Instructions for use
A uthor(s ) C ho,Y onggeun
C itation Hokkaido University Preprint S eries in Mathematics, 776: 1-64
Is s ue D ate 2006
D O I 10.14943/83926
D oc UR L http://hdl.handle.net/2115/69584
T ype bulletin (article)
F ile Information pre776.pdf
NAVIER-STOKES EQUATIONS
YONGGEUN CHO
Abstract. We study the Navier-Stokes equations for compressiblebarotropic
fluids in a bounded or unbounded domain Ω ofR3. The initial density may
vanish in an open subset of Ω or to be positive but vanish at space infinity. We first prove the local existence of solutions (ρ(j), u(j)) inC([0, T∗];H2(k−j)+3×
D1
0∩D2(k−j)+3(Ω)), 0≤j≤k, k≥1 under the assumptions that the data
satisfy compatibility conditions and that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of (k+1)-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions ofisentropiccompressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.
1. Introduction
We consider the following compressible Navier-Stokes equations describing the motion of a viscous compressible barotropic fluid in a domain Ω ofR3
ρt+ div(ρu) = 0 in (0, T)×Ω,
(1.1)
(ρu)t+ div(ρu⊗u) +L u+∇p=ρf in (0, T)×Ω,
(1.2)
Lu=−µ∆u−(λ+µ)∇divu, (1.3)
and the initial and boundary conditions
(ρ, u)|t=0= (ρ0, u0) in Ω, u= 0 on (0, T)×∂Ω,
(1.4)
(ρ−ρ∞, u)(t, x)→(0,0) as |x| → ∞, (t, x)∈(0, T)×Ω. (1.5)
Here we denote by ρ and uthe unknown density and velocity fields of the fluid, respectively. pis the pressure which is a smooth function of ρ. f denotes a given external force and the constantsµ,λare the viscosity coefficients. We assume that the viscosity coefficientsµandλsatisfy the natural physical restrictionsµ >0 and 3λ+ 2µ≥0 so that L=−µ∆−(λ+µ)∇div is a strongly elliptic operator. We assume thatT is a finite positive number and Ω is either a bounded domain inR3
with smooth boundary or a usual unbounded domain such as the whole spaceR3,
2000Mathematics Subject Classification. 35Q30, 76N10.
Key words and phrases. viscous compressible fluids, compressible Navier-Stokes equations, vacuum.
The author is JSPS Research Fellow.
the half spaceR2×R+ and an exterior domain with smooth boundary. If Ω is a
bounded domain (or the whole space), then the condition (1.5) at infinity (or the boundary condition in (1.4) respectively) is unnecessary and should be neglected. ρ∞is a nonnegative constant. For the simplicity of presentation, we always assume
thatρ∞= 0. When the density has positive lower bound on a domain, we mean it
by thatρ∞>0. In particular,ρ(x)→ρ∞as |x| → ∞for unbounded domain. For
the mathematical derivation of the equations (1.1) and (1.2), see [19] for instance. Throughout this paper, we use the following simplified notations for the homo-geneous and inhomohomo-geneous Sobolev spaces.
Lr=Lr(Ω), Dk, r={v∈L1loc(Ω) :|v|Dk,r <∞},
Wk, r=Lr∩Dk, r, Hk =Wk,2, Dk=Dk,2,
D01={v∈L6(Ω) :|v|D1
0 <∞ and v= 0 on ∂Ω},
H01=L2∩D01, |v|Dk,r =|∇kv|Lr and |v|D1
0 =|∇v|L2.
Hereafter we use the obvious notation
| · |X∩Y =| · |X+| · |Y for (semi-)normed spaces X, Y
and C denotes a generic positive constant depending only on the constants k (Sobolev space index), µ, λ, T and the norms of p= p(·) and f. For a detailed study of homogeneous Sobolev spaces, we refer the readers to the Galdi’s book [13]. In this paper, we study the local and global existence and the regularity for the initial boundary value problem (simply IBVP) (1.1)-(1.5). Our result is a sequel of those in [5] and [4], in which the existence and uniqueness of strong and classical solutions are considered under a general assumption on initial density thatρ0≥0.
In the present case, we consider a high regularity of local and global solutions in Sobolev space. There has been a large amount of literature on the existence and regularity related to IBVP (1.1)–(1.16). See [18, 19] and [9, 10, 11, 12, 16, 17] for the weak solutions, and [22, 15, 26, 20, 21, 8, 14, 25, 27, 28, 29, 23] and the recent works [3, 4, 5, 6] for strong solutions. For the finite time blowup of smooth solutions, see [2] and [30].
To treat the high regularity of solutions, we need to control the high order time derivatives of solutions. For this purpose it is necessary to assume the compatibil-ity conditions for the time derivatives of solutions at the initial time zero. More specifically, we denote (∂/∂t)kv, the k-th order time derivative of v, by v(k) and
differentiate the equation (1.2)k-times with respect tot. Then we have
ρu(k+1)+Lu(k)+∇p(k)
= X
0≤j≤k
µ
k j
¶
ρ(k−j)(f−u· ∇u)(j) (1.6)
−kρ(1)u(k)− X 1≤j≤k−1
µ
k j−1
¶
For the regularity ofu(j)at time zero we then define a (k+ 1)-coupled compatibility condition as follows: for some functions gj ∈D01∩D2(k−j)+3,1 ≤j ≤k+ 1, the
data satisfy that
(1.7) Lu0+∇p(ρ0) =ρ0(f(0)−u0· ∇u0−g1) in Ω
and
L(u(j)(0)) +∇p(j)(0)
=−jρ(1)(0)u(j)(0)− X 1≤l≤j−1
µ j l−1
¶
ρ(j+1−l)(0)u(l)(0)
+ X
0≤l≤j
µ j l
¶
ρ(j−l)(0)
f(l)− X 0≤m≤l
µ l m
¶
u(l−m)(0)· ∇u(m)(0)
−ρ0gj+1 in Ω
(1.8)
for all 1≤j≤k. Here we define p(k)(0) andρ(k)(0) recursively as follows:
p(j)(0) =
µ
∂ ∂t
¶j
(p(ρ))
¯ ¯ ¯ ¯ ¯
t=0
,
ρ(j+1)(0) =−div
X
0≤l≤j
µ j l
¶
ρ(j−l)(0)u(l)(0)
, 0≤j≤k−1.
We call the condition (1.7) zeroth compatibility condition and (1.8)j-th one. Under the smallness of condition onρ0, the solutionu(j)(0) to the mapping (g1,· · · , gk+1)7→
(u0, u(1)(0),· · ·, u(k)(0)) is unique. See Section 4 and 5 for the existence and
unique-ness.
In [5, 4], the authors used the zeroth compatibility condition (1.7) to prove the local existence of strong and classical solutions. See also [3]. We follow the similar strategy in [4] summarized as follows:
(1) linearization on bounded domain with initial density having positive lower bound,
(2) a priori estimates independent of domain size and lower bound of initial density, and domain expansion,
(3) construction of a sequence of approximate solutions to the linearized prob-lem,
(4) convergence of the sequence in a strong sense.
If we consider only the zeroth compatibility condition (1.7), then in the course of linearization (1) and domain expansion (2), it is usually necessary to reconstruct u0 satisfying (1.7) for a given g1 (see Lemma 9 in [4]). Since (1.7) is a single
elliptic equation, the existence ofu0 is well-known by the classical elliptic theory.
However, in the present case, the situation is quite different. We encounter the case that we should find (u0, u(1)(0),· · ·, u(k)(0)) satisfying (1.8) for a given pair
(g1,· · · , gk+1). In other words, we must find solutions of (k+ 1)-coupled elliptic
that the initial density is sufficiently small, then we can find solution successfully (see Lemma 4.2 below). From this solvability of (k+ 1)-coupled system, we can construct approximate solutions to the linearized problem, which is convergent to a solution of the original equations. For this purpose, we also have to construct a sequence of system of compatibility conditions and show the strong convergence of the sequence to the original system of compatibility conditions (1.7) and (1.8) which are possible under the smallness condition of the initial density. See Section 5 below.
Now we state the first main result.
Theorem 1.1. Assume that
ρ0∈L1∩H2k+3, ρ0≥0 in Ω, u0∈D10∩D2k+3,
f(j)∈C(0, T;H2(k−j)+1)∩L2(0, T;H2(k−j)+2) for 0≤j ≤k
f(k+1)∈L2(0, T;L2) (1.9)
and that the dataρ0, u0 andf satisfy the compatibility conditions (1.7)and (1.8). Assume further that |ρ0|L1∩H2k+3 is sufficiently small. Then there exist a time
T∗ ∈(0, T)and a unique solution (ρ, u) to the IBVP (1.1)–(1.5) such that for all
0≤j ≤k, ρ∈C([0, T∗];L1)
ρ(j)∈C([0, T
∗];H2(k−j)+3)∩L2(0, T∗;H2(k−j)+4),
u(j)∈C([0, T∗];D10∩D2(k−j)+3)∩L2(0, T∗;D2(k−j)+4)
u(k+1)∈L∞(0, T∗;D01)∩L2(0, T∗;D2),
√ρu(k+1)
∈L∞(0, T∗;L2).
(1.10)
Remark 1.2. If the initial density has positive lower bound, then the smallness condition of initial density can be removed. See Remark 3.3 and Section 4 below
Remark 1.3. For each k ≥ 1, one can observe a smoothing effect and apply the methods used to get a classical solution in [4] to the improvement of regularity slightly.
Now we study the global existence of radial solutions. The global existence of strong solution (k= 0) is known by H. Kim and H.J. Choe [7]. They showed the density is bounded on any finite time interval by using the radial symmetry and the effective viscous effect. The second main result of this paper is to show that every Sobolev norm of the solution is controlled by L∞ norm of density. See Theorem
6.2 below.
To do this, we need to estimate the L∞ norm of pressure p (see (6.9) below).
