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T itle E xistence result for heat-conducting viscous incompressible fluids with vacuum

A uthor(s ) C ho,Y onggeun; K im,Hyunseok

C itation Hokkaido University Preprint S eries in Mathematics, 742: 1-47

Is s ue D ate 2005

D O I 10.14943/83892

D oc UR L http://hdl.handle.net/2115/69550

T ype bulletin (article)

Existence result for heat-conducting viscous

incompressible fluids with vacuum

Yonggeun Cho

Department of Mathematics, Hokkaido University

Sapporo 060-0810, Japan

e-mail: [email protected]

Hyunseok Kim

School of mathematics, Korea Institute for Advanced Study

207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722 Korea

e-mail: [email protected], [email protected]

Abstract

We study the Navier-Stokes equations for heat-conducting incom-pressible fluids in a domain Ω⊂R3

whose viscosity, heat conduction coefficients and specific heat at constant volume are in general func-tions of density and temperature. We prove the local existence of the unique strong solution, provided the initial data satisfy a natural com-patibility condition. For the strong regularity, we do not assume the positivity of initial density; it may vanish in an open subset (vacuum) of Ω or decay at infinity when Ω is unbounded.

2000Mathematics Subject Classification. 35A05, 76D03

Key words and phrases. heat-conducting incompressible Navier-Stokes equations, strong

solutions, vacuum.

The first author was supported by Japan Society for the Promotion of Science under JSPS

1

Introduction

The governing system of equations for a heat-conducting viscous incompress-ible fluid is the following Navier-Stokes system of the fieldsρ(t, x), u(t, x) and θ(t, x) for (t, x)_∈(0, T)_×Ω_⊂R+×R3:

divu= 0, (1)

ρt+ div (ρu) = 0, (2)

(ρu)t+ div (ρu⊗u)−div (2µdu) +∇p=ρf, (3)

cv((ρθ)t+ div (ρuθ))−div (κ∇θ) = 2µ|du|2+ρh. (4) We consider the system (1)-(4) with the initial and boundary value condi-tions:

(ρ, u, θ)_|t=0 = (ρ0, u0, θ0) in Ω,

(u, θ) = (0,0) on ∂Ω_×[0, T],

(ρ(t, x), u(t, x), θ(t, x))_→(0,0,0) as _|x_{| → ∞}, (t, x)_∈(0, T)_×Ω. (5)

Here we denote by ρ, u, p and θ the unknown density, velocity, pressure and temperature fields for the fluid, respectively. The equations (1) and (2) imply the incompressiblity and mass conservation of fluid, respectively. The momentum equation (3) implies the balance of momentum. The tempera-ture equation (4) can be derived from the balance of energy:

(ρe)t+ div (ρeu)−div(κ∇θ) = 2µ|du|2+ρh

and the relationship cv = ∂e_∂θ, where e= e(ρ, θ) is the internal energy. We also denote by du the deformation tensor 1₂(_∇u+_∇tu), where _∇u is the gradient matrix(∂u_∂xj

i

)

of uand _∇tuits transpose. We assume that the vis-cosity coefficientµ=µ(ρ, θ), specific heat at constant volumecv =cv(ρ, θ) and heat conductivity κ = κ(ρ, θ) are positive functions of ρ and θ. The known fieldsf andh denote a heat source and external force per unit mass. Finally, (0, T)_×Ω is the time-space domain for the evolution of the fluid, whereT is a finite positive number and Ω is either a bounded domain inR3

with smooth boundary or an unbounded domain such as the whole space

Throughout this paper, we adopt the following simplified notations for homogeneous and inhomogeneous Sobolev spaces in Ω:

Lr =Lr(Ω), Dk, r =_{v_∈L1_loc(Ω) :_|v_|Dk,r <∞}, |v|_Dk,r =|∇kv|_Lr,

D₀1=_{v_∈L6(Ω) :_|v_|D1

0 <∞ and v= 0 on ∂Ω}, |v|D 1

0 =|∇v|L 2,

D₀1_{, σ}=_{v_∈D1₀ : divv= 0 in Ω_}, Dk=Dk,2, Wk, r =Lr_∩Dk, r, Hk =Wk,2, H₀1=L2_∩D1₀ and H₀1_{, σ} =L2_∩D₀1_{, σ}.

We also denote byH_σ−1 and D_σ−1 the dual space of H₀1_{, σ} and D₀1_{, σ} , respec-tively, with _⟨·,_·⟩ being the corresponding dual paring. A detailed study of homogeneous Sobolev spaces may be found in the book [10] by G. Galdi.

In this paper, we assume that the fluid flow satisfying (1)-(4) may have non-negative initial density. That is to say, we consider the incompressible fluid with an initial vacuum, a spatial domain whose interior is non-empty and in which the density of fluid vanishes identically. Thus if we assume the initial vacuum, from the continuity equation (2), we can observe that the vacuum evolves as time goes on and hence the parabolicity of the momentum equation (3) comes disappeared at any vacuum state. As a consequence, we cannot expect directly any high regularity or uniqueness from (3). To avoid this difficulty, it is necessary to introduce a suitable compatibility condition which shows how the initial fluid should behave near the vacuum.

For the global existence of weak solution to the problem (1),(2),(3) and (5), one may assume the condition √ρ0u0 ∈ L2. See [15, 23, 25] for a

constant viscosity coefficient and [19] for density-dependent one. But the uniqueness is still open even in two dimensional case.

As for the strong solution to the problem (1),(2),(3) and (5), H. Kim and H. J. Choe [8] used the condition

−µ∆u0+∇p0=ρ

1 2

0g and divu0= 0 in Ω (6)

for some (p0, g)∈H1×L2and they obtained the unique local strong solution

for the constant viscosity coefficient and bounded or whole space domain. In case of density-dependent viscosity, Y. Cho and H. Kim [5] used the condition

−div(2µ(ρ0)du0) +∇p0=ρ

1 2

for some (p0, g) ∈H1×L2. They also obtained the uniqueness and strong

solvability for 2 or 3 space dimensional bounded domain. It should be noted that the two conditions (6) and (7) turn out to be necessary and sufficient for the strong regularity. For other related topic to the vacuum, one can refer to [9, 21, 22], etc. Regarding the initial density with a positive lower bound, we refer the readers to [4, 13, 14, 18, 20] and the references therein. If the temperature equation (4) is under consideration, the situation is more complicated. The coupling between the equations (3) and (4) is deep-ened by the vacuum even in the case of constant coefficients. Thus we have to consider the relation between initial density and temperature as well as velocity near the vacuum. The authors in [6] first treated the problems with temperature equation for the polytropic compressible fluid with constant co-efficients and vacuum, and suggested a compatibility condition similar to (8) below to obtain the unique and strong solution. However, up to now, there have been few results on the uniqueness and regularity of heat-conducting incompressible or compressible Navier-Stokes equations with nonconstant coefficients. For the weak solvability to the problem (1)-(5), see the result of P. -L. Lions in Chapter 3 of [19]. For another related topic, we refer the readers to [1] in which a unique solvability is studied under a small data condition and uniform density condition but with coefficients depending on du andθ.

In this paper, we study the unique strong solutions to the initial bound-ary value problem (1)-(5) with initial vacuum and with slightly modified coefficients, and consider the minimal regularity of the strong solution in Sobolev sense. To do these, we first assume the natural extension of com-patibility condition (7) to the heat-conducting case as follows

−div (2µ0du0) +∇p0 =ρ

1 2

0g1,

−div (κ0∇θ0)−2µ0|du0|2=ρ

1 2

0g2

 

 in Ω (8)

for somep0 ∈H1 and (g1, g2) ∈L2. And further we assume the additional

condition such that

0< µ, cv, κ∈C1(R2),

µ=µ(ρ, ρθ), cv =cv(ρ, ρθ), κ=κ(ρ, ρθ).

for nonconstant coefficients. The following is our main result.

Theorem 1.1. Assume that the data(ρ0, u0, θ0, h, f)satisfies the regularity

condition

ρ0≥0, ρ0 ∈L

3

2 ∩H1∩W1, q, (u

0, θ0)∈D01∩D2, divu0= 0,

(h, f)_∈C([0, T];L2)_∩L2([0, T];Lq) and (ht, ft)∈L2(0, T;H−1)

for some 3 < q _≤ 6. Further assume the compatibility condition (8) and coefficient condition(9). Then there exist a small timeT∗ >0 and a unique strong solution (ρ, u, p, θ) to the initial boundary value problem for (1)–(5)

such that

ρ_∈C([0, T∗];L

3

2 ∩H1∩W1, q), ρ

t∈C([0, T∗];L

3

2 ∩Lq), (u, θ)_∈C([0, T∗];D01∩D2)∩L2(0, T∗;D2, q),

p_∈C([0, T∗];H1)∩L2(0, T∗, W1, q)

(ut, θt)∈L2(0, T∗;D01) and (√ρut,√ρθt)∈L∞(0, T∗;L2).

