ra=gα(T2−T1)h3νk(T2>T1) is the Rayleigh number.

This system is different from the nondimensional form in [12,13]. Encouraged by the results on the global existence and the regularities of the suitable weak solution in [12,13], and the related models, see [2,5,7,10,14, 15, 16, 17, 18], this system has also suitable weak solution by adopting Galerkin approximation method.

By using the asymptotic expansion methods of the singular perturbation theory and the Stokes operator [9,19,20,21,22], we construct an exact approximating solution and study the infinite Prandtl number limit Pr→∞ (i.e., ε→0), of Rayleigh-Bénard convection (1.1)-(1.6).

The main purpose of this paper is to show that the solutions of Boussinesq system for Rayleigh-Bénard convection converge to those of the infinite Prandtl number limit Pr→∞ (i.e., ε→0) model. It is a singular perturbation problem.

The rest of this paper is outlined as follows. The derivation of initial layer is stated in Section 2. The main convergence results are stated in section 3. The approximating solution is constructed and the properties of approximating solution are showed in section 4. The proofs of main convergence results are showed in Section 5. The conclusion is stated in Section 6.

2 The derivation of initial layer

In this section, formally, when ε=0, the 3-D Boussinesq system (1.1)-(1.4) turn into:

∇pO,0+1Eke3×uO,0=ΔuO,0+Rae3TO,0,(2.1)

∇⋅uO,0=0,(2.2)

∂tTO,0+(uO,0⋅∇)TO,0=0,(2.3)

uO,0|z=0,1=0,(2.4)

for (x,y,z,t)∈Q×(0,S), S>0.

Then we impose the initial condition of T^O,0 as follws:

TO,0(t=0)=T00(x,y,z),(x,y,z)∈Q,(2.5)

where T00(x,y,z) is the limit of T0ε(x,y,z) as ε → 0

We now turn to derive the boundary conditions of T^O,0.

Restricting (2.3) to z=0,1, one gets

∂tTO,0|z=0,1+(uO,0⋅∇)TO,0|z=0,1=0.(2.6)

Thus, plugging (2.4) into (2.6), we have

∂tTO,0|z=0,1=0.(2.7)

Due to the compatibility conditions, we deduce from (1.5), (1.6) and (3.1) (see below section 3) that,

T00(x,y,z)|z=0=1,T00(x,y,z)|z=1=0.(2.8)

In view of (2.5), (2.7) and (2.8), one gets

TO,0|z=0=1,TO,0|z=1=0.(2.9)

By comparing (1.5) with (2.9), a boundary layer of the the scalar temperature does not occur.

On the other hand, restricting (2.1), (2.2) and (2.4) to t=0, one gets

∇pO,0(t=0)+1Eke3×uO,0(t=0)=ΔuO,0(t=0)+Rae3TO,0(t=0),(2.10)

∇⋅uO,0(t=0)=0,(2.11)

uO,0|z=0,1(t=0)=0.(2.12)

The equation (2.10) is a stationary Stokes equation with rotation, we solve (2.10), (2.11) and (2.12) and know that the value u^O,0(t=0) is determined by the initial data T^O,0(t=0) of the temperature. But limε→0⁡u0ε(≠uO,0(t=0)) can be given arbitrarily and independently of T^O,0(t=0).

Thus, an initial layer occurs. We observe that the infinite Prandtl number limit of the Boussinesq system only has an initial layer, which is a singular perturbation problem.

Meanwhile, we get the infinite Prandtl number limit of the Boussinesq system as (2.1)-(2.5), namely,

∇pO,0+1Eke3×uO,0=ΔuO,0+Rae3TO,0,∇⋅uO,0=0,∂tTO,0+(uO,0⋅∇)TO,0=0,uO,0|z=0,1=0,TO,0(t=0)=T00(x,y,z).

3 Main convergence results

Assume that the initial data have an expansion up to the 1st order as follows

(uε,Tε)(t=0)=(u00+εu01+u0Eε,T00+εT01+T0Eε)(x,y,z),(3.1)

where u00,u01,T00,T01 are all C_∞(Q) functions, and u0Eε(x,y,z),T0Eε(x,y,z)∈C∞(Q) satisfy

∥(u0Eε,T0Eε)(x,y,z)∥L2(Q)≤Cε2,(3.2)

for some positive constant C independent of ε.

Theorem 3.1

Assume that (3.1) holds. Also, assume that u00, u01, T00, T01∈C∞(Q), satisfy the suitable compatibility conditions like u00∣z=0,1=0, ∇⋅u00=0, T00∣z=0=1, T00∣z=1=0, u01∣z=0,1=0, ∇⋅u01=0, T01∣z=0,1=0 etc. Then, as ε→0, for any 0<S<∞, we have the following convergence:

∥(uε−uappε,Tε−Tappε)∥L∞(0,S;L2(Q))≤Cε32,(3.3)

∥(uε−uappε,Tε−Tappε)∥L2(0,S;H1(Q))≤Cε32,(3.4)

where H¹(Q)=W^1,2(Q), for some positive constant C independent of ε.