This is possible in the case of the isentropic compressible Navier-Stokes equations with p = Aργ, A > 0 and γ > 1. Since from (1.1), the pressure p satisfies the
equation pt+∇p·u+γpdivu = 0. Hence for a high regularity for any γ > 1,
Navier-Stokes equations:
ρt+ div(ρu) = 0 in (0, T)×Ω,
(1.11)
pt+∇p·u+γpdivu= 0 in (0, T)×Ω,
(1.12)
(ρu)t+ div(ρu⊗u) +L u+∇p=ρf in (0, T)×Ω,
(1.13)
Lu=−µ∆u−(λ+µ)∇divu, (1.14)
and the initial and boundary conditions
(ρ, p, u)|t=0= (ρ0, p0, u0) in Ω, u= 0 on (0, T)×∂Ω,
(1.15)
(ρ, p, u)(t, x)→(0,0,0) as |x| → ∞, (t, x)∈(0, T)×Ω. (1.16)
In fact, the barotropic fluids satisfy the equationpt+∇p·u+ρp′(ρ)divu= 0 and
the initial pressurep(0) =p(ρ0)∈H2k+3. The isentropic case above is the one that
ρp′(ρ) =γp. Hence the local existence of IBVP (1.11)-(1.16) follows from Theorem
1.1, provided the initial pressurep0 of (1.15) is in H2k+3. Moreover, ifp0 =Aργ0,
then the solution pof (1.12) satisfies the equation of statep=Aργ. For this, see
the proof of Theorem 4 of [4].
Now let us introduce the second result. LetB(0, r),r >0 be the ball with radius rcentered at the origin.
Theorem 1.4. If (ρ0, u0, f)is a radially symmetric data satisfying the conditions in Theorem 1.1 withp0=Aργ0 ∈H2k+3 on an annular domainΩ =B(0, b)\B(0, a) with 0< a < b≤ ∞, then there exists a unique global radially symmetric solution
(ρ, p, u)of IBVP(1.11)–(1.16)satisfying the regularity (1.10)and the state equation
p=Aργwith Ω =B(0, b)\B(0, a).
Remark 1.5. In this isentropic case, dividing the density and pressure by a large constant, we can remove the smallness condition of initial density. But instead, we need a large viscosity coefficients.
This paper is organized as follows. In Section 2, we reduce the original problem to a linearized problem on a bounded domain under a positive lower bound assumption on the initial density. In Section 3, we prove uniform a priori estimates independent of domain size and lower bound of density. In Section 4, to prove Theorem 1.1 expanding domain size, we remove the lower bound restriction on the density. At this point, we solve the (k+1)-coupled elliptic system under a smallness assumption on the initial density. Then we prove Theorem 1.1 by constructing a sequence of approximate solutions and showing a strong convergence of it. Finally, Section 6 is devoted to proving Theorem 1.4.
2. Linearization
positivity of the initial density, we linearize the original equations on a bounded domain.
Let Ω be a bounded domain inR3 with smooth boundary, and we consider the following linear parabolic problem
(2.1)
ρwt+Lw=F in (0, T)×Ω,
w(0) =w0 in Ω, u= 0 on (0, T)×∂Ω,
whereρis a known scalar field in (0, T)×Ω such that
(2.2) ρ(j)∈C([0, T];H2(k−j)+3) and ρ≥δ on [0, T]×Ω
for any 0 ≤ j ≤ k, k ≥ 1 and for some constant δ > 0. Then applying the arguments in the proof of Lemma 2 and Lemma 3 in [4] and induction, one can show the following existence and regularity results on solutions to the linear parabolic problem (2.1). See also [27, 28, 29].
Lemma 2.1. Ifw0∈H01∩H2k+3,F(j)∈C([0, T];H2(k−j)+1)∩L2(0, T;H2(k−j)+2),
F(k+1)∈L2(0, T;L2)and
ρ(0)−1
(F(j)(0)−Lw(j)(0)− X 0≤l≤j−1
µ
j l
¶
ρ(j−l)(0)w(l+1)(0)
∈H01∩H2(k−j)+1
(2.3)
for all0≤j≤k,k≥1, then the solution walso satisfies
w(j)∈C([0, T];H2(k−j)+3) for all 0≤j≤k,
w(k+1)∈C([0, T];H01)∩L2(0, T;H2), and w(k+2)∈L2(0, T;L2).
Remark 2.2. From the elliptic regularity theory [1, 5], we see that the condition (2.3) meansw(j)(0)∈H01∩H2(k−j)+3.
Now we consider the following linear hyperbolic equation
ρt+ div(ρv) = 0 in (0, T)×Ω, ρ(0) =ρ0 in Ω,
(2.4)
wherev is a known vector field in (0, T)×Ω such that for some integerk≥1
v(j)∈C([0, T];D01∩D2(k−j)+3)∩L2(0, T;D2(k−j)+4)
for any 0≤j ≤k. Letρ(j+1)(0) be defined recursively by
ρ(j+1)(0) =−div
X
0≤l≤j
µ j l
¶
ρ(j−l)(0)v(l)(0)
, 0≤j ≤k−1.
Then we have
Lemma 2.3. Assume thatρ(j)(0)∈H2(k−j)+3 andρ
(i) there exists a unique solution ρto the problem (2.4) such that for all 0≤ j≤k
ρ(j)∈C([0, T];H2(k−j)+3),
(ii) the solutionsρ andpsatisfies the following estimate
|ρ(t)|H2k+3≤ |ρ0|H2k+3exp
µ
C
Z t
0 |
v(s)|D1
0∩D2k+4ds
¶
for0≤t≤T and finally,
(iii) the solutionρis represented by the formula
(2.5) ρ(t, x) =ρ0(U(0, t, x) ) exp ·
− Z t
0
divv(s, U(s, t, x) )ds
¸
,
whereU ∈C([0, T]×[0, T]×Ω)is the solution to the initial value problem
(2.6)
½ ∂
∂tU(t, s, x) =v(t, U(t, s, x) ), 0≤t≤T,
U(s, s, x) =x, 0≤s≤T, x∈Ω.
Proof. (i) withj= 0, (ii) and (iii) follow from Lemma 2.1 in [4]. And an induction
onj yields the case (i) withj≥1. ¤
To prove Theorem 1.1, we consider the following linearized problem
ρt+ div (ρv) = 0 in (0, T)×Ω,
(2.7)
ρut+Lu+∇p=ρ(f−v· ∇v) in (0, T)×Ω,
(2.8)
(ρ, u)|t=0= (ρ0, u0) in Ω, u= 0 on (0, T)×∂Ω, ,
(2.9)
wherev is a known vector field in (0, T)×Ω such that for 0≤j ≤k
v(j)∈C([0, T];D01∩D2(k−j)+3)∩L2(0, T;D2(k−j)+4)
and v(k+1)∈L∞(0, T;D01)∩L2(0, T;D2).
(2.10)
Recall again thatLu=−µ∆u−(λ+µ)∇divu.
First, from Lemma 2.1 and Lemma 2.2, we obtain an existence result for positive initial densities.
Lemma 2.4. LetΩbe a bounded domain inR3with smooth boundary. In addition to(1.9) and(2.10), we assume thatρ0≥δinΩ for some constantδ >0 and
(f−v· ∇v)(j)(0) +ρ−01 X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)(f−v· ∇v)(m)(0)
−ρ−01 X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)u(m)(0)−ρ−1 0
³
Lu(j)(0) +∇p(j)(0)´
∈H01∩H2(k−j)+1
for0≤j ≤k. Then there exists a unique solution(ρ, u)to the linearized problem
(2.7),(2.8) and(2.9) such that for all 0≤j≤k+ 1
ρ∈C(0, T];H2k+3),
ρ(j)∈C([0, T];H2(k−j)+2)∩L2(0, T;H2(k−j)+3) ),
u(j)∈C([0, T];H1
0∩H2(k−j)+3)∩L2(0, T;H2(k−j)+4),
and ρ≥δ on [0, T]×Ω (2.12)
for some constant δ >0.
Proof. The existence and regularity of a unique solutionρto the linear hyperbolic problem (2.7) and (2.9) were already proved in Lemma 2.3. To prove the remaining part of the lemma, let us defineF byρ(f −v· ∇v)− ∇p. Then
F(j)=ρ(f−v· ∇v)(j)+ X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(f−v· ∇v)(m)
− X
0≤m≤j−1 µ
j m
¶
ρ(j−m)u(m)− ∇p(j)
for 1≤j≤k. Then by virtue of (1.9), (2.10) and the regularity ofρ, we can easily show thatF(j)∈C([0, T];H2(k−j)+1)∩L2(0, T;H2(k−j)+2),F(k+1)∈L2(0, T;L2).