(10)

Adopting the arguments used in [5, 7, 8], one can easily show that the compatibility condition (8) also turns out to be necessary and sufficient for the strong regularity (10). The condition (9) is used to avoid some technical difficulties in estimating ∫₀t_|µt|2_L3ds and |µ−µ0|_L∞, which appear in the

basic energy estimates and are crucial in finding the local existence time. See Section 3 below. Until now, we don’t know whether the condition (9) is inevitable or not. It seems to be hard to remove the condition (9) in the presence of vacuum but worth while to try. But in case when ρ has a positive lower bound, one can assume that the coefficients are functions of ρ andθ not of ρθ.

The conditionL32 ∩H1 for initial density is necessary to prove the exis-tence and uniqueness in case of unbounded domains. See Section 4 below. In particular, the L32-condition is used for the regularity of the solution to the Stokes problem

where F = ρf ₋ρut−ρu· ∇u (see Section 2 below). For the regularity of (u, p), we need F _∈ D−1

σ the dual of D10, σ and hence ρut ∈ D−σ1. Since we assume the initial vacuum in an unbounded domain, we cannot expect ut ∈ L2 but in general ut ∈ D10, σ. Therefore L

3

2-condition is necessary. Furthermore, for higher regularity of solution of the above Stokes system (at leastD2 regularity for (u, θ) is necessary), it is inevitable to show that the density isW1, q _{and hence W}1, q _{condition for initial density is imposed.} For the details, see the proof of Proposition 2.9 below.

This paper is organized as follows. In Section 2, we show a new Sobolev embedding result and some regularity results for the solutions of linear trans-port equation, stationary Stokes and elliptic equations. In section 3, we prove a priori estimates for linearized problems under the assumptions that the initial density has positive lower bound and the domain is bounded. We show that the estimates are independent of the lower bound and the size of domain. In Section 4, we prove Theorem 1.1 by using the standard domain expansion and iteration of solutions to the linearized problems as mentioned in Section 2, which are possible thanks to the uniformity of a priori estimates on the lower bound of initial density and the size of domain.

The methods mentioned above can be applied directly to the compress-ible model with momentum (3) and energy (4) equations replaced by

p=p(ρ, ρθ),

(ρu)t+ div(ρu⊗u)−div(2µdu)− ∇(λdivu) +∇p=ρf,

cv((ρθ)t+ div (ρuθ)) + (γcvρθ−p)divu−div (κ∇θ)

= 2µ_|du_|2+λ(divu)2+ρh.

The factorγcvρθ−pfollows from a result of the first law of thermodynamics lawρ2∂e_∂ρ =p₋αKθθand the relationα ≡ −1_ρ

( ∂ρ ∂θ )

p = γcvρ

the case of the bounded domain we can remove the Stokes relation. We will however not present here the details of results on the above problem. Without Stokes like relation, it is much difficult to handle more general compressible fluids in an unbounded domain because it is hard to derive an elliptic estimate independent of the size of domain. This will be very interesting problem to try.

2

Preliminary results

Let Ω be a bounded domain, an exterior domain or the whole space, and let us define

R0 =R0(Ω) =

  

sup x∈∂Ω|

x_|+ 2 if ∂Ω is nonempty,

3 if ∂Ω is empty

(11)

and

ΩR={x∈Ω :|x|< R} for 2R0 < R≤ ∞.

Note that ΩR= Ω if Ω is bounded or R=∞, and ΩR is the intersection of an unbounded domain Ω and the open ballBR=BR(0) otherwise.

In this section, we derive some estimates in ΩR which are independent of the radiusR. These results will be crucial technical tools in proving our main theorem for unbounded domains via the method of domain expansions. Throughout the paper, we adapt the following notations for the sake of conciseness. For (semi)-normed spaces X and Y, we define

| · |X∩Y =| · |X+| · |Y.

Moreover, for a mapping _F from X into a space Y′ _{withY ⊂Y}′_{, we mean} by the inequality

|Fx_|Y ≤C|x|X (12) that ifx_∈X, then _Fx_∈Y and (12) holds for someC independent of x.

2.1 Sobolev embedding results

Lemma 2.1. Assume that 2R0 < R≤ ∞ and 3< q <∞. Then

|u_|L6_(Ω

R) ≤C|u|D10(ΩR), (13)

|u_|L6_(Ω

R)≤C|u|H1(ΩR), |u|_C0,1−3

q_(Ω R) ≤

C_|u_|W1,q_(Ω

R) (14)

and

|u_| C0,1−

3 q_(Ω

R)≤

C_|u_|D1

0(ΩR)∩D1,q(ΩR). (15)

Here C denotes a positive constant depending only on q and Ωbut indepen-dent of R and for0< α <1, C0,α_(Ω

R) is the H¨older space consisting of all scalar (or vector) fieldsu in ΩR such that

|u_|C0,α_(Ω

R)≡ sup x∈ΩR

|u(x)_|+ sup x,y∈ΩR, x̸=y

|u(x)₋u(y)_|

|x₋y_|α <∞.

Proof. The first inequality (13) is an immediate consequence of the corre-sponding Sobolev embedding inequality in the whole space becauseC∞

c (ΩR) is dense inD₀1(ΩR); see Theorem 6.1 in [10, Chapter II]. Moreover, since ΩR satisfies the cone condition uniformly onR, it follows from classical Sobolev embedding results (see [2, Chapter 5] for instance) that

|u_|L6_(Ω

R)≤C|u|H1(ΩR) and |u|C(ΩR)+|u|_C0,1−3

q_(Ω2 R0)≤

C_|u_|W1,q_(Ω R). Hence in order to prove (14), we have only to show that if u _∈ W1,q(ΩR), then

|u(x1)−u(x2)| ≤C|∇u|Lq_(Ω

R)|x1−x2|

1−3

q ₍₁₆₎

forx1, x2∈BR\B3

2R0 with|x1−x2| ≤1. Let us denote

r = |x1−x2|

2 and x=

x1+x2

2 .

Then since the setBr(x)∩BR is convex,

diam(Br(x)∩BR) = 2r and |Br(x)∩BR|> 1

2|Br(x)|= 1 2πr

3_,

it follows from Lemma 7.16 in [12] that

u(x)₋uBr(x)∩BR

_≤ 16

3π ∫

Br(x)∩BR

for allx_∈Br(x)∩BR, where

u_Br(x)∩BR = 1

|Br(x)∩BR| ∫

Br(x)∩BR

u(y)dy.

By virtue of H¨oler inequality, we deduce that ifx_∈Br(x)∩BR, then

∫

Br(x)∩BR

|∇u(y)_||x₋y_|−2dy_{≤ |∇}u_|Lq_(Ω R)

(∫

B2r(x)

|x₋y_|−q2q−1dy

)q−1 q

≤C_|∇u_|Lq_(Ω R)r

1−3 q.

Combining this and (17), we have

|u(x1)−u(x2)| ≤ |u(x1)−u|+|u(x2)−u| ≤C|∇u|Lq_(Ω R)r

1−3 q,

which implies (16) and thus the second inequality (14).

It remains to prove the last inequality (15). Letu_∈D1₀(ΩR)∩D1,q(ΩR). Then definingu by

u=u in ΩR and u= 0 outside ΩR,

we easily show that u _∈D1_(R3_)∩D1, q_(R3_{) and ∇u = 0 a.e. in R}3 _\Ω

R. Hence adapting the proof of (16), we deduce the classical result due to Morrey that uis H¨older continuous in R3 with exponentα= 1₋3_q and

|u(x)₋u(y)_{| ≤}C_|∇u_|Lq₍R3₎|x−y|α for x, y∈R3. (18) Moreover, it follows from (17) withr= 1 and R=_∞ that

|u(x)_{| ≤}C ∫

B1(x)

|u(y)_|dy+C ∫

B1(x)

|∇u(y)_||x₋y_|−2dy

for allx_∈R3. Hence in view of H¨older inequality again, we have

|u(x)_{| ≤}C(_|u_|L6_(B

1(x))+|∇u|Lq(B1(x))

)

for x_∈R3. (19) Now the inequality (15) follows immediately from (18), (19) and (13). This completes the proof of Lemma 2.1.

2.2 A linear transport equation

Let us consider the following linear hyperbolic problem

ρt+ div(ρv) = 0 in (0, T)×ΩR and ρ(0) =ρ0 in ΩR, (20) wherev is a known vector field in (0, T)_×ΩR such that

v_∈C([0, T];D1₀(ΩR)∩D2(ΩR))∩L2(0, T;D2,q(ΩR))

for some 3< q_≤6. Here it should be noted that vneed not be divergence-free in (0, T)_×ΩR. Using exactly the same arguments as in [6], we can prove the following existence and regularity results.