Remark 3.2

The functions uappε, Tappε, pappε are given in Section 3.

Remark 3.3

By standard method [20,23,24], we also formulate any m^th, m=0,1,2,..., order compatibility conditions.

Remark 3.4

Due to the assumption (3.2), we can get the optimal convergence rate by adding assumption

T00(x,y,z)=1nearz=0,andT00(x,y,z)=0nearz=1.(3.5)

Then we have the following theorem.

Theorem 3.5

Let the assumptions of Theorem 3.1 hold. Furthermore, assume that (3.5) hold. Then, as ε→+∞, for any 0<S<∞, we arrive the following convergence:

∥(uε−uappε,Tε−Tappε−ε2TI,2)∥L∞(0,S;L2(Q))≤Cε2,(3.6)

for some positive constant C independent of ε, where TI,2=TI,2(x,y,z,τ), τ=tε, is the solution of the following linear problem

∂τTI,2+(uO,0(t=0)∇)TI,1 +(uI,0∇)TO,1(t=0)+(uI,0∇)TI,1+(uI,0∇)∂tTO,0(t=0)τ+(uI,1∇)TO,0(t=0)=0,(x,y,z)∈Q,τ>0,(3.7)

TI,2(x,y,z,τ→∞)=0,(x,y,z)∈Q.(3.8)

Remark 3.6

The functions u^O,0, T^I,1, u^I,0, T^O,1, T^O,0, u^I,1 are given in Section 4.

4 Approximating solutions and the properties

In this section, we carry out the method of matched asymptotic expansions [25,26] and the two-time-scale approach [2,26]. We construct the approximating solution including the outer one away from t=0 and the initial layer expansion near t=0. We also derive the corresponding properties of this approximating solution.

Let u^ε,T^ε,p^ε be the global weak solution to (1.1)-(1.6) in the Leray’s sense. It is easy to see that

(uε,pε,Tε)(x,y,z,t)=∑i=0∞εi(uO,i(x,y,z,t)+uI,i(x,y,z,τ),pO,i(x,y,z,t)+pI,i(x,y,z,τ),TO,i(x,y,z,t)+TI,i(x,y,z,τ)),

where ε is the length of the initial layers, τ=tε is the fast time variable; (u^O,i,p^O,i,T^O,i)(x,y,z,t) are the outer functions for the velocity field, pressure and temperature field, respectively, which are independent of ε; (u^I,i,p^I,i,T^I,i)(x,y,z,τ) are the initial layer functions for the velocity field, pressure and temperature field, respectively. The initial layer functions satisfy: u^I,i,p^I,i,T^I,i decay to zero exponentially, as τ→ ∞.

We seek for the solutions of the system (1.1)-(1.6) having the approximating expansions as follows:

(uappε,pappε,Tappε)(x,y,z,t) =∑i=01εi(uO,i(x,y,z,t)+uI,i(x,y,z,τ),pO,i(x,y,z,t)+pI,i(x,y,z,τ),TO,i(x,y,z,t)+TI,i(x,y,z,τ)).(4.1)

We discuss in detail the construction of the outer and initial layer functions here as

(uappε,pappε,Tappε)=(uouε,pouε,Touε)(x,y,z,t)+(uIε,pIε,TIε)(x,y,z,τ),τ=tε.(4.2)

4.1 Outer functions

Away from the initial time t=0, the solution to the system (1.1)-(1.5) are expected to be the following expansions

(uouε,pouε,Touε)=∑i=01εi(uO,i,pO,i,TO,i)(x,y,z,t),(4.3)

with (u^O,i,p^O,i,T^O,i)(x,y,z,t) to be determined later.

Inserting (4.3) into the system (1.1)-(1.5), then by direct calculation and the matched asymptotic expansions, some equations do not hold and need to be added remainders as

ε[∂tuouε+(uouε⋅∇)uouε]+∇pouε+1Eke3×uouε=Δuouε+Rae3Touε+Rou,uε,(4.4)

∇⋅uouε=0,(4.5)

∂tTouε+(uouε⋅∇)Touε=εΔTouε+Rou,Tε,(4.6)

uouε|z=0,1=0,(4.7)

Touε|z=0,1=0,(4.8)

where the remainders Rou,uε and Rou,Tε satisfy the estimates

∥(Rou,uε,Rou,Tε)∥L∞(0,S;Hs(Q))≤Cε2,(4.9)

for any fixed S>0 and any s≥1.

Now, we first consider the coefficient of leading order O(ε⁰) in the outer equations. We set the coefficient of O(ε⁰) in the system (4.4)-(4.6) as zero and use the boundary conditions (4.7)-(4.8) and the initial data (2.5).

