Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 716451, 15 pages
doi:10.1155/2010/716451
Research Article
Local Smooth Solution and Non-Relativistic Limit
of Radiation Hydrodynamics Equations
Jianwei Yang,1 Shu Wang,2 and Yong Li2
1
College of Mathematics and Information Science, North China University of Water Resources and
Electric Power, Zhengzhou 450011, China
2
College of Applied Sciences, Beijing University of Technology, PingLeYuan100, Chaoyang District,
Beijing 100022, China
Correspondence should be addressed to Jianwei Yang, yjw@emails.bjut.edu.cn
Received 5 May 2010; Accepted 16 July 2010
Academic Editor: Donal O’Regan
Copyright q 2010 Jianwei Yang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We investigate a multidimensional nonisentropic radiation hydrodynamics model. We study the
local existence and the convergence of the nonisentropic radiation hydrodynamics equations via
the non-relativistic limit. The local existence of smooth solutions to both systems is obtained. For
well-prepared initial data, the convergence of the limit is rigorously justified by an analysis of
asymptotic expansion, an energy method, and an iterative scheme. We also establish uniform a
priori estimates with respect to .
1. Introduction
In this paper, we study a system of PDEs describing radiation-driven perfect compressible
flows, in particular in astrophysics cf. 1–4. Assuming that the radiative temperature and
the fluid temperature are equal, and that the gas is radiatively opaque so that the equilibrium
diffusion will be dealt with, and the mean free path of photons is much smaller than the
typical length of the flow, then, we can write the equations of radiation hydrodynamics
without radiative heat diffusivity in Rd , describing the conservation of mass, momentum
and energy, as see 2, 3, 5
∂t ρ div ρu 0,
1 4
∂t ρu div ρu ⊗ u ∇ p θ 0,
3
1
∂t E div E p θ4 u 0,
3
1.1
2
Boundary Value Problems
for x, t ∈ R3 × 0, T , T > 0, where ρ, u u1 , . . . , ud T , p, and θ denote the density, velocity,
thermal pressure, and absolute temperature, respectively, 8π 5 k4 /15h3 c3 > 0 is a radiation
constant, and c is the light speed, and
E
1 2
ρu ρe θ4
2
1.2
is the total energy, e eρ, θ is the internal energy, and u2 di1 u2i is the square of the
macroscopic velocity.
From 1.1 and 1.2, we see that the system includes both gas and radiative
contributions to flow dynamics. The quantities 1/3θ4 and θ4 represent the radiative
pressure and radiative energy density, respectively. To complete system 1.1, one needs
the equation of state for the pressure p pρ, θ. In this paper, for the purpose of our test
problems, we will limit our study to the polytropic ideal gases, namely: p Rρe γ − 1ρe
with γ > 1 being the specific heat ratio and e cV θ with cV being the specific heat; we assume
cV 1 without loss of generality.
We point out that if one assumes → 0 in 1.1, then system 1.1 reduces to the usual
inviscid Euler equations:
∂t ρ0 div ρ0 u0 0,
∂t ρ0 u0 div ρ0 u0 ⊗ u0 ∇p0 0,
∂t E0 div
1.3
E0 p0 u0 0,
which are nonisentropic and compressible Euler equations.
The aim of this paper is to justify rigorously the local existence of smooth solutions of
system 1.1 and the convergence of system 1.1 to this formal limit equations 1.3.
Concerning the non-relativistic limit c → ∞, that is, → 0, there are only partial
results. Indeed, we know that the phenomenon of non-relativistic is important in many
physical situations involving various nonequilibrium processes. For example, important
examples occur in inviscid radiation hydrodynamics 6, in quantum mechanics 7, in KleinGordon-Maxwell system 8, in Vlasov-Poisson system 9, in Euler equations 10, in EulerMaxwell equations 11, 12, and so on.
