˖ڍመڙጲ
http://www.paper.edu.cn
the boundary layer requires some special treatment in making estimates, cf. [27, 28]. In our
model (1), due to the presence of the second order derivative terms αψ
β
xx
and βθ
β
xx
, it seems
that we cannot directly use the standard energy method to deduce such some desired estimates
on ∥ψ
β
xx
(t)∥
H
1
and ∥θ
β
xx
(t)∥. However we found that if we can establish another some estimates
on ∥ψ
β
t
(t)∥
H
1
and ∥θ
β
t
(t)∥
H
1
instead, we can also obtain the desired estimates mentioned above.
Based on these observations, we can indeed show such some desired estimates by combining the
equations (1) with estimates on ∥ψ
β
t
(t)∥
H
1
and ∥θ
β
t
(t)∥
H
1
. In terms of the model (10), based on
the similar method, we establish further regularity estimates on ∥∂
xx
ψ
0
(t)∥
H
2
and ∥∂
xx
θ
0
(t)∥
H
2
by combining the equations (10) with estimates on ∥ψ
0
t
(t)∥
H
1
, ∥ψ
0
tt
(t)∥ and ∥θ
0
t
(t)∥
H
1
, where the
implicit boundary conditions (13) play an important role. It is noticed that, in contrast with
[13], the important point in this paper is that the restriction on the linear relation between the
parameters ν and β is removed. In addition, the convergence rates in W
1,∞
norm are improved
since the regularity on both solutions
ψ
β
, θ
β
and (ψ
0
, θ
0
) is improved.
The rest of this paper is arranged as follows. In Sections 2-4, we give the proof of Theorems
1.1-1.3 respectively. Precisely speaking, in Section 2, we establish a series of a priori estimates
on the solutions to (1)-(3) and obtain the global existence results. In Section 3, we devote
ourselves to the a priori estimates of the solution (ψ
0
(x, t), θ
0
(x, t)) of (10)-(12), obtain the
global existence and improve the regularity. Finally the convergence rates and the BL-thickness
are given in Section 4 as the diffusion parameter β → 0
+
.
1 Proof of Theorem 1.1
In this section, we devote ourselves to the a priori estimates of the solution
ψ
β
(x, t), θ
β
(x, t)
of (1)-(3) under the a priori assumption
N
1
(T ) = sup
0<t<T
ψ
β
, θ
β
(t)
2
2
6 ε
2
1
, (1)
where ε
1
is a positive constant satisfying 0 < ε
1
≪ 1, independent of β.
By Sobolev inequality, we have
ψ
β
, θ
β
(t)
W
1,∞
6 Cε
1
, (2)
which will be used later.
What follows will be a series of lemmas contributing to our desired estimates.
Lemma 2.1. Suppose that the assumptions in Theorem 1.1 hold. Then there exists a positive
constant C, independent of β, such that
1
0
ψ
β
2
+
θ
β
2
dx +
t
0
1
0
ψ
β
2
+
θ
β
2
+
ψ
β
x
2
+ β
θ
β
x
2
dxdτ ≤ C (3)
- 6 -
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