At leading order, (u^O,0,p^O,0,T^O,0) satisfy infinite Prandtl number system (2.1)-(2.5), namely,

∇pO,0+1Eke3×uO,0=ΔuO,0+Rae3TO,0,∇⋅uO,0=0,∂tTO,0+(uO,0⋅∇)TO,0=0,uO,0|z=0,1=0,TO,0(t=0)=T00(x,y,z).

Similarly, we consider the coefficient of the first order O(ε¹) in the outer equations. We set the coefficient of O(ε¹) in the system (4.4)-(4.6) as zero and use the boundary conditions (4.7)-(4.8).

At first order, (u^O,1,p^O,1,T^O,1) satisfy the following system:

∇pO,1+1Eke3×uO,1=ΔuO,1+Rae3TO,1−∂tuO,0−(uO,0⋅∇)uO,0,(4.10)

∇⋅uO,1=0,(4.11)

∂tTO,1+uO,0⋅∇TO,1+uO,1⋅∇TO,0=ΔTO,0,(4.12)

uO,1|z=0,1=0,(4.13)

TO,1(t=0)=T01(x,y,z)−TI,1(τ=0),(4.14)

where T^I,1(τ=0) will be determined later (see below (4.31)). The proof of (4.14) is complete by the initial data (3.1) (see below (4.33)).

The infinite Prandtl number rotating system (2.1)-(2.5) has stationary Stokes equations via a buoyancy force proportional to temperature coupled with heat advection of the temperature. The linearized infinite Prandtl number type rotating system (4.10)-(4.14) has Stokes equations via a buoyancy force proportional to temperature coupled with linearized heat advection of the temperature. Therefore, the existence of the smooth solutions is the same as the incompressible Stokes equations. We find that:

Proposition 4.1

Assume that TO,1(t=0)∈C∞(Q) satisfy the suitable compatibility conditions like T00|z=0=1, T00|z=1=0, TO,1(t=0)|z=0,1=0, etc. Then there exists a unique and global C_∞(Q×[0,+∞)) smooth solution to the system (2.1)-(2.5) and (4.10)-(4.14), respectively.

Proof

The proof of Proposition 4.1 is elementary and we omit it.

Now we turn to the construction of the initial layer functions.

4.2 Initial layer functions

Near t=0, we will approximate the solution uniformly up to t=0 by the two-scale expansions (4.2)

(uappε,pappε,Tappε)=(uouε,pouε,Touε)(x,y,z,t)+(uIε,pIε,TIε)(x,y,z,τ),τ=tε,

where (uouε,pouε,Touε) is given by (4.4)-(4.8) and

(uIε,pIε,TIε)=∑i=01εi(uI,i,pI,i,TI,i)(x,y,z,τ),(uI,i,pI,i,TI,i)(τ→+∞)=0.(4.15)

Inserting (4.2) into the system (1.1)-(1.6), due to the matched asymptotic expansions, some equations do not hold by direct calculation and need to be added remainders as

ε[∂tuappε+(uappε⋅∇)uappε]+∇pappε+1Eke3×uappε−Δuappε−Rae3Tappε

=ε[∂t(uouε+uIε)+((uouε+uIε)⋅∇)(uouε+uIε)]+∇(pouε+PIε)+1Eke3×(uouε+uIε)−Δ(uouε+uIε)−Rae3(Touε+TIε)

=Rou,uε+ε[∂tuIε+(uouε⋅∇)uIε+uIε⋅∇(uouε+uIε)]+∇pIε+1Eke3×uIε−ΔuIε−Rae3TIε,(4.16)

∇⋅uappε=∇⋅(uouε+uIε)=∇⋅uIε=∑i=01εi∇⋅uI,i,(4.17)

∂tTappε+(uappε⋅∇)Tappε−εΔTappε=∂t(Touε+TIε)+((uouε+uIε)⋅∇)(Touε+TIε)−εΔ(Touε+TIε)=Rou,Tε+∂tTIε+uouε⋅∇TIε+(uIε⋅∇)(Touε+TIε)−εΔTIε,(4.18)

uappε|z=0,1=(uouε+uIε)|z=0,1=uIε|z=0,1=∑i=01εiuI,i|z=0,1,(4.19)

Tappε|z=0=(Touε+TIε)|z=0=1+∑i=01εiTI,i|z=0,(4.20)

Tappε|z=1=(Touε+TIε)|z=1=∑i=01εiTI,i|z=1,(4.21)

uappε(t=0)=∑i=01εi(uO,i(t=0)+uI,i(τ=0)),(4.22)

Tappε(t=0)=∑i=01εi(TO,i(t=0)+TI,i(τ=0)).(4.23)

We use the Taylor series expansion

uO,i(x,y,z,t)=uO,i(x,y,z,ετ)=uO,i(x,y,z,0)+ε∂tuO,i(t=0)τ+⋯.

TO,i(x,y,z,t)=TO,i(x,y,z,ετ)=TO,i(x,y,z,0)+ε∂tTO,i(t=0)τ+⋯.