In this paper, we are interested in the nonrelativistic limit → 0 in the problem
1.1 for the radiation hydrodynamics equations. We prove the existence of smooth solutions
to the problem 1.1 and their convergence to the solutions of the compressible and
nonisentropic Euler equations in a time interval independent of . For this propose, we use
the method of iteration scheme and classical energy method. The convergence of the radiation
hydrodynamics equations to the compressible and nonisentropic Euler equations is achieved
through the energy estimates for error equations derived from 1.1 and it’s formal limit
equations 1.3.
The remainder of this paper is arranged as follows: In the next section, we give the
local smooth solutions to both system 1.1 and 1.3. Section 3 is devoted to justify the
convergence of 1.1 to 1.3. By formal analysis, we show that the leading profiles of the
density, velocity, and temperature with respect to satisfy a compressible nonisentropic
Euler equations, and their next order profiles satisfy the corresponding linearized equations.
Boundary Value Problems
3
The Cauchy problem for this nonisentropic Euler equations is solved in this section. The final
part is devoted to rigorously justifying the asymptotic expansion developed in Section 3 and
obtaining the convergence of solutions to the multidimensional compressible nonisentropic
Euler system in a time interval independent of .
Notations and Preliminary Results
1 Throughout this paper, ∇ ∇x is the gradient, α α1 , . . . , αd and β are multiindeices, and H s Rd denotes the standard Sobolev’s space in Rd , which is defined by Fourier
transform, namely, f ∈ H s Rd if and only if
s 2
2
2 f 2πd
1
Ff k < ∞,
|k|
s
1.4
k∈Zd
where Ffk Rd fxe−ikx dx is the Fourier transform of f ∈ H s Rd .
2 Also, we need the following basic Moser-type calculus inequalities see,
Klainerman and Majda 13, 14: for f, g, v ∈ H s and any nonnegative multi-index α, |α| ≤ s,
α
Dx fg 2 ≤ Cs f ∞ Dxs g 2 g ∞ Dxs f 2 , s ≥ 0,
L
L
L
L
L
α
s−1 s Dx fg − fDxα g 2 ≤ Cs Dx f ∞ g
D
, s ≥ 1,
g
f
D
∞
x
x
L
L
L
L2
2
L
Dxs AvL2 ≤ Cs
s
Dxs AvL∞ 1 ∇vL∞ s−1 Dxs vL2 ,
s ≥ 1.
1.5
1.6
1.7
j1
3 Sobolev’s inequality. For s > d/2,
f ∞ ≤ Cs f .
L
s
1.8
4 If s > d/2, then for f, g ∈ H s and |α| ≤ s,
α
Dx fg 2 ≤ Cs f g .
L
s
s
1.9
2. The Local Existence
In this section, we give our main result about local existence. For this purpose, we first rewrite
the system 1.1 as a symmetric hyperbolic system of first order. Then, we prove the local
existence and uniqueness of smooth solutions to the Cauchy problem for 1.1.
4
Boundary Value Problems
For smooth solutions, the system 1.1 can be rewritten as follows:
∂t ρ div ρu 0,
4θ3
Rθ
∇ρ R ∇θ 0,
∂t u u · ∇u ρ
3ρ
4/3 − 4Rθ4
∂t θ u · ∇θ Rθ div u 0.
ρ 4θ4
2.1
In fact, 2.1 is a non-relativistic, non-isotropic, and compressible Euler equations.
For convenience, we introduce the following two functions:
4θ3
f1 ρ, θ ,
3ρ
2.2
4/3 − 4Rθ4
f2 ρ, θ .
ρ 4θ4
Then, 2.1 can be rewriten as follows:
∂t ρ div ρu 0,
Rθ
∇ρ R f1 ∇θ 0,
ρ
∂t θ u · ∇θ Rθ f2 div u 0.
∂t u u · ∇u 2.3
Denote the vector and matrix
⎛
V ρ, u, θ
T
,
0
ρejT
0
⎞
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
Rθ
j V uj Id2×d2 ⎜
A
⎟,
e
0
R
f
e
j
1
j
⎟
⎜ ρ
⎟
⎜
⎠
⎝
T
0
Rθ f2 ej
0
2.4
where e1 , . . . , ed is the canonical basis of Rd and yi denotes the ith component of y ∈ Rd .
