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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

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
  • College of Mathematics and Systems Science, Shandong University of Science and Technology, Qian Wan Gang, Road 579, Huangdao District, Qingdao 266590, China
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Published Online: 2018-10-29 | DOI: https://doi.org/10.1515/math-2018-0094

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+uT=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+uεTε=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(T2T1)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 TO,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 TO,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 uO,0(t=0) is determined by the initial data TO,0(t=0) of the temperature. But limε0u0ε(uO,0(t=0)) can be given arbitrarily and independently of TO,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,T01C(Q), satisfy the suitable compatibility conditions like u00z=0,1=0,u00=0,T00z=0=1,T00z=1=0,u01z=0,1=0,u01=0,T01z=0,1=0 etc. Then, as ε→0, for any 0<S<∞, we have the following convergence:

(uεuappε,TεTappε)L(0,S;L2(Q))Cε32,(3.3)

(uεuappε,TεTappε)L2(0,S;H1(Q))Cε32,(3.4)

where H1(Q)=W1,2(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 mth, 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:

(uεuappε,TεTappεε2TI,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 uO,0, TI,1, uI,0, TO,1, TO,0, uI,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 uε,Tε,pε 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=0εi(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; (uO,i,pO,i,TO,i)(x,y,z,t) are the outer functions for the velocity field, pressure and temperature field, respectively, which are independent of ε; (uI,i,pI,i,TI,i)(x,y,z,τ) are the initial layer functions for the velocity field, pressure and temperature field, respectively. The initial layer functions satisfy: uI,i,pI,i,TI,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 (uO,i,pO,i,TO,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, (uO,0,pO,0,TO,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, (uO,1,pO,1,TO,1) satisfy the following system:

pO,1+1Eke3×uO,1=ΔuO,1+Rae3TO,1tuO,0(uO,0)uO,0,(4.10)

uO,1=0,(4.11)

tTO,1+uO,0TO,1+uO,1TO,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 TI,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×uappεΔuappεRae3Tappε

=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×uIεΔuIεRae3TIε,(4.16)

uappε=(uouε+uIε)=uIε=i=01εiuI,i,(4.17)

tTappε+(uappε)TappεεΔTappε=t(Touε+TIε)+((uouε+uIε))(Touε+TIε)εΔ(Touε+TIε)=Rou,Tε+tTIε+uouεTIε+(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)+εtuO,i(t=0)τ+.

TO,i(x,y,z,t)=TO,i(x,y,z,ετ)=TO,i(x,y,z,0)+εtTO,i(t=0)τ+.

Now we compare the coefficients of O(εi), 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 ∂τTI,0=0, which, together with TI,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 (uI,0,pI,0,TI,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)=u00uO,0(t=0),(uI,0,TI,1)(τ+)=0.(4.29)

Now, we turn to derive the initial and boundary conditions of TI,1.

Using (4.27) and the decay condition TI,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, (uI,0,pI,0,TI,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)=u00uO,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,1z=0,1=0,(4.36)

uI,1(τ=0)=u01uO,1(t=0),uI,1(τ+)=0.(4.37)

So, (uI,1,pI,1,TI,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,1z=0,1=0,uI,1(τ=0)=u01uO,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 (uI,0,pI,0) to the system (4.24)-(4.26), (4.28) and (4.29) and a unique and smooth solution (uI,1,pI,1,TI,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)Ceβτ,(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×uappεΔuappεRae3Tappε

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

+ε(τuI,1+pI,1+1Eke3×uI,1ΔuI,1Rae3TI,1

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

tTappε+(uappε)TappεεΔTappε

=Rou,Tε+ε1τTI,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)+τtuO,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)+τtuO,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,1TO,0(t=0)]+ε2[12τ2(uI,0)ttTO,0(θ5t)+τ(uI,1)tTO,0(θ6t)+τ(uI,0)tTO,1(θ7t)+τtuO,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)eβτ,RI,Tε(t)L(Q)Cε(τ+1)eβτ,(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)=(uεuappε,pεpappε,TεTappε).

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+Rae3TEεRou,uεRI,uε,(5.1)

uEε=0,(5.2)

tTEε+(uappε)TEε+(uEε)(Tappε+TEε)=εΔTEεRou,TεRI,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

12ddtTEεL2(Q)2=QεΔTEεTEεdxdydzQ(Rou,Tε+RI,Tε)TEεdxdydzQuappεTEεTEεdxdydzQuEε(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=ΓεTEεTEεndSεQ|TEε|2dxdydz=εQ|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|η1TEεL2(Q)2+C(η1)Rou,Tε+RI,TεL2(Q)2η1TEεL2(Q)2+C(η1)(Cε4+Cε2(τ+1)2e2βτ).(5.9)

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

|I2|η1TEεL2(Q)2+C(η1)Rou,Tε+RI,TεL2(Q)2η1TEεL2(Q)2+C(η1)(Cε4+Cε2(τ+1)2e2βτ).

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=Quappε(TEε)22dxdydz+Quappε(TEε)22dxdydz=0.(5.10)

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

I4=QuEεTappεTEεdxdydzQ(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

12ddtTEεL2(Q)2+εTEεL2(Q)2C(η1)(Cε4+Cε2(τ+1)2e2βτ)+η1TEεL2(Q)2+C(η2)TappεL(Q)2TEεL2(Q)2+η2uEεL2(Q)2.

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

ddtTEεL2(Q)2+2εTEεL2(Q)22TappεL(Q)2C(η2)TEεL2(Q)2+2η2uEεL2(Q)2+2C(η1)(Cε4+Cε2(τ+1)2e2βτ).(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+Rae3TEεRou,uεRI,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).