Now we compare the coefficients of O(ε_i), i≥0 in the resulting system and derive the systems satisfying the initial layer functions.

First taking the coefficient of O(ε⁻¹) in (4.18) as zero, we find ∂^τT^I,0=0, which, together with T^I,0(τ→+∞)=0 in (4.15), one yields

TI,0(x,y,z,τ)=0,(4.24)

this show that the temperature has no zero order initial layer.

Then, setting the coefficients of O(ε⁰) in (4.16)-(4.18) as zero, using (4.24) and requiring that the approximating solution satisfies the boundary and initial conditions (4.19) and (4.22), the initial layer functions (u^I,0,p^I,0,T^I,0) satisfy the system as

∂τuI,0+1Eke3×uI,0+∇pI,0=ΔuI,0,(4.25)

∇⋅uI,0=0,(4.26)

∂τTI,1+uI,0⋅∇(TO,0(t=0))=0,(4.27)

uI,0|z=0,1=0,(4.28)

uI,0(τ=0)=u00−uO,0(t=0),(uI,0,TI,1)(τ→+∞)=0.(4.29)

Now, we turn to derive the initial and boundary conditions of T^I,1.

Using (4.27) and the decay condition T^I,1(τ→+∞)=0 in (4.15), one gets

TI,1=−∫τ∞[uI,0⋅∇(TO,0(t=0))](s)ds.(4.30)

In fact, we restrict (4.30) to τ=0, and replace the right term of result by T¯I,1, that is,

TI,1(τ=0)=T¯I,1.(4.31)

We restrict (4.30) to z=0,1 and by using the boundary condition (4.28) we get

TI,1|z=0,1=0.(4.32)

Moreover, we deduce the initial condition (3.15) from (3.1) and (4.23) that

T01=TO,1(t=0)+TI,1(τ=0).(4.33)

So, (u^I,0,p^I,0,T^I,0) satisfy the system (4.24)-(4.26), (4.28) and (4.29) as

TI,0(x,y,z,τ)=0,∂τuI,0+1Eke3×uI,0+∇pI,0=ΔuI,0,∇⋅uI,0=0,uI,0|z=0,1=0,uI,0(τ=0)=u00−uO,0(t=0),uI,0(τ→+∞)=0.

Similarly, setting the coefficients of O(ε¹) in (4.16), (4.17), (4.19) and (4.22) as zero, and using the above method, we have

∂τuI,1+1Eke3×uI,1+∇pI,1

=ΔuI,1+Rae3TI,1−(uI,0⋅∇)uO,0(t=0)−(uO,0(t=0)⋅∇)uI,0−(uI,0⋅∇)uI,0,(4.34)

∇⋅uI,1=0,(4.35)

uI,1∣z=0,1=0,(4.36)

uI,1(τ=0)=u01−uO,1(t=0),uI,1(τ→+∞)=0.(4.37)

So, (u^I,1,p^I,1,T^I,1) satisfy the system (4.27) and (4.34)-(4.37) as

∂τTI,1+uI,0⋅∇(TO,0(t=0))=0,∂τuI,1+1Eke3×uI,1+∇pI,1=ΔuI,1+Rae3TI,1−(uI,0⋅∇)uO,0(t=0)−(uO,0(t=0)⋅∇)uI,0−(uI,0⋅∇)uI,0,∇⋅uI,1=0,uI,1∣z=0,1=0,uI,1(τ=0)=u01−uO,1(t=0),(uI,1,TI,1)(τ→+∞)=0.

Now we turn to state the exponentially decay properties of the initial layer functions.

Proposition 4.2

Let the assumptions of Theorem 3.1 hold. Then there exist a unique and smooth solution (u^I,0,p^I,0) to the system (4.24)-(4.26), (4.28) and (4.29) and a unique and smooth solution (u^I,1,p^I,1,T^I,1) to the system (4.27) and (4.34)-(4.37) satisfying the exponential decay to zero as τ→∞ , namely,

∥(uI,0,TI,1)(⋅,τ)∥Hs(Q)≤Ce−βτ,(4.38)

for some positive constants C,β and any s≥1.

Proof

The proof of Proposition 4.2 is elementary, see [18].

We summarize the approximating solution in the next subsection.