Moreover sinceF andusatisfies the condition (2.3) by (2.11), Lemma 2.1 allows us to deduce the existence and regularity of a unique solutionuto the linear parabolic problem (2.8) and (2.9). This completes the proof of Lemma 2.4. ¤
3. A priori estimates for positive density
Assume that ρ0, u0, v, f and Ω satisfy the hypotheses of Lemma 2.4. Then it
follows from Lemma 2.4 that there exists a unique smooth solution (ρ, u) to the linear problem (2.7), (2.8) and (2.9) satisfying the regularity (2.12). We derive somelocal (in time) a priori estimatesfor (ρ, u) which are independent of the lower boundδofρ0 and size of the domain Ω. Let us choose constantsε >0 andc0>1
such thatε≤c0
|ρ0|L1∩H2k+3< ε, 1 +
X
1≤j≤k+1 |gj|D1
0 < c0,
(3.1)
where g1 =f(0)−v(0)· ∇v(0)−ρ−01(Lu0+∇p0) =ut(0) ∈D01∩D2k+1 and for
1≤j ≤k
gj+1= (f−v· ∇v)(j)(0) +ρ−01 X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)(f−v· ∇v)(m)(0)
−ρ−01 X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)g
m+1−ρ−01 ³
Lgj+∇p(j)(0)
´
Assume for all 0≤j ≤k+ 1 that
(3.2) |v(j)(0)|D1
0∩D2(k−j)+3 ≤1 +c0,1,j,
and for all 0≤l≤kand 0≤j ≤k−l that
sup
0≤t≤T∗ ³
|v(j)(t)|D1 0∩D2l+1
´
+
Z T∗
0 |
v(j)(t)|2D2∩D2l+2dt≤1 +cl,2,j,
sup
0≤t≤T∗ ³
|v(j)(t)|D1 0∩D2l+2
´
+
Z T∗
0 ³
|v(j+1)(t)|2D1
0∩D2l+1+|v
(j)(t) |2D2l+3
´
dt≤1 +cl,3,j,
ess sup
0<t<T∗ ³
|v(j+1)(t)|D1
0∩D2l+1+|v
(j)(t) |D2l+3
´
+
Z T∗
0 ³
|v(j+1)(t)|2D2l+2+|v(j)(t)|2D2l+4
´
dt≤1 +cl,4,j
(3.3)
for some timeT∗∈(0, T) and constantscl,i,j such that
1≤cl,i,j ≤cl+1,i,,j for any i, j, l
cl,i,j≤cl,i+1,j for each fixed l, j,
and cl,i,j ≤cl,i,l+1 for each fixed l and for any i.
The constantscl,i,j’s and T∗ will be determined later and depend only on c0 and
the parameters ofC.
Lemma 3.1. For each 1≤j≤k+ 1
|ρ(j)(0)|
L32∩H2(k−j)+3 ≤Cεc j
0,1,j−1.
Proof. Using the equation
(3.4) ρ(j+1)=− X 0≤m≤j
µ
j m
¶
div(ρ(j−m)v(m)),
for any multi-index α= (α1, α2, α3) with|α|=α1+α2+α3≤2(k−(j+ 1)) + 3
we have
|Dαρ(j+1)(0)|L2
≤C X
0≤m≤j α1+α2=α
³
|∇Dα1ρ(j−m)(0)·Dα2v(m)(0)|L2
+|Dα1ρ(j−m)(0)divDα2v(m)(0)|L2
´
≤C X
0≤m≤j α1+α2=α
|ρ(j−m)(0)|H|α1|+2|v(m)(0)|D1
0∩D|α2|+1
≤C X
0≤m≤j α1+α2=α
|ρ(j−m)(0)|H2(k−j)+3|v(m)(0)|D1
whereDα= ∂|α|
∂xα1 1 ∂x
α2 2 ∂x
α3
3 . Since 0≤m≤j, 2(k−j) + 3 = 2(k−(j−m)) + 3−
2m)≤2(k−(j−m)) + 3 and 2(k−j) + 2≤2(k−m) + 2−2(j−m)≤2(k−m) + 2. Thus by the initial bound (3.1), we obtain
|ρ(j+1)(0)|H2(k−(j+1))+3
≤C X
0≤m≤j
|ρ(j−m)(0)|
H2(k−(j−m))+3|v(m)(0)|D1
0∩D2(k−m)+2
≤Cc0,1,j
X
0≤m≤j
|ρ(j−m)(0)|
H2(k−(j−m))+3.
Therefore the induction onj yields that
|ρ(j)(0)|H2(k−j)+3≤Cεcj0,1,j−1
for all 1≤j≤k+ 1. Similarly, we can treatL32 norm. ¤
Lemma 3.2. If εis sufficiently small, then we have X
0≤j≤k+1
|u(j)(0)|D1
0∩D2(k−j)+3 ≤Cc0.
Proof. We first consider the case 1≤j ≤k. Since gj ∈D01 satisfies the following
boundary value problem:
L(u(j)(0)) =ρ0(f−v· ∇v)(j)(0) + X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)(f−v· ∇v)(m)(0)
+ X
0≤m≤j−1 µ
j m
¶
ρ(j−m)(0)u(m+1)(0)− ∇p(j)(0)−ρ0gj+1,
we have
|u(j)|D1
0∩D2(k−j)+3≤C|u
(j) |D1
0+C|ρ0(f−v· ∇v)
(j)(0)
|H2(k−j)+1
+C X
0≤m≤j−1
|ρ(j−m)(0)(f −v· ∇v)(m)(0)|
H2(k−j)+1
+C X
0≤m≤j−1
|ρ(j−m)(0)u(m+1)(0)|
H2(k−j)+1
+C|p(j)(0)|H2(k−j)+2+C|ρ0gj+1|H2(k−j)+1
from the elliptic regularity results [1, 5], where C does not depend on the size of domain. We easily show that
|ρ0(f−v· ∇v)(j)(0)|H2(k−j)+1
≤C|ρ0|H2(k−j)+3|f(j)(0)|H2(k−j)+1
+C X
0≤m≤j
|ρ0|H2(k−j)+3|v(j−m)(0)|D1
0∩D2(k−j)+3|v
(m)|
D1
0∩D2(k−j)+2
Similarly, we have
X
0≤m≤j−1
|ρ(j−m)(0)(f−v· ∇v)(m)(0)|H2(k−j)+1
≤C X
0≤m≤j−1
|ρ(j−m)(0)|
H2(k−j)+3|(f−v· ∇v)(m)(0)|H2(k−j)+1 ≤Cεcj0+2,1,j−1,
|p(j)(0)|H2(k−j)+2 ≤Cεcj0,1,j−1,
|ρ0gj+1|H2(k−j)+1≤C|ρ0|H2(k−j)+3|gj+1|H2(k−j)+1≤Cε|gj+1|D1
0∩D2(k−(j+1))+3,
and
X
0≤m≤j−1
|ρ(j−m)(0)u(m+1)(0)|
H2(k−j)+1
≤C X
0≤m≤j
|ρ(j−m)(0)|
H2(k−j)+1|u(m+1)(0)|D1
0∩D2(k−j)+3
≤Cεcj0,1,j−1Aj−1+Cεc0,1,0|u(j)(0)|D1
0∩D2(k−j)+3,
where
Aj=
X
1≤m≤j
|u(m)(0)|D1
0∩D2(k−m)+3 and A0= 0.
Thus if we choose ε < 2M c1k+2 0,1,k
for some large M ≥ C, then since gj ∈ D10, we
obtain
Aj ≤C
X
1≤j≤k+1 |gj|D1
0+Cε|gj+1|D01∩D2(k−(j+1))+3
≤Cc0+Cε|gj+1|D1
0∩D2(k−(j+1))+3
for all 1≤j≤k. Since|gk+1|D1
0 ≤c0, we haveGk ≤Cc0.
Now using the energy estimate for u0, we easily get |u0|D1
0 ≤Cc0 for small ε.
By the another use of elliptic regularity result, we finally have forj = 0 that
|u0|D1
0∩D2k+3 ≤C|u0|D10+C|ρ0(f(0)−v(0)· ∇v(0))|H2k+1
+C|p0|H2k+2+C|ρ0g1|H2k+1
≤Cc0+Cεc20,1,0+Cεc0≤Cc0.
This completes the proof of the lemma. ¤
Remark 3.3. Since the estimate of Lemma 3.2 is stable under a small perturbation of gj, this lemma will be used for the reconstruction of initial data in Lemma 4.2
below. If the initial data has positive lower bound, then this lemma is not necessary.
Lemma 3.4. For any fixedδ >0 we have
inf
Ω ρ(t)≥C
−1δ,
and for each 1≤j≤k+ 1
|ρ(j)(t)|H2(k−j)+3 ≤Cεcj0,1,j−1,
Z t
0 |
ρ(k+2)(s)|2L2ds≤Cε2,
Proof. From Lemma 2.3, we recall that
|ρ(t)|H2k+3≤ |ρ0|H2k+3exp
µ
C
Z t
0 |
v(s)|D1
0∩D2k+4ds
¶
and
inf
Ω ρ(t)≥ ³
inf
Ω ρ0 ´
exp
µ −C
Z t
0 |
v(s)|D1
0∩D2k+4ds
¶
for 0≤t≤T. Hence from the observation that
Z t
0 |
v(s)|D1
0∩D2k+4ds≤t 1 2
µZ t
0 |
v(s)|2D1
0∩D2k+4ds
¶1 2
≤C(1 +ck,4,0)t+C((1 +ck,4,0)t)
1 2,
we obtain the desired estimate for j = 0. For the esimate of the case j ≥ 1, multiplying (3.4) byDαρ(j)for any multi-indexαwith|α| ≤2(k−j) + 3, we have
1 2
d dt|D
αρ(j) |2L2
=− Z
div (Dαρ(j)v)Dαρ(j)dx
− X
α1+α2=α α26= 0
Cα1, α2
Z
div (Dα1ρ(j)Dα2v(m))Dαρ(j)dx
− X
1≤m≤j α1+α2=α
Cj, α
Z
div (Dα1ρ(j−m)Dα2v(m))Dαρ(j)dx
= X
1≤i≤3
Ji
(3.5)
where
Cα1,α2= α!
α1!α2!
, Cj α=
µ
j m
¶
α! α1!α2!
.
Each term can be estimated as follows:
J1≤C|divv|D1 0∩D2|D
αρ
(j)|2L2,
J2≤C
X
α1+α2=α α26= 0
Z ³
|∇Dα1ρ(j)||Dα2v|+|Dα1ρ(j)||Dα2divv|´|Dαρ(j)|dx
≤C X
α1+α2=α
α26= 0
|v|D1
0∩D|α2|+2|ρ
(j)|
H|α1|+1|D αρ(j)|
L2,
J3≤C
X
1≤m≤j α1+α2=α
|ρ(j−m)|H|α1|+1|v(m)|D1
0∩D|α2|+2|D αρ(j)
Substituting all these estimate into (3.5) and integrating over (0, t), we have
|ρ(j)(t)|H2(k−j)+3≤ |ρ(j)(0)|H2(k−j)+3+C
Z t
0 |
divv|D1 0∩D2|D
αρ(j) |L2ds
+C X
α1+α2=α
α26= 0
Z t
0 |
v(s)|D1
0∩D|α2|+2|ρ
(j)(s)
|H|α1|+1ds
+C X
1≤m≤j α1+α2=α
Z t
0 |
ρ(j−m)|
H|α1|+2|v(m)|D1
0∩D|α2|+1ds .
