1 Introduction
In this paper, we study the generalized Forchheimer flows for slightly compressible fluids in porous media. These are non-Darcy flows which have a long history in physics and engineering, but their study has been neglected much in mathematics. The original Forchheimer equations are used to describe the fluids’ dynamics when the ubiquitous Darcy’s law is not applicable. They are nonlinear relations between the fluid’s velocity and gradient of pressure which were realized by Darcy [3] and Dupuit [8] in early works, formulated by Forchheimer [10, 11], and were studied more extensively afterward in physics and engineering; see [18, 26, 2, 19, 23] and the references therein. The mathematics of Darcy flows has been studied thoroughly for a long time dated back to the 1960s with a vast literature; see the treaty [25] for references. In contrast, Forchheimer flows were investigated mathematically much later in the 1990s. Even less are the mathematical works on compressible Forchheimer flows. The reader is referred to [1, 12, 13] for more information about this topic.
In fluid mechanics, there is a lack of theory to describe the physics of fluids in porous media for high Reynolds numbers or complicated geometry. Generalized Forchheimer equations were proposed in order to cover a large class of fluid flows formulated from experiments. They are studied numerically in [6, 20] and mathematically in our previous works [1, 12, 13, 14, 16]. For compressible fluids, they form a new class of degenerate parabolic equations with their own characteristics compared to other models of porous medium equations. Among a small number of papers on these flows, recent work [16] is focused on studying the pressure and its time derivative in the space L∞, the pressure gradient in Ls for s∈[1,∞) and the pressure’s Hessian in L2-δ for δ∈(0,1). However, it requires the so-called strict degree condition (SDC), that is, the degree of the Forchheimer polynomial is less than 4n-2, where n is the spatial dimension. Another related paper [14], in contrast, does not require SDC, but the analysis is mainly for pressure in the space Ls, pressure gradient in the space L2-a for a specific number a∈(0,1), and pressure’s time derivative in L2. Our current work is to unite the two approaches and develop them further without any restrictions on the Forchheimer polynomials. Specifically, we consider the initial boundary value problem (IBVP) for the pressure in a bounded domain with time-dependent Dirichlet boundary data. The interior W1,∞ and W2,2 norms of the solutions are estimated, particularly for large time. Such estimates for degenerate parabolic equations usually require much work; see, e.g., [24, 21, 22, 7]. Improving on [16], we adapt and refine techniques by De Giorgi [4], Ladyzhenskaya–Uraltseva [17] and also DiBenedetto [5] for parabolic equations, and combine them with those used for Navier–Stokes equations [9]. These techniques are utilized successfully here thanks to the special structure of our equation.
The paper is organized as follows: We recall the model, equations and basic facts in Section 2. In Section 3, we derive the uniform Gronwall-type estimates that sharpen the previous results in [12, 14] especially for large time. In Section 4, the interior L∞-estimates for pressure are established by using Lα-based De Giorgi iteration with sufficiently large α. This is different from [16] which is based on the L2-estimate and, hence, requires SDC. As a result, no restrictions on the Forchheimer polynomials are needed for our estimates. In Section 5, we first use Ladyzhenskaya–Uraltseva-type imbedding and iteration to estimate the pressure gradient in Ls for any s≥1. Our estimates only require the initial data to be in the spaces W1,2-a and Lα for a fixed α. When s is large, this requirement is much less than the W1,s condition in [16]. In Section 5.2, we derive the W1,∞ estimates for the pressure, which were not previously studied in [14, 16]. We make use of equation (5.20) for the pressure gradient, which it is highly nonlinear but has a special structure. Thanks to this and previous W1,s (s<∞) bounds, we utilize the De Giorgi technique to obtain the L∞-estimates for the pressure gradient. Section 6 contains the L∞-estimates for the time derivative of the pressure, while Section 7 contains new L2-bounds for the pressure’s Hessian. It is noteworthy that, thanks to the uniform Gronwall-type inequalities in Section 3, the asymptotic bounds obtained as time goes to infinity depend only on the asymptotic behavior of the boundary data. This paper is focused primarily on the interior estimates. The issue of the solutions’ estimates on the entire domain is addressed in another work of ours [15], which makes use of some fundamental estimates and techniques from the current paper and also requires more regularity for the boundary data.
2 Background
Consider a fluid in a porous medium occupying a bounded domain U with boundary Γ=∂U in the space ℝn. For physics problem n=3, but here we consider any n≥2. Let x∈ℝn and t∈ℝ be the spatial and time variables. The fluid flow has velocity v(x,t)∈ℝn, pressure p(x,t)∈ℝ and density ρ(x,t)∈[0,∞).
The generalized Forchheimer equations studied in [1, 12, 13, 14, 16] are of the form
(2.1)
g
(
|
v
|
)
v
=
-
∇
p
,
where g(s)≥0 is a function defined on [0,∞). When g(s)=α,α+βs,α+βs+γs2,α+γmsm-1, where α, β, γ, m, γm are empirical constants, we have Darcy’s law, Forchheimer’s two-term, three-term, and power laws, respectively.
In this paper, we study the case when the function g in (2.1) is a generalized polynomial with non-negative coefficients, that is,
g
(
s
)
=
a
0
s
α
0
+
a
1
s
α
1
+
⋯
+
a
N
s
α
N
for
s
≥
0
,
where N≥1, the powers α0=0<α1<⋯<αN are fixed real numbers (not necessarily integers) and the coefficients a0,a1,…,aN are non-negative with a0>0 and aN>0. This function g(s) is referred to as the Forchheimer polynomial in equation (2.1), and αN is the degree of g.
From (2.1) one can solve v implicitly in terms of ∇p and derive a nonlinear version of Darcy’s equation:
(2.2)
v
=
-
K
(
|
∇
p
|
)
∇
p
,
where the function K:[0,∞)→[0,∞) is defined by
(2.3)
K
(
ξ
)
=
1
g
(
s
(
ξ
)
)
with
s
=
s
(
ξ
)
≥
0
satisfying
sg
(
s
)
=
ξ
for
ξ
≥
0
.
In addition to (2.1), we have the continuity equation
(2.4)
ϕ
∂
ρ
∂
t
+
∇
⋅
(
ρ
v
)
=
0
,
where the number ϕ∈(0,1) is the constant porosity. Also, for slightly compressible fluids, the equation of state is
(2.5)
d
ρ
d
p
=
ρ
κ
with
κ
=
const
.
>
0
.
From (2.2), (2.4) and (2.5) one derives a scalar equation for the pressure:
(2.6)
ϕ
∂
p
∂
t
=
κ
∇
⋅
(
K
(
|
∇
p
|
)
∇
p
)
+
K
(
|
∇
p
|
)
|
∇
p
|
2
.
On the right-hand side of (2.6), the constant κ is very large for most slightly compressible fluids in porous media [18], hence we neglect its second term and by scaling the time variable, we study the reduced equation
(2.7)
∂
p
∂
t
=
∇
⋅
(
K
(
|
∇
p
|
)
∇
p
)
.
Note that this reduction is commonly used in engineering.
Our aim is to study the IBVP for equation (2.7) in a bounded domain. Here, afterward U is a bounded open connected subset of ℝn, n=2,3,…, with C2 boundary Γ=∂U. Throughout, the Forchheimer polynomial g(s) is fixed. The following number is frequently used in our calculations:
a
=
α
N
1
+
α
N
∈
(
0
,
1
)
.
The function K(ξ) in (2.3) has the following properties (cf. [1, 12]): it is decreasing in ξ mapping ξ∈[0,∞) onto (0,1a0], and
(2.8)
C
1
(
1
+
ξ
)
a
≤
K
(
ξ
)
≤
C
2
(
1
+
ξ
)
a
,
(2.9)
C
3
(
ξ
2
-
a
-
1
)
≤
K
(
ξ
)
ξ
2
≤
C
2
ξ
2
-
a
,
(2.10)
-
a
K
(
ξ
)
≤
K
′
(
ξ
)
ξ
≤
0
,
where C1, C2, C3 are positive constants depending on g. As in previous works, we use the function H(ξ) defined by
H
(
ξ
)
=
∫
0
ξ
2
K
(
s
)
𝑑
s
for
ξ
≥
0
.
It satisfies K(ξ)ξ2≤H(ξ)≤2K(ξ)ξ2, thus, by (2.9), also
(2.11)
C
3
(
ξ
2
-
a
-
1
)
≤
H
(
ξ
)
≤
2
C
2
ξ
2
-
a
.
The following parabolic Poincaré–Sobolev inequalities are needed for our study. For each T>0 denote QT=U×(0,T). We define a threshold exponent
α
*
=
a
n
2
-
a
.
Lemma 2.1.
Assume
(2.12)
α
≥
2
𝑎𝑛𝑑
α
>
α
*
.
Let
(2.13)
p
=
α
(
1
+
2
-
a
n
)
-
a
.
Then
(2.14)
∥
u
∥
L
p
(
Q
T
)
≤
C
(
1
+
δ
T
)
1
p
[
[
u
]
]
,
where δ=1 in general, δ=0 in case u vanishes on the boundary ∂U, and
[
[
u
]
]
=
ess
sup
[
0
,
T
]
∥
u
(
t
)
∥
L
α
(
U
)
+
(
∫
0
T
∫
U
|
u
(
x
,
t
)
|
α
-
2
|
∇
u
(
x
,
t
)
|
2
-
a
𝑑
x
𝑑
t
)
1
α
-
a
.
In the case U=BR, a ball of radius R, inequality (2.14) holds with
(2.15)
[
[
u
]
]
=
R
n
p
-
n
α
ess
sup
[
0
,
T
]
∥
u
(
t
)
∥
L
α
(
B
R
)
+
R
n
p
-
n
-
(
2
-
a
)
α
-
a
(
∫
0
T
∫
B
R
|
u
(
x
,
t
)
|
α
-
2
|
∇
u
(
x
,
t
)
|
2
-
a
𝑑
x
𝑑
t
)
1
α
-
a
and the constant C independent of R.
The proof of Lemma 2.1 is given in Appendix A. The next result is a particular embedding with spatial weights from [16, Lemma 2.4] (see inequality (2.28) with m=2 there).
Lemma 2.2 (cf. [16, Lemma 2.4]).
Given W(x,t)>0 on QT, let r be a number that satisfies
(2.16)
2
n
n
+
2
<
r
<
2
.
Set
ϱ
=
ϱ
(
r
)
:=
4
(
1
-
1
r
*
)
.
Then
∥
u
∥
L
ϱ
(
Q
T
)
≤
C
[
[
u
]
]
2
,
W
;
T
{
δ
T
1
ϱ
+
ess
sup
t
∈
[
0
,
T
]
(
∫
U
W
(
x
,
t
)
-
r
2
-
r
χ
supp
u
(
x
,
t
)
𝑑
x
)
2
-
r
ϱ
r
}
,
where δ=1 in general, δ=0 in case u vanishes on the boundary ∂U, and
[
[
u
]
]
2
,
W
;
T
=
ess
sup
[
0
,
T
]
∥
u
(
t
)
∥
L
2
(
U
)
+
(
∫
0
T
∫
U
W
(
x
,
t
)
|
∇
u
(
x
,
t
)
|
2
𝑑
x
𝑑
t
)
1
2
.
The following is a generalization of the convergence of fast decay geometry sequences in [17, Chapter II, Lemma 5.6]. It will be used in the De Giorgi iterations.
Lemma 2.3 (cf. [16, Lemma A.2]).
Let {Yi}i=0∞ be a sequence of non-negative numbers satisfying
Y
i
+
1
≤
∑
k
=
1
m
A
k
B
k
i
Y
i
1
+
μ
k
,
i
=
0
,
1
,
2
,
…
,
where Ak>0, Bk>1 and μk>0 for k=1,2,…,m. Let
B
=
max
{
B
k
:
1
≤
k
≤
m
}
𝑎𝑛𝑑
μ
=
min
{
μ
k
:
1
≤
k
≤
m
}
.
If
Y
0
≤
min
{
(
m
-
1
A
k
-
1
B
-
1
μ
)
1
μ
k
:
1
≤
k
≤
m
}
,
then
lim
i
→
∞
Y
i
=
0
.
3 Uniform Gronwall-Type Estimates
We study the following IBVP for p(x,t):
(3.1)
{
∂
p
∂
t
=
∇
⋅
(
K
(
|
∇
p
|
)
∇
p
)
in
U
×
(
0
,
∞
)
,
p
(
x
,
0
)
=
p
0
(
x
)
in
U
,
p
(
x
,
t
)
=
ψ
(
x
,
t
)
on
Γ
×
(
0
,
∞
)
.
In order to deal with the non-homogeneous boundary condition, the data ψ(x,t) with x∈Γ and t>0 is extended to a function Ψ(x,t) with x∈U¯ and t≥0. Throughout, our results are stated in terms of Ψ instead of ψ. Nonetheless, corresponding results in terms of ψ can be retrieved as performed in [12]. The function Ψ is always assumed to have adequate regularities for all calculations in this paper.