Lemma 2.3. Assume that

ρ0 ∈L

3 2(Ω

R)∩H1(ΩR)∩W1,q(ΩR) and ρ0 ≥0 in ΩR. Then

(i) there exists a unique solution ρ to the problem (20) such that

ρ_∈C([0, T];L32(Ω

R)∩H1(ΩR)∩W1,q(ΩR)), (ii) the solution ρ satisfies the following estimate

|ρ(t)_| L32_(Ω

R)∩H1(ΩR)∩W1,q(ΩR)≤ |

ρ0|

L32_(Ω

R)∩H1(ΩR)∩W1,q(ΩR)

×exp

( C

∫ t

0 |∇

v(s)_|H1_(Ω

R)∩D1,q(ΩR)ds

)

for 0_≤t_≤T and finally,

(iii) the solution ρ is represented by the formula

ρ(t, x) =ρ0(U(0, t, x) ) exp

[

−

∫ t

0

divv(s, U(s, t, x) )ds ]

, (21)

where U _∈C([0, T]_×[0, T]_×ΩR) is the solution to the initial value problem

{ _∂

∂tU(t, s, x) =v(t, U(t, s, x) ), 0≤t≤T, U(s, s, x) =x, 0_≤s_≤T, x_∈ΩR.

Remark 2.4. It follows immediately from the equation in (20) that

ρt∈C([0, T];L 3 2(Ω

R)∩Lq(ΩR)).

Proof. The lemma except the result (ii) has been well-known in case when ΩR is a bounded domain: see [26] for instance. A very related estimate to that in (ii) was obtained in [6, 7] by means of a standard energy method. Moreover the case of unbounded domains can be reduced to that of bounded domains using a regularization technique and a cut-off technique. We omit its detailed proof of the lemma and refer the readers to our previous papers [6, 7].

2.3 The nonhomogeneous Stokes equations

Next, we consider the boundary value problem for the nonhomogeneous Stokes equations

−div (2µdu) +_∇p=F, divu= 0 in ΩR, (23) u= 0 on ∂ΩR and u(x)→0 as |x| → ∞, x∈ΩR, (24)

whereF and µare known vector and scalar fields in ΩR such that

F _∈D−1(ΩR), µ, µ−1 ∈L∞(ΩR) and µ≤µ≤µ in ΩR (25) for some constantsµ andµ with 0< µ_≤1_≤µ.

By weak solutions to the problem (23) and (24), we mean either

Definition 2.5. A vector field u _∈ D1_{0, σ}(ΩR) is a weak solution to the boundary value problem (23) and (24) provided that

∫

2µdu: dv dx = _⟨F, v_⟩ for all v_∈D₀1_{, σ}(ΩR). (26) or

Definition 2.6. A pair (u, p) _∈ D1_{0, σ}(ΩR)×L20(ΩR) of vector and scalar fields in ΩRis aweak solution to the boundary value problem (23) and (24) provided that

∫

2µdu: dv dx₋ ∫

Here the spaceL2₀(ΩR) is defined by

L2₀(ΩR) = { {

p_∈L2(ΩR) :∫_ΩRp dx= 0 }

if ΩR is bounded,

L2(ΩR) otherwise.

These two definitions are actually equivalent. If (u, p) _∈ D1_{0, σ}(ΩR)× L2₀(ΩR) is a weak solution to (23) and (24) in the sense of Definition 2.6, then it is obvious that u is a weak solution in the sense of Definition 2.5. The converse is an immediate consequence of the following result.

Lemma 2.7. Let u_∈D1_{0, σ}(ΩR) be a weak solution to the problem(23) and (24) in the sense of Definition 2.5. Then there exists a unique scalar field p_∈L2

0(ΩR) such that(u, p) is a weak solution to (23) and(24) in the sense of Definition2.6. This field p is called the pressure associated with u.

Proof. Let us define a bounded linear functional_F on D₀1(ΩR) by

F(v) =

∫

2µdu: dv dx _{− ⟨}F, v_⟩ for all v_∈D1₀(ΩR). (28)

Then since_F vanishes identically onD1_{0, σ}(ΩR), it follows from Theorem 5.2 and Corollary 5.1 in [10, Chapter III] that there exists a unique scalar field p_∈L2₀(ΩR) such that

F(v) =

∫

pdivv dx for all v_∈D1₀(ΩR). (29)

Note that the statement (29) holds if and only if (u, p) satisfies (27). This completes the proof of Lemma 2.7.

The existence of a unique weak solution to the problem (23) and (24) can be easily proved using the Riesz representation theorem and Lemma 2.7. In fact, the left hand side of (26) defines an inner product on the Hilbert space D₀1_{, σ}(ΩR) because |dv|L2

(ΩR) = |∇v|L2(ΩR) = |v|D10,σ(ΩR) for all v_∈D1

Lemma 2.8. For each F _∈ D−1(ΩR), there exists a unique weak solution (u, p) _∈ D1

0,σ(ΩR)×L02(ΩR) to the boundary value problem (23) and (24). Moreover we have the the following estimate

|u_|D1

0(ΩR)+|p|L2(ΩR) ≤C µ µ

−1

|F_|D−1_(Ω

R). (30)

Proof. It remains to derive the estimate (30). Takingv =u _∈D1_{0, σ}(ΩR) in (26), we first have

∫

2µ_|du_|2dx = _⟨F, u_{⟩ ≤ |}F_|D−1_(Ω

R)|u|D01(ΩR). But since µ _≥ µ > 0 in ΩR and |du|L2

(ΩR) = |∇u|L2(ΩR) = |u|D10(ΩR), we readily deduce that

|u_|D1

0(ΩR)≤(2µ)

−1_|F|

D−1_(Ω

R). (31)

To derive the estimate forp, we need to show that there is a vector field v inD₀1(ΩR) such that

divv=p in ΩR and |v|D1

0(ΩR) ≤C|p|L2(ΩR). (32) To show this, we adapt the proof of Theorem 3.4 in [10, Chapter III]. See also [17]. Let us extendptoBR by zero outside ΩR.1 Then sincep∈L20(BR), it follows from a classical result due to Bogovskiˇi [3] (see also Theorem 3.1 in [10, Chapter III]) that there exists a vector fieldv1 ∈D10(BR) such that

divv1 =p in BR and |v1|D1

0(BR)≤C|p|L2(ΩR). Moreover, we observe that

∫

∂Ω

v1·ν dσ=−

∫

BR0\Ω

divv1dx=−

∫

BR0\Ω

p dx= 0.

Hence by virtue of a corollary (Exercise 3.4) of Theorem 3.1 in [10, Chapter III], there exists a vector fieldv2 such that

divv2 = 0 in Ω2R0, v2= 0 on ∂B2R0, v2 =−v1 on ∂Ω

v2∈D1(Ω2R0) and |v2|D1(Ω2R0) ≤C|div(φv1)|L2(Ω2R0),

(33)

1

B∞=R

3

whereφ_∈C_c∞(B2R0) is a cut-off function withφ= 1 in BR0. We extendv2 to ΩRby zero outside Ω2R0. Then from (33), we easily deduce that

divv2 = 0 in ΩR, v2= 0 on ∂BR, v2=−v1 on ∂Ω

v2 ∈D1(ΩR) and |v2|D1

(ΩR) ≤C|v1|D01(BR).

It is now easy to show that if we define v by v =v1 +v2, then v satisfies

(32). Hence from (28), (29) and (31), we deduce that

∫

p2dx = _F(v) =

∫

2µdu: dv dx _{− ⟨}F, v_⟩

≤ C(µ_|u_|D1

0(ΩR)+|F|D−1(ΩR)

)

|v_|D1 0(ΩR)

≤ C µ µ−1_|F_|D−1

(ΩR)|p|L2(ΩR). This completes the proof of Lemma 2.8.

One of the main purposes of this section is to prove the following higher regularity estimates for weak solutions to the problem (23) and (24).

Proposition 2.9. Let (u, p)_∈D1

0, σ(ΩR)×L20(ΩR)be a weak solution to the boundary value problem (23) and(24). Assume in addition to (25) that

∇µ_∈L3(ΩR)∩Lq(ΩR) for some q ∈(3,6]. Then we have the following regularity estimates:

|∇u_|H1

(ΩR)+|p|H1(ΩR) ≤CM(µ) N

|F_|D−1

(ΩR)∩L2(ΩR). (34)

and

|∇u_|W1, q_(Ω

R)+|p|W1, q(ΩR) ≤CM(µ) N_|F|

D−1

(ΩR)∩L2(ΩR)∩Lq(ΩR), (35)

where

M(µ) =µ µ−1(1 +_|∇µ_|L2_(Ω

R)∩Lq(ΩR)

)

and N =N(q)>1.

Proof. Our proof consists of three steps.

Step 1: We first consider the case that ΩRis bounded and µ= 1 identi-cally in ΩR. In this case, (23) reduces to the classical Stokes equations

We will show that if (u, p) _∈H₀1(ΩR)∩L20(ΩR) is a weak solution of (36), then

|∇u_|W1,r_(Ω

R)+|p|W1,r(ΩR)≤C|F|D−1(ΩR)∩L2(ΩR)∩Lr(ΩR). (37) for 2 _≤ r _≤ 6. To show this, let F _∈ Lr_(Ω

R) for some r ∈ [2,6]. Then it follows from a standard regularity result thatu _∈W2,r(ΩR), p∈W1,r(ΩR) and _|u_|W2,r_(Ω

R)+|p|W1,r(ΩR) ≤ CR|F|Lr(ΩR) for some constant CR but de-pending on R. An R-independent estimate can be derived using a cut-off technique and Poincar´e inequality in Ω2R0. That is, adapting the arguments in [7, Section 5] and [16, Section 3], we can show that

|∇u_|W1,r_(Ω

R)+|p|W1,r(ΩR)≤C

(

|F_|Lr_(Ω

R)+|∇u|Lr(ΩR)+|p|Lr(ΩR)

) . (38)

A detailed derivation of (38) is omitted. Combining (30) and (38), we im-mediately obtain the estimate (37) forr = 2. Assume that 2< r_≤6. Then in view of Sobolev inequality, we deduce from (38) and (37) withr= 2 that

|∇u_|W1,r_(Ω

R)+|p|W1,r(ΩR) ≤ C

(

|F_|Lr_(Ω

R)+|∇u|Lr(ΩR)+|p|Lr(ΩR)

)

≤ C(_|F_|Lr_(Ω

R)+|∇u|H1(ΩR)+|p|H1(ΩR)

)

≤ C_|F_|D−1

(ΩR)∩L2(ΩR)∩Lr(ΩR). This proves the regularity estimate (37).