Since 2(k−j) + 5≤2k+ 3 for j≥1,
2(k−j) + 5≤2(k−(j−m)) + 5−2m≤2(k−(j−m)) + 3 for m≥1
and 2(k−j) + 4≤2(k−m) + 4−2(j−m)≤2(k−m) + 4 for m≤j,
we obtain
|ρ(j)(t)|H2(k−j)+3≤Cεcj0,1,,j−1+Cc1,2,0
Z t
0 |
ρ(j)(s)|H2(k−j)+3ds
+C max
0≤m≤j−1 µ
sup
0≤s≤t|
ρ(m)(s)|
H2(k−m)+3
¶
(1 +ck,4,0)
1 2t12.
Hence the Gronwall’s inequality yields that ift≤(1 +ck,4,0)−1
|ρ(j)(t)|H2(k−j)+3≤C
µ
εcj0,1,j−1+ max 0≤m≤j−1
µ
sup
0≤s≤t|
ρ(m)(s)|H2(k−m)+3
¶¶
.
Using the induction onj again, we have
|ρ(j)(t)|H2(k−j)+3≤Cεcj0,1,j−1
for allt∈[0,min(T∗, T1)]. Similarly (more easily), we can estimate|ρ(j)|L1.
Finally we have
|ρ(k+2)|L2≤C
X
0≤m≤k+1 ³
|∇ρ(k+1−m)·v(m)|L2+|ρ(k+1−m)v|L2
´
≤C X
1≤m≤k+1
|ρ(k+1−m)|H2|∇v(m)|L2+C|ρ(k+1)|H1|v|D1 0∩D2
≤Cεck0,1,k−1c0,4,k+Cck0,+11,kc0,3,0
and hence Z
t
0 |
ρ(k+2)(s)|2L2ds≤Cε2.
¤
Using the equationpt+∇p·u+ρp′(ρ)divu= 0, by the similar estimates as in
the proof of Lemma 3.1 and Lemma 3.4 we have
Lemma 3.5. For any 0≤j≤k+ 1, we have
for0≤t≤min(T∗, T1)whereT1= (1 +ck,4,0)−(2k+6), wherep(0)is the value ofp at point zero.
The three lemmas below, Lemma 3.6, 3.7 and 3.8, are similar to those in [4]. But since they are needed for a priori estimates for high regularity, we revisit the proof of them.
Lemma 3.6. Z t
0 | √ρu
t(s)|2L2ds+|u(t)|2D1 0 ≤Cc
2 0,
Z t
0 |
u(s)|2
D2ds≤Cc30
for0≤t≤min(T∗, T1).
Proof. Multiplying the equation (2.8) byutand integrating over Ω, we obtain
Z
ρ|ut|2dx+
1 2
d dt
Z
µ|∇u|2+ (λ+µ)(divu)2dx
=− Z
∇p·utdx+
Z
ρ(f−v· ∇v)·utdx.
(3.6)
Using the conditonε≤c0and Lemma 3.4 together with (3.3), we can estimate the
second term of the right hand side in (3.14) as follows:
Z
ρ(f−v· ∇v)·utdx≤ |ρ| 1 2
L∞|f −v· ∇v|L2|√ρut|L2
≤C|ρ|L∞ ³
|f|2
L2+|v|4D1 0∩D2
´
+1 2|
√ρu
t|2L2
≤Cc0c40,3,0+
1 2|
√ρu
t|2L2.
To estimate the first term, we observe from Lemma 3.5 that
− Z
∇p·utdx=
Z
(p−p(0)) divutdx
= d dt
Z
(p−p(0)) divu dx− Z
ptdivu dx,
Z
(p−p(0) divu dx≤C|p(ρ)−p(0)|2
L2+
µ 4|∇u|
2
L2 ≤Cc20+
µ 4|∇u|
2
L2
and
− Z
ptdivu dx≤ |pt|L22+|∇u|2L2 ≤Cc20c20,1,0+|∇u|2L2.
Hence integrating (3.14) in time over (0, t), we have
Z t
0 | √ρu
t(s)|2L2ds+|∇u(t)|2L2
≤Cc20 ¡
1 +|∇u0|2L2
¢
+Cc20c40,3,0t+C Z t
0 |∇
u(s)|2L2ds
for 0≤t ≤min(T∗, T1). Therefore, in view of Gronwall’s inequality, we conclude
that Z
t
0 | √ρu
Moreover, since for eacht∈(0, T),u=u(t)∈D1
0∩D2 is a solution of the elliptic
system
Lu=−∇p+ρ(f−v· ∇v)−ρut in Ω,
it follows from the elliptic regularity result in [1, 5] that
|u|D2≤C
³
| − ∇p+ρ(f −v· ∇v)−ρut|L2+|u|D1 0
´
≤C³c20,3,0+c
1 2
0|√ρut|L2
´
and thus
Z t
0 |
u(s)|2
D2ds≤Cc30 for 0≤t≤min(T∗, T1).
This completes the proof of Lemma 3.6. ¤
Lemma 3.7.
|√ρut(t)|L2+|u(t)|D2+
Z t
0 ³
|ut(s)|2D1
0+|u(s)|
2
D3
´
ds≤Cc30c
3 2
0,2,0c
1 2
0,3,0
for0≤t≤min(T∗, T1).
Proof. We differentiate (2.8) with respect totand have
(3.7) ρutt+Lut+∇pt=ρ(f−v· ∇v)t+ρt(f−v· ∇v−ut).
Multiplying this byutand integrating over Ω, we obtain
1 2
d dt
Z
ρ|ut|2dx+
Z
µ|∇ut|2+ (λ+µ)(divut)2dx
=
Z µ
−∇pt+ρ(f −v· ∇v)t+ρt(f−v· ∇v−
1 2ut)
¶ ·utdx.
(3.8)
To estimate each term in the right hand side of (3.8), we follow the arguments in [3, 5, 6]; we first apply the standard inequalities such as H¨older, Sobolev and Young’s inequalities and Lemma 3.4.
− Z
∇pt·utdx=
Z
ptdivutdx≤C|pt|2L2+
µ 8|∇ut|
2
L2 ≤Cc20c20,1,0+
µ 8|∇ut|
2
L2,
Z
ρft·utdx≤ |ft|L2|ρ| 1 2
L∞|√ρut|L2 ≤ |ft|L22+Cc0|√ρut|2L2,
− Z
ρ(v· ∇v)t·utdx≤C|ρ| 1 2 L∞|vt|D1
0|v|D10|
√ρu
t|L3
≤C|ρ|34 L∞|vt|D1
0|v|D10|
√ρu
t| 1 2 L2|∇ut|
1 2 L2
≤η−2C|ρ|3L∞|v|4D1 0|
√ρu
t|2L2+η|vt|2D1 0 +
µ 8|∇ut|
2
L2
≤η−2Cc70,2,0|√ρut|2L2+η|vt|2D1 0+
µ 8|∇ut|
2
Z
ρt(f−v· ∇v)·utdx≤C|ρt|H1
³
|f|L2+|v|2D1 0∩D2
´ |∇ut|L2
≤Cc20c20,1,0 ¡
|f|2L2+c40,3,0
¢
+µ 8|∇ut|
2
L2
≤Cc80,3,0+
µ 8|∇ut|
2 L2 and finally − Z ρt µ 1 2|ut|
2 ¶
dx=
Z
div(ρv)
µ
1 2|ut|
2 ¶
dx
≤ Z
ρ|v||ut||∇ut|dx
≤C|ρ|34 L∞|v|D1
0|
√ρu
t| 1 2 L2|∇ut|
3 2 L2
≤Cc70,2,0|√ρut|2L2+
µ 8|∇ut|
2
L2.
Hereη∈(0,1) is a small number. Substituting these estimates into (3.8) and taking η= (1 +c0,3,0)−1, we have
d dt
Z
ρ|ut|2dx+µ
Z
|∇ut|2dx
≤C¡|ft|2L2+c80,3,0
¢
+Cc9 0,3,0|
√ρu
t|2L2+ (1 +c0,3,0)−1|vt|2D1 0
(3.9)
for 0≤t≤min(T∗, T1). On the other hand, sinceut(0) =g2, it follows that
(3.10) |√ρut(0)|L2+|ut(0)|D1 0 ≤Cc
3 0.
Hence integrating (3.9) over (0, t), we also have
|√ρut(t)|2L2+
Z t
0 |∇
ut(s)|2L2ds≤C
¡
c3
0+c80,3,0t ¢
+Cc9 0,3,0
Z t
0 | √ρu
t(s)|2L2ds.
Therefore, in view of Gronwall’s inequality, we conclude that
|√ρut(t)|2L2+
Z t
0 |
ut(s)|2D1
0ds≤Cc
3
0 for 0≤t≤min(T∗, T1).
Moreover, since for eacht∈(0, T),u=u(t)∈D1
0∩D3 is a solution of the elliptic
system
Lu=−∇p+ρ(f−v· ∇v)−ρut in Ω,
it follows from the elliptic regularity result in [1, 5] that
|u(t)|D2 ≤Cc20(1 +|v· ∇v|L2)
≤Cc20 ³
1 +|v|32 D1
0|v| 1 2 D1
0∩D2
´
≤Cc20c
3 2
0,2,0c
1 2
0,3,0
and
Z t
0 |
u(s)|2D3ds≤Cc20
Z t
0 ³
1 +|v(s)|4D1
0∩D2+|ut(s)|
2
D1 0
´
ds≤Cc50
Lemma 3.8.
|ut(t)|2D1
0+|u(t)|
2
D3+
Z t
0 ¡
|√ρutt(s)|2L2+|ut(s)|2D2+|u(s)|2D4
¢
ds≤Cc70c120,3,0
for0≤t≤min(T∗, T1).