It is proved in [14, Section 3] that (3.1) possesses a weak solution p(x,t) for all t>0. It has, in fact, more regularity in spatial and time variables; see [5]. For the current study, we assume that the solution p(x,t) has sufficient regularities both in x and t variables such that our calculations hereafter can be performed legitimately.
In this section, we obtain improved estimates for solutions of (3.1). We emphasize the asymptotic estimates as t→∞ in terms of the asymptotic behavior of Ψ(x,t).
Notation.
Hereafter, the symbol C is used to denote a positive number independent of the initial and boundary data and the time variables t, T0, T; it may depend on many parameters, namely, exponents and coefficients of the polynomial g, the spatial dimension n and domain U, other involved exponents α, s, etc. in calculations. The value of C may vary from place to place, even on the same line.
For partial derivative notation, we will alternatively use ∂p∂t=∂tp=pt and ∂p∂xm=∂mp=pxm.
The Lebesgue norm ∥⋅∥Ls means ∥⋅∥Ls(U).
For a function f(x,t) we denote by f(t) the function x→f(x,t).
For α≥1 we define
A
(
α
,
t
)
=
[
∫
U
|
∇
Ψ
(
x
,
t
)
|
α
(
2
-
a
)
2
𝑑
x
]
2
(
α
-
a
)
α
(
2
-
a
)
+
[
∫
U
|
Ψ
t
(
x
,
t
)
|
α
𝑑
x
]
α
-
a
α
(
1
-
a
)
for t≥0, and
A
(
α
)
=
lim sup
t
→
∞
A
(
α
,
t
)
and
β
(
α
)
=
lim sup
t
→
∞
[
A
′
(
α
,
t
)
]
-
.
Also, define for α>0 the number
α
^
=
max
{
α
,
2
,
α
*
}
.
Whenever β(α) is in use, it is understood that the function t→A(α,t) belongs to C1((0,∞)).
For a function f:[0,∞)→ℝ we denote by Envf a continuous and increasing function F:[0,∞)→ℝ such that F(t)≥f(t) for all t≥0.
Let p(x,t) be a solution to IBVP (3.1). Denote p¯=p-Ψ and p¯0=p0-Ψ(⋅,0). We recall relevant results from [14] below.
Theorem 3.1 (cf. [14, Theorem 4.3]).
Let α>0.
For all
t
≥
0
we have
(3.2)
∫
U
|
p
¯
(
x
,
t
)
|
α
𝑑
x
≤
C
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
^
𝑑
x
+
[
Env
A
(
α
^
,
t
)
]
α
^
α
^
-
a
)
.
If
A
(
α
^
)
<
∞
, then
(3.3)
lim sup
t
→
∞
∫
U
|
p
¯
(
x
,
t
)
|
α
𝑑
x
≤
C
(
1
+
A
(
α
^
)
α
^
α
^
-
a
)
.
If
β
(
α
^
)
<
∞
, then there is
T
>
0
such that
(3.4)
∫
U
|
p
¯
(
x
,
t
)
|
α
𝑑
x
≤
C
(
1
+
β
(
α
^
)
α
^
α
^
-
2
a
+
A
(
α
^
,
t
)
α
^
α
^
-
a
)
for all
t
≥
T
.
For gradient and time derivative estimates we denote
G
1
(
t
)
=
∫
U
|
∇
Ψ
(
x
,
t
)
|
2
𝑑
x
+
[
∫
U
|
Ψ
t
(
x
,
t
)
|
r
0
𝑑
x
]
2
-
a
r
0
(
1
-
a
)
+
[
∫
U
|
Ψ
t
(
x
,
t
)
|
r
0
𝑑
x
]
1
r
0
,
G
2
(
t
)
=
∫
U
|
∇
Ψ
t
(
x
,
t
)
|
2
𝑑
x
+
∫
U
|
Ψ
t
(
x
,
t
)
|
2
𝑑
x
,
G
3
(
t
)
=
G
1
(
t
)
+
G
2
(
t
)
,
G
4
(
t
)
=
G
3
(
t
)
+
∫
U
|
Ψ
t
t
(
x
,
t
)
|
2
𝑑
x
,
with r0=n(2-a)(2-a)(n+1)-n. For t≥0 recall from [14, (4.20)] and from [12, (3.25)] that
(3.5)
∫
0
t
∫
U
H
(
|
∇
p
(
x
,
τ
)
|
)
𝑑
x
𝑑
τ
≤
C
∫
U
p
¯
0
2
(
x
)
𝑑
x
+
C
∫
0
t
G
1
(
τ
)
𝑑
τ
,
(3.6)
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
+
∫
0
t
∫
U
|
p
¯
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
≤
∫
U
[
H
(
|
∇
p
0
(
x
)
|
)
+
p
¯
0
2
(
x
)
]
𝑑
x
+
C
∫
0
t
G
3
(
τ
)
𝑑
τ
.
Let α≥2^. For t>0, applying [14, Theorem 4.5] with t0=t and following [14, Remark 4.8] to replace 2^ by α, we have
∫
U
|
p
¯
t
(
x
,
t
)
|
2
𝑑
x
≤
C
{
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
t
-
1
(
∫
U
|
p
¯
0
(
x
)
|
2
+
H
(
|
∇
p
0
(
x
)
|
)
d
x
+
∫
0
t
G
3
(
τ
)
𝑑
τ
)
}
(3.7)
+
C
∫
0
t
e
-
d
0
(
t
-
τ
)
(
[
Env
A
(
α
,
τ
)
]
α
α
-
a
+
G
4
(
τ
)
)
𝑑
τ
,
where d0>0 is independent of α. Since EnvA(α,t) is increasing in t, and G3≤G4, we simplify (3.7) as
(3.8)
∫
U
|
p
¯
t
(
x
,
t
)
|
2
𝑑
x
≤
C
(
1
+
t
-
1
)
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
∫
U
H
(
|
∇
p
0
(
x
)
|
)
𝑑
x
+
[
Env
A
(
α
,
t
)
]
α
α
-
a
+
∫
0
t
G
4
(
τ
)
𝑑
τ
)
.
Below are improved estimates for a large time t by using uniform Gronwall-type inequalities.
Lemma 3.2.
For t≥1 we have
(3.9)
∫
t
-
1
t
∫
U
H
(
|
∇
p
(
x
,
τ
)
|
)
𝑑
x
𝑑
τ
≤
C
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
+
C
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
,
(3.10)
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
+
1
2
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
C
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
+
C
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
,
(3.11)
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
C
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
+
C
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
.
Proof.
The proof follows from [13, Theorems 4.4 and 4.5].
Proof of (3.9). Using [12, Lemma 3.1, (3.4)], we have
(3.12)
d
d
t
∫
U
p
¯
2
(
x
,
t
)
𝑑
x
≤
-
C
∫
U
H
(
|
∇
p
|
)
𝑑
x
+
C
G
1
(
t
)
.
Dropping the negative term on the right-hand side of (3.12), and integrating it from t-1 to t, we obtain (3.9).
Proof of (3.10). Using [12, Lemma 3.3, (3.17)] with ε=1, we have
(3.13)
d
d
t
∫
U
H
(
|
∇
p
|
)
𝑑
x
≤
-
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
+
∫
U
H
(
|
∇
p
|
)
𝑑
x
+
C
G
2
(
t
)
.
Let s∈[t-1,t]. Integrating (3.13) from s to t, we have
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
+
∫
s
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
∫
U
H
(
|
∇
p
|
)
(
x
,
s
)
𝑑
x
+
∫
s
t
∫
U
H
(
|
∇
p
|
)
𝑑
x
𝑑
τ
+
C
∫
s
t
G
2
(
τ
)
𝑑
τ
.
Thus,
(3.14)
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
+
∫
s
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
∫
U
H
(
|
∇
p
|
)
(
x
,
s
)
𝑑
x
+
∫
t
-
1
t
∫
U
H
(
|
∇
p
|
)
𝑑
x
𝑑
τ
+
C
∫
t
-
1
t
G
2
(
τ
)
𝑑
τ
.
Integrating (3.14) in s from t-1 to t and using (3.9), we obtain
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
+
∫
t
-
1
t
∫
s
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
𝑑
s
≤
2
∫
t
-
1
t
∫
U
H
(
|
∇
p
|
)
𝑑
x
𝑑
τ
+
C
∫
t
-
1
t
G
2
(
τ
)
𝑑
τ
(3.15)
≤
C
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
+
C
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
+
C
∫
t
-
1
t
G
2
(
τ
)
𝑑
τ
.
Observe that
∫
t
-
1
t
∫
s
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
𝑑
s
=
∫
t
-
1
t
∫
t
-
1
τ
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
s
𝑑
τ
≥
∫
t
-
1
2
t
∫
t
-
1
τ
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
s
𝑑
τ
≥
1
2
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
.
Therefore, (3.10) follows from (3.15).
Proof of (3.11). From [12, Proposition 3.11, (3.37)] we have
d
d
t
∫
U
p
¯
t
2
𝑑
x
≤
-
C
∫
U
K
(
|
∇
p
|
)
|
∇
p
t
|
2
𝑑
x
+
C
∫
U
|
∇
Ψ
t
|
2
𝑑
x
-
∫
U
p
¯
t
Ψ
t
t
𝑑
x
.
Dropping the first term on the right-hand side and using Cauchy’s inequality give
(3.16)
d
d
t
∫
U
p
¯
t
2
𝑑
x
≤
C
∫
U
|
∇
Ψ
t
|
2
𝑑
x
+
1
2
∫
U
|
p
¯
t
|
2
𝑑
x
+
1
2
∫
U
|
Ψ
t
t
|
2
𝑑
x
.
For s∈[t-12,t], integrating (3.16) from s to t gives
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
∫
U
p
¯
t
2
(
x
,
s
)
𝑑
x
+
C
∫
s
t
∫
U
(
|
∇
Ψ
t
|
2
+
|
Ψ
t
t
|
2
)
𝑑
x
𝑑
τ
+
1
2
∫
s
t
∫
U
|
p
¯
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
,
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
∫
U
p
¯
t
2
(
x
,
s
)
𝑑
x
+
C
∫
t
-
1
t
∫
U
(
|
∇
Ψ
t
|
2
+
|
Ψ
t
t
|
2
)
𝑑
x
𝑑
τ
+
1
2
∫
t
-
1
2
t
∫
U
|
p
¯
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
.
Integrating the last inequality in s from t-12 to t yields
1
2
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
5
4
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
s
)
𝑑
x
𝑑
s
+
C
∫
t
-
1
t
∫
U
(
|
∇
Ψ
t
|
2
+
|
Ψ
t
t
|
2
)
𝑑
x
𝑑
τ
.
Using (3.10) for the first integral on the right-hand side, we have
(3.17)
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
C
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
+
C
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
+
C
∫
t
-
1
t
∫
U
(
|
∇
Ψ
t
|
2
+
|
Ψ
t
t
|
2
)
𝑑
x
𝑑
τ
.
Using the fact ∫U|∇Ψt|2𝑑x≤CG2(t)≤CG3(t), we obtain (3.11) from (3.17). The proof is complete. ∎
Combining the above, we have the following specific estimates which will be used conveniently in subsequent sections.
Corollary 3.3.
Let α≥2^.
For
t
≥
1
we have
(3.18)
∫
t
-
1
t
∫
U
H
(
|
∇
p
(
x
,
τ
)
|
)
𝑑
x
𝑑
τ
≤
C
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
[
Env
A
(
α
,
t
)
]
α
α
-
a
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
,
(3.19)
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
C
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
[
Env
A
(
α
,
t
)
]
α
α
-
a
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
.
If
A
(
α
)
<
∞
, then
(3.20)
lim sup
t
→
∞
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
≤
C
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
,
(3.21)
lim sup
t
→
∞
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
C
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
,
(3.22)
lim sup
t
→
∞
∫
t
-
1
t
∫
U
H
(
|
∇
p
(
x
,
τ
)
|
)
𝑑
x
𝑑
τ
≤
C
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
,
(3.23)
lim sup
t
→
∞
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
C
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that one has for all
t
≥
T
that
(3.24)
∫
U
H
(
|
∇
p
(
x
,
t
)
|
)
𝑑
x
≤
C
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
,
(3.25)
∫
U
p
¯
t
2
(
x
,
t
)
𝑑
x
≤
C
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
,
(3.26)
∫
t
-
1
t
∫
U
H
(
|
∇
p
(
x
,
τ
)
|
)
𝑑
x
𝑑
τ
≤
C
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
,
(3.27)
∫
t
-
1
2
t
∫
U
p
¯
t
2
(
x
,
τ
)
𝑑
x
𝑑
τ
≤
C
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
.
Proof.
Note that α^=α.