4.3 Approximating solution

With outer functions and initial layer functions defined in section 4.1 and 4.2, one gets

ε[∂tuappε+(uappε⋅∇)uappε]+∇pappε+1Eke3×uappε−Δuappε−Rae3Tappε

=Rou,uε+(∂τuI,0+∇pI,0+1Eke3×uI,0−ΔuI,0−Rae3TI,0)

+ε(∂τuI,1+∇pI,1+1Eke3×uI,1−ΔuI,1−Rae3TI,1

+(uI,0⋅∇)uO,0(t=0)+(uO,0(t=0)⋅∇)uI,0+(uI,0⋅∇)uI,0)+RI,uε,(4.39)

∂tTappε+(uappε⋅∇)Tappε−εΔTappε

=Rou,Tε+ε−1∂τTI,0+(∂τTI,1+uI,0⋅∇(TO,0(t=0))

+(uO,0(t=0)+uI,0)⋅∇TI,0+RI,Tε,(4.40)

where the remainders RI,uε and RI,Tε, caused by the initial layer, are given exactly by

RI,uε=ε2[τ(uI,0⋅∇)∂tuO,0(θ1t)+τ∂tuO,0(θ2t)⋅∇uI,0+(uO,0(t=0)⋅∇)uI,1+(uO,1⋅∇)uI,0+uI,0⋅∇(uO,1+uI,1)+uI,1⋅∇(uO,0(t=0)+uI,0)]+ε3(τ(uI,1⋅∇)∂tuO,0(θ3t)+τ∂tuO,0(θ4t)⋅∇uI,1+(uO,1⋅∇)uI,1+uI,1⋅∇(uO,1+uI,1),0<θi<1,i=1,2,3,4,(4.41)

RI,Tε=ε[τ(uI,0⋅∇)∂tTO,0(t=0)+uO,0(t=0)⋅∇TI,1+uI,0⋅∇(TO,1(t=0)+TI,1)+uI,1⋅∇TO,0(t=0)]+ε2[12τ2(uI,0⋅∇)∂ttTO,0(θ5t)+τ(uI,1⋅∇)∂tTO,0(θ6t)+τ(uI,0⋅∇)∂tTO,1(θ7t)+τ∂tuO,0(θ8t)⋅∇TI,1+(uO,1⋅∇)TI,1+uI,1⋅∇(TO,1+TI,1)−ΔTI,1],0<θi<1,i=5,6,7,8.(4.42)

Hence, the previous computations show that (uappε,pappε,Tappε) solves the following initial-boundary problem:

ε[∂tuappε+(uappε⋅∇)uappε]+∇pappε+1Eke3×uappε=Δuappε+Rae3Tappε+Rou,uε+RI,uε,(4.43)

∇⋅uappε=0,(4.44)

∂tTappε+(uappε⋅∇)Tappε=ΔTappε+Rou,Tε+RI,Tε,(4.45)

uappε|z=0,1=0,(4.46)

Tappε|z=0=1,(4.47)

Tappε|z=1=0,(4.48)

(uappε,Tappε)(t=0)=(u00+εu01,T00+εT01),(4.49)

where the remainders Rou,uε, Rou,Tε satisfy the estimate (4.9) and RI,uε, RI,Tε defined by (4.41) and (4.42) respectively satisfy the the following estimate

∥RI,uε(t)∥L∞(Q)≤Cε2(τ+1)e−βτ,∥RI,Tε(t)∥L∞(Q)≤Cε(τ+1)e−βτ,(4.50)

for some positive constant C and β and for any t∈[0,S] and any fixed S>0. The estimate (4.50) can easily be obtained by the definitions of RI,uε,RI,Tε and the decay estimate (4.38).

We now turn to the proofs of convergence results.

5 The proofs of main convergence results

Without loss of generality, we denote C by a positive generic constant independent of ε. Noting that C may depend upon S for any fixed S>0. Let t∈[0,S]. We use the standard L²-energy method to prove Theorems 3.1 and 3.5.

5.1 The proof of Theorem 3.1

In this subsection we assume that (3.1) holds and define error functions

(uEε,pEε,TEε)=(uε−uappε,pε−pappε,Tε−Tappε).

Step 1. Combining (1.1)-(1.6) and (4.43)-(4.49), (uEε,pEε,TEε) satisfy the following equations

ε[∂tuEε+(uappε⋅∇)uEε+(uEε⋅∇)(uappε+uEε)]+∇pEε+1Eke3×uEε=ΔuEε+Rae3TEε−Rou,uε−RI,uε,(5.1)

∇⋅uEε=0,(5.2)

∂tTEε+(uappε⋅∇)TEε+(uEε⋅∇)(Tappε+TEε)=εΔTEε−Rou,Tε−RI,Tε,(5.3)

uEε|z=0,1=0,(5.4)

TEε|z=0,1=0,(5.5)

uEε(t=0)=u0Eε(x,y,z),TEε(t=0)=T0Eε(x,y,z).(5.6)

Step 2. Taking the L²-inner product of temperature error equation (5.3) with TEε and integrating over Q with respect to (x,y,z) yield

12ddt∥TEε∥L2(Q)2=∫QεΔTEεTEεdxdydz−∫Q(Rou,Tε+RI,Tε)TEεdxdydz−∫Quappε⋅∇TEεTEεdxdydz−∫QuEε⋅∇(Tappε+TEε)TEεdxdydz=:I1+I2+I3+I4.(5.7)

We first estimate I₁ by Green’s first formula and the boundary condition (5.5). We have that

I1=∮∮ΓεTEε∂TEε∂ndS−ε∫Q|∇TEε|2dxdydz=−ε∫Q|∇TEε|2dxdydz,(5.8)

where Γ is the boundary surface.