Proof. Multiplying (3.7) byutt and integrating over Ω, we have
Z
ρ|utt|2dx+
1 2
d dt
Z
µ|∇ut|2+ (λ+µ)(divut)2dx
=
Z
(−∇pt+ρ(f −v· ∇v)t+ρt(f −v· ∇v−ut))·uttdx
(3.11)
We can estimate the first two terms in the right hand side of (3.11) as follows:
− Z
∇pt·uttdx=
Z
ptdivuttdx=
d dt
Z
ptdivutdx−
Z
pttdivutdx
≤ dtd Z
ptdivutdx+|ptt|2L2+|∇ut|2L2
and
Z
ρ(f−v· ∇v)t·uttdx≤C|ρ| 1 2 L∞
³
|ft|L2+|v|D1
0∩D2|vt|D10
´
|√ρutt|L2
≤Cc0 ³
|ft|2L2+c20,3,0|vt|2D1 0
´
+1 2|
√ρu
tt|2L2.
To estimate the last term, we observe that
Z
ρt(f −v· ∇v)·uttdx
= d dt
Z
ρt(f −v· ∇v)·utdx−
Z
ρtt(f−v· ∇v)·utdx
− Z
ρt(f −v· ∇v)t·utdx
and
− Z
ρtut·uttdx=−d
dt
Z
ρt
µ
1 2|ut|
2 ¶
dx+
Z
ρtt
µ
1 2|ut|
2 ¶
dx.
Then by virtue of Lemma 3.4, we obtain
− Z
ρtt(f−v· ∇v)·utdx≤C|ρtt|L2
³
|f|H1+|v|2D1 0∩D2
´
|∇ut|L2
≤Cc40,3,0|ρtt|2L2+|∇ut|2L2,
− Z
ρt(f−v· ∇v)t·utdx≤C|ρt|L3
³
|ft|L2+|v|D1
0∩D2|vt|D10
´ |∇ut|L2
≤Cc40,3,0 ³
|ft|2L2+c23|vt|2D1 0
´
and
Z
ρtt
µ
1 2|ut|
2 ¶
dx=− Z
div(ρtv+ρvt)
µ
1 2|ut|
2 ¶
dx
≤ Z
(|ρt||v|+ρ|vt|)|ut||∇ut|dx
≤Cc30,3,0|∇ut|2L2+Cc 3 4
0|vt|D1 0|
√ρu
t| 1 2 L2|∇ut|
3 2 L2
≤Cc3
0,3,0|∇ut|2L2+ (1 +c0,3,0)−1|vt|2D1 0|
√ρu
t|L2|∇ut|L2
≤Cc30,3,0|ut|2D1
0 + (1 +c0,3,0) −1
|vt|2D1 0
¡
|√ρut|2L2+|∇ut|2L2
¢
.
Substituting all the above estimates into (3.11), we have
Z
ρ|utt|2dx+
d dt
Z
µ|∇ut|2+ (λ+µ)(divut)2dx
≤ dtd Z ¡2ptdivut+ 2ρt(f −v· ∇v)·ut−ρt|ut|2¢dx
+C³|ptt|2L2+c04,3,0|ρtt|2L2+c40,3,0|ft|2L2+c60,3,0|vt|2D1 0+c
3
0,3,0|ut|2D1 0
´
(3.12)
+|vt|2D1 0|
√ρu
t|2L2+ (1 +c0,3,0)−1|vt|2D1 0|∇ut|
2
L2
for 0≤t≤min(T∗, T1). Now let us define a function Λ by
Λ(t) =Z ¡µ|∇ut|2+ (λ+µ)(divut)2¢(t)dx
−Z ¡2ptdivut+ 2ρt(f−v· ∇v)·ut−ρt|ut|2¢(t)dx.
Then it follows from Lemma 3.4, Lemma 3.7 and (3.10) that
|Λ| ≤C³|∇ut|2L2+|pt|2L2+|ρt|2L3|f−v· ∇v|2L2+|ρ|3L∞|v|4D1 0|
√ρu
t|2L2
´
≤C|∇ut|2L2+Cc70c80,3,0,
Λ≥C−1|∇u
t|2L2−Cc70c80,3,0 and |Λ(0)| ≤Cc70c80,3,0.
Hence integrating (3.12) over (0, t) and using Lemma 3.4 and Lemma 3.7, we deduce that
Z t
0 | √ρu
tt(s)|2L2ds+|∇ut(t)|2L2
≤Cc70c120,3,0+ Z t
0
C(1 +c0,3,0)−1|vt|2D1
0|∇ut(s)|
2
L2ds
for 0≤t ≤min(T∗, T1). Therefore, in view of Gronwall’s inequality, we conclude
that Z
t
0 | √ρu
tt(s)|2L2ds+|ut(t)|2D1 0 ≤Cc
7 0c120,3,0
for 0≤t ≤min(T∗, T1). Moreover, since Lu=−∇p+ρ(f −v· ∇v−ut) in Ω, it
follows from the elliptic regularity result that
Z t
0 |
ut(s)|2D2ds+|u(t)|2D3 ≤Cc70c120,3,0 for 0≤t≤min(T∗, T1).
Lemma 3.9. For1≤j ≤k+ 1 we have
Z t
0 |
√ρu(j+1)
|2L2ds+ max
1≤m≤j
µ
sup
0≤s≤t|
u(m)(s)|2D1 0
¶
≤Ccj0(2,3,j−j+6)1
µ
1 + max
1≤m≤j
µ
sup
0≤s≤t|
√ρu(m)(s)|2
L2
¶¶
.
Proof. From (1.6) we deduce that for all 1≤j≤k+ 1
ρu(j+1)+Lu(j)+∇p(j)
= X
0≤m≤j
µ
j m
¶
ρ(j−m)(f −v· ∇v)(m) (3.13)
−jρ(1)u(j)− X 1≤m≤j−1
µ
j m−1
¶
ρ(j+1−m)u(m).
Multiplying (3.13) byu(j+1) and integrating over Ω, we have
Z
ρ|u(j+1)|2dx+1 2
d dt
Z ³
µ|∇u(j)|2+ (λ+µ)(divu(j))2´dx
=− Z
∇p(j)·u(j+1)dx
+ X
0≤m≤j
µ j m
¶
ρ(j−m)(f−v· ∇v)(m)·u(j+1)dx
−j
Z
ρ(1)u(j)·u(j+1)dx
− X
1≤m≤j−1 µ
j m−1
¶
ρ(j+1−m)u(m)·u(j+1)dx. (3.14)
To estimate the first term, we observe from Lemma 3.5 that
− Z
∇p(j)·u(j+1)dx=
Z
p(j)divu(j+1)dx
= d dt
Z
p(j)divu(j)dx− Z
p(j+1)divu(j)dx
Z
p(j)divu(j)dx≤C|p(j)|2L2+
µ 8|∇u
(j)
|2L2 ≤Cc
2j+2 0,1,j−1+
µ 8|∇u
(j) |2L2
and
− Z
To estimate the second term of the right hand side in (3.14), we rewrite it as follows:
X
0≤m≤j
µ
j m
¶
ρ(j−m)(f−v· ∇v)(m)·u(j+1)dx
= X
0≤m≤j−1 µ
j m
¶ Z
ρ(j−m)(f−v· ∇v)(m)·u(j+1)dx
+
Z
ρ(f −v· ∇v)(j)·u(j+1)dx.
Using Lemma 3.4 together with (3.3), we have
Z
ρ(f −v· ∇v)(j)·u(j+1)dx≤ |ρ|12
L∞|(f−v· ∇v)(j)|L2|√ρu(j+1)|L2
≤C|ρ|L∞|(f −v· ∇v)(j)|2
L2+
1 2|
√ρu(j+1) |2L2
≤Cc50,3,j +
1 2|
√ρu(j+1) |2L2,
X
0≤m≤j−1 µ
j m
¶ Z
ρ(j−m)(f−v· ∇v)(m)·u(j+1)dx
= X
0≤m≤j−1 µ
j m
¶
d dt
Z
ρ(j−m)(f−v· ∇v)(m)·u(j)dx
− X
0≤m≤j−1 µ
j m
¶ Z ³
ρ(j+1−m)(f−v· ∇v)(m)
+ρ(j−m)(f−v· ∇v)(m+1)´·u(j)dx.
Since for all 0≤m≤j−1,
|ρ(j−m)|H1 ≤Cc0j+1,1,j−1 and |(v· ∇v)(m)|L2∩L3 ≤Cc20,3,j−1,
we have
X
0≤m≤j−1 µ
j m
¶ Z
ρ(j−m)(f −v· ∇v)(m)·u(j)dx
≤ X
0≤m≤j−1 µ
j m
¶
|ρ(j−m)|L3|(f−v· ∇v)(m)|L3|∇u(j)|L2
≤C X
0≤m≤j−1
|ρ(j−m)|2H1|(f−v· ∇v)(m)|2L2+
µ 8|∇u
(j) |2L2
≤Cc20j,3+6,j−1+
µ 8|∇u
On the other hand, we obtain
X
0≤m≤j−1 µ
j m
¶ Z ³
ρ(j+1−m)(f −v· ∇v)(m)
+ρ(j−m)(f −v· ∇v)(m+1)´·u(j)dx
≤Cc20j,3+6,j−1+c04,3,j−1|ρ(j+1)|2L2
+C X
0≤m≤j−1
|ρ(j−m)|2H1|(f −v· ∇v)(m)|2L2+|∇u(j)|2L2
≤Cc20j,3+6,j−1+Cc40,3,j−1|ρ(j+1)|2L2+|∇u(j)|2L2.