Thus, we can rewrite the system 2.3 as follows:
∂t V d
j V ∂xj V 0.
A
2.5
j1
We will study the Cauchy problem for 2.5 together with the initial data
V x, 0 V0 x,
x ∈ Rd .
2.6
Boundary Value Problems
5
It is not difficult to see that the equations of V in 2.5 are symmetrizable and
hyperbolic. If we introduce the d 2 × d 2 matrix
⎛
ρ−1
⎜
⎜
⎜
⎜ 0
A0 V ⎜
⎜
⎜
⎝
0
0
ρ
Rθ
0
0
⎞
⎟
⎟
⎟
0
⎟
⎟,
⎟
ρ R f1 ⎟
⎠
Rθ Rθ f2 2.7
j V are symmetric for all 1 ≤ j ≤ d.
which is positive definite for 1, then Aj V A0 V A
Note that for smooth solutions, 2.3 is equivalent to that of 2.5.
Noticing the above facts and using the standard iteration techniques of local existence
theory for symmetrizable hyperbolic system see 15, we have the following.
Theorem 2.1. Assume that V0 ∈ H s , s > d/21, V0 x ∈ G1 , G1 ⊂⊂ G {V : ρ, θ ≥ C1 > 0}, and
C1 is a positive constant. Then there exists a time interval 0, T with T > 0, such that 2.5 and 2.6
have a unique solution V x, t ∈ C1 Rd ×0, T , with V x, t ∈ G2 , G2 ⊂⊂ G for x, t ∈ Rd ×0, T .
Furthermore, V ∈ C0, T , H s ∩ C1 0, T , H s−1 , and T depends on , V0 s and G1 .
3. Asymptotic Analysis
3.1. Formal Asymptotic Expansions
Let ρ , u , θ be the smooth solution to the system 2.3. In this section, we are going to
study the formal expansions of ρ , u , θ as → 0. To this end, we assume that initial data
ρ0 , u0 , θ0 have the asymptotic expansion with respect to :
ρ0 x m
j ρj x m1 ρm1
x,
j0
u0 x m
j uj x m1 um1 x,
3.1
j0
θ0 x m
j θj x m1 θm1
x.
j0
Then, we take the following ansatz:
ρ x, t j ρj x, t,
j≥0
j uj x, t,
u x, t j≥0
θ x, t j θj x, t,
j≥0
3.2
6
Boundary Value Problems
in terms of for the solutions to the system 2.3. Substituting the expansion 3.2 into the
system 2.3, we have the following.
1 The leading terms p0 , u0 , θ0 satisfy the following problem:
∂t ρ0 div ρ0 u0 0,
Rθ0
∂t u0 u0 · ∇ u0 0 ∇ρ0 R∇θ0 0,
ρ
∂t θ0 Rθ0 div u0 u0 · ∇ θ0 0,
3.3
ρ0 , u0 , θ0 t 0 ρ0 , u0 , θ0 .
These are nonisentropic and compressible Euler equations of ideal fluids. In fact, 3.3 is
equivalent to 1.3.
2 For any j ≥ 1, the profiles ρj , uj , θj satisfy the following problem for linearized
equations:
∂t ρj j
div ρk uj−k 0,
k0
∂t uj j j−1
uk · ∇ uj−k R θj ∇ ln ρ0 θ0 ∇ ln ρ0 ρj ∇θj g1 0,
k0
∂t θj j k0
uk · ∇ θj−k R
j
j−1
θk div uj−k g2
3.4
0,
k0
ρj , uj , θj t 0 ρj , uj , θj ,
j−1
where gi0 0 i 1, 2 for j ≥ 1. In fact, gi i 1, 2 depends only on {ρk , uk , θk }k≤j−1 and
can be obtained from the following relation:
⎛
⎞
⎞⎞
⎛
⎞ ⎤
R d ⎣⎝⎝ 0
j j⎠
⎝ρ0 j ρj⎠⎠ f1 ⎝ρ0 j ρj , θ0 j θj⎠⎦
∇
ln
θ
θ
j
j! d
j≥1
j≥1
j≥1
j≥1
⎡⎛⎛
j−1
g1
j
− R θj ∇ ln ρ0 θ0 ∇ ln’ρ0 ρj ,
j−1
g2
⎞
⎛
⎞ ⎤
⎡ ⎛
1 dj ⎣ ⎝ 0 j j 0 j j ⎠
0
j j ⎠ ⎦
⎝
div u u f2 ρ ρ , θ θ
j! dj
j≥1
j≥1
j≥1
0
.
