We have the estimate

B1≥ 1

Proof. The boundary condition (4.31) forωimplies that

N−1

The term I2 can be controlled by Cauchy’s inequality directly. First, re-call the definition of hi,0 in (4.31). Then summing by parts gives I₂

=

which implies thatI1can be, after summing by parts, expressed as

I1= 25

The first term on the right-hand side of (4.44) is estimated directly, while for the remaining terms we apply Cauchy’s inequality, giving

25

−1

The combination ofI₁andI₂then gives

N−1

whereBψ was defined in (4.40). Moreover, (4.39) is a direct consequence of (4.48). The treatment of the other three boundary terms is exactly the same.

This completes the proof of Lemma 4.4.

To complete the estimate ofB1we need to control

1+^h₆²D²_y this case. The methodology we adopt here is similar to that used in [21], i.e., control the local terms by global terms via elliptic regularity.

Lemma 4.5 For anyψthat vanishes on the boundary, we have

D²_xψ²1+ D²_yψ²1≤ 9

Proof. Given the homogeneous boundary conditionψ_i,j |=0, we perform a Sine transformation of{ψ_i,j}in both the x and y directions, i.e.,

which in turn implies that

N−1

On the other hand, direct calculation along with the fact that−4

h² ≤f_k, g ≤ 0 shows that

|f_k+g+h²

6 f_kg|²≥ | 1+h²

6 g

f_k|²+ | 1+h²

6f_k

g|², (4.57)

|fk+g+h²

6fkg|²≥

f_k²+g²−2 9fkg

≥ 8

9(f_k²+g²) . (4.58)

Combining (4.55)–(4.57) gives (4.50). Estimate (4.49) can be argued in a

similar fashion. Lemma 4.5 is proven.

The combination of Lemmas 4.4 and 4.5 results in the estimate B1≥ 1

3h²Bψ −3

4ω²1−Ch⁹. (4.59)

An estimate forB2 can be derived in a similar fashion, which we only briefly outline. Consider first the expression

iD_x²ψ_i,1D_x²ω_i,0inB2. Once again the boundary condition (4.31) for ω leads to examining

iD²_xψi,1

D_x²ω_i,0in two parts,I₃andI₄, given by I₃= 1

h²

N−1 i=1

D_x²ψ_i,1

8D_x²ψ_i,1−3D_x²ψ_i,2+8

9D_x²ψ_i,3−1 8D_x²ψ_i,4

,

I₄=

N−1 i=1

D²_xψ_i,1D_x²hi,0. (4.60)

The estimate ofI₃andI₄is similar to that ofI₁andI₂, respectively. Repeating the arguments in the proof of Lemma 4.4, we arrive at (omitting the details)

B2≥ − 1

144h⁴D²_yD_x²ψ²1−Ch⁹. (4.61)

On the other hand, the fact thatD²_yD_x²ψ1≤ 4

h²D_x²ψ1andD_y²D_x²ψ1≤ 4

h²D_y²ψ1implies D_y²D_x²ψ²1= 1

2(D²_yD_x²ψ²1+ D_y²D_x²ψ²1)≤ 8

h⁴D²_xψ²1+ 8

h⁴D_y²ψ²1

≤ 9 h⁴ω²1, (4.62)

where in the last step we applied (4.49) in Lemma 4.5. Substituting (4.62) into (4.61), we arrive at

B2≥ − 1

16ω²1−Ch⁹. (4.63)

Finally,B3can be controlled by applying Cauchy’s inequality (we only consider here the term 1

6ψ1,1ω0,0) 1

6ψ_1,1ω_0,0≥ − 1 12

ψ_1,1²

h² − 1

12h²ω²_0,0≥ − 1 12

ψ_1,1²

h² −Ch¹⁰ψ_e²_C⁸, (4.64)

where in the last step we used the fact that|ω_0,0| ≤Ch⁴ψ_eC⁸ by our con-struction of in section 3.1. The first term on the right-hand side of (4.64) can be absorbed into theBψterm, giving

B3≥ − 1

12h²Bψ−Ch⁹. (4.65)

The combination of (4.63), (4.65), and Lemma 4.4 shows that B ≥

−13

16ω²1−h⁸, whose substitution into (4.37) is exactly (4.36). This

com-pletes the proof of Proposition 4.3.

The estimates for the linearized convection terms in (4.32) are given in the following proposition. The proof is similar to that of Proposition 4.3 and the details are left to interested readers.