We estimate the third term of (3.14) as follows:
−j
Z
ρ(1)u(j)·u(j+1)dx=−j
2 d dt
Z
ρ(1)|u(j)|2dx+j
2
Z
ρ(2)|u(j)|2dx,
−j2 Z
ρ(1)|u(j)|2dx≤C
Z
ρ|v||u(j)||∇u(j)|dx
≤C|ρ|34 L∞|v|D1
0|
√ρu(j) |12
L2|∇u
(j) |32
L2
≤Cc70,2,0|√ρu(j)|2L2+
µ 8|∇u
(j) |2L2,
and
j 2
Z
ρ(2)|u(j)|2dx=−2j Z
(ρ(1)v+ρv(1))·(u(j)· ∇u(j))dx
≤C(|ρ(1)|H1|v|D1
0+|ρ|L3|v
(1) |D1
0)|∇u
(j) |2L2
≤Cc30,3,0(1 +|v(1)|D1 0)|∇u
(j) |2L2.
Finally, we have for the last term
− X
1≤m≤j−1 µ
j m−1
¶ Z
ρ(j+1−m)u(m)·u(j+1)dx
=− X
1≤m≤j−1 µ
j m−1
¶
d dt
Z
ρ(j+1−m)u(m)·u(j)dx
+ X
1≤m≤j−1 µ
j m−1
¶ Z
(ρ(j+2−m)u(m)+ρ(j+1−m)u(m+1))·u(j)dx,
− X
1≤m≤j−1 µ
j m−1
¶ Z
ρ(j+1−m)u(m)·u(j)dx
= X
1≤m≤j−1 µ
j m−1
¶ Z
(ρv)(j−m)(u(j)· ∇u(m)+u(m)· ∇u(j))dx
≤C X
1≤m≤j−1
|(ρv)(j−m)|L23|∇um|2L2+
µ 8|∇u
(j) |2L2
≤Cc20j,2+4,j−1 X
1≤m≤j−1
|∇u(m)|2
L2+
µ 8|∇u
(j)|2
X
1≤m≤j−1 µ
j m−1
¶ Z
(ρ(j+2−m)u(m)+ρ(j+1−m)u(m+1))·u(j)dx
=− X
1≤m≤j−1 µ
j m−1
¶ Z
(ρv)(j+1−m)(u(j)· ∇u(m)+u(m)· ∇u(j))dx
+ X
1≤m≤j−1 µ
j m−1
¶ Z
ρ(j+1−m)u(m+1)·u(j)dx
≤C X
1≤m≤j−1
|(ρv)(j+1−m)|2
L3|∇u(m)|2L2+|∇u(j)|2L2
+C X
2≤m≤j−1
|ρ(j+2−m)|2
L32|∇u
(m)
|2L2+|∇u(j)|2L2−
Z
ρ(2)|u(j)|2dx
≤C³c20j,3+4,j−1+c 2j+4 0,1,j−1
´ X
1≤m≤j−1
|∇u(m)|2L2
+
Z
(|ρ(1)||v|+ρ|v(1)|)|∇u(j)||u(j)|dx
≤Cc20j,3+4,j−1 X
1≤m≤j−1
|∇u(m)|2L2+Cc30,3,0(1 +|∇v(1)|L2)|∇u(j)|2L2.
Now integrating (3.14) over (0, t)⊂(0, T1), we have Z t
0 |
√ρu(j)(s)|2ds+|∇u(j)(t)|
L2
≤Cc20j,3+6,j−1+Cc70,2,0|√ρu(j)(t)|2L2+Cc20j,3+4,j−1
X
1≤m≤j−1
|∇u(m)|2L2
+C
Z t
0 |
p(j+1)(s)|2L2ds+Cc40,3,j−1
Z t
0 |
ρ(j+1)(s)|2L2ds
+Cc20j,3+4,j−1 X
1≤m≤j−1 Z t
0 |∇
u(m)|2L2ds
+Cc3 0,3,0
Z t
0
(1 +|v(1)(s)|
D1 0)|∇u
(j)(s)|2
L2ds.
Let us define functionsIj andIIj(t) by
Ij(t) = max
1≤m≤j
µ
sup
0≤s≤t|∇
u(m)|2L2
¶
,
IIj(t) = max
1≤m≤j
µ
sup
0≤s≤t|
√ρu(j)(s) |2L2
¶
.
Then
|∇u(j)|2L2≤Cc20j,3+6,j−1+Cc70,2,0|√ρu(j)(t)|2L2
+C(c02,j3+4,j−1+c02,j3+4,j−1t)Ij−1(t)
+Cc30,3,0 Z t
0
(1 +|v(1)(s)|D1 0)|∇u
Therefore Gronwall’s inequality implies that
Ij(t)≤Cc02,j3+6,j−1+Cc 7
0,2,0IIj(t) +C(j−1)c20j,3+4,j−1Ij−1(t)
(3.15)
for 0≤t≤T1. Then by induction onj we have
Ij(t)≤C
³
c20j,3+6,j−1+c70,2,0IIj(t)
´
׳1 +c20j,3+4,j−1+c 2j+4 0,3,j−1c
2(j−1)+4 0,3,j−2 +· · ·
+c20j,3+4,j−1c
2(j−1)+4
0,3,j−2 · · ·c20·,2+43,1 ´
and hence
Ij(t)≤Cc0(j−,3j−1)(1j+6) ³
c20j,3+6,j−1+c70,2,0IIj(t)
´
. ¤
Lemma 3.10. For each1≤j≤k+ 1 we have
max
1≤m≤j
µ
sup
0≤s≤t|
√ρu(m)(s) |2L2
¶ ≤Cc20, Z t
0 ³
|√ρu(j+1)(s)|2L2+|u(j)(s)|2D2
´
ds
+ max
1≤m≤j
µ
sup
0≤s≤t|∇
u(m)(s)|2L2
¶
≤Cc(0j,3+1)(2,j−1j+6).
for0≤t≤min(T∗, T2), whereT2= min ³
((k+ 1)c(0k,3+2)(2,k k+8))−1, T 1
´ .
Proof. Multiplying (3.13) byu(j)and integrating over Ω, we have
1 2
d dt
Z
ρ|u(j)|2dx+µ|∇u(j)|2
L2+ (λ+µ)
Z
(divu(j))2dx
=− Z
∇p(j)·u(j)dx
+ X
0≤m≤j
µ
j m
¶ Z
ρ(j−m)(f−v· ∇v)(m)·u(j)dx
−j
Z
ρ(1)|u(j)|2dx− X 1≤m≤j−1
µ
j m−1
¶
ρ(j+1−m)u(m)·u(j)dx. (3.16)
We estimate each term of right hand side of (3.16) as follows:
− Z
∇p(j)·u(j)dx=
Z
p(j)divu(j)dx≤C|p(j)|2L2+
µ 8|∇u
X
0≤m≤j
µ
j m
¶ Z
ρ(j−m)(f−v· ∇v)(m)·u(j)dx
= X
0≤m≤j−1 µ
j m
¶ Z
ρ(j−m)(f−v· ∇v)(m)·u(j)dx
+
Z
ρ(f−v· ∇v)(j)·u(j)dx
≤C X
0≤m≤j−1
|ρ(j−m)|2L2|(f−v· ∇v)m|2L3+
µ 8|∇u
(j) |2L2
+|ρ|L∞|(f−v· ∇v)(j)|2
L2+|√ρu(j)|2L2
≤C(c20j,3+6,j−1+c30,3,j) +
µ 8|∇u
(j)
|2L2+|√ρu(j)|2L2,
−j
Z
ρ(1)|u(j)|2dx= 2j
Z
ρv·(u(j)· ∇u(j))dx
≤Cc30,3,0|√ρu(j)|2L2+
µ 8|∇u
(j) |2L2,
and
X
1≤m≤j−1 µ
j m−1
¶ Z
ρ(j+1−m)u(m)·u(j)dx
= X
1≤m≤j−1 µ
j m−1
¶ Z
(ρv)(j−m)(u(m)· ∇u(j)+u(j)· ∇u(m))dx
≤C X
1≤m≤j−1
|(ρv)(j−m)|L3|∇u(m)||∇u(j)|L2
≤Cc20j,3+4,j−1 X
1≤m≤j−1
|∇u(m)|2L2+
µ 8|∇u
(j) |2L2.
Substituting all these estimate into (3.16), we have
d dt
Z
ρ|u(j)|2dx+µ|∇u(j)|2L2
≤C(c20j,1+2,j−1+c 2j+6
0,3,j−1+c30,3, j) +Cc03,3,0|√ρu(j)|2L2
+Cc20j,3+4,j−1 X
1≤m≤j−1
|∇u(m)|2
L2.
(3.17)
Recall that Ij(t) = max1≤m≤j
¡
sup0≤s≤t|∇u(m)(s)|2L2
¢
. Then integrating (3.17) over (0, t), we have for t∈[0, T1]
Z
ρ|u(j)|2dx+µZ
t
0 |∇
u(j)|2
L2ds
≤ |√ρu(j)(0)|2L2+C+Cc
2j+4
0,3,j−1jtIj−1(t) +Cc30,3,0 Z t
0 | √ρu(j)
|2L2ds.
From the observation that
|√ρu(j)(0)|L2 =|√ρ0gj|L2 ≤C|ρ0|
L32|gj|D10 ≤Cc
we deduce that
|√ρu(j)(t)|2
L2 ≤Cc20+Cc02j,3+4,j−1jtIj−1(t)
and hence
(3.18) max
1≤m≤j
µ
sup
0≤s≤t|
√ρu(m) |2L2
¶
≤Cc20+Cc 2j+4
0,3,j−1jtIj−1(t))
for 0≤t≤T1. Therefore from Lemma 3.9, we conclude that
(3.19) Ij(t)≤Cc0(j,3+1)(2,j−1j+6)+Cjc
(j+1)(2j+6)
0,3,j−1 tIj−1(t).