(i) For (3.18), resp. (3.19), we combine (3.9), resp. (3.10), with inequality
(3.28)
∫
U
p
¯
2
(
x
,
t
-
1
)
𝑑
x
≤
∫
U
(
1
+
|
p
¯
(
x
,
t
-
1
)
|
α
)
𝑑
x
,
and the estimate (3.2) for t-1.
(ii) We combine the estimates in Lemma 3.2 with (3.28) and the limit (3.3).
(iii) This part is similar to part (ii) with the use of (3.4) in place of (3.3). ∎
4 Interior L∞-Estimates for Pressure
In this section, we estimate the L∞-norm of the pressure. Hereafter, α is a number that satisfies (2.12). We use κj to denote a number of exponents that depend on α. Let
(4.1)
κ
0
=
α
(
1
+
2
-
a
n
)
-
a
,
κ
1
=
1
δ
1
,
κ
2
=
1
δ
2
,
where δ1=1-ακ0 and δ2=(1-aα)(1-α*α). Note that δ1,δ2∈(0,1), hence κ1,κ2>1.
We start with estimating the L∞-norm in terms of the Lα-norm.
Theorem 4.1.
Let U′⋐U. If T0≥0, T>0 and θ∈(0,1), then
(4.2)
sup
[
T
0
+
θ
T
,
T
0
+
T
]
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
T
)
κ
1
κ
0
(
1
+
(
θ
T
)
-
1
)
κ
1
α
-
a
(
1
+
∥
p
∥
L
α
(
U
×
(
T
0
,
T
0
+
T
)
)
)
κ
2
.
Proof.
Without loss of generality, we assume T0=0. For k≥0 define p(k)=max{p-k,0} and denote by χk(x,t) the characteristic function of the set suppp(k). Let ϕ1(x) and ϕ2(t) be cut-off functions with ϕ1=1 on U′, ϕ1=0 on a neighborhood of ∂U, and ϕ2(0)=0. Let ζ(x,t)=ϕ1(x)ϕ2(t). Multiplying the first equation in (3.1) by |p(k)|α-1ζ2 and integrating over U, we have
1
α
d
d
t
∫
U
|
p
(
k
)
|
α
ζ
2
𝑑
x
+
(
α
-
1
)
∫
U
K
(
|
∇
p
(
k
)
|
)
|
∇
p
(
k
)
|
2
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
=
2
α
∫
U
|
p
(
k
)
|
α
ζ
ζ
t
𝑑
x
+
2
∫
U
K
(
|
∇
p
(
k
)
|
)
(
∇
p
(
k
)
⋅
∇
ζ
)
|
p
(
k
)
|
α
-
1
ζ
𝑑
x
.
By Cauchy’s inequality, we have for the last integral that
2
|
K
(
|
∇
p
(
k
)
|
)
(
∇
p
(
k
)
⋅
∇
ζ
)
|
p
(
k
)
|
α
-
1
ζ
|
≤
α
-
1
2
K
(
|
∇
p
(
k
)
|
)
|
∇
p
(
k
)
|
2
|
p
(
k
)
|
α
-
2
ζ
2
+
C
K
(
|
∇
p
(
k
)
|
)
|
p
(
k
)
|
α
|
∇
ζ
|
2
.
Combining the above, we obtain
1
α
d
d
t
∫
U
|
p
(
k
)
|
α
ζ
2
𝑑
x
+
α
-
1
2
∫
U
K
(
|
∇
p
(
k
)
|
)
|
∇
p
(
k
)
|
2
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
≤
C
∫
U
|
p
(
k
)
|
α
ζ
|
ζ
t
|
𝑑
x
+
C
∫
U
K
(
|
∇
p
(
k
)
|
)
|
p
(
k
)
|
α
|
∇
ζ
|
2
𝑑
x
.
Using (2.9), we then have
d
d
t
∫
U
|
p
(
k
)
|
α
ζ
2
𝑑
x
+
∫
U
|
∇
p
(
k
)
|
2
-
a
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
≤
C
∫
U
|
p
(
k
)
|
α
ζ
|
ζ
t
|
𝑑
x
+
C
∫
U
K
(
|
∇
p
(
k
)
|
)
|
p
(
k
)
|
α
|
∇
ζ
|
2
𝑑
x
+
C
∫
U
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
.
Using the boundedness of K(⋅) in the second to last integral, and applying Young’s inequality to the last terms yield
d
d
t
∫
U
|
p
(
k
)
|
α
ζ
2
𝑑
x
+
∫
U
|
∇
p
(
k
)
|
2
-
a
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
≤
C
∫
U
|
p
(
k
)
|
α
(
ζ
2
+
ζ
|
ζ
t
|
+
|
∇
ζ
|
2
)
𝑑
x
+
C
∫
U
χ
k
ζ
2
𝑑
x
.
For t∈[0,T], integrating from 0 to T and taking the supremum in t, we obtain
sup
[
0
,
T
]
∫
U
|
p
(
k
)
|
α
ζ
2
𝑑
x
+
∫
0
T
∫
U
|
∇
p
(
k
)
|
2
-
a
|
p
(
k
)
|
α
-
2
ζ
2
𝑑
x
𝑑
t
(4.3)
≤
C
∫
0
T
∫
U
|
p
(
k
)
|
α
(
ζ
2
+
ζ
|
ζ
t
|
+
|
∇
ζ
|
2
)
𝑑
x
𝑑
t
+
C
∫
0
T
∫
U
χ
k
ζ
2
𝑑
x
𝑑
t
.
Let x0 be any given point in U¯′. Denote ρ=dist(U¯′,∂U)>0. Let M0>0 be fixed, which will be determined later. For i≥0 define
(4.4)
k
i
=
M
0
(
1
-
2
-
i
)
,
t
i
=
θ
T
(
1
-
2
-
i
)
,
ρ
i
=
1
4
ρ
(
1
+
2
-
i
)
.
Then t0=0<t1<⋯<θT and ρ0=ρ2>ρ1>⋯>ρ4>0. Note that
lim
i
→
∞
t
i
=
θ
T
and
lim
i
→
∞
ρ
i
=
ρ
4
.
Let Ui={x:∥x-x0∥<ρi}; then Ui+1⋐Ui for i=0,1,2,…. For i,j≥0 we denote
(4.5)
𝒬
i
=
U
i
×
(
t
i
,
T
)
and
A
i
,
j
=
{
(
x
,
t
)
∈
𝒬
j
:
p
(
x
,
t
)
>
k
i
}
.
For each 𝒬i we use a cut-off function ζi(x,t) with ζi≡1 in 𝒬i+1 and ζi≡0 on QT∖𝒬i, and
(4.6)
|
(
ζ
i
)
t
|
≤
C
t
i
+
1
-
t
i
=
C
2
i
+
1
θ
T
and
|
∇
ζ
i
|
≤
C
ρ
i
-
ρ
i
+
1
=
C
2
i
+
1
4
ρ
for some C>0. Applying (4.3) with k=ki+1 and ζ=ζi gives
sup
[
0
,
T
]
∫
U
|
p
(
k
i
+
1
)
|
α
ζ
i
2
𝑑
x
+
∫
0
T
∫
U
|
∇
p
(
k
i
+
1
)
|
2
-
a
|
p
(
k
i
+
1
)
|
α
-
2
ζ
i
2
𝑑
x
𝑑
t
(4.7)
≤
∫
0
T
∫
U
|
p
(
k
i
+
1
)
|
α
(
ζ
i
2
+
ζ
i
|
ζ
i
t
|
+
|
∇
ζ
i
|
2
)
𝑑
x
𝑑
t
+
C
∫
0
T
∫
U
χ
k
i
+
1
ζ
i
2
𝑑
x
𝑑
t
.
Define
F
i
:=
sup
[
t
i
+
1
,
T
]
∫
U
i
+
1
|
p
(
k
i
+
1
)
|
α
𝑑
x
+
∫
t
i
+
1
T
∫
U
i
+
1
|
∇
p
(
k
i
+
1
)
|
2
-
a
|
p
(
k
i
+
1
)
|
α
-
2
𝑑
x
𝑑
t
.
Then (4.7) yields
F
i
≤
∫
t
i
T
∫
U
|
p
(
k
i
+
1
)
|
α
(
ζ
i
2
+
ζ
i
|
ζ
i
t
|
+
|
∇
ζ
i
|
2
)
𝑑
x
𝑑
t
+
C
(
∫
t
i
T
∫
U
i
χ
k
i
+
1
𝑑
x
𝑑
t
)
(4.8)
≤
C
4
i
(
(
θ
T
)
-
1
+
1
)
∥
p
(
k
i
+
1
)
∥
L
α
(
A
i
+
1
,
i
)
α
+
C
|
A
i
+
1
,
i
|
.
Since
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
≥
∥
p
(
k
i
)
∥
L
α
(
A
i
+
1
,
i
)
≥
(
k
i
+
1
-
k
i
)
|
A
i
+
1
,
i
|
1
α
,
we have
(4.9)
|
A
i
+
1
,
i
|
≤
(
k
i
+
1
-
k
i
)
-
α
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
α
≤
C
2
α
i
M
0
-
α
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
α
.
This and (4.8) imply
F
i
≤
C
4
i
(
1
+
(
θ
T
)
-
1
)
∥
p
(
k
i
+
1
)
∥
L
α
(
A
i
+
1
,
i
)
α
+
C
2
α
i
M
0
-
α
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
α
(4.10)
≤
C
2
α
i
(
1
+
(
θ
T
)
-
1
+
M
0
-
α
)
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
α
.
Note that κ0 is the exponent defined in (2.13). Applying Lemma 2.1 to the domain Ui, the interval [ti+1,T] and the function p(ki+1) with the use of (2.15), we have
(4.11)
∥
p
(
k
i
+
1
)
∥
L
κ
0
(
A
i
+
1
,
i
+
1
)
=
∥
p
(
k
i
+
1
)
∥
L
κ
0
(
𝒬
i
+
1
)
≤
C
(
1
+
T
)
1
κ
0
(
F
i
1
α
+
F
i
1
α
-
a
)
.
Note that the constant C depends on ρ. Hölder’s inequality gives
∥
p
(
k
i
+
1
)
∥
L
α
(
A
i
+
1
,
i
+
1
)
≤
∥
p
(
k
i
+
1
)
∥
L
κ
0
(
A
i
+
1
,
i
+
1
)
|
A
i
+
1
,
i
+
1
|
1
α
-
1
κ
0
≤
∥
p
(
k
i
+
1
)
∥
L
κ
0
(
A
i
+
1
,
i
+
1
)
|
A
i
+
1
,
i
|
1
α
-
1
κ
0
.
It follows from this, (4.11), (4.10), and (4.9) that
∥
p
(
k
i
+
1
)
∥
L
α
(
A
i
+
1
,
i
+
1
)
≤
C
(
1
+
T
)
1
κ
0
(
F
i
1
α
+
F
i
1
α
-
a
)
|
A
i
+
1
,
i
|
1
α
-
1
κ
0
≤
C
(
1
+
T
)
1
κ
0
B
i
{
(
1
+
(
θ
T
)
-
1
+
M
0
-
α
)
1
α
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
+
(
1
+
(
θ
T
)
-
1
+
M
0
-
α
)
1
α
-
a
∥
p
(
k
i
)
∥
L
α
(
A
i
)
α
α
-
a
}
M
0
-
1
+
α
κ
0
∥
p
(
k
i
)
∥
L
2
(
A
i
,
i
)
1
-
α
κ
0
,
where B=2α. Now selecting M0≥1, we have
∥
p
(
k
i
)
∥
L
α
(
A
i
+
1
,
i
+
1
)
≤
C
(
1
+
T
)
1
κ
0
B
i
{
(
1
+
(
θ
T
)
-
1
)
1
α
M
0
-
1
+
α
κ
0
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
2
-
α
κ
0
+
(
1
+
(
θ
T
)
-
1
)
1
α
-
a
M
0
-
1
+
α
κ
0
∥
p
(
k
i
)
∥
L
α
(
A
i
,
i
)
α
α
-
a
+
1
-
α
κ
0
}
.
Denote Yi=∥p(ki)∥Lα(Ai,i). Then
Y
i
+
1
≤
C
B
i
(
D
1
Y
i
1
+
δ
1
+
D
2
Y
i
1
+
ν
1
)
,
where δ1>0 is defined in (4.1), ν1=αα-a-ακ0>0, and
D
1
=
(
1
+
T
)
1
κ
0
(
1
+
(
θ
T
)
-
1
)
1
α
M
0
-
δ
1
and
D
2
=
(
1
+
T
)
1
κ
0
(
1
+
(
θ
T
)
-
1
)
1
α
-
a
M
0
-
δ
1
.
Take M0 sufficiently large such that
(4.12)
Y
0
≤
C
min
{
D
1
-
1
δ
1
,
D
2
-
1
ν
1
}
.
Then limi→∞Yi=0 by Lemma 2.3. Thus, consequently, ∫θTT∫Bρ/4(x0)|p(M0)|α𝑑x𝑑t=0, that is,
p
(
x
,
t
)
≤
M
0
a.e. in
B
ρ
/
4
(
x
0
)
×
(
θ
T
,
T
)
.