Proposition 4.6 Assume a-priori that the error functions for the velocity field and temperature satisfy

uL^∞ ≤h², θL^∞ ≤h². (4.66)

Then we have

1+h² 12_h

ψ , L2

1

≤C₁∇hψ²2+ν

8ω²1+h⁸, (4.67)

where C1= C(1+ ueC⁰)²

ν +C(2+ ueC¹)²+CueC⁵.

In addition, by Cauchy’s inequality and the boundary condition for the temperature error functionθ in (4.31), we have the estimate of the gravity

term

1+ h²

12_h

ψ , D_x

1−h²

12(D²_y−D²_x) θ

1

≤C(ψ²1+ ∇hθ²2)+Ch¹⁰, (4.68)

In (4.33), the local truncation error term−θ ,g¹ can be controlled by Cauchy’s inequality. The technique used in Lemma 4.6 can be applied here for the estimate of the linearized temperature convection term, i.e., the assumed a-priori assumption (4.66) leads to

θ ,L1¹

≤C2θ²1+1

2κ∇hθ²2+h⁸, (4.69)

whereC2= C(1+ ueC⁰)²

κ .

Next, we apply the technique demonstrated in section 3.1 for the sta-bility analysis of the long-stencil discretization of the one-dimensional heat equation to the temperature diffusion term.

Proposition 4.7 We have

− θ ,

_h− h²

12(D_x⁴+D_y⁴) θ

1

≥ 1

2∇hθ²2−h⁸. (4.70)

Proof. The proof of (4.70) is just the two-dimensional version of the stability analysis in section 3. Sinceθ vanishes on the boundary, we have

− θ ,

_h−h²

12(D_x⁴+D⁴_y) θ

1

= ∇hθ²2+h²

12D²_xθ²1+h²

12D²_yθ²1+h² 12B, (4.71)

where Barises from the boundary terms, which after summation by parts, can be written as

B=

N−1 i=1

θi,1(D²_yθ )i,0+

N−1 i=1

θ_i,N−1(D_y²θ )_i,N +

N−1

j=1

θ1,j(D_x²θ )0,j

+

N−1 j=1

θ_N−1,j(D_x²θ )_N,j. (4.72)

We focus on the first term appearing on the right-hand side of (4.72); the other three boundary terms can be treated similarly. Applying the boundary condition forθ at the “ghost points” as in (4.31),(D_y²θ )i,0can be written as

(D²_yθ )_i,0= 1 h²

5

11θ_i,1− 4

11θ_i,2+ 3

11θ_i,3+ei,0

(4.73) ,

which is analogous to (3.7) except for the local error termei,0 (defined in (4.31)), whose product withθ_i,1 can be controlled by Cauchy’s inequality.

Alternatively, we can rewrite the right-hand side of (4.73), as we did in sec-tion 3, as

(D²_yθ )_i,0= −2

11(D_y²θ )_i,1+ 1

11(D_y²θ )_i,2+ei,0

h² . (4.74)

The aim here is the control of local terms by global terms. Applying Cauchy’s inequality to each term in (4.74) leads to

θ_i,1(D²_yθ )_i,0≥ − 2²

4·11²·h²θ_i,1² −h²(D²_yθ_i,1)²− 1²

4·11²·h²θ_i,1² −h²(D_y²θ_i,2)²

− 1

4h²θ_i,1² −e²_i,0 h²

≥ − 1

2h²θ_i,1² −h²(D_y²θ_i,1)²−h²(D_y²θ_i,2)²−e²_i,0 h² . (4.75)

Here the arguments in section 4 can be repeated: the first term appearing above can be controlled by∇hθ²2, since it will be multiplied by h²

12, resulting in a term greater than 1

2∇hθ²2; the second and third terms can be controlled by D_y²θ²2; while the last term can be controlled by

h² 12

N−1 i=1

e²_i,0

h² ≤N h²·Ch¹⁰θ_e²_C⁶· 1

h² ≤Ch⁹θ_e²_C⁶, (4.76)

where we used the fact thath= 1

N. (4.70) then follows.

Finally, the combination of (4.32)–(4.37), and (4.67)–(4.70) gives us 1

2 dE

dt +1 2

d

dtθ²1≤Ch⁸+Cf²1+Cg²1+C1∇hψ²2

+C2θ²1−κ

2∇hθ²2. (4.77)