Step 2. Next we prove the proposition for the case that ΩR is bounded and µ _∈ C1(ΩR). Let (u, p) ∈ H01(ΩR)×L20(ΩR) be a weak solution to (23) and (24) with F _∈ Lr(ΩR) for r = 2 or q. Then it follows from a regularity result in [11] that u _∈ W2,r_(Ω

R) and p ∈ W1,r(ΩR). Hence it remains to derive (34) and (35). From now on, we drop ΩR and write Lr instead of Lr(ΩR) for simplicity. The same notations are also adapted to Sobolev spaces.

We begin with rewriting (23) as

−∆u+_∇p˜= ˜F and divu= 0 in Ω, (39)

where ˜p = µ−1_{p and ˜F =µ}−1_{(F + 2∇µ·du−p˜∇µ). Using the estimate}

(30), H¨older inequality and Sobolev inequalities, we have

|F˜_|D−1 ≤ Cµb −2(1 +|∇µ|_L3) (|F|_D−1+|∇u|_L2 +|p|_L2)

and

|F˜_|L2 ≤C µ−1(|F|_L2 +| |∇µ||∇u| |_L2+|p˜∇µ|_L2)

≤C µ−1_|F_|L2+C µ−2|∇µ|_Lq

(

|∇u_| L

2q

q−2 +|p|_L 2q q−2

)

≤C µ−1_|F_|L2+C µ−2|∇µ|_Lq

(

|∇u_|1−

3 q L2 |∇u|

3 q

H1+|p|

1−3 q L2 |p|

3 q H1

)

≤C µ−1_|F_|L2+η−1CM(µ) 2q q−3(|∇u|

L2+|p|

L2) +η(|∇u|

H1 +|p| H1)

≤Cη−1M(µ) 2q q−3|F|

D−1

∩L2 +η(|∇u|

H1+|p| H1)

for any small constant η _∈ (0,1). Hence it follows from the estimate (37) that

|∇u_|H1 +|∇p˜|_L2 ≤C|F˜|_D−1_∩L2

≤Cη−1M(µ)q3q−3|F| D−1

∩L2 +η(|∇u|

H1 +|p| H1).

(40)

On the other hand, we observe that

|p_|H1 ≤ |p˜∇µ|_L2+|µ∇p˜|_L2 +|p|_L2

≤Cµ−1_|∇µ_|Lq|p|

1−3 q L2 |p|

3 q

H1+Cµ|∇p˜|L2+|p|L2

≤CM(µ)q−q3

|F_|D−1 +Cµ|∇p˜|_L2+ 1 2|p|H1

and so

|p_|H1 ≤CM(µ) q q−3|F|

D−1 +Cµ|∇p˜|_L2.

Substituting this into (40) and choosingη = (2Cµ)−1_{, we deduce that}

|∇u_|H1 +|p|_H1 ≤CM(µ) 5q q−3

|F_|D−1_∩L2. (41)

This proves the first estimate (34).

To derive the second one, we use the estimate (37) again to deduce that

|∇u_|W1,q+|∇p˜|_Lq ≤C|F˜|_D−1

∩L2

∩Lq. (42)

By virtue of (41), H¨older inequality and Sobolev inequalities, we obtain

|F˜_|D−1_∩L2 ≤CM(µ) 5q q−3|F|

and

|F˜_|Lq ≤Cµ−1(|F|_Lq+| |∇µ||∇u| |_Lq+|p˜∇µ|_Lq)

≤Cµ−1_|F_|Lq+Cµ−2|∇µ|_Lq(|∇u|_L∞+|p|_L∞)

≤Cµ−1_|F_|Lq+Cµ−2|∇µ|_Lq

(

|∇u_|1_H−1γ|∇u| γ

W1, q +|p|

1−γ H1 |p|

γ W1, q

)

≤Cη−1M(µ)N_|F_|D−1_∩L2_∩Lq +η(|∇u|_W1,q+|p|_W1,q)

for someγ =γ(q)_∈(0,1) and N =N(q)>1. Similarly, we have

|p_|W1,q≤ |p˜∇µ|_Lq +|µ∇p˜|_Lq+|p|_Lq

≤Cµ−1_|∇µ_|Lq|p|1−γ H1 |p|

γ

W1,q+Cµ|∇p˜|Lq+|p|Hq

≤CM(µ)N_|F_|D−1

∩L2+Cµ|∇p˜|_Lq+ 1 2|p|W1,q

and so

|p_|W1,q≤CM(µ)N|F|_D−1

∩L2+Cµ|∇p˜|_Lq.

Substituting these estimates into (42) and choosingη appropriately, we eas-ily derive the second estimate (35).

Step 3. Finally we prove the propositioin without any restriction. In Step 2, we proved the proposition under the additional assumption that ΩR is bounded and µ ∈ C1(ΩR). But since the estimates (34) and (35) are independent of theC1-regularity ofµ, a simple regularization argument allows us to prove the lemma for the case that ΩR is bounded. Hence to complete the proof, it remains to prove the proposition for the case that Ω is unbounded andR=_∞. This can be deduced from the uniform estimates (34) and (35) on R < _∞ using the method of domain expansions. Let (u, p) _∈ D₀1_,σ(Ω)_∩L2₀(Ω) be a weak solution to the problem (23) and (24) withR=_∞ and F _∈D−1_(Ω)∩Lr_{(Ω) forr = 2 or q. Then since}

M(µ_|ΩR)≤M(µ)

and _|F_|ΩR|D−1(ΩR)∩L2(ΩR)∩Lr(ΩR) ≤ |F|D−1(Ω)∩L2(ΩR)∩Lr(Ω)

there exists a unique weak solution (uR, pR) _∈ H₀1_,σ(ΩR)×L20(ΩR) to the problem (23) and (24), which satisfies the uniform estimate

|uR_|D1

0(ΩR)+|∇u R

|W1, r_(Ω R)+|p

R

|W1,r_(Ω R)

≤CM(µ)N_|F_|D−1

(Ω)∩L2

(Ω)∩Lr_(Ω). (43) Extending (uR_{, p}R_{) to Ω by zero outside Ω}

R, we find that (uR, pR)∈D01,σ(Ω)× L2₀(Ω). Then adapting the proof of Proposition 6 in [8] (see also the proof of Lemma 4.1 below), we easily show that

uR_→u in D₀1_,σ(Ω) as R_{→ ∞}, which implies immediately that

∇pR_{→ ∇}p in D_loc−1(Ω) as R_{→ ∞}. Hence from (43), we deduce that

|∇u_|W1, r_(Ω)+|∇p|_W1,r_(Ω) ≤CM(µ)N|F|_D−1

(Ω)∩L2

(Ω)∩Lr_(Ω). Combining this estimate and (30), we complete the proof of Lemma 2.9.

Remark 2.10. In our previous paper [5], we derived similar estimates to

(34)and (35). But they depend on the radiusR because Poincar´e inequality in ΩR was used in an essential way.

2.4 The stationary heat conduction equation

Adapting the previous arguments, we can also prove the similar regularity results on the boundary value problem for the stationary heat conduction equation

−div (κ_∇θ) =G in ΩR

θ= 0 on ∂ΩR and θ(x)→0 as |x| → ∞, x∈ΩR, (44)

whereGand κ are known scalar fields in ΩR such that

Proposition 2.11. For each G _∈ D−1(ΩR), there exists a unique weak solutionθ_∈D1

0(ΩR) to the boundary value problem(44), which satisfies the estimate

|θ_|D1

0(ΩR)≤Cκ

−1_|G|

D−1

(ΩR).

Moreover, if κ satisfies the additional regularity

∇κ_∈L3(ΩR)∩Lq(ΩR) for some q∈(3,6], then we have the following regularity results:

|∇θ_|H1

(ΩR) ≤CM(κ) N

|G_|D−1

(ΩR)∩L2(ΩR)

and

|∇θ_|W1, q_(Ω

R)≤CM(κ) N

|G_|D−1

(ΩR)∩L2(ΩR)∩Lq(ΩR),

where

M(κ) =κ−1(1 +_|∇κ_|L3_(Ω

R)∩Lq(ΩR)

)

and N =N(q)>1.