Since Y0≤∥p∥Lα(QT), condition (4.12) is met if ∥p∥Lα(QT)≤Cmin{D1-1/δ1,D2-1/ν1}. This, in turn, is satisfied if
M
0
≥
C
(
1
+
T
)
1
κ
0
δ
1
{
(
1
+
(
θ
T
)
-
1
)
1
α
δ
1
∥
p
∥
L
α
(
Q
T
)
+
(
1
+
(
θ
T
)
-
1
)
1
(
α
-
a
)
δ
1
∥
p
∥
L
α
(
Q
T
)
ν
1
δ
1
}
.
Note that ν1δ1=1δ2=κ2>1. Combining this and condition M0≥1, we choose
M
0
=
C
(
1
+
T
)
κ
1
κ
0
(
1
+
θ
T
)
-
1
)
κ
1
α
-
a
(
1
+
∥
p
∥
L
α
(
Q
T
)
)
κ
2
with an appropriate positive constant C. Using a finite covering of U¯′, we obtain
p
(
x
,
t
)
≤
M
0
a.e. in
U
′
×
(
θ
T
,
T
)
.
Repeating the argument for -p instead of p, we obtain |p(x,t)|≤M0 a.e. in U′×[θT,T], and hence (4.2) follows. The proof is complete. ∎
Combining Theorem 4.1 with the estimates in Section 3, we obtain the following specific estimates for the L∞-norm.
Theorem 4.2.
Let U′⋐U.
If
t
∈
(
0
,
1
)
, then
(4.13)
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
κ
1
α
-
a
(
1
+
∥
p
¯
0
∥
L
α
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
2
.
If
t
≥
1
, then
(4.14)
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
2
.
If
A
(
α
)
<
∞
, then
(4.15)
lim sup
t
→
∞
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
2
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that
(4.16)
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
∥
A
(
α
,
⋅
)
∥
L
α
α
-
a
(
t
-
1
,
t
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
2
for all t≥T.
Proof.
(i) Note that α=α^. Let t∈(0,1). Applying (4.2) for T=t and θ=12, we have
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
t
-
κ
1
α
-
a
)
(
1
+
∥
p
∥
L
α
(
U
×
(
0
,
t
)
)
κ
2
)
≤
C
t
-
κ
1
α
-
a
(
1
+
∥
p
¯
∥
L
α
(
U
×
(
0
,
t
)
)
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
2
.
Using (3.2) to estimate ∥p¯∥Lα(U×(0,t)) and the fact that EnvA(α,t) is increasing in t, we obtain (4.13).
For t≥1, applying (4.2) with T0=t-1, T=1 and θ=12, we obtain
(4.17)
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∥
p
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
κ
2
)
≤
C
(
1
+
∥
p
¯
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
2
.
Again, using inequality (3.2) to estimate ∥p¯∥Lα(U×(t-1,t)), we obtain (4.14).
(ii) From (4.17) we have
lim sup
t
→
∞
∥
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
lim sup
t
→
∞
∥
p
¯
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
2
.
By (3.3), we have
lim sup
t
→
∞
∥
p
¯
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
α
≤
lim sup
t
→
∞
∫
U
|
p
¯
(
x
,
t
)
|
α
𝑑
x
≤
C
(
1
+
A
(
α
)
)
α
α
-
a
.
Thus (4.15) follows.
(iii) By (3.4), we have for large t that
(4.18)
∫
t
-
1
t
∫
U
|
p
¯
(
x
,
τ
)
|
α
𝑑
x
𝑑
τ
≤
C
(
1
+
β
(
α
)
α
α
-
2
a
+
∫
t
-
1
t
A
(
α
,
τ
)
α
α
-
a
𝑑
τ
)
.
From (4.17) and (4.18) we obtain (4.16). ∎
It is worth mentioning that the quantities on the right-hand sides of estimates (4.15) and (4.16) only depend on the boundary data’s large time behavior. This feature will be re-established in many of our estimates below for the gradient, time derivative and Hessian of the pressure.
5 Interior Estimates for Pressure Gradient
We will first estimate the pressure gradient in Ls-norm (for s<∞) in Section 5.1, and then in L∞-norm in Section 5.2.
5.1 Ls-Estimates
The following imbedding lemma from [16] is a suitable extension of [17, p. 93, Lemma 5.4].
Lemma 5.1 (cf. [16, Lemma 3.2]).
For each s≥1 there exists a constant C>0 depending on s such that for each smooth cut-off function ζ(x)∈Cc∞(U) the following inequality holds:
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
ζ
2
d
x
≤
C
max
supp
ζ
|
p
|
2
[
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
-
2
|
∇
2
p
|
2
ζ
2
d
x
+
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
|
∇
ζ
|
2
d
x
]
for every sufficiently regular function p(x) such that the right-hand side is well-defined.
We establish the basic step for the Ladyzhenskaya–Uraltseva iteration.
Lemma 5.2.
For each s≥0 if T0≥0, T>0, and ζ(x,t) is a smooth function with ζ(⋅,T0)=0 and suppζ(⋅,t)⊂U for all t∈[T0,T0+T], then
sup
[
T
0
,
T
0
+
T
]
∫
U
|
∇
p
(
x
,
t
)
|
2
s
+
2
ζ
2
𝑑
x
+
∫
T
0
T
0
+
T
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
2
|
∇
p
|
2
s
ζ
2
𝑑
x
𝑑
t
(5.1)
≤
C
∫
T
0
T
0
+
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
|
∇
ζ
|
2
𝑑
x
𝑑
t
+
C
∫
T
0
T
0
+
T
∫
U
|
∇
p
|
2
s
+
2
ζ
|
ζ
t
|
𝑑
x
𝑑
t
.
Proof.
Without loss of generality, assume T0=0. Multiplying equation (3.1) by -∇⋅(|∇p|2sζ2∇p), integrating the resultant over U and using integration by parts, we obtain
1
2
s
+
2
d
d
t
∫
U
|
∇
p
|
2
s
+
2
ζ
2
𝑑
x
=
-
∫
U
∂
j
(
K
(
|
∇
p
|
)
∂
i
p
)
∂
i
(
|
∇
p
|
2
s
∂
j
p
ζ
2
)
d
x
+
1
s
+
1
∫
U
|
∇
p
|
2
s
+
2
ζ
ζ
t
𝑑
x
.
Calculated as in [16, Lemma 3.1], the above equation is rewritten as
1
2
s
+
2
d
d
t
∫
U
|
∇
p
|
2
s
+
2
ζ
2
𝑑
x
=
-
∫
U
[
∂
y
l
(
K
(
|
y
|
)
y
i
)
|
y
=
∇
p
∂
j
∂
l
p
]
∂
j
∂
i
p
|
∇
p
|
2
s
ζ
2
𝑑
x
-
2
∫
U
[
∂
y
l
(
K
(
|
y
|
)
y
i
)
|
y
=
∇
p
∂
j
∂
l
p
]
∂
j
p
|
∇
p
|
2
s
ζ
∂
i
ζ
d
x
-
2
s
∫
U
[
∂
y
l
(
K
(
|
y
|
)
y
i
)
|
y
=
∇
p
∂
j
∂
l
p
]
∂
j
p
(
|
∇
p
|
2
s
-
2
∂
i
∂
m
p
∂
m
p
)
ζ
2
𝑑
x
+
1
s
+
1
∫
U
|
∇
p
|
2
s
+
2
ζ
ζ
t
𝑑
x
.
We denote the four terms on the right-hand side by I1, I2, I3, and I4. It follows from the calculations in [16, Lemma 3.1] that
I
1
≤
-
(
1
-
a
)
∑
j
∫
U
K
(
|
∇
p
|
)
|
∇
(
∂
j
p
)
|
2
|
∇
p
|
2
s
ζ
2
𝑑
x
,
|
I
2
|
≤
2
(
1
+
a
)
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
|
∇
p
|
2
s
+
1
ζ
|
∇
ζ
|
𝑑
x
,
and
I
3
≤
-
2
(
1
-
a
)
s
∫
U
K
(
|
∇
p
|
)
|
∇
(
1
2
|
∇
p
|
2
)
|
2
|
∇
p
|
2
s
-
2
ζ
2
𝑑
x
≤
0
.
Combining these estimates with Young’s inequality, we find that
1
2
s
+
2
d
d
t
∫
U
|
∇
p
|
2
s
+
2
ζ
2
𝑑
x
+
1
-
a
2
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
2
|
∇
p
|
2
s
ζ
2
𝑑
x
(5.2)
≤
C
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
|
∇
ζ
|
2
𝑑
x
+
1
s
+
1
∫
U
|
∇
p
|
2
s
+
2
ζ
|
ζ
t
|
𝑑
x
.
Inequality (5.1) follows directly by integrating (5.2) from 0 to T. ∎
Next, we reduce estimates for the W1,s-norm, with large s, down to W1,2-a and L∞ norms.
Proposition 5.3.
Let U′⋐V⋐U, T0≥0, T>0, and θ∈(0,1). If s≥2, then
∫
T
0
+
θ
T
T
0
+
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
(5.3)
≤
C
(
1
+
(
θ
T
)
-
1
)
s
-
2
(
1
+
sup
[
T
0
+
θ
T
/
2
,
T
0
+
T
]
∥
p
∥
L
∞
(
V
)
2
)
s
-
2
⋅
∫
T
0
T
0
+
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
and
sup
t
∈
[
T
0
+
θ
T
,
T
0
+
T
]
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
(5.4)
≤
C
(
1
+
(
θ
T
)
-
1
)
s
+
a
-
1
(
1
+
sup
[
T
0
+
θ
T
/
2
,
T
0
+
T
]
∥
p
∥
L
∞
(
V
)
2
)
s
-
2
+
a
⋅
∫
T
0
T
0
+
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
,
where the constant C>0 is independent of T0, T and θ.
Proof.
Without loss of generality, assume T0=0. First, we prove a more general version of (5.3).
Claim. For 0<θ′<θ<1 we have
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
(5.5)
≤
C
(
1
+
[
(
θ
-
θ
′
)
T
]
-
1
)
s
-
2
(
1
+
sup
[
θ
′
T
,
T
]
∥
p
∥
L
∞
(
V
)
2
)
s
-
2
⋅
∫
0
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
,
where the constant C>0 is independent of T, θ and θ′.
Note that (5.5) holds trivially when s=2. Hence, we will only prove (5.5) for s>2 in a number of steps.
Step 1. Assume ζ(x,t) to be the cut-off function with ζ(x,t)=0 for (x,t)∉V×(θ′T,T], and ζ(x,t)=1 for (x,t)∈U′×(θT,T]. Applying Lemma 5.1 with s+1 in place of s, we have for s≥0 that
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
4
ζ
2
𝑑
x
𝑑
t
≤
C
max
supp
ζ
|
p
|
2
[
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
|
∇
2
p
|
2
ζ
2
d
x
d
t
+
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
|
∇
ζ
|
2
d
x
d
t
]
.
Let N0=supV×[θ′T,T]|p|2. Using (5.1) to estimate the first integral on the right-hand side, we find that
(5.6)
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
4
ζ
2
𝑑
x
𝑑
t
≤
C
N
0
[
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
|
∇
ζ
|
2
𝑑
x
𝑑
t
+
∫
0
T
∫
U
|
∇
p
|
2
s
+
2
ζ
|
ζ
t
|
𝑑
x
𝑑
t
]
.
By the boundedness of the function K(ξ) and its property (2.8), we have
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
≤
C
|
∇
p
|
2
s
+
2
≤
C
K
(
|
∇
p
|
)
(
1
+
|
∇
p
|
a
)
|
∇
p
|
2
s
+
2
≤
C
K
(
|
∇
p
|
)
(
1
+
|
∇
p
|
2
s
+
2
+
a
)
(5.7)
≤
C
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
+
a
)
.
Hence, inequality (5.6) leads to
(5.8)
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
4
ζ
2
𝑑
x
𝑑
t
≤
C
N
0
∫
0
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
+
a
)
(
|
∇
ζ
|
2
+
ζ
|
ζ
t
|
)
𝑑
x
𝑑
t
.
Step 2. Let s=0 in (5.6). For ε>0 applying Cauchy’s inequality to the last integral of (5.6) gives
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
4
ζ
2
𝑑
x
𝑑
t
≤
ε
N
0
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
4
ζ
2
𝑑
x
𝑑
t
+
C
ε
-
1
N
0
∫
0
T
∫
U
K
(
|
∇
p
|
)
-
1
ζ
t
2
𝑑
x
𝑑
t
(5.9)
+
C
N
0
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
|
∇
ζ
|
2
𝑑
x
𝑑
t
.