Integrating in time results in E+ θ²1≤C

T 0

(f²1+ g²1) dt+2C2

T 0

θ²1dt +2C₁

T 0

∇hψ²2dt+CT h⁸. (4.78)

It can be seen that

∇hψ²2≤3(∇hψ²2− h²

12_hψ²1−h²

6D_xD_yψ²1) , (4.79)

sinceψvanishes on the boundary, which along with (4.78) implies that

∇hψ²2+ θ²1≤C T

0

(f²1+ g²1) dt +CC₁ T

0

∇hψ²2dt +CC₂

T 0

θ²1dt+CT h⁸. (4.80)

By Gronwall’s inequality we then have

∇hψ²2+ θ²1≤Cexp

CC1T +CC2T ^T

0

f(·, s)²1+ g(·, s)²1ds +CT h⁸

≤CT h⁸exp

CC₁T +CC₂T (4.81)

ue²_C^7,α(1+ ueC⁵)²+ θ_e²_C5ue²_C5

+θ_e²_C⁶+CT h⁸

. Thus, we have proven

u(·, t )−ue(t )L²+ θ (·, t )−θ_e(t )L²

≤Ch⁴

ueC^7,α(1+ ueC⁵)+ θ_eC⁵ueC⁵+ θ_eC⁶

·exp CT

ν (1+ ueC¹)²+CT

κ (1+ ueC⁰)² . (4.82)

Using the inverse inequality, we have

uL^∞ ≤Ch³. (4.83)

At this point, we can introduce a standard concept which asserts that (4.66) will never be violated ifhis small enough, and Theorem 1.1 is proven.

5 Convergence proof of Theorem 1.2

The numerical scheme with the Neumann boundary condition (1.4), namely (2.7), (2.16), and (2.26), is analyzed in this section. For simplicity of pre-sentation we set θf = 0 in which case the one-sided extrapolation of the temperature at the boundary is given by (2.27).

The consistency analysis of the momentum equation is the same as that presented in section 4. We denote by ψ_e,ue, andω_e the exact solutions of (1.1)–(1.2), and (1.4), and extend ψe smoothly to [−δ,1+δ]². Then let _i,j =ψ_e(x_i, y_j)for−2 ≤i, j ≤N +2. The approximated velocity pro-filesU andV, and the vorticity profile are given by (4.1) and (4.2)–(4.5),

respectively. Lemmas 4.1 and 4.2, along with the estimate for the time march-ing term in (4.18), remain valid. The fourth order approximation (4.22) for the constructed vorticity on the boundary is also preserved.

Regarding the temperature variable, instead of substituting the exact solu-tion into the numerical scheme, a careful construcsolu-tion of an approximated temperature profile is performed by adding anO(h⁴)correction term to θ_e to satisfy the truncation error fully to fourth order. The reason for this proce-dure is to avoid the loss of accuracy near the boundary which would result from a direct truncation error estimate. To be more precise, we construct the approximate temperature fieldas

=θ_e+h⁴θ , (5.1)

in which the correction functionθ satisfies the Poisson equation θ =C¹,

(5.2a)

with the Neumann boundary condition

∂_yθ (x,0)= 1

80∂_y⁵θ_e(x,0) , ∂_yθ (x,1)= 1

80∂_y⁵θ_e(x,1) ,

∂_xθ (0, y)= 1

80∂_x⁵θ_e(0, y) , ∂_xθ (1, y)= 1

80∂_x⁵θ_e(1, y) . (5.2b)

The scalarC¹(a function of timet) is chosen as

C¹dx =

∂

∂θ

∂n dn in order to maintain consistency with the Neumann boundary condition, i.e.,

C¹= 1

| | ¹

0

− 1

80∂_y⁵θe(x,0)+ 1

80∂_y⁵θe(x,1) dx +

1 0

−1

80∂_x⁵θ_e(0, y)+ 1

80∂_x⁵θ_e(1, y) dy (5.3) .