Next, we estimate the integral term I₂ by virtue of Hӧlder inequality, Young inequality and the estimates (4.9), (4.50). We get that

|I2|≤η1∥TEε∥L2(Q)2+C(η1)∥Rou,Tε+RI,Tε∥L2(Q)2≤η1∥TEε∥L2(Q)2+C(η1)(Cε4+Cε2(τ+1)2e−2βτ).(5.9)

Here η₁ is a small constant, C(η₁)>0 is a constant, independent of ε. We have used the estimate

|I2|≤η1∥TEε∥L2(Q)2+C(η1)∥Rou,Tε+RI,Tε∥L2(Q)2≤η1∥TEε∥L2(Q)2+C(η1)(Cε4+Cε2(τ+1)2e−2βτ).

Then, we estimate the integral term I₃ by divergence formula, divergence theorem, (4.44) and the boundary condition (4.46), (5.5) as follows

I3=−∫Quappε⋅∇(TEε)22dxdydz=−∫Q∇⋅uappε(TEε)22dxdydz+∫Q∇⋅uappε(TEε)22dxdydz=0.(5.10)

Similarly, we estimate the last integral term I₄ by using same method in estimating I₃. We obtain that

I4=−∫QuEε⋅∇TappεTEεdxdydz−∫Q(uEε⋅∇)TEεTEεdxdydz=−∫Q(uEε⋅∇)TappεTEεdxdydz≤|−∫Q(uEε⋅∇)TappεTEεdxdydz|(5.11)

where we have used Hӧlder inequality, Young inequality, the properties of the approximating solution, (5.2) and (5.5). η₂ is a small constant, C(η₂)>0 is a constant, independent of ε.

Finally, inserting the estimates derived in (5.8)-(5.11) into (5.7) leads to the inequality

12ddt∥TEε∥L2(Q)2+ε∥∇TEε∥L2(Q)2≤C(η1)(Cε4+Cε2(τ+1)2e−2βτ)+η1∥TEε∥L2(Q)2+C(η2)∥∇Tappε∥L∞(Q)2∥TEε∥L2(Q)2+η2∥uEε∥L2(Q)2.

With the help of Poincaré inequality and taking η₁ to be sufficiently small but independent of ε, one gets

ddt∥TEε∥L2(Q)2+2ε∥∇TEε∥L2(Q)2≤2∥∇Tappε∥L∞(Q)2C(η2)∥TEε∥L2(Q)2+2η2∥uEε∥L2(Q)2+2C(η1)(Cε4+Cε2(τ+1)2e−2βτ).(5.12)

Step 3. Similarly, testing the velocity equation (5.1) by uEε and integrating over Q with respect to (x,y,z).

∫Q(ε[∂tuEε+(uappε⋅∇)uEε+(uEε⋅∇)(uappε+uEε)]+∇pEε+1Eke3×uEε)uEεdxdydz=∫Q(ΔuEε+Rae3TEε−Rou,uε−RI,uε)uEεdxdydz.(5.13)

First, we deal with the left-hand side terms of (5.13) by divergence formula, divergence theorem, the approximating solution’s property (4.38), (4.44), (5.2) and the boundary condition (4.46), (5.4).

∫Qε∂tuEεuEεdxdydz=ε2ddt∥uEε∥L2(Q)2,∫Qε(uappε⋅∇)uEεuEεdxdydz=∫Qε∇⋅uappε(uEε)22dxdydz−∫Qε∇⋅uappεuEε22dxdydz=0,

∫Qε(uEε⋅∇)(uappε+uEε)uEεdxdydz=∫Qε(uEε⋅∇)uappεuEεdxdydz+∫Qε∇⋅uEε(uEε)22dxdydz−∫Qε∇⋅uEε(uEε)22dxdydz=∫QεuEε⋅∇uappεuEεdxdydz≤|∫QεuEε⋅∇uappεuEεdxdydz|≤ε∥∇uappε∥L∞(Q)∥uEε∥L2(Q)2,∫Q∇pEεuEεdxdydz=∫Q∇⋅(pEεuEε)dxdydz−∫Q∇⋅uEεpEεdxdydz=0,∫Q1Eke3×uEεuEεdxdydz=∫Q1Ek(−u2Eε,u1Eε,0)(u1Eε,u2Eε,u3Eε)Tdxdydz=0.

Next, we deal with the right-hand side terms of (5.13) as follows:

∫QΔuEεuEεdxdydz=∫Q∑i=13(∂xx+∂yy+∂zz)uiEεuiEεdxdydz=∫01∫02π∑i=13∂xuiEεuiEε|x=0x=2π−∫02π(∂xuiEε)2dxdydz+∫01∫02π∑i=13∂yuiEεuiEε|y=0y=2π−∫02π(∂yuiEε)2dydxdz+∫02π∫02π∑i=13∂zuiEεuiEε|z=0z=1−∫01(∂zuiEε)2dzdxdy=−∫Q(∇uEε)2dxdydz,

where we use uEε=(u1Eε,u2Eε,u3Eε) and the boundary condition (5.4).