Here, the cut-off function ζ satisfies |∇ζ|≤C and 0≤ζt≤C[(θ-θ′)T]-1. Taking ε=(2(N0+1))-1 in (5.9) and using Young’s inequality yield
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
4
𝑑
x
𝑑
t
≤
C
(
N
0
+
1
)
N
0
(
(
θ
-
θ
′
)
T
)
-
2
∫
θ
′
T
T
∫
U
(
1
+
|
∇
p
|
)
a
𝑑
x
𝑑
t
+
C
N
0
∫
θ
′
T
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
𝑑
x
𝑑
t
≤
C
(
1
+
N
0
)
2
(
1
+
1
(
θ
-
θ
′
)
T
)
2
∫
θ
′
T
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
.
This implies (5.5) when s=4.
Step 3. When s∈(2,4), let β be a number in (0,1) such that 1s=1-β2+β4. Then, by the interpolation inequality we have
(
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
)
1
s
≤
(
∫
θ
T
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
𝑑
x
𝑑
t
)
1
-
β
2
(
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
4
𝑑
x
𝑑
t
)
β
4
.
Note that βs4=s2-1. Using (5.9) to estimate the last double integral, we obtain
(5.10)
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
≤
C
(
1
+
1
(
θ
-
θ
′
)
T
)
s
-
2
(
1
+
N
0
)
s
-
2
∫
θ
′
T
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
.
This implies (5.5) for s∈(2,4). Therefore, we have proved (5.5) for s∈(2,4].
Step 4. When s>4, let m be the positive integer that satisfies s-42-a≤m<s-42-a+1. Then
s
-
m
(
2
-
a
)
∈
(
2
+
a
,
4
]
.
Then let {Uk}k=0m and {Vk}k=0m be two families of open subsets of U such that
U
′
=
U
0
⋐
V
0
⋐
U
1
⋐
V
1
⋐
U
2
⋐
V
2
⋐
⋯
⋐
U
m
-
1
⋐
V
m
-
1
⋐
U
m
⋐
V
m
=
V
⋐
U
.
For each k=0,1,…,m+1 let θk=θ-(θ-θ′)km+1. Note that θ=θ0>θ1>θ2>⋯>θm>θm+1=θ′ and θk-θk+1=θ-θ′m+1. For 1≤k≤m let ζk(x,t) be a smooth cut-off function which is equal to one on Uk×(tk,T] and zero on QT∖Vk×(tk-1,T], and satisfies |∇ζk|≤Cm and 0≤ζk,t≤Cm[(θ-θ′)T]-1. Here and in the following calculations in this proof, Cm denotes a generic positive constant depending on all Uk and Vk, k=1,2,…,m.
We still denote N0=supV×[θ′T,T]|p|2. Applying (5.8) to V0, U1, θ1 in place of V, U, θ, we have
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
≤
C
m
(
1
+
[
(
θ
-
θ
′
)
T
]
-
1
)
N
0
∫
θ
1
T
T
∫
U
1
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
s
-
(
2
-
a
)
)
𝑑
x
𝑑
t
=
C
m
(
1
+
[
(
θ
-
θ
′
)
T
]
-
1
)
N
0
[
T
+
∫
θ
1
T
T
∫
U
1
K
(
|
∇
p
|
)
|
∇
p
|
s
-
(
2
-
a
)
𝑑
x
𝑑
t
]
.
Above, we used the fact that supVk×[θk-1T,T]|p|2≤N0 for 1≤k≤m. Applying (5.8) recursively to Uk⋐Vk⋐Uk+1 and θk>θk+1, we obtain
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
≤
C
m
T
∑
i
=
1
m
d
i
+
C
d
m
∫
θ
m
T
T
∫
U
m
K
(
|
∇
p
|
)
|
∇
p
|
s
-
m
(
2
-
a
)
𝑑
x
𝑑
t
≤
C
m
d
m
[
T
+
∫
θ
m
T
T
∫
U
m
K
(
|
∇
p
|
)
|
∇
p
|
s
-
m
(
2
-
a
)
𝑑
x
𝑑
t
]
,
where d=(1+[(θ-θ′)T]-1)(1+N0). Since s-m(2-a)∈(2,4], estimating the last integral by (5.10) gives
(5.11)
∫
θ
T
T
∫
U
′
K
(
|
∇
p
|
)
|
∇
p
|
s
𝑑
x
𝑑
t
≤
C
d
m
T
+
C
d
s
-
m
(
2
-
a
)
-
2
+
m
∫
θ
′
T
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
𝑑
x
𝑑
t
.
Since d≥1 and m,s-m(2-a)-2+m≤s-2, the desired estimate (5.5) follows from (5.11). This completes the proof of (5.5) for all s≥2.
The inequality (5.3) then is an obvious consequence of (5.5) with θ′=θ2.
Now, we turn to the proof of (5.4). For s>2-a it follows from (5.1) that
sup
[
0
,
T
]
∫
U
|
∇
p
|
s
ζ
2
𝑑
x
≤
C
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
s
|
∇
ζ
|
2
𝑑
x
𝑑
t
+
C
∫
0
T
∫
U
|
∇
p
|
s
ζ
|
ζ
t
|
𝑑
x
𝑑
t
≤
C
∫
0
T
∫
U
|
∇
p
|
s
(
|
∇
ζ
|
2
+
ζ
|
ζ
t
|
)
𝑑
x
𝑑
t
.
Similar to (5.7), we have |∇p|s≤C(1+K(|∇p|)|∇p|s+a), and hence
(5.12)
sup
[
0
,
T
]
∫
U
|
∇
p
|
s
ζ
2
𝑑
x
≤
C
∫
0
T
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
s
+
a
)
(
|
∇
ζ
|
2
+
ζ
|
ζ
t
|
)
𝑑
x
𝑑
t
.
Let V′ be an open set such that U′⋐V′⋐V. Choose ζ(x,t) such that ζ=0 for t≤3θT4 or x∉V′, and ζ=1 for (x,t)∈U′×[θT,T]. Then we have from (5.12) that
sup
[
θ
T
,
T
]
∫
U
|
∇
p
|
s
𝑑
x
≤
C
(
1
+
(
θ
T
)
-
1
)
∫
3
θ
T
/
4
T
∫
V
′
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
s
+
a
)
𝑑
x
𝑑
t
.
To estimate the last integral we apply (5.5) with the parameters s, θ′, θ, U′ being replaced by s+a, θ2, 3θ4, V′, respectively. Therefore, we obtain
sup
[
θ
T
,
T
]
∫
U
′
|
∇
p
|
s
𝑑
x
≤
C
{
(
1
+
(
θ
T
)
-
1
)
s
+
a
-
1
(
1
+
N
0
)
s
+
a
-
2
∫
0
T
∫
U
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
d
x
d
t
}
,
hence proving (5.4). The proof is complete. ∎
Now, we combine Proposition 5.3 with the estimates in Section 3 to express the bounds in terms of the initial and boundary data.
Theorem 5.4.
Let U′⋐U and s≥2. If t∈(0,2), then
(5.13)
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
t
-
μ
1
(
1
+
∥
p
¯
0
∥
L
α
)
μ
2
+
2
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
μ
2
⋅
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
,
where
(5.14)
μ
1
=
μ
1
(
α
,
s
)
:=
[
1
+
2
κ
1
α
-
a
]
(
s
+
a
-
2
)
+
1
𝑎𝑛𝑑
μ
2
=
μ
2
(
α
,
s
)
:=
2
κ
2
(
s
-
2
+
a
)
.
If t≥2, then
(5.15)
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
μ
2
+
α
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
μ
2
+
α
⋅
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
Proof.
Let t∈(0,2). Applying (5.4) to T0=0, T=t and θ=12, and using (4.13), (3.5) and relation (2.11), we obtain
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
t
-
μ
1
⋅
(
1
+
∥
p
¯
0
∥
L
α
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
2
κ
2
(
s
-
2
+
a
)
⋅
(
1
+
∥
p
¯
0
∥
L
2
2
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
.
Then (5.13) follows. Let t≥2. Applying (5.4) with T0=t-1, T=1 and θ=12, then combining it with (4.14) and (3.18), we obtain
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
∥
p
¯
0
∥
L
α
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
μ
2
⋅
(
1
+
∥
p
¯
0
∥
L
α
α
+
Env
A
(
α
,
t
)
α
α
-
a
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
Then (5.15) follows. ∎
For large time estimates, we have the following result.
Theorem 5.5.
Let U′⋐U, let s≥2 and let μ2 be defined as in Theorem 5.4.
If
A
(
α
)
<
∞
, then
(5.16)
lim sup
t
→
∞
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
μ
2
+
α
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that for all
t
>
T
we have
(5.17)
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
2
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
μ
2
+
α
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
Proof.
Let t≥1. Applying (5.4) with T0=t-1, T=1 and θ=12 yields
(5.18)
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
sup
[
t
-
1
,
t
]
∥
p
∥
L
∞
(
V
)
2
)
s
-
2
+
a
⋅
∫
t
-
1
t
∫
U
(
1
+
K
(
|
∇
p
|
)
|
∇
p
|
2
)
𝑑
x
𝑑
t
.
(i) Taking the limit superior as t→∞ and using (4.15) and (3.22) give
lim sup
t
→
∞
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
μ
2
⋅
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
Then estimate (5.16) follows.
(ii) Combining (5.18) with (4.16) and (3.26), we have
∫
U
′
|
∇
p
(
x
,
t
)
|
s
𝑑
x
≤
C
{
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
2
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
}
μ
2
⋅
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
.
Then (5.17) follows. ∎
5.2 L∞-Estimates
In this subsection, we obtain interior L∞-estimates for the gradient of pressure. For each m=1,2,…,n denote um=pxm and u=(u1,u2,…,un)=∇p. We have
(5.19)
∂
u
m
∂
t
=
∂
m
(
∇
⋅
(
K
(
|
u
|
)
u
)
)
=
∇
⋅
(
K
(
|
u
|
)
∂
m
u
)
+
∇
⋅
[
K
′
(
|
u
|
)
∑
i
u
i
∂
m
u
i
|
u
|
u
]
.
Since ∂ium=∂mui, we have
∂
m
u
=
(
∂
m
u
1
,
…
,
∂
m
u
n
)
=
(
∂
1
u
m
,
…
,
∂
n
u
m
)
=
∇
u
m
and
∑
i
u
i
∂
m
u
i
=
∑
i
u
i
∂
i
u
m
=
u
⋅
∇
u
m
.
Therefore, we rewrite (5.19) as
(5.20)
∂
u
m
∂
t
=
∇
⋅
(
K
(
|
u
|
)
∇
u
m
)
+
∇
⋅
[
K
′
(
|
u
|
)
u
⋅
∇
u
m
|
u
|
u
]
.
We will apply De Giorgi’s technique to equation (5.20). In the following, we fix a number s0 such that r=s0 satisfies (2.16). Note that s0*>2. We will also use sj for j≥1 to denote some exponents that depend on s0 but are independent of α. Let
s
1
=
(
1
-
2
s
0
*
)
-
1
>
1
.
Theorem 5.6.
Let U′⋐V⋐U. For any T0≥0, T>0 and θ∈(0,1), if t∈[T0+θT,T0+T], then
(5.21)
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
(
θ
T
)
-
1
)
s
1
+
1
2
λ
s
1
2
∥
∇
p
∥
L
2
(
V
×
(
T
0
+
θ
T
/
2
,
T
0
+
T
)
)
,
where
(5.22)
λ
=
λ
(
T
0
,
T
,
θ
;
V
)
=
(
∫
T
0
+
θ
T
/
2
T
0
+
T
∫
V
(
1
+
|
∇
p
|
)
a
s
0
2
-
s
0
𝑑
x
𝑑
t
)
2
-
s
0
s
0
and the constant C>0 is independent of T0, T and θ.
Proof.
Without loss of generality, assume T0=0. Fix m∈{1,2,…,n}. We will show for t∈[θT,T] that
(5.23)
∥
p
x
m
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
(
θ
T
)
-
1
)
s
1
+
1
2
λ
s
1
2
∥
p
x
m
∥
L
2
(
V
×
(
T
0
+
θ
T
/
2
,
T
0
+
T
)
)
.
Let ζ(x,t)=ϕ(x)φ(t) be a cut-off function with φ(t)=0 for t≤θT/2, and suppϕ⊂U. For k≥0 we define
u
m
(
k
)
=
max
{
u
m
-
k
,
0
}
,
S
k
(
t
)
=
{
x
∈
U
:
u
m
(
k
)
(
x
,
t
)
≥
0
}
,
and denote by χk(x,t) the characteristic function of Sk(t).
Multiplying (5.20) by um(k)ζ2 and integrating over U, we have
1
2
d
d
t
∫
U
|
u
m
(
k
)
|
2
ζ
2
𝑑
x
=
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
-
∫
U
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
2
ζ
2
𝑑
x
-
2
∫
U
K
(
|
u
|
)
(
∇
u
m
(
k
)
⋅
∇
ζ
)
u
m
(
k
)
ζ
𝑑
x
(5.24)
-
∫
U
1
|
u
|
K
′
(
|
u
|
)
(
u
⋅
∇
u
m
(
k
)
)
(
u
⋅
∇
(
u
m
(
k
)
ζ
2
)
)
𝑑
x
.