3

A priori estimates for a linearized problem

To prove Theorem 1.1, we consider the following linearized problem

divu= 0 in (0, T)_×Ω, (46)

ρt+ div (ρv) = 0 in (0, T)×Ω, (47)

(ρu)t+ div (ρv⊗u)−div(2µ0du) +∇p

= div (2(µ₋µ0)dv) +ρf in (0, T)×Ω, (48)

cv((ρθ)t+ div (ρθv))−div(κ0∇θ)

= div ((κ₋κ0)∇η) + 2µ|dv|2+ρh in (0, T)×Ω, (49)

(ρ, u, θ)_|t=0 = (ρ0, u0, θ0) in Ω, (u, θ) = (0,0) on (0, T)×∂Ω, (50)

(ρ(t, x), u(t, x), θ(t, x))_→(0,0,0) as _|x_{| → ∞}, (t, x)_∈(0, T)_×Ω, (51)

where we write

µ=µ(ρ, ρη), cv =cv(ρ, ρη), κ=κ(ρ, ρη),

for simplicity. Throughout this section, we assume that the dataρ0,u0,θ0,

f,h satisfy

ρ0 ≥0, ρ0 ∈L

3

2 ∩H1∩W1, q, u

0 ∈D10,σ∩D2, θ0∈D01∩D2,

(h, f)_∈C([0, T];L2)_∩L2([0, T];Lq), (ht, ft)∈L2(0, T;H−1), (52)

−div (2µ0du0) +∇p0 =ρ

1 2

0g1 and −div (κ0∇θ0)−2µ0|dv(0)|2=ρ

1 2

0g2

for someq _∈(3,6], p0 ∈H1 and (g1, g2)∈L2. We assume further that the

pair (v, η) of known vector and scalar fields satisfies

(v, η)_∈C([0, T];D₀1_∩D2)_∩L2(0, T;D2, q), (vt, ηt)∈L2(0, T;D01). (53)

Here we emphasize again thatv need not be divergence-free in (0, T)_×Ω. First, we prove an existence result for the problem (46)-(50) for the case thatρ0 is bounded below away from zero and Ω is a bounded domain.

Lemma 3.1. Let Ω be a bounded domain in R3 with smooth boundary. In addition to(52)and(53), we assume thatρ0 ≥δ inΩfor some constant δ >

0. Then there exists a unique solution (ρ, u, p, θ) to the linearized problem

(46)-(50) such that

ρ_∈C([0, T∗];W1, q), (u, θ)∈C([0, T∗];H01∩H2)∩L2(0, T∗;W2, q), p_∈C([0, T∗];H1)∩L2(0, T∗, W1, q), ρt∈C([0, T∗];Lq), (54) (ut, θt)∈C([0, T];L1)∩L2(0, T∗;H01) and ρ≥δ in (0, T)×Ω

for some constant δ >0.

Assume thatρ0,u0,θ0,f,h,v,ηand Ω satisfy the hypotheses of Lemma

3.1. Then it follows from Lemma 3.1 that there exists a unique strong solution (ρ, u, p, θ) to the linear problem (46)-(50) satisfying the regularity (54). The purpose of this section is to derive some local (in time) a priori estimates for (ρ, u, p, θ) which are independent of the lower bound δ of ρ0

and the size of Ω. For this purpose, we choose a fixed constant c0 >1 so

that

c0 ≥1 +|ρ0|

L32_∩H1_∩W1, q +|(u0, θ0)|D01+|(g1, g2)|L 2

and assume that

|(v(0), η(0))_|D1

0 ≤2c0, sup

0≤t≤T∗

|v(t)_|D1 0 +

∫ T∗

0

(

|vt(t)|2_D1

0 +|v(t)|

2

D2, q

)

dt_≤2c1,

sup

0≤t≤T∗ (

|v(t)_|D2+|η(t)| D1

0∩D 2

)

+

∫ T∗

0

(

|ηt(t)|2_D1

0 +|η(t)|

2

D2, q

)

dt_≤2c2,

sup

0≤t≤T∗

|η(t)₋η(0)_|D1 0∩D

1,4 ≤2c−1

2 and

∫ T∗

0 |

ρ0ηt(t)|_L22dt≤2c−₂6 (55)

for some constants c1, c2 and T∗ with 1 < c0 ≤ c1 ≤ c2 and 0 < T∗ ≤ T, which will be determined later and depend only on c0 and the parameters

ofC. Throughout this and next sectoins, we denote byC a generic positive constant depending only on the fixed constants q, T, _|µ_|C1_(R2₎, |κ|_C1_(R2₎,

|cv|C1

(R2

) and the norms of f and h. Moreover, M = M(·) denotes a

generic increasing continuous function from [1,_∞) to [1,_∞) which depends only on the parameters of C. We also adopt the simplified notation µ(t) = µ(ρ(t), ρ(t)η(t)), etc.

3.1 Estimates for the density ρ

We first derive estimates for the density ρ, which is the solution to the hyperbolic problem (47) and (50). By virtue of Lemma 2.3, we have

|ρ(t)_| L32_∩H1

∩W1, q ≤ |ρ0|_L32_∩H1

∩W1, qexp

( C

∫ t

0 |∇

v_|H1_∩D1,qds

)

for 0_≤t_≤T. Then since

∫ t

0 |∇

v_|H1_∩D1,qds≤t 1 2

(∫ t

0 |∇

v_|2_H1

∩D1,qds

)1 2

≤Cc2t+C(c2)t)

1

2, (57)

it follows from (56) and (47) that

|ρ(t)_|

L32_∩H1_∩W1, q ≤Cc0 and |ρt(t)|_L32_∩Lq ≤Cc

2

2 (58)

for 0_≤t_≤min(T∗, T1), where T1 =c−21. On the other hand, from Lemma

2.3, we also deduce that

ρ(t, x) =ρ0(U(0, t, x) ) exp

[

−

∫ t

0

divv(s, U(s, t, x) )ds ]

, (59)

whereU _∈C([0, T]_×[0, T]_×Ω) is the solution to the initial value problem

{ _∂

∂tU(t, s, x) =v(t, U(t, s, x)), 0≤t≤T U(s, s, x) =x, 0_≤s_≤T, x_∈Ω.

In view of the classical embedding resultW1, q ֒_→C0,1−

3

q_{, we have}

|ρ0(U(0, t, x) )−ρ0(x)| ≤C|ρ0|W1, q|U(0, t, x)−x|1− 3 q_,

while

|U(0, t, x)₋x_{| ≤} ∫ t 0 ∂

∂sU(s, t, x) ds≤ ∫ t 0 |

v(s)_|L∞ds≤c

2t.

Hence using the estimate (57), we obtain

|ρ(t, x)₋ρ0(x)|

≤

(ρ0(U(0, t, x) )−ρ0(x)) exp

(

−

∫ t

0

divv(s, U(s, t, x) )ds)

+ρ0(x)

1−exp (

−

∫ t

0

divv(s, U(s, t, x) )ds)

≤C_|ρ0|W1, q

(

|U(0, t, x)₋x_|1−

3 q +

∫ t

0 |∇

v(s)_|L∞ds ) ×exp ( C ∫ t 0 |∇

v(s)_|L∞ds )

≤Cc0

(

(c2t)1−

3 q +c

2t+ (c2t)

1 2

)

for 0_≤t_≤min(T∗, T1) andx∈Ω. Therefore, taking

T2= min(T1, c−2α1) with α1= max

(

42q₋3 q₋3 ,100

) ,

we conclude that

|ρ(t)₋ρ0|L∞ ≤Cc−40

2 for 0≤t≤min(T∗, T2). (61)

Using this and (58), we also conclude that

|ρ(t)₋ρ0|L3

∩L6 ≤Cc−20

2 for 0≤t≤min(T∗, T2). (62)

3.2 Estimates for the coefficients µ, κ and cv

Note thatµ, κand cv are positive C1-functions of (ρ, ρη). But by virtue of Lemma 2.1, (55) and (61), we have

|η(t)₋η(0)_|L∞ ≤C|η(t)−η(0)|

D1 0∩D

1,4 ≤Cc−1

2

and

|ρ(t)η(t)₋ρ(0)η(0)_|L∞ ≤ |(ρ(t)−ρ₀)η(t)|_L∞+|ρ₀(η(t)−η(0))|_L∞ ≤Cc−1

2

for 0_≤t_≤min(T∗, T2). Using this observation together with (58) and (61),

we easily show that

M(c0)−1 ≤µ(t), κ(t), cv(t)≤M(c0),

|(µ(t)₋µ0, κ(t)−κ0)|L∞ ≤M(c

0)c−21,

|(µ(t), κ(t), cv(t))|D1

∩D1,q1 ≤M(c₀),

|(µ(t), κ(t), cv(t))|D1, q ≤M(c₀)c₂

(63)

for 0 _≤ t _≤ min(T∗, T2), where q1 = min(q,4). Moreover, in view of (58)

and (62), we have

∫ t

0 |

(µt(s), κt(s),(cv)t(s))|2_L3ds

≤M(c0)

∫ t

0

(

|ρt|2L3 +|ρ_t|_L23|η|2_L∞+|ρηt|2

L3

) ds

≤M(c0)

( c4₂t+

∫ t

0

(

|ρ₋ρ0|2_L6|∇ηt|_L22 +|ρ0|_L∞|ρ₀η_t|

L2|∇η_t| L2) ds

)

≤M(c0)

( c−₂2+

∫ t

0

(

c−₂3_|∇ηt|2L2+|ρ₀η_t|_L2|∇η_t|_L2

) ds

for 0_≤t_≤min(T∗, T2). Hence it follows from (55) that

∫ t

0 |

(µt(s), κt(s),(cv)t(s))|2L3ds≤M(c₀)c−₂2 (64)

for 0_≤t_≤min(T∗, T2).