A Schauder estimate applied to the Poisson equation (5.2) gives θC^m,α ≤CθeC^m+4,α, for m≥2. (5.4)

The reason for taking the boundary condition forθ in (5.2b) will become apparent later. A local Taylor expansion for the exact temperature fieldθ_eat

points near the boundaryy =0 gives

(θ_e)_i,−1=(θ_e)_i,1−h³

3 ∂_y³θ_e(x_i,0)−h⁵

60∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC⁷

=(θ_e)i,1+ h³

3κ (ω_e)i,0(∂_xθ_e)i,0−h⁵

60∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC⁷, (θ_e)_i,−2=(θ_e)_i,2−8h³

3 ∂_y³θ_e(x_i,0)−32h⁵

60 ∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC⁷

=(θ_e)_i,2+8h³

3κ(ω_e)_i,0(∂_xθ_e)_i,0−32h⁵

60 ∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC⁷, (5.5)

due to the no-flux boundary condition forθ_e and the derivation for∂_y³θ_e as given in (2.25) by applying the original PDE on the boundary. The insertion of the boundary conditions given by (5.2b) into a Taylor expansion ofθ, along with the Schauder estimateθC³ ≤Cθ_eC^7,αgiven by (5.4), gives

θ_i,−1=θi,1−2h∂yθi,0+O(h³)∂_y³θi,0

=θ_i,1−₄₀^h ∂_y⁵θ_e(x_i,0)+O(h³)θ_eC^7,α, θ_i,−2=θi,2−4h∂yθi,0+O(h³)∂_y³θi,0

=θ_i,2−₂₀^h ∂_y⁵θ_e(x_i,0)+O(h³)θ_eC^7,α. (5.6)

The combination of (5.5) and (5.6) results in an estimate for=θe+h⁴θ given by

_i,−1=_i,1+ h³

3κ (ω_e)_i,0(∂_xθ_e)_i,0−h⁵

24∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC^7,α, _i,−2=_i,2+8h³

3κ (ω_e)_i,0(∂_xθ_e)_i,0−7h⁵

12 ∂_y⁵θ_e(x_i,0)+O(h⁷)θ_eC^7,α. (5.7)

Similar results can be obtained at the other three boundary segments, namely y =1,x =0, andx=1. Note that theO(h⁵)coefficients of_i,−1and_i,−2 have the ratio 1 : 14. This will be needed for the error analysis of the inner product of the temperature with the diffusion term in the temperature equa-tion. This crucial point is the reason for the choice of the boundary condition forθ in (5.2b).

A direct consequence of the Schauder estimate (5.4) is given by θW^2,∞( )≤CθC^2,α ≤CθeC^6,α,

(5.8)

in which · W^m,∞( ) represents the maximum value, at grids points, of the given function up tom-th order finite-difference, over the domain . As a result, we have

−θ_eW^2,∞( ) =h⁴θW^2,∞( )≤Ch⁴θ_eC^6,α. (5.9)

The combination of (5.9) and a local Taylor expansion forθ_e gives the esti-mates

Dx

1+ h²

12(D_y²−D_x²)

=Dx

1+h²

12(D²_y−D²_x)

θe+O(h⁴)θeC^6,α

= 1+h²

12

∂xθe+O(h⁴)θeC^6,α, (5.10)

D_x

1−h² 6 D²_x

=∂_xθ_e+O(h⁴)θ_eC^6,α, D_y

1−h²

6 D²_y

=∂_yθ_e+O(h⁴)θ_eC^6,α, (5.11)

_h−h²

12(D_x⁴+D⁴_y)

=θ_e+O(h⁴)θ_eC^6,α,

UD_x

1−h² 6D_x²

=u_e∂_xθ_e+O(h⁴)ueC⁵θ_eC^6,α, (5.12)

VD_x

1−h² 6D_x²

=v_e∂_yθ_e+O(h⁴)ueC⁵θ_eC^6,α. (5.13)

Moreover, taking the time derivative of (5.2) leads to a Poisson equation for∂_tθ, namely

(∂_tθ )=∂_tC¹, (5.14a)

with the Neumann boundary conditions

∂y(∂tθ )(x,0)= 1

80(∂t∂_y⁵θe)(x,0) , ∂y(∂tθ )(x,1)= 1

80(∂t∂_y⁵θe)(x,1) ,

∂x(∂tθ )(0, y)= 1

80(∂t∂_x⁵θe)(0, y) , ∂x(∂tθ )(1, y)= 1

80(∂t∂_x⁵θe)(1, y) . (5.14b)

Similarly, the value of∂_tC¹is given by

∂_tC¹= 1

| | −

1 0

1

80(∂_t∂_y⁵θ_e)(x,0)+ 1

80(∂_t∂_y⁵θ_e)(x,1) dx +

1 0

− 1

80(∂t∂_x⁵θe)(0, y)+ 1

80(∂t∂_x⁵θe)(1, y) dy . (5.15)