By the product rule,
-
1
|
u
|
K
′
(
|
u
|
)
(
u
⋅
∇
u
m
(
k
)
)
(
u
⋅
∇
(
u
m
(
k
)
ζ
2
)
)
=
-
1
|
u
|
K
′
(
|
u
|
)
(
u
⋅
∇
u
m
(
k
)
)
(
u
⋅
∇
u
m
(
k
)
ζ
2
+
2
u
m
(
k
)
ζ
u
⋅
∇
ζ
)
.
It follows from property (2.10) that
-
1
|
u
|
K
′
(
|
u
|
)
(
u
⋅
∇
(
u
m
(
k
)
)
2
ζ
2
)
≤
a
K
(
|
u
|
)
|
u
|
2
|
u
|
2
|
∇
u
m
(
k
)
|
2
ζ
2
=
a
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
2
ζ
2
and
-
2
|
u
|
K
′
(
|
u
|
)
u
2
u
m
(
k
)
ζ
∇
u
m
(
k
)
⋅
∇
ζ
≤
2
|
u
|
K
′
(
|
u
|
)
|
u
|
2
|
∇
u
m
(
k
)
|
|
u
m
(
k
)
|
ζ
|
∇
ζ
|
≤
C
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
|
u
m
(
k
)
|
ζ
|
∇
ζ
|
.
Then we obtain from (5.24) that
1
2
d
d
t
∫
U
|
u
m
(
k
)
|
2
ζ
2
𝑑
x
≤
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
-
(
1
-
a
)
∫
U
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
2
ζ
2
𝑑
x
+
C
∫
U
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
|
u
m
(
k
)
|
ζ
|
∇
ζ
|
𝑑
x
.
Applying Cauchy’s inequality to the last term in the previous inequality yields
1
2
d
d
t
∫
U
|
u
m
(
k
)
|
2
ζ
2
𝑑
x
≤
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
-
1
-
a
2
∫
U
K
(
|
u
|
)
|
∇
u
m
(
k
)
|
2
ζ
2
𝑑
x
+
C
∫
U
K
(
|
u
|
)
|
u
m
(
k
)
|
2
|
∇
ζ
|
2
𝑑
x
.
Since
|
ζ
∇
u
m
(
k
)
|
2
=
|
∇
(
u
m
(
k
)
ζ
)
-
u
m
(
k
)
∇
ζ
|
2
≥
1
2
|
∇
(
u
m
(
k
)
ζ
)
|
2
-
|
u
m
(
k
)
∇
ζ
|
2
,
we obtain
(5.25)
1
2
d
d
t
∫
U
|
u
m
(
k
)
ζ
|
2
𝑑
x
+
1
-
a
4
∫
U
K
(
|
u
|
)
|
∇
(
u
m
(
k
)
ζ
)
|
2
𝑑
x
≤
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
+
C
∫
U
K
(
|
u
|
)
|
u
m
(
k
)
|
2
|
∇
ζ
|
2
𝑑
x
.
Integrating (5.25) from 0 to t for t∈[0,T], and then taking the supremum in t give
(5.26)
max
[
0
,
T
]
∫
U
|
u
m
(
k
)
ζ
|
2
𝑑
x
+
C
∫
0
T
∫
U
K
(
|
u
|
)
|
∇
(
u
m
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
≤
∫
0
T
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
𝑑
t
+
C
∫
0
T
∫
U
K
(
|
u
|
)
|
u
m
(
k
)
|
2
|
∇
ζ
|
2
𝑑
x
𝑑
t
.
Let
(5.27)
ν
2
=
4
(
1
-
1
s
0
*
)
>
2
.
Applying Lemma 2.2 to the function um(k)ζ which vanishes on the boundary, weight W=K(|u|) and exponents r=s0 and ϱ=ϱ(s0)=ν2, we have
∥
u
m
(
k
)
ζ
∥
L
ν
2
(
Q
T
)
≤
C
[
ess
sup
t
∈
[
0
,
T
]
∥
u
m
(
k
)
ζ
∥
L
2
(
U
)
+
(
∫
0
T
∫
U
K
(
|
u
|
)
|
∇
(
u
m
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
)
1
2
]
⋅
[
∫
0
T
∫
supp
ζ
K
(
|
u
|
)
-
s
0
2
-
s
0
𝑑
x
𝑑
t
]
2
-
s
0
ν
2
s
0
.
Using (2.9), we have
K
(
|
u
|
)
-
s
0
2
-
s
0
≤
C
(
1
+
|
u
|
)
a
s
0
2
-
s
0
,
hence
(5.28)
∥
u
m
(
k
)
ζ
∥
L
ν
2
(
Q
T
)
≤
C
λ
1
ν
2
[
max
t
∈
[
0
,
T
]
∫
U
|
u
m
(
k
)
ζ
|
2
𝑑
x
+
∫
0
T
∫
U
K
(
|
u
|
)
|
∇
(
u
m
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
]
1
2
.
By (5.28), (5.26) and the boundedness of the function K(⋅), we find that
∥
u
m
(
k
)
ζ
∥
L
ν
2
(
Q
T
)
≤
C
λ
1
ν
2
(
max
[
0
,
T
]
∫
U
|
u
m
(
k
)
ζ
|
2
𝑑
x
+
C
∫
0
T
∫
U
K
(
|
u
|
)
|
∇
(
u
m
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
)
1
2
(5.29)
≤
C
λ
1
ν
2
(
∫
0
T
∫
U
|
u
m
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
𝑑
t
+
∫
0
T
∫
U
|
u
m
(
k
)
|
2
|
∇
ζ
|
2
𝑑
x
𝑑
t
)
1
2
.
Here, we use the same notation x0, ρ, M0, ki, ρi, Ui, 𝒬i, and ζi as introduced in the proof of Theorem 4.1 from (4.4) to (4.6). Also, the sets Ai,j are defined by (4.5) with p being replaced by um.
Define Fi=∥um(ki+1)ζi∥Lν2(Ai+1,i). Applying (5.29) with k=ki+1 and ζ=ζi gives
(5.30)
F
i
≤
C
λ
1
ν
2
{
∫
0
T
∫
U
|
u
m
(
k
i
+
1
)
|
2
ζ
i
|
(
ζ
i
)
t
|
𝑑
x
𝑑
t
+
∫
0
T
∫
U
|
u
m
(
k
i
+
1
)
|
2
|
∇
ζ
i
|
2
𝑑
x
𝑑
t
}
1
2
.
Using (4.6), we obtain
F
i
≤
C
2
i
λ
1
ν
2
(
1
+
1
θ
T
)
1
2
∥
u
m
(
k
i
+
1
)
∥
L
2
(
A
i
+
1
,
i
)
≤
C
2
i
(
1
+
1
θ
T
)
1
2
λ
∥
u
m
(
k
i
)
∥
L
2
(
A
i
,
i
)
.
Since ν2>2, it follows from Hölder’s inequality that
∥
u
m
(
k
i
+
1
)
ζ
i
∥
L
2
(
A
i
+
1
,
i
+
1
)
≤
∥
u
m
(
k
i
+
1
)
ζ
i
∥
L
ν
2
(
A
i
+
1
,
i
+
1
)
|
A
i
+
1
,
i
+
1
|
1
2
-
1
ν
2
(5.31)
≤
∥
u
m
(
k
i
+
1
)
ζ
i
∥
L
ν
2
(
A
i
+
1
,
i
+
1
)
|
A
i
+
1
,
i
|
1
2
-
1
ν
2
≤
C
F
i
|
A
i
+
1
,
i
|
1
2
-
1
ν
2
.
Note that ∥um(ki)∥L2(Ai,i)≥∥um(ki)∥L2(Ai+1,i)≥(ki+1-ki)|Ai+1,i|1/2. Thus,
(5.32)
|
A
i
+
1
,
i
|
≤
(
k
i
+
1
-
k
i
)
-
2
∥
u
m
(
k
i
)
∥
L
2
(
A
i
)
2
≤
C
4
i
M
0
-
2
∥
u
m
(
k
i
)
∥
L
2
(
A
i
,
i
)
2
.
Then it follows from (5.31), (5.30) and (5.32) that
∥
u
m
(
k
i
+
1
)
∥
L
2
(
A
i
+
1
,
i
+
1
)
≤
C
2
i
(
1
+
1
θ
T
)
1
2
λ
1
ν
2
∥
u
m
(
k
i
)
∥
L
2
(
A
i
)
2
i
-
2
i
ν
2
M
0
-
1
+
2
ν
2
∥
u
m
(
k
i
)
∥
L
2
(
A
i
,
i
)
1
-
2
ν
2
≤
C
4
i
(
1
+
1
θ
T
)
1
2
λ
1
ν
2
M
0
-
1
+
2
ν
2
∥
u
m
(
k
i
)
∥
L
2
(
A
i
,
i
)
2
-
2
ν
2
.
Denote ν3=1-2ν2. Let Yi=∥um(ki)∥L2(Ai,i), B=4 and D=C(1+1θT)1/2λ1/ν2M0-ν3. We obtain
Y
i
+
1
≤
D
B
i
Y
i
1
+
ν
3
for all
i
≥
0
.
We now determine M0 so that Y0≤D-1/ν3B-1/ν32. This condition is met if
M
0
≥
C
[
λ
1
ν
2
(
1
+
1
θ
T
)
1
2
]
1
ν
3
Y
0
=
C
λ
s
1
2
(
1
+
1
θ
T
)
1
+
s
1
2
Y
0
.
Since Y0=∥um(k0)∥L2(A0,0)≤∥um∥L2(V×(θT/2,T)), it suffices to choose M0 as
M
0
=
C
λ
s
1
2
(
1
+
1
θ
T
)
s
1
+
1
2
∥
u
m
∥
L
2
(
V
×
(
θ
T
/
2
,
T
)
)
.
Then Lemma 2.3 gives limi→∞Yi=0. Hence,
∫
θ
T
T
∫
B
(
x
0
,
ρ
/
4
)
|
u
m
(
M
0
)
|
2
𝑑
x
𝑑
t
=
0
.
Thus, um(x,t)≤M0 a.e. in B(x0,ρ4)×(0,T). Replacing um and u by -um and -u, respectively, and using the same argument, we obtain |um(x,t)|≤M0 a.e. in B(x0,ρ4)×(0,T). Now, covering U′ by finitely many such balls B(x0,ρ4), we come to the conclusion
(5.33)
|
u
m
(
x
,
t
)
|
≤
M
0
a.e. in
U
′
×
(
θ
T
,
T
)
.
By the choice of M0, we obtain from (5.33) that
|
u
m
(
x
,
t
)
|
≤
C
(
1
+
1
θ
T
)
s
1
+
1
2
λ
s
1
2
∥
u
m
∥
L
2
(
V
×
(
θ
T
/
2
,
T
)
)
for all m=1,…,n. Then (5.23) follows. ∎
We will combine Theorem 5.6 with the high integrability of ∇p in Section 5.1 to obtain the L∞-estimates. Let
s
2
=
max
{
2
,
a
s
0
2
-
s
0
}
,
s
3
=
s
1
(
2
-
s
0
)
s
0
+
1
,
κ
3
=
(
s
2
+
a
-
2
)
(
1
+
2
κ
1
α
-
a
)
,
κ
4
=
1
+
s
1
+
κ
3
s
3
,
κ
5
=
κ
2
(
s
2
-
2
+
a
)
.
Theorem 5.7.
If t∈(0,2), then
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
κ
4
2
(
1
+
∥
p
¯
0
∥
L
α
)
s
3
(
κ
5
+
1
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
s
3
κ
5
(5.34)
⋅
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
s
3
2
.
If t≥2, then
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
s
3
(
κ
5
+
α
2
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
s
3
(
κ
5
+
α
2
)
(5.35)
⋅
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
s
3
2
.
Proof.
First, we have from (5.21) that
(5.36)
sup
[
T
0
+
θ
T
,
T
0
+
T
]
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
(
θ
T
)
-
1
)
1
+
s
1
2
(
∫
T
0
+
θ
T
/
2
T
0
+
T
∫
V
(
1
+
|
∇
p
|
)
s
2
𝑑
x
𝑑
t
)
s
3
2
.
Let t∈(0,2). Applying (5.36) with T0=0, T=t and θ=12, we obtain
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
1
+
s
1
2
(
∫
t
/
4
t
∫
V
(
1
+
|
∇
p
|
)
s
2
𝑑
x
𝑑
t
)
s
3
2
.
We apply (5.13) with s=s2 and U′=V. Note from the formulas in (5.14) that
(5.37)
μ
1
(
α
,
s
2
)
=
κ
3
+
1
and
μ
2
(
α
,
s
2
)
=
2
κ
5
.