Remark 3.2. If µ, κ andcv areC1-functions of ρ and η, then ∫ t

0 |

(µt(s), κt(s),(cv)t(s))|2_L3ds≤M(c0)

∫ t

0

(

|ρt(s)|2_L3 +|ηt(s)|2_L3

) ds

≤M(c0)

(

c−₂2+ ˜C ∫ t

0 |

ηt(s)|2_D1 0ds

)

≤M(c0) ˜Cc22

for0_≤t_≤min(T∗, T2), whereC˜ is a constant depending also on the size of

Ω. However, we have failed to derive an analogue of(64), which is necessary in particular to estimate the third term of the right hand side of(66).

3.3 Estimates for the velocity u

In view of (47), we can rewrite (48) as

ρut+ρv· ∇u−div(2µ0du) +∇p= div (2(µ−µ0)dv) +ρf. (65)

Differentiating this with respect tot, we have

ρutt+ρv· ∇ut−div(2µ0dut) +∇pt

=₋ρtut−(ρv)t· ∇u+ div (2(µ−µ0)dv)t+ (ρf)t.

We multiplying this by ut and integrating over Ω. Then by virtue of (47), we derive

1 2

d dt

∫

ρ_|ut|2dx+ ∫

2µ0|dut|2dx

=₋

∫

ρt|ut|2dx− ∫

((ρv)t· ∇u)·utdx (66)

−

∫

Using H¨older and Sobolev inequalities together with (58), (63) and (64), we can estimate each term of the right hand side of (66) as follows.

−

∫

ρt|ut|2dx= ∫

div (ρv)_|ut|2dx=−2 ∫

(ρv_{· ∇}ut)·utdx

≤ |ρ_|

1 2

L∞|v|L∞|∇u_t|

L2|√ρu_t|_L2

≤Cc0c22|√ρut|2_L2 + 1 8µ|ut|

2

D1 0,

−

∫

((ρv)t· ∇u)·utdx=− ∫

(ρtv· ∇u+ρvt· ∇u)·utdx

≤C_|ρt|L3|v|_L∞|u|

D1 0|ut|D

1

0+C|ρ| 1 2

L∞|vt|_D1

0|∇u|L

3|√ρu_t|_L2

≤Cc6_2|u_|2_D1

0 +Cc0ε|vt|

2

D1 0|

√_ρu

t|2L2+ε−1|∇u|2_L3 + 1 8µ|ut|

2

D1 0

≤Cc0

(

c6₂+ε_|vt|2_D1 0 +ε

−2) (

|√ρut|2_L2 +|u|2 D1

0

)

+_|∇u_|2_H1 + 1 8µ|ut|

2

D1 0,

−

∫

2 ((µ₋µ0)dv)_t:∇utdx

≤C(_|µt|L3|∇v|

H1 +|µ−µ₀|_L∞|∇v

t|L2)|∇u_t| L2

≤C(c2_2|µt|2_L3+|µ−µ₀|2_L∞|vt|2

D1 0

)

+1 8µ|ut|

2

D1 0 and

⟨(ρf)t, ut⟩ = ⟨ρtf, ut⟩+⟨ft, ρut⟩

≤C_|ρt|L3|f| L2|u_t|

D1

0 +|ft|H

−1|ρu_t| H1

0

≤ C(c4_2|f_|2_L2+c2₀|ft|2_H−1

)

+_|√ρut|2_L2 + 1 8µ|ut|

2

D1 0.

Hereε >0 is a small constant andµ= infΩµ0. Substituting all the estimates

into (66), takingε=c−₂1<1 and observing that µ_≥M(c0)−1, |µ−µ0|L∞ ≤M(c

0)c−21 and |dut|L2 =|∇u_t|

L2 =|u_t| D1

0, we deduce that

d dt|

√_ρu

t|2_L2+M(c₀)−1|u_t|2 D1

0

≤M(c0)

(

c6₂+c−₂1_|vt|2_D1 0

) (

1 +_|√ρut|2_L2 +|u|2 D1

0

)

+C(c2_2|µt|_L23+c2₀|f_t|2

H−1 +|∇u|2 H1

Hence integrating this over (τ, t) and using (64), we have

|√ρut(t)|L22 +M(c₀)−1

∫ t

τ | ut|2_D1

0ds

≤M(c0) +|√ρut(τ)|2_L2+C

∫ t

τ |∇

u_|2_H1ds (67)

+M(c0)

∫ t

τ (

c6₂+c−₂1_|vt|2_D1 0

) (

1 +_|√ρut|2_L2 +|u|2 D1

0

) ds

for 0< τ _≤t_≤min(T∗, T2). On the other hand, since ut is divergence-free in (0, T)_×Ω, it follows from (65) that

∫

ρ_|ut|2dx

=

∫

(₋ρv_{· ∇}u+ div (2µ0du)− ∇p+ div (2(µ−µ0)dv) +ρf)·utdx

=

∫

(₋ρv_{· ∇}u+ div (2µ0du)− ∇p0+ div (2(µ−µ0)dv) +ρf)·utdx

and

∫

ρ_|ut|2dx≤C∫ (ρ|v|2|∇u|2+ρ|f|2)dx+ ∫

ρ−1_|div (2µ0du)− ∇p0|2dx

+C ∫

ρ−1_|div (2(µ₋µ0)dv)|2dx.

Noting thatρ_≥δ >0,ρ, µ_∈C([0, T∗];W1, q), u∈C([0, T∗];H2) and

div (2µ0du(t))→div (2µ0du0) =∇p0+ρ

1 2

0g1 in L2 as t→0,

we thus have

lim sup τ→0 |

√_ρu

t(τ)|2_L2

≤C_|ρ0|L∞ (

|∇v(0)_|2_H1|u₀|2 D1

0 +|f(0)|L 2

)

+_|g1|2_L2 ≤Cc5₀. Lettingτ _→0 in (67) and using the fact that

|∇u(t)_|2_L2 ≤C|∇u₀|2+C

∫ t

0 |∇

we obtain

(

1 +_|√ρut(t)|_L22+|u(t)|2 D1 0 ) + ∫ t 0 |

ut|2_D1 0ds

≤M(c0)

(

1 +

∫ t

0 |∇

u_|2_H1ds

+

∫ t

0

(

c6₂+c−₂1_|vt|2_D1 0

) (

1 +_|√ρut|2_L2 +|u|2 D1

0

) ds

)

for 0_≤t_≤min(T∗, T2). Hence, in view of Gronwall’s inequality, we deduce

that

(

|√ρut(t)|L22+|u(t)|2 D1 0 ) + ∫ t 0 |

ut|2_D1

0ds≤M(c0)

(

1 +

∫ t

0 |∇

u_|2_H1ds

)

(68)

for 0 _≤ t _≤ min(T∗, T2). To estimate |∇u|H1, we observe that for each

t_∈[0, T∗], u=u(t)∈D10∩D2 is a solution of the Stokes equations

−div (2µ0du) +∇p=F and divu= 0 in Ω,

where

F =ρ(f₋ut−v· ∇u) + div (2(µ−µ0)dv)

satisfies

|F_|D−1_∩L2 ≤C|ρ| 1 2 L32_∩L∞|

√_ρu

t|L2+C|ρ|_L3_∩L∞ (

|v_|D1 0|∇u|L

3 +|f|_L2

)

+C_|µ₋µ0|L∞|∇v|

H1+C|∇(µ−µ₀)·dv|_L2 and

|F_|Lq ≤C|ρ| 1 2 L∞|

√_ρu

t|L2 +C|ρ|_L∞|u

t|D1

0 +C|ρ|L

∞|∇v|

H1|∇u|_H1 +C_|ρ_|L∞|f|

H1 +C|µ−µ₀|_L∞|∇2v|

Lq+C|∇(µ−µ₀)|_Lq|∇v|_L∞

Hence by virtue of Proposition 2.9 and (63), we have

|(_∇u, p)_|H1 ≤M(c₀)|F| D−1_∩L2

≤M(c0)

(

1 +_|√ρut|L2 +|v| D1

0|∇u|L 3

)

+C (

|µ₋µ0|L∞|∇v|

H1 +C|∇(µ−µ₀)|_Lq1|∇v| L

2q1 q1−2

)

≤M(c0)c1

(

1 +_|√ρut|L2 +|∇u| 1 2 L2|∇u|

1 2

H1 +|∇v|

1−3 q1 L2 |∇v|

3 q1 H1

and thus

|(_∇u, p)_|H1 ≤M(c₀)c2

1

( c

3 q1

2 +|√ρut|L2 +|u| D1

0

) .