0
3.5
Boundary Value Problems
7
3.2. Determination of Formal Expansions
3.2.1. Preliminary
From 3.4, we know that once ρ0 , u0 , θ0 are solved from the problem 3.3, ρ1 , u1 , θ1 are
solutions to the following problem for a linearized equations:
∂t ρ1 div ρ0 u1 ρ1 u0 0,
∂t u1 u0 · ∇ u1 u1 · ∇ u0 R θ1 ∇ ln ρ0 θ0 ∇ln ρ0 ρ1 ∇θ1 f10 0,
∂t θ1 u0 · ∇ θ1 u1 · ∇ θ0 Rθ0 div u1 Rθ1 div u0 f20 div u0 f30 u0 · ∇ θ0 0,
ρ1 , u1 , θ1 t 0 ρ1 , u1 , θ1 ,
3.6
where
f10 , f20 f1 , f2 |ρ,θρ0 ,θ0 .
3.7
Inductively, suppose that pk , uk , θk k≤j−1 are solved already for some j ≥ 2, from 3.4, we
know that pj , uj , θj satisfy the following linear problem:
∂t ρj j
div ρk uj−k 0,
k0
∂t uj j j−1
uk · ∇ uj−k R θj ∇ ln ρ0 θ0 ∇ ln ρ0 ρj ∇θj −g1 ,
k0
3.8
j
j j−1
∂t θj R θk div uj−k uk · ∇ θj−k −g2 ,
k0
k0
ρj , uj , θj t 0 ρj , uj , θj .
Thus, in order to determine the profiles ρ , u , θ , we require to solve the nonlinear problem
3.3 for ρ0 , u0 , θ0 and the linear system 3.8.
3.2.2. Existence and Uniqueness of Solution ρ0 , u0 , θ 0 Obviously, 3.3 are nonisentropic and compressible Euler equations. Thus, we recall
the following the classical result on the existence of sufficiently regular solutions of the
compressible Euler equations, see 15.
8
Boundary Value Problems
Proposition 3.1. Assume that ρ0 , u0 , θ0 ∈ H s1 ∩ L∞ Rd with ρ0 , θ0 ≥ C1 > 0 and s > d/2 1.
Then, there is a finite time T ∈ 0, ∞, depending on the H s and L∞ norms of the initial data, such
that the Cauchy problem 3.3 has a unique bounded smooth solution ρ, u, θ ∈ C0, T ; H s1 ∩
C1 0, T ; H s .
3.2.3. Existence and Uniqueness of Solution ρj , uj , θ j for j ≥ 1
Now, let us briefly describe the solvability of ρj , uj , θj for any j ≥ 1 from the problem 3.3
and 3.8 provided that we have known ρk , uk , θk k≤j−1 already. Thus, ρj , uj , θj satisfy the
following linear system:
j−1
∂t ρj div ρ0 uj ρj u0 − div ρk uj−k ,
k0
j−1
∂t uj u0 · ∇ uj uj · ∇ u0 R θj ∇ ln ρ0 θ0 ∇ ln ρ0 ρj ∇θj G1 ,
j−1
∂t θj R θ0 div uj θj div u0 u0 · ∇ θj uj · ∇ θ0 G2 ,
3.9
ρj , uj , θj t 0 ρj , uj , θj ,
where
j−1
G1
j−1
−g1
−
j−1 uk · ∇ uj−k ,
k0
j−1
G2
j−1
j−1 j−1
uk · ∇ θj−k .