We obtain
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
1
+
s
1
2
{
t
⋅
t
-
(
κ
3
+
1
)
(
1
+
∥
p
¯
0
∥
L
α
)
2
(
κ
5
+
1
)
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
2
κ
5
⋅
(
1
+
∫
0
t
G
1
(
τ
)
d
τ
)
}
s
3
2
≤
C
t
-
1
+
s
1
+
κ
3
s
3
2
(
1
+
∥
p
¯
0
∥
L
α
)
s
3
(
κ
5
+
1
)
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
s
3
κ
5
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
s
3
2
.
Then (5.34) follows. Let t≥2. Applying (5.36) with T0=t-34, T=34 and θ=23 yields
(5.38)
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∫
t
-
1
2
t
∫
V
|
∇
p
|
s
2
𝑑
x
𝑑
t
)
s
3
2
.
Thanks to (5.15) with s=s2, we obtain (5.35). ∎
Combining (5.38) with Theorem 5.5, we have the following asymptotic estimates.
Theorem 5.8.
The following asymptotic estimates hold:
If
A
(
α
)
<
∞
, then
lim sup
t
→
∞
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
s
3
(
κ
5
+
α
2
)
(5.39)
⋅
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
s
3
2
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that when
t
>
T
, we have
∥
∇
p
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
3
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
3
,
t
)
)
)
s
3
(
κ
5
+
α
2
)
(5.40)
⋅
(
1
+
∫
t
-
2
t
G
1
(
τ
)
𝑑
τ
)
s
3
2
.
6 Interior Estimates for Time Derivative of Pressure
In this section, we estimate the L∞-norm of pt(x,t) for t>0. Let q=pt. Then
(6.1)
∂
q
∂
t
=
∇
⋅
(
K
(
|
∇
p
|
)
∇
p
)
t
.
Using (6.1), we first derive a local-in-time estimate for the L∞-norm of pt.
Proposition 6.1.
Let U′⋐V⋐U. If T0≥0, T>0 and θ∈(0,1), then
(6.2)
sup
[
T
0
+
θ
T
,
T
0
+
T
]
∥
p
t
∥
L
∞
(
U
′
)
≤
C
λ
s
1
2
(
1
+
(
θ
T
)
-
1
)
s
1
+
1
2
∥
p
t
∥
L
2
(
U
×
(
T
0
,
T
0
+
T
)
)
,
where s1 and λ=λ(T0,T,θ;V) are defined in Theorem 5.6, and the constant C>0 is independent of T0, T and θ.
Proof.
Without loss of generality, assume T0=0. For k≥0 let q(k)=max{q-k,0}, Sk(t)={x∈U:q(x,t)>k} and let χk(x,t) be the characteristic function of the set {(x,t)∈U×(0,T):q(x,t)>k}. On Sk(t), we have (∇p)t=∇q=∇q(k).
Let ζ=ζ(x,t) be the cut-off function on U×[0,T] satisfying ζ(⋅,0)=0 and ζ(⋅,t) having compact support in U. We will use the test function q(k)ζ2, noting that ∇(q(k)ζ2)=ζ[∇(q(k)ζ)+q(k)∇ζ]. Multiplying (6.1) by q(k)ζ2 and integrating the resultant on U, we get
1
2
d
d
t
∫
U
|
q
(
k
)
ζ
|
2
𝑑
x
=
∫
U
|
q
(
k
)
|
2
ζ
ζ
t
𝑑
x
-
∫
U
(
K
(
|
∇
p
|
)
)
t
∇
p
⋅
[
∇
(
q
(
k
)
ζ
)
+
q
(
k
)
∇
ζ
]
ζ
𝑑
x
(6.3)
-
∫
U
K
(
|
∇
p
|
)
(
∇
p
)
t
⋅
[
∇
(
q
(
k
)
ζ
)
+
q
(
k
)
∇
ζ
]
ζ
𝑑
x
.
For the last integral of (6.3) put z=ζ[∇(q(k)ζ)+q(k)∇ζ]. We have
(
∇
p
)
t
⋅
z
=
ζ
∇
q
(
k
)
⋅
[
∇
(
q
(
k
)
ζ
)
+
q
(
k
)
∇
ζ
]
=
|
∇
(
q
(
k
)
ζ
)
|
2
-
|
q
(
k
)
∇
ζ
|
2
.
For the second integral on the right-hand side of (6.3), taking into account (2.10), we have
|
(
K
(
|
∇
p
|
)
)
t
∇
p
⋅
z
|
=
|
K
′
(
|
∇
p
|
)
|
|
∇
p
⋅
∇
p
t
|
|
∇
p
|
|
∇
p
⋅
z
|
≤
a
K
(
|
∇
p
|
)
|
∇
q
|
|
z
|
.
Note that
|
∇
q
|
|
z
|
=
|
ζ
∇
q
¯
(
k
)
|
|
∇
(
q
(
k
)
ζ
)
+
q
(
k
)
∇
ζ
|
≤
{
|
∇
(
q
(
k
)
ζ
)
|
+
|
q
(
k
)
|
|
∇
ζ
|
}
2
=
|
∇
(
q
(
k
)
ζ
)
|
2
+
2
|
q
(
k
)
|
|
∇
ζ
|
|
∇
(
q
(
k
)
ζ
)
|
+
|
q
(
k
)
∇
ζ
|
2
,
and, applying Cauchy’s inequality to the second to last term, we have
a
|
∇
q
|
|
z
|
≤
a
|
∇
(
q
(
k
)
ζ
)
|
2
+
1
-
a
2
|
∇
(
q
(
k
)
ζ
)
|
2
+
2
a
2
1
-
a
|
q
¯
(
k
)
∇
ζ
|
2
+
a
|
q
(
k
)
∇
ζ
|
2
.
It follows from the above calculations that
d
d
t
∫
U
|
q
(
k
)
ζ
|
2
𝑑
x
+
(
1
-
a
)
∫
U
K
(
|
∇
p
|
)
|
∇
(
q
(
k
)
ζ
)
|
2
𝑑
x
≤
2
∫
U
|
q
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
+
C
∫
U
K
(
|
∇
p
|
)
|
q
(
k
)
∇
ζ
|
2
𝑑
x
.
Integrating this inequality from 0 to T, we obtain
max
[
0
,
T
]
∫
U
|
q
(
k
)
ζ
|
2
𝑑
x
+
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
(
q
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
≤
C
[
∫
0
T
∫
U
|
q
(
k
)
|
2
ζ
|
ζ
t
|
𝑑
x
𝑑
t
+
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
q
(
k
)
∇
ζ
|
2
𝑑
x
𝑑
t
]
≤
C
∫
0
T
∫
U
|
q
(
k
)
|
2
(
ζ
|
ζ
t
|
+
|
∇
ζ
|
2
)
𝑑
x
𝑑
t
.
The last inequality uses the fact that the function K(⋅) is bounded above. Applying Lemma 2.2 to q(k)ζ with W=K(|∇p|) and ϱ=ν2 defined by (5.27), we have
∥
q
(
k
)
ζ
∥
L
ν
2
(
Q
T
)
≤
C
λ
1
ν
2
⋅
{
max
[
0
,
T
]
∫
U
|
q
(
k
)
ζ
|
2
𝑑
x
+
∫
0
T
∫
U
K
(
|
∇
p
|
)
|
∇
(
q
(
k
)
ζ
)
|
2
𝑑
x
𝑑
t
}
1
2
,
and hence
∥
q
(
k
)
ζ
∥
L
ν
2
(
Q
T
)
≤
C
λ
1
ν
2
{
∫
0
T
∫
U
|
q
(
k
)
|
2
(
|
ζ
t
|
ζ
+
|
∇
ζ
|
2
)
𝑑
x
𝑑
t
}
1
2
.
This is similar to inequality (5.29). Then, following arguments of Theorem 5.6 applied for q(k) instead of um(k), we obtain (6.2). The proof is complete. ∎
The next theorem contains the estimates in terms of the initial and boundary data for all t>0. Let
κ
6
=
1
+
s
1
+
κ
3
(
s
3
-
1
)
,
κ
7
=
(
s
3
-
1
)
(
κ
5
+
1
)
+
1
,
κ
8
=
(
s
3
-
1
)
κ
5
,
κ
9
=
(
s
3
-
1
)
(
κ
5
+
α
2
)
+
α
2
.
Theorem 6.2.
Let U′⋐U. For t∈(0,2) we have
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
κ
6
2
(
1
+
∥
p
¯
0
∥
L
α
)
κ
7
(
1
+
∫
U
H
(
|
∇
p
0
(
x
)
|
)
𝑑
x
)
1
2
(6.4)
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
8
(
1
+
∫
0
t
G
3
(
τ
)
𝑑
τ
)
s
3
2
.
For t≥2 we have
(6.5)
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
κ
9
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
κ
9
⋅
(
1
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
s
3
2
.
Proof.
Let t∈(0,2); apply (6.2) with T0=0, T=t and θ=12. By (5.22) and (5.3), we have
λ
s
1
2
=
(
∫
t
/
2
t
∫
V
(
1
+
|
∇
p
|
)
a
s
0
2
-
s
0
𝑑
x
𝑑
τ
)
(
2
-
s
0
)
s
1
2
s
0
≤
(
∫
t
/
2
t
∫
V
(
1
+
|
∇
p
|
)
s
2
𝑑
x
𝑑
τ
)
ν
4
2
,
where
(6.6)
ν
4
=
(
2
-
s
0
)
s
1
s
0
=
s
3
-
1
.
By (5.13) and relation (5.37) we have
λ
s
1
2
≤
C
{
t
⋅
t
-
(
κ
3
+
1
)
(
1
+
∥
p
¯
0
∥
L
α
)
2
(
κ
5
+
1
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
2
κ
5
⋅
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
}
ν
4
2
≤
C
t
-
κ
3
ν
4
2
(
1
+
∥
p
¯
0
∥
L
α
)
(
κ
5
+
1
)
ν
4
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
5
ν
4
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
.
Combining this with (6.2) and (3.6) gives
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
λ
s
1
2
t
-
1
+
s
1
2
∥
p
t
∥
L
2
(
U
×
(
0
,
t
)
)
≤
C
t
-
κ
3
ν
4
-
(
1
+
s
1
)
2
(
1
+
∥
p
¯
0
∥
L
α
)
(
κ
5
+
1
)
ν
4
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
5
ν
4
⋅
(
1
+
∫
0
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
⋅
(
∫
U
[
H
(
|
∇
p
0
(
x
)
|
)
+
p
¯
0
2
(
x
)
]
𝑑
x
+
∫
0
t
G
3
(
τ
)
𝑑
τ
+
∫
0
t
∫
U
|
Ψ
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
)
1
2
.
Therefore,
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
t
-
κ
6
2
(
1
+
∥
p
¯
0
∥
L
α
)
(
κ
5
+
1
)
ν
4
+
1
(
1
+
∫
U
H
(
|
∇
p
0
(
x
)
|
)
𝑑
x
)
1
2
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
5
ν
4
(
1
+
∫
0
t
G
3
(
τ
)
𝑑
τ
)
ν
4
+
1
2
.
We obtain (6.4).
Now consider t≥2. We apply (6.2) with T0=t-1, T=1 and θ=12. Then by (5.22) and (5.15) we have
λ
s
1
2
≤
C
(
∫
t
-
1
t
∫
V
(
1
+
|
∇
p
|
)
s
2
𝑑
x
𝑑
τ
)
ν
4
2
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
ν
4
(
κ
5
+
α
2
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
⋅
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
.
Combining this with (6.2) and (3.19) gives
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
λ
s
1
2
∥
p
t
∥
L
2
(
U
×
(
t
-
1
2
,
t
)
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
ν
4
(
κ
5
+
α
2
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
⋅
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
⋅
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
[
Env
A
(
α
,
t
)
]
α
α
-
a
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
1
2
.
Thus,
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
ν
4
(
κ
5
+
α
2
)
+
α
2
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
+
α
2
⋅
(
1
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
ν
4
2
+
1
2
,
and we obtain (6.5). The proof is complete. ∎
For a large time or asymptotic estimates we have the following theorem.
Theorem 6.3.
Let U′⋐U.
If
A
(
α
)
<
∞
, then
(6.7)
lim sup
t
→
∞
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
9
⋅
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
s
3
2
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that for all
t
>
T
there holds
(6.8)
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
3
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
3
,
t
)
)
)
κ
9
⋅
(
1
+
∫
t
-
2
t
G
3
(
τ
)
𝑑
τ
)
s
3
2
.
Proof.
(i) For large t we apply (6.2) with T0=t-1, T=1 and θ=12. We have
(6.9)
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
λ
s
1
2
∥
p
t
∥
L
2
(
U
×
(
t
-
1
2
,
t
)
)
≤
C
(
∫
t
-
1
t
∫
V
(
1
+
|
∇
p
|
)
s
2
𝑑
x
𝑑
τ
)
ν
4
2
∥
p
t
∥
L
2
(
U
×
(
t
-
1
2
,
t
)
)
,
where ν4 is defined by (6.6). Take the limit superior and using (5.16) with s=s2, and (3.23), we obtain
lim sup
t
→
∞
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
⋅
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
t
-
1
t
∫
U
|
Ψ
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
+
lim sup
t
→
∞
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
1
2
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
+
α
2
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
ν
4
2
+
1
2
.