∫QRae3TEεuEεdxdydz≤|∫QRae3TEεuEεdxdydz|≤η3∥uEε∥L2(Q)2+C(η3)Ra2∥TEε∥L2(Q)2,

and

−∫Q(Rou,uε+RI,uε)uEεdxdydz≤|∫Q(Rou,uε+RI,uε)uEεdxdydz|≤η4∥uEε∥L2(Q)2+C(η4)∥Rou,uε+RI,uε∥L2(Q)2≤η4∥uEε∥L2(Q)2+C(η4)(Cε4+Cε4(τ+1)2e−2βτ),

where we have used Hӧlder inequality, Young inequality and the estimates (4.9), (4.50). Here η_i>0,i=3,4 is a small constant, C(η_i)>0 is a constant, independent of ε.

Then, putting the above derivation equations into (5.13), we obtain that

ε2ddt∥uEε∥L2(Q)2+∥∇uEε∥L2(Q)2≤ε∥∇uappε∥L∞(Q)∥uEε∥L2(Q)2+η3∥uEε∥L2(Q)2+C(η3)Ra2∥TEε∥L2(Q)2+η4∥uEε∥L2(Q)2+C(η4)(Cε4+Cε4(τ+1)2e−2βτ).

With the help of the Poincaré inequality, restricting ε to be sufficiently small such that ε∥∇uappε∥L∞(Q)≤Cε≤14 and taking η₃, η₄ to be sufficiently small (η3+η4=14) but independent of ε, one gets

εddt∥uEε∥L2(Q)2+∥∇uEε∥L2(Q)2≤2C(η3)Ra2∥TEε∥L2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e−2βτ),(5.14)

that is,

εddt∥uEε∥L2(Q)2+∥uEε∥L2(Q)2≤2C(η3)Ra2∥TEε∥L2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e−2βτ),

i.e.,

ddt(etε∥uEε∥L2(Q)2)≤[2C(η3)Ra2∥TEε∥L2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e−2βτ)]ε−1etε.(5.15)

Integrating (5.15) with respect to t over [0,t] for any t∈[0,S] and any fixed S>0, one gets

∥uEε(t)∥L2(Q)2≤∥uEε(t=0)∥L2(Q)2+2C(η3)Ra2∥TEε(t)∥L∞(0,t;L2(Q))2+2C(η4)Cε4.(5.16)

Step 4. Then, combing (5.12) and (5.14), using the Poincaré inequality and restricting η₂ to be sufficiently small independent of ε yield

ddt∥TEε∥L2(Q)2+εddt∥uEε∥L2(Q)2+2ε∥∇TEε∥L2(Q)2+∥∇uEε∥L2(Q)2≤2∥∇Tappε∥L∞(Q)2C(η2)∥TEε∥L2(Q)2+2η2∥uEε∥L2(Q)2+2C(η1)(Cε4+Cε2(τ+1)2e−2βτ)+2C(η3)Ra2∥TEε∥L2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e−2βτ)≤C1∥TEε∥L2(Q)2+C2(ε4+ε2(τ+1)2e−2βτ)≤C1∥TEε∥L2(Q)2+C1ε∥uEε∥L2(Q)2+C2(ε4+ε2(τ+1)2e−2βτ),(5.17)

where C1=2∥∇Tappε∥L∞(Q)2C(η2)+2C(η3)Ra2, C₂=2C(η₁)C+2C(η₄)C.

So,

ddt(∥TEε∥L2(Q)2+ε∥uEε∥L2(Q)2)≤C1(∥TEε∥L2(Q)2+ε∥uEε∥L2(Q)2)+C2(ε4+ε2(τ+1)2e−2βτ).

Using Gronwall’s lemma and the assumption (3.2) yield

∥TEε∥L2(Q)2+ε∥uEε∥L2(Q)2≤e∫0tC1dξ[∥TEε(t=0)∥L2(Q)2+ε∥uEε(t=0)∥L2(Q)2+∫0tC2(ε4+ε2(τ+1)2e−2βτ)dξ]≤Cε3,(5.18)

where we have used the estimate

∫0t(τ+1)2e−2βτdξ≤Cε.

We deduce from (5.18) that

∥TEε(t)∥L∞(0,S;L2(Q))2≤Cε3,(5.19)

and

∥uEε(t)∥L∞(0,S;L2(Q))2≤Cε2.