−g2 − R θk div uj−k −
k0
3.10
k0
It is not difficult to see that the system 3.9 can be rewritten as a symmetrizable
hyperbolic system. Thus, by the standard existence theory of local smooth solutions of
symmetrizable hyperbolic equations see 15, we have
Proposition 3.2. Let T0 ∈ 0, T , and assume that ρj , uj , θj ∈ H s ∩ L∞ , s > d/2 1. Then, there
exists a time interval 0, T0 , such that 3.9 or 3.8 has a unique smooth solution ρj , uj , θj ∈
∩1i0 Ci 0, T0 , H s−i Rd .
Remark 3.3. In particular, if the initial data is C∞ , the solution of 3.9 or 3.8 belongs to
C∞ 0, T0 × Rd .
4. Convergence to Compressible Euler Equations
In this section, we are devoted to prove the convergence of system 2.3 to compressible Euler
equations.
Boundary Value Problems
9
4.1. Derivation of Error Equations
For any fixed integers m ≥ 1 and s0 > d/2 2, set
ρa,m
x, t m
j ρj x, t,
j0
ua,m x, t m
j uj x, t,
4.1
j0
θa,m
x, t m
j θj x, t,
j0
with ρj , uj , θj being given by Proposition 3.2. From the asymptotic analysis of Section 3.1,
, ua,m , θa,m
satisfy the following problem:
we know that ρa,m
∂t ρa,m
div ρa,m
ua,m Rρ ,
Rθa,m
∂t ua,m ua,m · ∇ ua,m ∇ρa,m
R f1a,m ∇θa,m
Ru ,
ρa,m
Rθa,m
f2a,m div ua,m ua,m · ∇ θa,m
Rθ ,
∂t θa,m
4.2
m
, ua,m , θa,m
j ρj , uj , θj x, 0,
|t0 ρa,m
j0
where
, θa,m
,
f1a,m , f2a,m f1 , f2 ρa,m
4.3
and the remainders Rρ , Ru , and Rθ satisfy
sup Rρ , Ru , Rθ 0≤t≤T0
H s0
< Mm ,
4.4
for some constant M > 0 independent of .
Now, we let ρ , u , θ be the smooth solution to the system 2.3 and denote
, u − ua,m , θ − θa,m
.
N , U , Θ ρ − ρa,m
4.5
10
Boundary Value Problems
Obviously, N , U , Θ satisfy the following problem:
U −Rρ ,
∂t N div N U ua,m ρa,m
R Θ θa,m
∇N R f1 ∇Θ U · ∇ua,m
∂t U U ua,m · ∇ U N ρa,m
Θ − RN θa,m
Rρa,m
f1 − f1a,m ∇θa,m
−Ru ,
∇ρa,m
ρa,m N ρa,m
f2 div U U ua,m · ∇Θ
∂t Θ R Θ θa,m
RΘ f2 − f2a,m div ua,m U · ∇θa,m
−Rθ ,
N , U , Θ |t0 N0 , U0 , Θ0 ,
4.6
where
, Θ θa,m
f1 , f2 f1 , f2 N ρa,m
,
N0 , U0 , Θ0 ρ0 − ρa,m
x, 0, u0 − ua,m x, 0, θ0 − θa,m
x, 0 .
4.7
Set
V N , U , Θ T ,
⎛
T
N ρa,m
ej
0
0
⎞
⎟
⎜
⎟
⎜ ⎟
⎜ R Θ θa,m
⎟
⎜
e
0
R
f
e
j
j
⎟
⎜
1
j V U ua,m Id2×d2 ⎜ N ρa,m
A
⎟,
⎟
⎜
⎟
⎜
⎟
⎜
T
⎠
⎝
0
R Θ θa,m f2 ej
0
⎛
⎞
N div ua,m U · ∇ρa,m
⎟
⎜
⎟
⎜
⎟
⎜
⎜U · ∇u Rρa,mΘ − RN θa,m ∇ρ ⎟
⎟
⎜
a,m
a,m
H1 V ⎜
⎟,
ρ N ρa,m
⎟
⎜
⎟
⎜
⎟
⎜
⎠
⎝
RΘ div ua,m U · ∇θa,m
⎛
0
⎞
⎟
⎜ ⎟
⎜
H2 V ⎜ f1 − f1a,m ∇θa,m
⎟,
⎠
⎝
f2 − f2a,m div ua,m
V |t0 N0 , U0 , Θ0 .