QεtuEεuEεdxdydz=ε2ddtuEεL2(Q)2,Qε(uappε)uEεuEεdxdydz=Qεuappε(uEε)22dxdydzQεuappεuEε22dxdydz=0,

Qε(uEε)(uappε+uEε)uEεdxdydz=Qε(uEε)uappεuEεdxdydz+QεuEε(uEε)22dxdydzQεuEε(uEε)22dxdydz=QεuEεuappεuEεdxdydz|QεuEεuappεuEεdxdydz|εuappεL(Q)uEεL2(Q)2,QpEεuEεdxdydz=Q(pEεuEε)dxdydzQuEεpEεdxdydz=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=Qi=13(xx+yy+zz)uiEεuiEεdxdydz=0102πi=13xuiEεuiEε|x=0x=2π02π(xuiEε)2dxdydz+0102πi=13yuiEεuiEε|y=0y=2π02π(yuiEε)2dydxdz+02π02πi=13zuiEεuiEε|z=0z=101(zuiEε)2dzdxdy=Q(uEε)2dxdydz,

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

QRae3TEεuEεdxdydz|QRae3TEεuEεdxdydz|η3uEεL2(Q)2+C(η3)Ra2TEεL2(Q)2,

and

Q(Rou,uε+RI,uε)uEεdxdydz|Q(Rou,uε+RI,uε)uEεdxdydz|η4uEεL2(Q)2+C(η4)Rou,uε+RI,uεL2(Q)2η4uEεL2(Q)2+C(η4)(Cε4+Cε4(τ+1)2e2βτ),

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

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

ε2ddtuEεL2(Q)2+uEεL2(Q)2εuappεL(Q)uEεL2(Q)2+η3uEεL2(Q)2+C(η3)Ra2TEεL2(Q)2+η4uEεL2(Q)2+C(η4)(Cε4+Cε4(τ+1)2e2βτ).

With the help of the Poincaré inequality, restricting ε to be sufficiently small such that εuappεL(Q)Cε14 and taking η3, η4 to be sufficiently small (η3+η4=14) but independent of ε, one gets

εddtuEεL2(Q)2+uEεL2(Q)22C(η3)Ra2TEεL2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e2βτ),(5.14)

that is,

εddtuEεL2(Q)2+uEεL2(Q)22C(η3)Ra2TEεL2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e2βτ),

i.e.,

ddt(etεuEεL2(Q)2)[2C(η3)Ra2TEεL2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e2βτ)]ε1etε.(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)Ra2TEε(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

ddtTEεL2(Q)2+εddtuEεL2(Q)2+2εTEεL2(Q)2+uEεL2(Q)22TappεL(Q)2C(η2)TEεL2(Q)2+2η2uEεL2(Q)2+2C(η1)(Cε4+Cε2(τ+1)2e2βτ)+2C(η3)Ra2TEεL2(Q)2+2C(η4)(Cε4+Cε4(τ+1)2e2βτ)C1TEεL2(Q)2+C2(ε4+ε2(τ+1)2e2βτ)C1TEεL2(Q)2+C1εuEεL2(Q)2+C2(ε4+ε2(τ+1)2e2βτ),(5.17)

where C1=2TappεL(Q)2C(η2)+2C(η3)Ra2, C2=2C(η1)C+2C(η4)C.

So,

ddt(TEεL2(Q)2+εuEεL2(Q)2)C1(TEεL2(Q)2+εuEεL2(Q)2)+C2(ε4+ε2(τ+1)2e2βτ).

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

TEεL2(Q)2+εuEεL2(Q)2e0tC1dξ[TEε(t=0)L2(Q)2+εuEε(t=0)L2(Q)2+0tC2(ε4+ε2(τ+1)2e2βτ)dξ]Cε3,(5.18)

where we have used the estimate

0t(τ+1)2e2βτdξCε.

We deduce from (5.18) that

TEε(t)L(0,S;L2(Q))2Cε3,(5.19)

and

uEε(t)L(0,S;L2(Q))2Cε2.

Inserting (5.19) into (5.16) yields

uEε(t)L(0,S;L2(Q))2Cε3.(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

2ε0tTEεL2(Q)2dξ+0tuEεL2(Q)2dξCε3.

So that,

TEε(t)L2(0,S;H1(Q))2Cε2,(5.21)

and

uEε(t)L2(0,S;H1(Q))2Cε3.(5.22)

Here H1(Q)=W1,2(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 TI,2, we define it as RI,T2ε.

We define TI,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 TI,1 give

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

which, together with (5.23) and the boundary condition (uI,0,uI,1)|z=0,1=0, yields to the boundary condition (5.24).

By the exponential decay of the initial layer functions (uI,0,uI,1,TI,1), it follows that

(TI,2)(,τ)Hs(Q)Ceγτ,(5.25)

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

Step 2. Now set T~Eε=TEεε2TI,2=TεTappεε2TI,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,2Rou,uεRI,uε,(5.26)

uEε=0,(5.27)

tT~Eε+(uappε)T~Eε+(uEε)(Tappε+ε2TI,2+T~Eε)=εΔT~EεRou,TεRI,T2εε2(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 TI,2 and the definition of RI,T2ε, one has the following estimates on the remainder Rou,TεRI,T2εε2(uappε)TI,2+ε2ΔTI,2 appearing in (5.28):

Rou,TεRI,T2εε2(uappε)TI,2+ε2ΔTI,2L2(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 limε→0(uε,Tε) are not equal to (uO,0,TO,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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About the article

Received: 2018-03-21

Accepted: 2018-07-11

Published Online: 2018-10-29


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1145–1160, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0094.

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© 2018 Fan et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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