Thus if we chooset≤((k+ 1)c0(k,3+2)(2,k k+8))−1≤(jc(j+1)(2j+6)
0,3,j−1 )−1, then using (3.18),
(3.19) and Lemma 3.6, we can prove the lemma by induction. ¤
Lemma 3.11. For each1≤j≤k, we have
|u(j)(t)|D2+
Z t
0 ³
|u(j+1)(s)|2D1 0+|u
(j)(s) |2D3
´
ds≤Cc(0j,2+1)(2,j j+7)c 1 2
0,3,j,
|u(j+1)(t)|D1 0+|u
(j)(t) |D3
+
Z t
0 ³
|u(j+1)(s)|2D2+|u(j)(s)|2D4
´
ds≤Cc(0j,+3)(23,j j+8)
for0≤t≤min(T∗, T2).
Proof. Since for eacht,u(j)(t) is a solution to the elliptic boundary value problem:
Lu(j)= X
0≤m≤j
µ
j m
¶
ρ(j−m)(f−v· ∇v)(m)
− X
1≤m≤j
µ
j m
¶
ρ(j+1−m)u(m)− ∇p(j)−ρu(j+1),
from the elliptic regularity (see [5]) we have
|u(j)|D1
0∩D2≤C|u
(j) |D1
0+C
X
0≤m≤j
|ρ(j−m)(f−v· ∇v)(m)|L2
+C X
1≤m≤j
|ρ(j+1−m)u(m)|L2+|∇p(j)|L2+|ρu(j+1)|L2
≤C|u(j)|D1 0+C
X
0≤m≤j
|ρ(j−m)|H2|(f−v· ∇v)(m)|L2
+C X
1≤m≤j
|ρ(j+1−m)|H1|∇u(m)|L2+C|p(j)|H1
+C|ρ|12
L∞|√ρu(j+1)|L2.
Using the bound (3.3), Lemma 3.4 and Lemma 3.10, we can easily show that for 1≤j ≤k
|u(j)(t)|
D1
0∩D2 ≤Cc
(j+1)(2j+7) 0,3,j +Cc
3 2
0,2,jc 1 2
for 0≤t≤min(T∗, T2). Further, we have
|u(j+1)|
D2 ≤C|u(j+1)|D1 0+C
X
0≤m≤j+1
|ρ(j+1−m)|
H2|(f−v· ∇v)(m)|L2
+C X
1≤m≤j
|ρ(j+2−m)|
H1|∇u(m)|L2+C|p(j+1)|H1
+C|ρ|12
L∞|√ρu(j+2)|L2
and hence
Z t
0 |
u(j+1)(s)|2D2ds
≤C X
0≤m≤j+1 Z t
0 |
ρ(j+1−m)(s)|2
H2|(f−v· ∇v)(m)|2L2ds
+C X
1≤m≤j+1 Z t
0 ³
|ρ(j+2−m)|2H1|∇u(m)|2L2+|p(j+1)|2H1
´
ds
+C
Z t
0 |
ρu(j+2)|2
L2ds≤Cc
(j+3)(2j+8) 0,3,j .
(3.20)
On the other hand, using the elliptic regularity [1, 5], we obtain
|u(j)|
D3 ≤C|u(j)|D1 0+C
X
0≤m≤j
|ρ(j−m)(f−v· ∇v)(m)|
H1
+C X
1≤m≤j
|ρ(j+1−m)u(m)|
H1+C|p(j)|H2+C|ρu(j+1)|H1
≤Cc(0j,+1)(23,j−1j+6)+ X
0≤m≤j
|ρ(j−m)|H2(f−v· ∇v)(m)|H1
+C X
1≤m≤j
|ρ(j+1−m)|H2|∇u(m)|L2
+Ccj0+1,1,j−1+|ρ|H2|∇u(j+1)|L2
≤C³c(0j,3+1)(2,j−1j+6)+cj0+1,1,j−1(1 +c2 0,3,j)
+cj0+1,1,j−1c0(j,3+1)(2,j−1j+6)+c0(j,3+2)(2,j j+8)´
And finally we have
|u(j)|D4 ≤C|u(j)|D1 0+C
X
0≤m≤j
|ρ(j−m)(f−v· ∇v)(m)|H2
+C X
1≤m≤j
|ρ(j+1−m)u(m)|
H2+C|p(j)|H3+C|ρu(j+1)|H2
≤Cc(0j,+1)(23,j−1j+6)+ X
0≤m≤j
|ρ(j−m)|
H2|(f−v· ∇v)(m)|H2
+C X
1≤m≤j
|ρ(j+1−m)|H2|u(m)|D1 0∩D2
+Ccj0+1,1,j−1+C|ρ|H2|u(j+1)|D1 0∩D2.
and therefore by (3.20)
Z t
0 |
u(j)(s)|2D4ds≤Cc
(j+3)(2j+8) 0,3,j
for 0≤t≤min(T∗, T2). This completes the proof of lemma. ¤
Lemma 3.12. For alll andj with1≤l≤k and0≤j≤k−l, we have
|u(j)(t)|
D1
0∩D2l+1≤Cc
(j+2l+1)(2j+2l+6)
l−1,3,j
|u(j)(t)|D1
0∩D2l+2≤Cc
(j+2l+2)(2j+2l+7)
l,2,j
|u(j)(t)|D1
0∩D2l+3≤Cc
(j+2l+3)(2j+2l+8)
l,3,j
for0≤t≤min(T∗, T3), whereT3= min(c−(2k+9)
2
k,4,0 , T2).
Proof. For the proof in case whenj= 0, we regardcl,i,−1ascl,i,0. From the elliptic
regularity, we have
|u(j)|D1 0∩D2l+1
≤C|u(j)|D1 0+C
X
0≤m≤j
|ρ(j−m)(f−v· ∇v)(m)|H2l−1
+C X
1≤m≤j
|ρ(j+1−m)u(m)|H2l−1+C|p(j)|H2l+C|ρu(j+1)|L2l−1
≤Cc(0j,+1)(23,j−1j+6)+C X
0≤m≤j
X
α1+α2=α |α| ≤2l−1
|Dα1ρ(j−m)|
H2|Dα2(f−v· ∇v)(m)|L2
+C X
1≤m≤j
X
α1+α2=α
|α| ≤2l−1
|Dα1ρ(j+1−m)Dα2u(m)|L2
+Ccj0+1,1,j−1+C|ρ|H2l|u(j+1)|D1 0∩D2l−1
≤Cc(0j,+1)(23,j−1j+6)+Cc3
l−1,3,j+Ccj0+1,1,j−11≤m≤jmax+1|u(m)|D1
0∩D2(l−1)+1
≤Cc(l−j+1)(21,3,jj+6)+Ccj0+1,1,j−1Ml−1,j+1,
whereMl,n(t) = max0≤m≤n|u(m)|D1
0∩D2l+1 forn≥0. Thus we have
Ml,j(t)≤Ccl−(j+1)(21,3,jj+6)+Cc j+1
for all 0≤t≤min(T∗, T2). Since by Lemma 3.10
M0,j+1= max 0≤m≤j+1|u
(m) |D1
0 ≤Cc
(j+2)(2j+8) 0,3,j ,
we obtain
M1,j ≤Cc(0j,3+3)(2,j j+8).
By induction, we deduce that
Ml,j(t)≤Ccl−(j+21,3l,j+1)(2j+2l+6).
Now let us define a functionNl,nby
Nl,n(t) = max
1≤m≤n|u(m)(t)|D 1 0∩D2l+2.
Then we similarly have
|u(j)|D1
0∩D2l+2≤Cc
(j+1)(2j+6) 0,3,j−1 +Cc
3
l,2,j+Cc j+1
0,1,j−1Nl−1,j+1
≤Cc(l,j2+1)(2,j j+6)+Cc0j+1,1,j−1Nl−1,j+1
and hence
Nl,j(t)≤Ccl,(j2+1)(2,j j+6)+Cc j+1
0,1,j−1Nl−1,j+1(t)
for all 0≤t≤min(T∗, T2). Since by Lemma 3.11
N0,j+1= max 0≤m≤j+1|u
(m) |D1
0∩D2≤Cc
(j+2)(2j+9) 0,2,j+1 c
1 2
0,3,j+1≤Cc
(j+3)(2j+9) 0,3,j+1 ,
we obtain
N1,j ≤Cc(1j,2+1)(2,j j+6)+Cc
(j+4)(2j+9) 0,3,j+1 ≤Cc
(j+4)(2j+9) 1,2,j .
Hence by induction we have
Nl,j(t)≤Ccl,(j2+2,jl+2)(2j+2l+7)
for all 0≤t≤min(T∗, T2).
From the similar argument as above results, we can easily deduce that
|u(j)|
D1
0∩D2l+3≤Cc
(j+2l+3)(2j+2l+8)
l,3,j ,
|u(j+1)|
D1
0∩D2l+1≤Cc
(j+2l+2)(2j+2l+8)
l−1,3,j+1 , Z t
0 ³
|u(j+1)|2D1
0∩D2l+2+|u
(j) |2D1
0∩D2l+4
´
≤Cc(l,j3+2,jl+3)(2j+2l+8)
for all 0≤t≤min(T∗, T3). ¤
Summarizing the estimates up to now, we have the following.