⎛
Rρ
⎞
⎜ ⎟
⎜ ⎟
Rc −⎜Ru ⎟,
⎝ ⎠
Rθ
4.8
Boundary Value Problems
11
Thus, the problem 4.6 for the unknown V can be rewritten as
∂t V 3
Aj V ∂xj V H1 V −H2 V R ,
4.9
j1
V |t0
V0 N0 , U0 , Θ0 .
It is easy to see that the equations of V in 4.6 are symmetrizable and hyperbolic if
we introduce
⎛
⎞
R
2
⎜ ⎜ N ρa,m
⎜
⎜
1
⎜
0
A0 V ⎜
Id×d
c
Θ θa,m
⎜
⎜
⎜
⎝
0
0
0
0
0
R f1 Θ θa,m
R Θ θa,m
f2 ⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠
4.10
j V which is positively definite. When N ρa,m
, Θ θa,m
≥ C > 0 for 1, then A
A0 V Aj V c are symmetric for all 1 ≤ j ≤ d.
4.2. Proof of Convergence
Obviously, the existence and uniqueness of smooth solutions of 2.3 are equivalent to that of
4.6 or 4.9. Then, in order to rigorously justify the convergence of 2.3 to 1.3, it suffices
to obtain their uniform estimates with respect to the light speed c. This will be done by using
iteration techniques for the symmetrizable hyperbolic problem.
More amply, we solve the nonlinear problem 4.9 by the following iteration for linear
problems cf. 15:
∂t V ,k1 d
Aj V ,k ∂xj V ,k1 H1 V ,k −H2 V ,k c−3 R ,
j1
4.11
V ,k1 |t0 V0 ,
with
4.12
V ,0 x, t V0 x.
To study the problem 4.9 and 4.11, we introduce the Sobolev’s norms:
⎛
V tl ⎝
|α|≤l
⎞1/2
∂αx V t2L2 Rd ⎠
,
V tl,T sup V tl ,
l ∈ N∗ .
4.13
0≤t≤T
The key point for proving the convergence as → 0 is the following a priori estimate.
12
Boundary Value Problems
Lemma 4.1. Let s and l be two integers such that d/2 1 < l ≤ s. Assume that
V ≤ M1 m ,
0 l
4.14
for some constant M1 > 0 independent of . Then, there exist constants M2 > 0, M3 > 0, 0 > 0, and
T1 ∈ 0, T0 , such that for all ≤ 0 the solutions V ,k of 4.11 satisfy
,k V l,T1
∂t V ,k ≤ M2 m ,
l−1,T1
∀k ∈ N,
≤ M3 m ,
∀k ∈ N.
4.15
4.16
Proof. Let α ∈ Nd with |α| ≤ l. We define Vα,k by Vα,k ∂αx V ,k ; thus, it is not difficult to know
that α ∈ Nd satisfies the following problem:
d
j V ,k ∂xj Vα,k1 R,k
A0 V ,k ∂t Vα,k1 A0 V ,k A
α ,
j1
4.17
Vα,k1 |t0 ∂αx V0 ,
where R,k
α is defined by
,k
α
,k
,k
c
V
∂
−H
V
−
H
V
R
A
R,k
0
1
2
α
x
d
A0 V ,k Aj V ,k Vα,k1 − ∂αx Aj V ,k V ,k1 .
4.18
j1
Estimates 4.15-4.16 are obviously true for k 0 with any T1 > 0. By induction on
k, assume 4.15-4.16 hold for some k ≥ 1 where M2 > 0 and T1 > 0 are to b fixed, and we
want to obtain 4.15 for k 1, that is:
,k1 V
l,T1
≤ M2 m ,
4.19
which implies, together with 4.111 , that
∂t V ,k1 l−1,T1 ≤ M3 m .