Inserting (5.19) into (5.16) yields

∥uEε(t)∥L∞(0,S;L2(Q))2≤Cε3.(5.20)

Inserting (5.18) into (5.17) and integrating (5.17) with respect to t over [0,t] for any t∈[0,S] any fixed S>0 yield

2ε∫0t∥∇TEε∥L2(Q)2dξ+∫0t∥∇uEε∥L2(Q)2dξ≤Cε3.

So that,

∥TEε(t)∥L2(0,S;H1(Q))2≤Cε2,(5.21)

and

∥uEε(t)∥L2(0,S;H1(Q))2≤Cε3.(5.22)

Here H¹(Q)=W^1,2(Q), the estimates (5.19)-(5.22) yield to (3.3)-(3.5) in Theorem 3.1.

The proof of Theorem 3.1 is complete.

Obviously, the convergence rate O(ε32) is not optimal one, so we derive the optimal convergence rate in the next subsection.

5.2 The proof of Theorem 3.5

First we assume that (3.1) and (3.6) hold.

Step 1. We cancel the order O(ε) term of (4.42) to get the optimal convergence rate, and regard new result as RI,T1ε in the remainder. Moreover, by virtue of another initial layer function T^I,2, we define it as RI,T2ε.

We define T^I,2 to be the solution of the system (3.8)-(3.9), which can be solved by

TI,2=∫τ∞[(uO,0(t=0)∇)TI,1+(uI,0∇)TO,1(t=0)+(uI,0∇)TI,1+(uI,0∇)∂tTO,0(t=0)τ+(uI,1∇)TO,0(t=0)](s)ds.(5.23)

Thus,

TI,2|z=0,1=0.(5.24)

In fact, the assumption (3.6) and the definition (4.30) of T^I,1 give

TI,1=0,nearz=0,1,

which, together with (5.23) and the boundary condition (u^I,0,u^I,1)|_z=0,1=0, yields to the boundary condition (5.24).

By the exponential decay of the initial layer functions (u^I,0,u^I,1,T^I,1), it follows that

∥(TI,2)(⋅,τ)∥Hs(Q)≤Ce−γτ,(5.25)

for some positive constants C,Γ and any s≥1.

Step 2. Now set T~Eε=TEε−ε2TI,2=Tε−Tappε−ε2TI,2..

Then (uEε,pEε,T~Eε) satisfies the following error equations

ε[∂tuEε+(uappε⋅∇)uEε+(uEε⋅∇)(uappε+uEε)]+∇pEε+1Eke3×uEε=ΔuEε+Rae3T~Eε+Rae3ε2TI,2−Rou,uε−RI,uε,(5.26)

∇⋅uEε=0,(5.27)

∂tT~Eε+(uappε⋅∇)T~Eε+(uEε⋅∇)(Tappε+ε2TI,2+T~Eε)=εΔT~Eε−Rou,Tε−RI,T2ε−ε2(uappε⋅∇)TI,2+ε2ΔTI,2,(5.28)

uEε|z=0.1=0,(5.29)

T~Eε|z=0,1=0,(5.30)

uEε(t=0)=u0Eε(x,y,z),T~Eε(t=0)=T0Eε(x,y,z)−ε2TI,2(τ=0).(5.31)

Step 3. Using the decay property (5.25) of T^I,2 and the definition of RI,T2ε, one has the following estimates on the remainder −Rou,Tε−RI,T2ε−ε2(uappε⋅∇)TI,2+ε2ΔTI,2 appearing in (5.28):

∥−Rou,Tε−RI,T2ε−ε2(uappε⋅∇)TI,2+ε2ΔTI,2∥L2(Q)≤Cε2,(5.32)

where the estimate (5.32) is much better than the estimate in (5.9).

Now we use the estimate (5.32) in subsection 5.1, and can derive the optimal convergence rate O(ε²) in Theorem 3.5 by using the method in the proof of Theorem 3.1.

The proof of Theorem 3.5 is complete.

Conclusion

In this paper, we have used matched asymptotic expansion analysis to study the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit, which involves initial layers. It is a singular perturbation problem. We have derived the convergence of the solution of the Boussinesq system for Rayleigh-Bénard convection to that of the infinite Prandtl number limit system by adopting the effective approximating expansion.

The boundary value of the limit lim_ε→0(u^ε,T^ε) are not equal to (u^O,0,T^O,0), due to the initial and boundary conditions effect, the boundary layer occurs. This need an extra correction term of boundary layer with two fast variables. We will discuss it in the future.

The initial data satisfies a higher order correction in powers of ε, then the similar higher-order correction result in powers of ε can be obtained in the same way. We leave it for further investigation.

Acknowledgement

This work is supported by National Natural Science Foundation of China (No. 11771031, 11531010) and by Natural Science Fund of Qinghai Province (2017-ZJ-908).

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Received: 2018-03-21

Accepted: 2018-07-11

Published Online: 2018-10-29

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0094

Keywords for this article

Boussinesq system; Rayleigh-Bénard convection; Infinite Prandtl number limit; Initial layers; Asymptotic expansion; Two-time-scale approach

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