We obtain (6.7).
(ii) Using (6.9), (5.17) and (3.27), we obtain
∥
p
t
(
t
)
∥
L
∞
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
3
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
3
,
t
)
)
)
ν
4
(
κ
5
+
α
2
)
(
1
+
∫
t
-
2
t
G
1
(
τ
)
𝑑
τ
)
ν
4
2
⋅
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
∫
U
|
Ψ
t
(
x
,
τ
)
|
2
𝑑
x
𝑑
τ
+
∫
t
-
1
t
G
3
(
τ
)
𝑑
τ
)
1
2
.
Therefore, (6.8) follows.∎
7 Interior Estimates for Pressure’s Hessian
In this section, we estimate the L2-norm of the Hessian ∇2p=(pxixj)i,j=1,2,…,n.
Lemma 7.1.
Let U′⋐V⋐U. For t>0 we have
(7.1)
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
(
1
+
∥
∇
p
(
t
)
∥
L
∞
(
V
)
)
a
(
∫
U
[
|
∇
p
(
x
,
t
)
|
2
-
a
+
|
p
t
(
x
,
t
)
|
2
]
𝑑
x
)
1
2
.
Proof.
From (5.2) of Lemma 5.2 with ζ=ζ(x) being a cut-off function in space, we have
(7.2)
1
2
s
+
2
d
d
t
∫
U
|
∇
p
|
2
s
+
2
ζ
2
𝑑
x
+
1
-
a
2
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
2
|
∇
p
|
2
s
ζ
2
𝑑
x
≤
C
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
s
+
2
|
∇
ζ
|
2
𝑑
x
.
Clearly,
(7.3)
1
2
s
+
2
d
d
t
∫
U
|
∇
p
|
2
s
+
2
ζ
2
𝑑
x
=
-
∫
U
p
t
∇
⋅
(
|
∇
p
|
2
s
∇
p
ζ
2
)
𝑑
x
.
Combining (7.2) and (7.3) with s=0, we have
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
2
ζ
2
𝑑
x
≤
C
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
|
∇
ζ
|
2
𝑑
x
+
C
∫
U
|
p
t
∇
⋅
(
∇
p
ζ
2
)
|
𝑑
x
.
Since
|
p
t
∇
⋅
(
∇
p
ζ
2
)
|
≤
|
p
t
|
|
∇
2
p
|
ζ
2
+
2
|
p
t
|
|
∇
p
|
ζ
|
∇
ζ
|
≤
1
2
K
(
|
∇
p
|
)
|
∇
2
p
|
2
ζ
2
+
C
K
(
|
∇
p
|
)
|
∇
p
|
2
|
∇
ζ
|
2
+
C
|
p
t
|
2
K
-
1
(
|
∇
p
|
)
ζ
2
,
and, by (2.9), K-1(ξ)≤C(1+ξ)a we find that
(7.4)
∫
U
K
(
|
∇
p
|
)
|
∇
2
p
|
2
ζ
2
𝑑
x
≤
C
∫
U
K
(
|
∇
p
|
)
|
∇
p
|
2
|
∇
ζ
|
2
𝑑
x
+
C
∫
U
|
p
t
|
2
(
1
+
|
∇
p
|
)
a
ζ
2
𝑑
x
.
Constructing an appropriate ζ in (7.4) with ζ≡1 on U′ and suppζ⊂V, we obtain
(7.5)
∫
U
′
K
(
|
∇
p
|
)
|
∇
2
p
|
2
𝑑
x
≤
C
(
1
+
∥
∇
p
∥
L
∞
(
V
)
)
a
∫
V
[
|
∇
p
|
2
-
a
+
|
p
t
|
2
]
𝑑
x
.
For x∈U′ we have
(7.6)
K
(
|
∇
p
(
x
,
t
)
|
)
≥
C
(
1
+
|
∇
p
(
x
,
t
)
|
)
-
a
≥
C
(
1
+
∥
∇
p
(
t
)
∥
L
∞
(
V
)
)
-
a
Combining (7.5) and (7.6), we obtain (7.1). ∎
Now, we combine Lemma 7.1 with the estimates in Section 3 and Section 5.2 to obtain particular bounds for the Hessian. Let
s
4
=
a
s
3
+
1
,
κ
10
=
a
κ
4
+
1
,
κ
11
=
a
s
3
(
κ
5
+
α
2
)
+
α
2
,
κ
12
=
a
s
3
κ
5
+
α
2
.
Theorem 7.2.
Let U′⋐U.
If
t
∈
(
0
,
2
)
, then
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
t
-
κ
10
2
(
1
+
∥
p
¯
0
∥
L
α
)
κ
11
(
1
+
∫
U
H
(
|
∇
p
0
(
x
)
|
)
𝑑
x
)
1
2
(7.7)
⋅
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
κ
12
(
1
+
∫
0
t
G
4
(
τ
)
𝑑
τ
)
s
4
2
.
If
t
≥
2
, then
(7.8)
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
κ
11
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
κ
11
⋅
(
1
+
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
s
4
2
.
If
A
(
α
)
<
∞
, then
(7.9)
lim sup
t
→
∞
∥
∇
2
p
∥
L
2
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
κ
11
⋅
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
s
4
2
.
If
β
(
α
)
<
∞
, then there is
T
>
0
such that for
t
>
T
there holds
(7.10)
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
3
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
3
,
t
)
)
)
κ
11
⋅
(
1
+
∫
t
-
2
t
G
4
(
τ
)
𝑑
τ
)
s
4
2
.
Proof.
(i) Using (7.1), (5.34), (3.6), and (3.8), we obtain
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
t
-
a
κ
4
2
(
1
+
∥
p
¯
0
∥
L
α
)
a
s
3
(
κ
5
+
1
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
0
,
t
)
)
)
a
s
3
κ
5
⋅
(
1
+
∫
0
t
G
1
(
τ
)
d
τ
)
a
s
3
2
{
t
-
1
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
d
x
+
∫
U
H
(
|
∇
p
0
(
x
)
|
)
d
x
+
[
Env
A
(
α
,
t
)
]
α
α
-
a
+
∫
U
|
Ψ
t
(
x
,
t
)
|
2
d
x
+
∫
0
t
G
4
(
τ
)
d
τ
)
}
1
2
.
Note that α≥2 and
∫
U
|
Ψ
t
(
x
,
t
)
|
2
𝑑
x
≤
C
(
∫
U
|
Ψ
t
(
x
,
t
)
|
α
𝑑
x
)
2
α
≤
C
A
(
α
,
t
)
2
(
1
-
a
)
α
-
a
≤
C
(
1
+
A
(
α
,
t
)
α
α
-
a
)
.
Then (7.7) follows.
(ii) Using (7.1), (5.35), (3.10), (3.11), and (3.2), we obtain
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
(
1
+
∥
p
¯
0
∥
L
α
)
a
s
3
(
κ
5
+
α
2
)
(
1
+
[
Env
A
(
α
,
t
)
]
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
2
,
t
)
)
)
a
s
3
(
κ
5
+
α
2
)
⋅
(
1
+
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
a
s
3
2
⋅
(
1
+
∫
U
|
p
¯
0
(
x
)
|
α
𝑑
x
+
[
Env
A
(
α
,
t
-
1
)
]
α
α
-
a
+
∫
U
|
Ψ
t
(
x
,
t
-
1
)
|
2
𝑑
x
+
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
1
2
.
Then (7.8) follows.
(iii) Using (7.1), (5.39), (3.20), and (3.21), we obtain
lim sup
t
→
∞
∥
∇
2
p
∥
L
2
(
U
′
)
≤
C
(
1
+
A
(
α
)
1
α
-
a
+
lim sup
t
→
∞
∥
Ψ
∥
L
α
(
U
×
(
t
-
1
,
t
)
)
)
a
s
3
(
κ
5
+
α
2
)
⋅
(
1
+
lim sup
t
→
∞
∫
t
-
1
t
G
1
(
τ
)
𝑑
τ
)
a
s
3
2
⋅
(
1
+
A
(
α
)
α
α
-
a
+
lim sup
t
→
∞
∫
U
|
Ψ
t
(
x
,
t
)
|
2
𝑑
x
+
lim sup
t
→
∞
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
)
1
2
.
Then (7.9) follows.
(iv) Using (7.1), (5.40), (3.24), and (3.25), we obtain
∥
∇
2
p
(
t
)
∥
L
2
(
U
′
)
≤
C
(
1
+
β
(
α
)
1
α
-
2
a
+
sup
[
t
-
3
,
t
]
A
(
α
,
⋅
)
1
α
-
a
+
∥
Ψ
∥
L
α
(
U
×
(
t
-
3
,
t
)
)
)
a
s
3
(
κ
5
+
α
2
)
⋅
(
1
+
∫
t
-
2
t
G
1
(
τ
)
𝑑
τ
)
a
s
3
2
⋅
(
1
+
β
(
α
)
α
α
-
2
a
+
A
(
α
,
t
-
1
)
α
α
-
a
+
∫
t
-
1
t
G
4
(
τ
)
𝑑
τ
+
∫
U
|
Ψ
t
(
x
,
t
)
|
2
𝑑
x
)
1
2
.
Then (7.10) follows. ∎
A Appendix
Proof of Lemma 2.1.
In this proof, we use the notation ∥⋅∥Lp to denote ∥⋅∥Lp(U). First, we have the following Poincaré–Sobolev inequality:
(A.1)
∥
ϕ
∥
L
r
*
≤
C
∥
∇
ϕ
∥
L
r
+
C
δ
∥
ϕ
∥
L
1
.
Let r=2-a and ϕ=|u|m, where m=α-a2-a≥1. Applying (A.1) to ϕ, we obtain
(A.2)
∥
u
∥
L
m
r
*
≤
C
(
∫
U
|
u
|
α
-
2
|
∇
u
|
2
-
a
𝑑
x
)
1
α
-
a
+
C
δ
∥
u
∥
L
m
≤
C
(
∫
U
|
u
|
α
-
2
|
∇
u
|
2
-
a
𝑑
x
)
1
α
-
a
+
C
δ
∥
u
∥
L
α
.
The last inequality comes from Hölder’s inequality and the fact that α>m. Consider a number p∈(α,mr*). Let θ∈(0,1) such that
(A.3)
1
p
=
θ
α
+
1
-
θ
m
r
*
.
Combining the interpolation inequality and (A.2) yields
∥
u
∥
L
p
p
≤
∥
u
∥
L
α
θ
p
∥
u
∥
L
m
r
*
(
1
-
θ
)
p
≤
C
∥
u
∥
L
α
θ
p
{
(
∫
U
|
u
|
α
-
2
|
∇
u
|
2
-
a
𝑑
x
)
1
α
-
a
+
C
δ
∥
u
∥
L
α
}
(
1
-
θ
)
p
≤
C
δ
∥
u
∥
L
α
p
+
∥
u
∥
L
α
θ
p
(
∫
U
|
u
|
α
-
2
|
∇
u
|
2
-
a
𝑑
x
)
(
1
-
θ
)
p
α
-
a
.
In the preceding inequality, selecting (1-θ)p=α-a, letting u=u(x,t), and integrating in t from 0 to T give
∫
0
T
∥
u
(
t
)
∥
L
p
p
𝑑
t
≤
C
δ
T
ess
sup
[
0
,
T
]
∥
u
∥
L
α
p
+
C
ess
sup
[
0
,
T
]
∥
u
∥
L
α
θ
p
∫
0
T
∫
U
|
u
|
α
-
2
|
∇
u
|
2
-
a
𝑑
x
𝑑
t
≤
C
δ
T
[
[
u
]
]
p
+
C
[
[
u
]
]
θ
p
[
[
u
]
]
(
1
-
θ
)
p
=
C
(
1
+
δ
T
)
[
[
u
]
]
p
.
Therefore, (2.14) follows. It remains to calculate p and check sufficient conditions. Note that θp=p-α+a, and from (A.3) we derive
1
=
θ
p
α
+
(
1
-
θ
)
p
m
r
*
=
p
-
α
+
a
α
+
α
-
a
m
r
*
.
Solving this for p gives p=α+(α-a)(1-αmr*). Simple calculations yield p in the form of (2.13). It is elementary to verify that the second condition in (2.12) guarantees α<p<mr*.
In case U is a ball BR(x0), by the scaling y=x-x0R for x∈BR(x0) and applying (2.14) for B1(0), we obtain inequality (2.14) for BR(x0) with [[u]] defined by (2.15). ∎
Acknowledgements
The authors would like to thank Tuoc Phan and Akif Ibragimov for their suggestions and helpful discussions.
References
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