Therefore, substituting this into (68) and applying Gronwall’s inequality again, we conclude that

|√ρut(t)|_L22 +|u(t)|2 D1

0+

∫ t

0 |

ut(s)|2_D1

0ds≤M(c0),

|(_∇u(t), p(t))_|H1 ≤M(c₀)c2

1c

3 q1

2

(69)

for 0_≤t_≤min(T∗, T2). Moreover, it follows also from Proposition 2.9, (63)

and (69) that

|(_∇u, p)_|W1, q

≤M(c0)|F|D−1_∩L2_∩Lq

≤M(c0)

( c2₁c

3 q1

2 +|√ρut|L2+|u_t| D1

0 +|∇v|H 1|∇u|

H1 +|f| H1

)

+M(c0) (|µ−µ0|L∞|v|

D2, q +|∇(µ−µ₀)|_Lq|∇v|_L∞)

≤M(c0)

(

c4₂+_|f_|H1 +|u_t| D1

0 +c

−1

2 |v|D2, q +c2₂|∇v|1−γ H1 |∇v|

γ W1, q

)

for someγ =γ(q)_∈(0,1). Hence taking

T3 =cα₂2 with α2 = min

(

α1,6−2γ

1₋γ )

,

we deduce that

∫ t

0

(_|u(s)_|2_D2, q +|p(s)|2_D1,q)ds≤M(c0)(1 +c6₂−2γt1−γ)≤M(c0) (70)

for 0_≤t_≤min(T∗, T3).

3.4 Estimates for the temperature θ

Differentiating (49) in time, we obtain

cvρθtt−div(κ0∇θt)

Then multiplying this byθt and integrating over Ω, we have 1 2 d dt ∫

cvρ|θt|2dx+ ∫

κ0|∇θt|2dx

=

∫ (

−1₂(cv)tρ|θt|2+(2µ|dv|2)_tθt )

dx₋ ∫ ₁

2cvρt|θt|

2_{dx (71)}

−

∫

(cvρv· ∇θ)tθtdx− ∫

((κ₋κ0)∇η)_t· ∇θtdx+⟨(ρh)t, θt⟩.

Using (58), (63), (64) and (69), we can estimate each term of the right hand side in (71) as follows.

∫ (

−1

2(cv)tρ|θt|

2₊(_2µ|dv|2)

tθt )

dx

≤M(c0)(|(cv)t|L3|√ρθ_t|_L2+|µ_t|_L3|∇v|2

L4 +|∇v|_L3|∇v_t|_L2

)

|θt|D1 0

≤M(c0)(|(cv)t|2L3|√ρθ_t|2_L2 +c₁c3₂|µ_t|2_L3 +c₁c₂|∇v_t|2_L2

)

+1 8κ|θt|

2

D1 0,

−

∫ ₁

2cvρt|θt|

2_dx= 1

2

∫

div (ρv)cv|θt|2dx

≤

∫

ρ_|v_||∇cv||θt|2dx+ ∫

ρ_|v_|cv|θt||∇θt|dx

≤C_|ρ_|

1 2 L∞|v|D1

0|∇cv|L

q|√ρθ_t| L

3q

2q−3|∇θt|L

2 +C|ρ| 1 2

L∞|v|L∞|

√_ρθ

t|L2|∇θ_t| L2

≤M(c0)c1c22|√ρθt| 3 2−

3 q L2 |∇θt|

1 2+

3 q

L2 +M(c0)c2|

√_ρθ

t|L2|∇θ_t|_L2

≤M(c0)(c1c22)

4q 3q−6|√ρθ

t|2_L2 + 1 8κ|θt|

2

D1 0,

−

∫

(cvρv· ∇θ)tθtdx

≤M(c0)(c22|(cv)t|2L3 +c6₂+ε|∇v_t|2_L2

) (

|∇θ_|2_L2+|√ρθ_t|2_L2

)

+ε−1_|∇θ_|2_L3 + 1 8κ|θt|

2

D1 0,

∫

((κ₋κ0)∇η)t· ∇θtdx≤C(c22|κt|2_L3 +c−2

2 |ηt|2D1 0) +

1 8κ|θt|

2

D1 0 and

⟨(ρh)t, θt⟩ ≤C(c24+c20|ht|2_H−1

)

+_|√ρθt|2_L2 + 1 8κ|θt|

2

Hereε >0 is a small number andκ= infΩκ0. Substituting all the estimates

into (71) and takingε=c−₁1, we have d

dt ∫

cvρ|θt|2dx+κ ∫

|∇θt|2dx

≤M(c0)

((

c2_2|(cv)t|2_L3+c 4q q−2

2 +c−12|∇vt|2_L2 +c6₂

) (

|√ρθt|2_L2+|θ|2 D1

0

)

+_|∇θ_|2_H1+c1c3₂|µt|2_L3 +c1c2|∇vt|2_L2 +c2₂|κt|2_L3+c4₂+|ht|2_H−1+c−₂2|ηt|2 D1

0

)

Hence integrating over (τ, t), we derive an analogue of (67):

|√ρθt(t)|2L2 +

∫ t

τ | θt|2_D1

0ds

≤M(c0)

( c3₁c2+

∫ t

τ |∇ θ_|2_H1ds

)

+_|√ρθt(τ)|2_L2

+M(c0)

∫ t

τ (

c2_2|(cv)t|2L3 +c 4q q−2

2 +c−12|∇vt|2L2

) (

|√ρθt|2L2 +|θ|2 D1

0

) ds

for 0< τ _≤t_≤min(T∗, T3). Then observing that

−div(κ0∇θ(t))→µ0|dv(0)|2+ρ

1 2

0g2 in L2 as t→0

and

|∇θ(t)_|L2 ≤C|∇θ₀| L2 +C

∫ t

0 |∇

θt|2L2ds for 0≤t≤T_∗, we easily deduce that

|√ρθt(t)|L22 +|θ(t)|2 D1

0 +

∫ t

0 |

θt|2_D1

0 ds≤M(c0)

( c3₁c2+

∫ t

τ |∇ θ_|2_H1ds

)

+M(c0)

∫ t

τ (

c2_2|(cv)t|2_L3 +c 4q q−2

2 +c−12|∇vt|2_L2

) (

|√ρθt|2_L2 +|θ|2 D1

0

) ds

for 0_≤t_≤min(T∗, T3). Hence in view of Gronwall’s inequality, we have

|√ρθt(t)|_L22 +|θ(t)|2 H1

0 +

∫ t

0 |

θt|2_H1

0 ds≤M(c0)

( c3₁c2+

∫ t

τ |∇ θ_|2_H1ds

)

(72)

for 0_≤t_≤min(T∗, T3). On the other hand, since each θ=θ(t)∈D10∩D2

is a solution of the elliptic equation

where G= ₋cvρ(θt+v· ∇θ) + div ((κ−κ0)∇η) + 2µ|dv|2+ρh, it follows

from Proposition 2.11 that

|∇θ_|W1, r ≤M(c₀)|G|_D−1_∩L2_∩Lr for r= 2, q. Combining this result and (72), we can easily show that

|√ρθt(t)|_L22+|∇θ(t)|2 H1+

∫ t

0

(

|θt(s)|2_D1

0 +|θ(s)|

2

D2, q

)

ds_≤M(c0)c41c

3 q1

2 (73)

for 0_≤t_≤min(T∗, T3). As a consequence, we have

|θ(t)₋θ0|D1 0 ≤

( t

∫ t

0 |

θt|2_D1 0ds

)1 2

≤M(c0)(c52t)

1

2 ≤M(c

0)c−28 (74)

and thus

|θ(t)₋θ0|D1,4 ≤C|θ(t)−θ₀| 1 4 D1

0|

θ(t)₋θ0|

3 4

D2 ≤M(c0)c

−2

2 (75)

for 0_≤t_≤min(T∗, T3). Moreover, since

∫ t

0 |

ρ0θt|2_L2ds≤2

∫

(_|(ρ₋ρ0)θt|2_L2 +|ρθ_t|2 L2)ds

≤C sup

0≤s≤t|

ρ(s)₋ρ0|L3

∫ t

0 |

θt|2_D1

0ds+Cc 1 2

0t sup 0≤s≤t|

√_ρθ

t(s)|2L2,

it follows from (62) and (73) that

∫ t

0 |

ρ0θt(s)|2_L2ds≤M(c₀)c−₂7 for 0≤t≤min(T_∗, T₃). (76)

3.5 Conclusion

From (58), (69), (73), (74), (75) and (76), it follows that if 0 _≤ t _≤ min(T∗, T3), then there exists a constant M(c0)≥1 depending increasingly

onc0 such that

|u(t)_|D1 0 +

∫ t

0

(

|ut|2_D1 0 +|u|

2

D2, q +|∇p|2_Lq

)

ds_≤M(c0),

|u(t)_|D2 +|θ(t)| D1

0∩D 2 +

∫ t

0

(

|θt|2_D1 0 +|θ|

2

D2, q

)

ds_≤M(c0)c41c

3 q1

and

|θ(t)₋θ0|D1 0∩D1,

4 ≤M(c₀)c−2

2 ,

∫ t

0 |

ρ0θt(s)|2_L2ds≤M(c0)c−₂7,

|ρ(t)_|

L32_∩H1_∩W1, q +|ρt(t)|_L32_∩Lq +|p(t)|H 1

+_|(√ρut,√ρθt)(t)|L2 +

∫ t

0 |

p(s)_|2_W1,qds≤M(c0)c41c

3 q1

2 .