Proposition 3.13. If(ρ, u)is a solution to the linearized problem(2.7)-(2.10)with the initial densityρ0> δ the known vector fieldv satisfying(3.1),(3.2) and(3.3), then for sufficiently smallε, there exists a time interval[0, T∗]such that the velocity
usatisfies the following estimates that for all0≤j≤k+ 1
(3.21) |u(j)(0)|D1
and for all0≤l≤k and0≤j ≤k−l thatinf(0,T∗)×Ωρ(t, x)≥C−1δ.
sup
0<t<T∗ ³
|ρ(t)|L1+|(ρ(j)(t),(p−p(0))(j)(t))|H2(k−j)+3
´
+
Z T∗
0 |
ρ(k+2)(s)|2
L2dt≤εc0,1,j
sup
0≤t≤T∗ ³
|u(j)(t)|D1 0∩D2l+1
´
+
Z T∗
0 |
u(j)(t)|2D2∩D2l+2dt≤cl,2,j,
sup
0≤t≤T∗ ³
|u(j)(t)|D1 0∩D2l+2
´
+
Z T∗
0 ³
|u(j+1)(t)|2D1
0∩D2l+1+|u
(j)(t) |2D2l+3
´
dt≤cl,3,j,
ess sup
0<t<T∗ ³
|u(j+1)(t)|
D1
0∩D2l+1+|u
(j)(t)|
D2l+3
´
+
Z T∗
0 ³
|u(j+1)(t)|2D2l+2+|u(j)(t)|2D2l+4
´
dt≤cl,4,j,
ess sup
0≤t≤T∗
|√ρu(j+1)(t)|L2+
Z T∗
0 |
√ρu(j+2)(t)
|2L2dt≤c0,4,j.
(3.22)
Proof. From Lemma 3.4, we see the lower bound ofρby choosingc0,1,j ≥Ccj0,1,j−1.
If we let the constantc0,1,j beCc0for allj, by Lemma 3.2 the estimate (3.21) follows
immediately.
For the estimate (3.22), we consider the following three cases: (1) l = 0 and j= 0, (2) l= 0 and 1≤j≤k, (3) l≥1 and 0≤j≤k−l.
Case (1): l= 0 andj= 0.
From Lemma 3.6 to Lemma 3.8, it follows that
|u(t)|D1 0+
Z t
0 |
u(s)|2D2ds≤Cc30,
|u(t)|D2+
Z t
0 ³
|ut(s)|2D1
0+|u(s)|
2
D3
´
ds≤Cc3 0c
3 2
0,2,0c
1 2
0,3,0,
|ut(t)|D1
0+|u(t)|D3+
Z t
0 ¡
|ut(s)|2D2+|u(s)|2D4
¢
ds≤Cc70c120,3,0,
|√ρut(t)|L2+
Z t
0 | √ρu
tt(s)|2L2ds≤Cc70c120,3,0
for 0≤t≤min(T∗, T3). Therefore, defining the constantsc0,i,0’s andT∗ by
(3.23) c0,1,0=Cc0, c0,2,0=Cc30, c0,3,0=C2c60c30,2,0, c0,4,0=Cc70c120,3,0
and
we conclude that
sup
0≤t≤T∗
|u(t)|D1 0+
Z T∗
0 |
u(t)|2D2dt≤c0,2,0,
sup
0≤t≤T∗
|u(t)|D2+
Z T∗
0 ³
|ut(t)|2D1
0 +|u(t)|
2
D3
´
dt≤c0,3,0,
ess sup
0≤t≤T∗ ³
|ut(t)|D1
0+|u(t)|D3
´
+
Z T∗
0 ¡
|ut(t)|2D2+|u(t)|2D4
¢
dt≤c0,4,0,
ess sup
0≤t≤T∗
(|√ρut(t)|L2) +
Z T∗
0 | √ρu
tt(t)|2L2dt≤c0,4,0.
(3.25)
Case (2): l= 0 and 1≤j≤k.
From Lemma 3.9 to Lemma 3.11, we obtain that
|u(j)(t)|D1 0+
Z t
0 |
u(j)(s)|2D2ds≤Cc
(j+1)(2j+6) 0,3,j−1 ,
|u(j)(t)|D2+
Z t
0 ³
|u(j+1)|2D1
0+|u(j)|
2
D3
´
ds≤Cc(0j,2+1)(2,j j+7)c12
0,3,j
|u(j+1)(t)|D1
0+|u(j)(t)|D3
+
Z t
0 ¡
|u(j+1)(s)|2D2+|u(j)(s)|2D4
¢
ds≤Cc(0j,+3)(23,j j+8).
(3.26)
Thus if we define the constantsc0,i,j by
c0,2,j =Cc0(j,3+1)(2,j−1j+6), c0,3,j =C2c0(2,2j,j+2)(2j+7), c0,4,j =Cc(0j,3+3)(2,j j+8),
then we have
sup
0≤t≤T∗
|u(j)(t)|
D1 0+
Z T∗
0 |
u(j)(t)|2
D2dt≤c0,2,j,
sup
0≤t≤T∗
|u(j)(t)|D2+
Z T∗
0 ³
|u(j+1)(t)|2D1 0+|u
(j)(t) |2D3
´
dt≤c0,3,j,
ess sup
0≤t≤T∗ ³
|u(j+1)(t)|
D1 0+|u
(j)t)|
D3
´
+
Z T∗
0 ³
|u(j+1)(t)|2D2+|u(j)(t)|2D4
´
dt≤c0,4,j,
ess sup
0≤t≤T∗ ³
|√ρu(j+1)(t)|L2
´
+
Z T∗
0 | √
ρu(j+2)(t)|L22dt≤c0,4,j.
(3.27)
From Lemma 3.12, we easily show that
|u(j)(t)|
D1
0∩D2l+1+
Z t
0 |
u(j)(s)|2
D1
0∩D2l+2ds
≤Cc(l−j+21,3l,j+1)(2j+2l+6),
|u(j)(t)|D1
0∩D2l+2+
Z t
0 ³
|u(j+1)|2D1
0∩D2l+1+|u
(j) |2D1
0∩D2l+3
´
ds
≤Cc(l,j2+2,jl+2)(2j+2l+7) |u(j+1)(t)|D1
0∩D2l+1+|u
(j)(t) |D1
0∩D2l+3
+
Z t
0 ³
|u(j+1)(s)|2D1
0∩D2l+2+|u
(j)(s) |2D1
0∩D2l+4
´
ds
≤Cc(l,j3+2,jl+3)(2j+2l+8). (3.28)
and hence if we choose the constantscl,i,j such that
cl,2,j =Ccl−(j+21,3l,j+1)(2j+2l+6),
cl,3,j =Ccl,(j2+2,jl+2)(2j+2l+7),
cl,4,j =Ccl,(j3+2,jl+3)(2j+2l+8),
then we finally have
sup
0≤t≤T∗
|u(j)(t)|D1
0∩D2l+1+
Z T∗
0 |
u(j)(t)|2D1
0∩D2l+2dt≤cl,2,j,
sup
0≤t≤T∗
|u(j)(t)|D1 0∩D2l+2
+
Z T∗
0 ³
|u(j+1)(t)|2
D1
0∩D2l+1+|u
(j)(t)|2
D1 0∩D2l+3
´
dt≤cl,3,j,
ess sup
0≤t≤T∗ ³
|u(j+1)(t)|D1
0∩D2l+1+|u(j)t)|D10∩D2l+3
´
+
Z T∗
0 ³
|u(j+1)(t)|2D1
0∩D2l+2+|u(j)(t)|
2
D1 0∩D2l+4
´
dt≤cl,4,j.
(3.29)
Combining all the results above and Lemma 3.4, we complete the proof of
propo-sition. ¤
4. Linear problem with nonnegative density
In this section, we establish the existence and local time boundedness of solution to the linearized problem with nonnegative initial density on a general domain.
The following is the concerning linearized problem.
ρt+ div (ρv) = 0 in (0, T)×Ω,
(4.1)
ρut+Lu+∇p=ρ(f−v· ∇v) in (0, T)×Ω,
(4.2)
(ρ, u)|t=0= (ρ0, u0) in Ω, u= 0 on (0, T)×∂Ω,
(4.3)
We assume that the initial data satisfy the bound and compatibility conditions that
(4.5) ρ0≥0, |ρ0|L1∩H2k+3≤
ε 2, 2 +
X
1≤j≤k+1 |gj|D1
0 ≤c0,
and
(4.6) L(u(j)(0)) =jdiv(ρ
0v(0))u(j)(0) +Gj−1+Hj− ∇p(j)(0)−ρ0gj+1
for somegj ∈D01∩D2(k−j)+3 and all 0≤j≤k, where
Gj−1= 0 for j= 0,1,
Gj−1=− X
1≤m≤j−1 µ
j m
¶
ρ(j+1−m)(0)u(m)(0) for j≥2,
Hj =
X
0≤m≤j
µ
j m
¶
ρ(j−m)(0)(f−v· ∇v)(m)(0).
We also assume that the known v satisfies the bound conditions that for all 0≤j ≤k+ 1,
(4.7) |v(j)(0)|
D1
0∩D2(k−j)+3 ≤c0,1,j,
and for all 0≤l≤kand 0≤j ≤k−l,
sup
0≤t≤T∗ ³
|v(j)(t)|D1 0∩D2l+1
´
+
Z T∗
0 |
v(j)(t)|2D2∩D2l+2dt≤cl,2,j,
sup
0≤t≤T∗ ³
|v(j)(t)|D1 0∩D2l+2
´
+
Z T∗
0 ³
|v(j+1)(t)|2D1
0∩D2l+1+|v
(j)(t) |2D2l+3
´
dt≤cl,3,j,
(4.8)
ess sup
0<t<T∗ ³
|v(j+1)(t)|
D1
0∩D2l+1+|v
(j)(t)|
D2l+3
´
+
Z T∗
0 ³
|v(j+1)(t)|2D2l+2+|v(j)(t)|2D2l+4
´
dt≤cl,4,j
for some timeT∗∈(0, T) and constantscl,i,j such that
1≤c0≤cl,i,j ≤cl+1,i,,j for any i, j, l,
cl,i,j≤cl,i+1,j for each fixed l, j,
and cl,i,j ≤cl,i,l+1 for each fixed l and for any i.
Proposition 4.1. If the data ρ0, u0, f and the vector fieldv satisfy the conditions
(4.5),(4.6),(4.7)and(4.8)for some sufficiently small constantε, some fixedc0and
cl,i,j appearing in Proposition 3.13, then there exists a timeT∗ >0 and a unique
solution (ρ, u) to the linearized problem(4.1)−(4.4)satisfying the estimate (3.21)
and(3.22).