4.20
In what follows we let Mi i ≥ 4 be various positive constants independent of , k ∈ N, M2 ,
and M3 .
Boundary Value Problems
13
Equation 4.15 implies that the matrix A0 V ,k is positively definite, uniformly with
respect to , k ∈ N, M2 , and
,k V ∂t V ,k ≤ 1,
l,T1
l−1,T1
≤ 1,
4.21
for all ≤ 0 with some 0 > 0. Because l > d/2 1, we obtain
d
div Aj V ,k ∂t A0 V ,k ∂xj Aj V ,k ,
4.22
j1
satisfying
div Aj V ,k L∞ Rd ×0,T1 ≤ M4 .
4.23
Employing the classical energy estimate of symmetric hyperbolic equations to the
problem 4.171 , we can obtain
sup Vα,k1 0≤t≤T1
L2 Rd ≤ M5 e
α ∂x V 2 d 0 L R M5 T1
T1
,k Rα τ
L2 Rd 0
4.24
dτ .
By the definition of R,k
α in 4.18, the classical Moser-type inequality 1.5, 1.7, and
Sobolev’s embedding lemma with l > d/2 1, we deduce that
T1
,k Rα τ
0
L2 Rd dτ ≤ CM2 T1 m V ,k1 l,T1
.
4.25
Here the constant CM2 > 0 may depend on M2 . Now, substituting 4.25 into 4.24 and
using 4.14, one gets
,k1 Vα
L2 Rd ≤ M5 eM5 T1 M1 CM2 T1 m M5 eM5 T1 CM2 T1 Vα,k1 L2 Rd .
4.26
Now, we choose T1 > 0 such that
eM5 T1 ≤ 2,
CM2 T1 ≤ 1,
M5 eM5 T1 CM2 T1 ≤
1
2
4.27
Then
c,k1 V
l,T1
≤ M2 m ,
with M2 4M5 M1 1. The proof of Lemma 4.1 is complete.
Returning to the problem 2.3 and 4.6, we conclude the following.
4.28
14
Boundary Value Problems
Theorem 4.2. For any fixed integers s > d/2 1, suppose that
m
ρ , u , θ − j ρj , uj , θj ≤ Mm .
0 0 0
j0
4.29
s
Then the solution of 2.3 satisfies
m
j
j
j
j
ρ , u , θ − ρ , u , θ j0
≤ Mm ,
4.30
s,T1
where M > 0 is a constant independent of .
Proof. First, the uniform estimates 4.15-4.16, together with 4.11, yield the bound of
the sequence {V ,k }k∈N in L∞ 0, T1 , H s Rd ∩ W 1,∞ 0, T1 , and H s−1 Rd . Then Aubin’s
lemma implies that {V ,k }k∈N is compact in C0, T1 , C1 Rd . Hence, up to a subsequence,
{V ,k }k∈N convergence to some V in the space C0, T1 , C1 Rd as k → ∞. Combining
this with the bound results 4.15-4.16, we have V ∈ C0, T1 , H s Rd ∩ Lip0, T1 , and
H s−1 Rd . Furthermore, a similar argument as in 15 see Theorem 2.1b gives V ∈
C1 0, T1 , H s−1 Rd . Passing to the limit k → ∞ in the system 4.11 shows that V is a
classical solution to the problem 4.6. The uniqueness implies the convergence of the whole
sequence {V ,k }k∈N to V .
Finally, the estimate 4.30 can be easily derived from the estimate 4.24. This ends
the proof of Theorem 4.2.
Acknowledgments
The authors cordially acknowledge partial support from the National Science Foundation of
China Grant no. 10771099,10901011, the Beijing Science Foundation Grant no. 1082001,
Beijing Municipal Commission of Education, Funding Project for Academic Human
Resources Development in Institutions of High Learning Under the Jurisdiction of Beijing
Municipality PHR200906103, and Research Initiation Project for High-level Talents no.
201025 of North China University of Water Resources and Electric Power.
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