Therefore, choosingc1, c2 and T∗ so that

c1=M(c0), c2=M(c0)c41c

3 q1

2 and 0< T∗ ≤T∗∗≡min(T∗, T4), (77)

we conclude that

sup

0≤t≤T∗

|u(t)_|D1 0 +

∫ T∗

0

(_|ut(t)|D1

0 +|u(t)|

2

D2, q)dt≤c1,

sup

0≤t≤T∗

(_|u(t)_|D2+|θ(t)| D1

0∩D 2) +

∫ T∗

0

(_|θt(t)|2_D1

0 +|θ(t)|

2

D2, q)dt≤c2,

sup

0≤t≤T∗

|θ(t)₋θ(0)_|D1 0∩D

1,4 ≤c−1

2 ,

∫ T∗

0 |

ρ0θt(t)|L22dt≤c−₂6,

sup

0≤t≤T∗

(_|ρ(t)_|

L32_∩H1_∩W1, q +|ρt(t)|_L32_∩Lq+|p(t)|H 1)

+ sup

0≤t≤T∗

|(√ρut,√ρθt)(t)|L2 +

∫ T∗

0 |

p(t)_|2_W1,qdt≤c2.

(78)

Remark 3.3. It should be pointed out that the constant C doesn’t depend on the lower bound δ of the initial density ρ0 and further, the radius R in

case when Ω is the intersection of an unbounded domain and a large open ball with radiusR.

4

Proof of Theorem 1.1

In this section, we provide a complete proof of Theorem 1.1. Let (ρ0, u0, θ0)

be a given initial data satisfying the hypotheses of Theorem 1.1. To prove the theorem, we construct a sequence _{(ρk_{, u}k_{, p}k_{, θ}k_)}

k≥1 of approximate

4.1 The construction of _{(ρk_{, u}k_{, p}k_{, θ}k_)} k≥1

First, let u0 be the solution in C([0,_∞);D1_{0 ∩}D2) _∩L2(0,_∞;D3) of the Stokes equations

u0_{t −}∆u0+_∇p0 = 0 and divu0 = 0 in (0,_∞)_×Ω with the initial data u0_{(0) = u}

0. We also denote by θ0 the

solution inC([0,_∞);D1_0∩D2)_∩L2(0,_∞;D3) of the linear parabolic equation θ_t0₋∆θ0 = 0 in (0,_∞)_×Ω with the initial data θ0(0) =θ0. It is easy to

show that

sup

0≤t≤1|

u0(t)_|D1 0∩D

2+

∫ 1 0

(

|u0_t(t)_|2_D1 0 +|u

0_(t)|2

D3

)

dt_≤C(1+_|u0|2_D1 0∩D

2) (79)

and

sup

0≤t≤1|

θ0(t)_|D1 0∩D

2+

∫ 1

0

(

|θ_t0(t)_|2_D1 0 +|θ

0_(t)

|2D3

)

dt_≤C(1 +_|θ0|2_D1 0∩D

2). (80)

Next, let us define c0 by

c0 = 2 +|ρ0|

L32_∩H1_∩W1, q +|(u0, θ0)|D10 +|(g1, g2)|L 2,

and we choose the positive constantsc1,c2 andT∗∗as in (77), which depend only onc0 and the parameters ofC. Then since u0 ∈D10∩D2 is a solution

of the stationary Stokes equations

−div (2µ0du0) +∇p0 =ρ

1 2

0g1 and divu0 = 0

in Ω, it follows immediately from Proposition 2.9 that

|u0|D1 0∩D

2 ≤M(c₀) (81)

for some increasing function M =M(_·). Similarly, using Proposition 2.11, we easily deduce that

|θ0|D1 0∩D

2 ≤M(c₀). (82)

By virtue of (77), (79), (80), (81) and (82), we may assume without loss of generality that

sup

0≤t≤1|

(u0, θ0)(t)_|D1 0∩D

2+

∫ 1 0

(_|(u0_t, θ0_t)(t)_|2_D1 0+|(u

0_{, θ}0_)(t)|2

Moreover, since θ0 _∈ C([0,_∞);D₀1_∩D2), θ_t0 _∈ L2(0,_∞;D₀1) and ρ0 ∈ L3,

there is a small timeT∗ ∈(0, T∗∗) such that

sup

0≤t≤T∗

|θ0(t)₋θ0|D1 0∩D

1,4 ≤c−1

2 and

∫ T∗

0 |

ρ0θt0(t)|L22dt≤c−₂6. (84)

Our construction of the sequence _{(ρk, uk, pk, θk)_}k≥1 is based on the

following key lemma to the proof of Theorem 1.1.

Lemma 4.1. Let (v, η) be a pair of vector and scalar fields satisfying the regularity(53) with T replaced byT∗. Assume further that (v, η) satisfies

(v(0), η(0)) = (u0, θ0),

sup

0≤t≤T∗

|v(t)_|D1 0 +

∫ T∗

0

(

|vt(t)|D1

0 +|v(t)|

2

D2, q

)

dt_≤c1,

sup

0≤t≤T∗ (

|v(t)_|D2 +|η(t)| D1

0∩D 2

)

+

∫ T∗

0

(

|ηt(t)|2_D1

0 +|η(t)|

2

D2, q

)

dt_≤c2,

sup

0≤t≤T∗

|η(t)₋η(0)_|D1 0∩D

1,4 ≤c−1

2 and

∫ T∗

0 |

ρ0ηt(t)|L22dt≤c−₂6. (85)

Then there exists a unique strong solution(ρ, u, p, θ)to the linearized problem

(47)–(51) in [0, T∗]satisfying the estimate (78) as well as the regularity

ρ_∈C([0, T∗];L

3

2 ∩H1∩W1, q), ρ

t∈C([0, T∗];L

3

2 ∩Lq), (u, θ)_∈C([0, T∗];D₀1_∩D2)_∩L2(0, T∗;D2, q),

p_∈C([0, T∗];H1)_∩L2(0, T∗, W1, q),

(ut, θt)∈L2(0, T∗;D01) and (√ρut,√ρθt)∈L∞(0, T∗;L2).

(86)

Proof. Let φ _∈ C_c∞(B1) be a smooth cut-off function such that φ = 1 in

B_1/2, and we define

φR(x) =φ(x/R), vR(t, x) =φR(x)v(t, x) and ηR(t, x) =φR(x)η(t, x)

for (t, x) _∈ [0, T∗]×Ω and 2R0 < R < ∞, where R0 ≥ 3 is the constant

to (v, η) and (vt, ηt) in C([0, T];D01∩D2)∩L2(0, T;D2, q) and L2(0, T;D01),

respectively.

For each R > 2R0, let (uR0, pR0) ∈

(

D1_0∩D2)(ΩR) ×H1(ΩR) be the solution to the Stokes equations

−div(2µR₀duR₀) +_∇pR₀ = (ρR₀)12gR

1 and divuR0 = 0 in ΩR, (87) where

ρR₀ =ρ0+R−3, µR0 =µ(ρR0, ρR0ηR(0)) and (ρR0)

1 2gR

1 =ρ

1 2

0g1.

Then extending (uR₀, pR₀) to Ω by zero outside ΩR, we will show that uR_{0 →}u0 in D10,σ(Ω) as R→ ∞. (88)

To show this, we first observe from (8) and (87) that

∫

Ω

2µR_0|duR_0|2dx=

∫

Ω

ρ

1 2

0g1·uR0 dx=

∫

Ω

2µ0du0 : duR0 dx. (89)

But since_|ρR

0 −ρ0|

L32_∩H1

∩W1,q ≤CR

−1 _and

|µR_{0 −}µ0|L∞ ≤M(c

2)(R−3(1 +|η(0)|L∞) +|ηR(0)−η(0)|

L∞ )

, (90)

it follows immediately from (89) that _{uR

0} is bounded in D01,σ(Ω). Hence there exists a sequence_{Rj},Rj → ∞, such that{uR₀j}converges weakly in D₀1_,σ(Ω) to a limitu∞₀ . Moreover in view of (90), we deduce from (88) that u∞

0 ∈D01,σ(Ω) is a weak solution of the Stokes equations

−div(2µ0du∞0 ) +∇p∞0 =ρ

1 2

0g1 and divu∞0 = 0 in Ω

with some pressure p∞

0 . Since u0 ∈ D10,σ(Ω) is also a weak solution of the same equations, it follows from Lemma 2.7 that u∞₀ = u0 in Ω. Then by

virtue of (88) and (90), we easily show that_{uRj

0 }converges strongly to u0

in D1

0,σ(Ω). Since the above argument also shows that every subsequence of _{uR_0} has a subsequence converging in D₀1_,σ(Ω) to the same limit u0,