A Schauder estimate applied to the Poisson equation (5.14) reads ∂_tθC^m,α ≤C∂_tθ_eC^m+4,α ≤C

ueC^m+4,αθ_eC^m+5,α+ θ_eC^m+6,α

, for m≥2,

(5.16)

in which the original temperature transport equation∂tθe+(ue·∇)θe =κθe

was used. It can be seen that (5.16) amounts to

∂_t=∂_tθ_e+O(h⁴)

ueC^6,αθ_eC^7,α+ θ_eC^8,α

(5.17) .

The combination of (4.10)–(4.13), (4.18)–(4.19), (5.10), and the original momentum equation results in

∂_t +D_x

1+h² 6D_y²

(U )+D_y

1+h² 6 D²_x

(V )

− h²

12_h(UD_x +VD_y )−gD_x

1+h²

12(D²_y−D²_x)

=ν

_h+h² 6D_x²D_y²

+f , (5.18)

where|f| ≤Ch⁴ueC^7,α(1+ ueC⁵)+Ch⁴θ_eC^6,α. Similarly, the com-bination of (5.11)–(5.13), (5.17), and the original temperature equation gives

∂t+UDx

1−h²

6D_x²

+VDy

1−h²

6D_y²

=κ

h− h²

12(D_x⁴+D_y⁴)

+g, (5.19)

where|g| ≤Ch⁴(ueC^6,αθ_eC^7,α+ θ_eC^8,α).

The error functions at the computational grid points(x_i, y_j), 0≤i, j ≤N are defined in the same way as in (4.28). Subtracting (5.18)–(5.19) from the

numerical scheme (2.7), (2.16), and (2.27), gives

in which the linearized convection error termsL1andL2are given by (4.31), and the local truncation error terms satisfy

|g| ≤Ch⁴(ueC^6,αθeC^7,α+ θeC^8,α) , from (5.7), and using the approximations (4.22) and (5.11), that for the tem-perature field

Subtracting (5.23) from (2.27), we arrive at θi,−1=θi,1+ h³

where the ratio 1 : 14. Such a ratio is a crucial point in the error analysis of the temperature diffusion term in (5.20), which will be established in detail in Proposition 5.1.

The estimate of the error functions in the system (5.20)–(5.22) and (5.24) is similar to that in section 4. Multiplying the vorticity error equation by

− 1+ h²

12h

ψat interior grid points 1 ≤ i, j ≤ N −1 gives the same identity as in (4.32)

(5.25)

The energy estimate of the temperature error is different from the Dirichlet boundary condition case since the temperature field is updated at every grid point 0 ≤ i, j ≤ N. Taking the, 3inner product (see the definition in (3.13)) of the temperature error equation withθ gives

1

which is also the same as (4.33) except for the difference of inner product and L²norms. Again, this is due to the fact that the temperature field is updated at every grid points 0≤i, j ≤N.

An estimate for (5.25) is the same as that for (4.32). The identities (4.34)–

(4.35), Propositions 4.3 and 4.6, and (4.68) are still valid. More precisely,

−

provided the a-priori assumption (4.66) is satisfied, along withC₁as intro-duced after (4.67).

Similar to (4.69), the linearized temperature convection term can be controlled by

An estimate of the temperature diffusion term in (4.69) is outlined below.

Its proof relies on the stability analysis given in section 3.2 and some error estimates.

Proposition 5.1 We have

−

Proof. Summing by parts under the inner product , ³ and using the boundary extrapolation (5.24) gives

θ ,

_hh²

12(D⁴_x+D_y⁴) θ

3

= −∇hθ²2−h²

12D²_xθ²3−h²

12D²_yθ²3+B. (5.33)

The boundary termBcan be decomposed as B=B¹+B²+B³+B⁴, (5.34a)

in whichB¹, corresponding to the boundary term alongy =0, reads

B¹= 1 12

N−1

i=1

θ_i,1(θ_i,−1−θ_i,1)+1 2θ_i,0

(θ_i,−2−θ_i,2)−16(θ_i,−1−θ_i,1)

= 1 12

N−1

i=1

θ_i,1 h³

3κq^b_i −h⁵

24r^b_i +e^b1_i +1

2θi,0

8h³

3κq^b_i −7h⁵

12 r^b_i +e^b2_i −16(h³

3κq^b_i −h⁵

24r^b_i +e^b1_i )

. (5.34b)

It should be noted that the derivation of (5.34b) comes from the formula for

θ_i,−1andθ_i,−2 in (5.24a). The definitions ofq^b_i, r^b_i, e^b1_i ,e^b2_i were given in (5.24b). The boundary terms alongy =1,x =0, andx =1 can be similarly presented. In more detail,B¹can be simplified as

B¹= h³ 36κ

N−1 i=1

q^b_i(θi,1−4θi,0)+ 1 12

N−1 i=1

e^b1_i (θi,1−8θi,0)+1 2e^b2_i θi,0

+ h⁵ 288

N−1 i=1

r^b_i(θi,0−θi,1)

≡I₁^b+I₂^b+I₃^b. (5.35)

The first termI₁^bcan be controlled by using the form ofq^b_i as in (5.24b), namely

h³

N−1 i=1

q^b_iθi,1=h³

N−1 i=1

θi,1ωi,0Dx(1−h² 6 D²_x)θi,0

+h³

N−1

i=1

θi,1 i,0Dx(1−h² 6D_x²)θi,0

≤CθW^1,∞∇hψ2θ3+Ch L^∞θ3∇hθ2+Ch¹⁰ (5.36)

≤C(θ_eC^6,α+1)(∇hψ²2+ θ²3)+Ch(ueC⁵ +1) (θ²3 + ∇hθ²2)+Ch¹⁰,

in which the first inequality comes from the boundary formula forωi,0 in (5.22). The second inequality results from the estimate (5.9), (4.6), and the a-priori assumption (4.66). A similar result can be obtained forh³N−1

i=1 q^b_i θ_i,0. Then we arrive at

I₁^b≤C(θ_eC^6,α+1)(∇hψ²2+ θ²3)+Ch(ueC⁵+1) (θ²3 + ∇hθ²2)+Ch¹⁰.

(5.37)

The termI₂^bcan be controlled by Cauchy’s inequality and the estimate (5.24b), giving

I₂^b≤C ^N−1

i=1

e^b1_i θ_i,1 +C

^N−1

i=1

e^b1_i θ_i,0

≤Cθ²3+Ch¹⁰. (5.38)

What remains is the estimate of I₃^b. As can be seen, the detailed esti-mates forθ_i,−1andθ_i,−2in (5.24), which shows that theO(h⁵)coefficients of θ_i,−1,θ_i,−2 have the ratio 1 : 14, allows the term I₃^b to be written as

h⁵ 288

N−1

i=1 r^b_i(θ_i,0−θ_i,1). That is crucial to implement the error analysis below.

An application of Cauchy’s inequality shows that I₃^b = h⁵

288

N−1 i=1

r^b_i(θi,0−θi,1)≤ 1 288h²

N−1

i=1

(θ_i,0−θ_i,1)² h² + 1

288h¹⁰

N−1 i=1

r²_i . (5.39)

It is observed that the first term appearing above can be absorbed into∇hθ²2. Meanwhile, we note thatr^b_i =∂_y⁵θ_e(x_i,0), which is a bounded quantity on y =0. Then we get

I₃^b= h⁵ 288

N−1 i=1

r^b_i(θ_i,0−θ_i,1)≤ 1

288∇hθ²2+Ch⁹. (5.40)

The combination of (5.37)–(5.38) and (5.40) leads to B¹≤C(θ_eC^6,α+1)(∇hψ²2+ θ²3)+ 1

288∇hθ²2+1 8h⁸. (5.41)

The other three boundary terms B²,B³, B⁴ can be treated similarly. As a result, we arrive at

B≤C(θ_eC^6,α+1)(∇hψ²2+ θ²3)+ 1

72∇hθ²2+1 2h⁸. (5.42)

The insertion of (5.42) into (5.33) implies (5.32). Proposition 5.1 is proven.

By the combination of (5.25)–(5.32) we have the following inequality

1 2

dE dt +1

2 d

dtθ²3≤Ch⁸+C(f²3+ g²3)+C1∇hψ²2+C2θ²3

−κ

8∇hθ²2. (5.43)

The proof of Theorem 1.2 can be carried out by using a similar argument as in (4.78)–(4.83). The details are left to the interested reader.

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