Global existence of classical solutions for isentropic compressible Navier–Stokes equations with small initial density

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ScienceDirect J. Differential Equations 259 (2015) 6830–6850 www.elsevier.com/locate/jde

Global existence of classical solutions for isentropic compressible Navier–Stokes equations with small initial density ✩ Jinju Qian ∗ , Junning Zhao School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, PR China Received 6 March 2015; revised 2 August 2015 Available online 1 September 2015

Abstract In this paper we establish the global existence of classical solutions to the Cauchy problem for the 3-D isentropic compressible Navier–Stokes equations with smooth initial data that are small density but possibly large energy, which could be either vacuum or non-vacuum. © 2015 Elsevier Inc. All rights reserved. MSC: 76N10; 35M10 Keywords: Compressible N-S equations; Small initial density; Global classical solutions

1. Introduction The motion of a viscous isentropic compressible fluid occupying a domain  ⊂ R3 is governed by the compressible Navier–Stokes equations 

ρt + div(ρu) = 0, (ρu)t + div(ρu ⊗ u) − μu − (μ + λ)∇(divu) + ∇P (ρ) = 0,

✩

This work is partially supported by the Natural Science Foundation of China (Grant No. 11371297).

* Corresponding author.

E-mail addresses: [email protected] (J. Qian), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.jde.2015.08.007 0022-0396/© 2015 Elsevier Inc. All rights reserved.

(1.1)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

6831

where ρ ≥ 0, u = (u1 , u2 , u3 ) and P = aρ γ (a > 0, γ > 1) are the fluid density, velocity and pressure, respectively. The constant viscosity coefficients μ and λ satisfy the physical restrictions 3 μ > 0, μ + λ ≥ 0. 2

(1.2)

Let  = R3 . We look for the solutions, (ρ(x, t), u(x, t)), to the Cauchy problem for (1.1) with the far field behavior: u(x, t) → 0, ρ(x, t) → 0, as |x| → ∞,

(1.3)

(ρ, u)|t=0 = (ρ0 , u0 ), x ∈ R3 .

(1.4)

and initial data,

Much efforts have been devoted to study the global existence and behavior of solutions to (1.1). The one dimensional problem has been studied extensively by many people (see [1–4] and the references therein). For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in [3,5–9] in the absence of vacuum and in [1,10–12] for the case that the initial density need not be positive and may vanish in an open set. The global classical solutions were first obtained by Matsumura and Nishida [13] for initial data close to a nonvacuum equilibrium in some Sobolev space H s . Later, Hoff [10,11] studied the problem for discontinuous initial data. For the existence of solutions for arbitrary data (the far field is vacuum, that is ρ˜ = 0), the major breakthough is due to Lions [14] (see also Feireisl [15]), where he proved the global existence of weak solutions, defined as solutions with finiteenergy, when the adiabatic exponent γ is suitably large (i.e. γ > 3/2). The main restriction on initial data is that the initial energy is finite, so that, the density vanishes at far fields, or even has compact support. However, the uniqueness and regularity of such weak solutions are still open. Recently, Huang, Li and Xin [16] establish the global well-posedness of classical solutions with large oscillations and vacuum to the Cauchy problem (1.1)–(1.4) under the assumption that the initial energy is suitably small. The result obtained in [16] is an important advance in the study of compressible Navier–Stokes equations. Lately using the idea in [16], Deng, Zhang and Zhao [17] establish the global existence and uniqueness of classical solutions to the Cauchy problem of (1.1)–(1.4) under the assumption that the viscosity coefficient μ is large enough. In this paper we are interested in studying the global existence of classical solutions to Cauchy problem (1.1)–(1.4) with large initial energy, which could be either vacuum or non-vacuum. Basing on the ideals in [16], we establish the global existence of classical solutions with general initial energy under the assumption that the upper bound of the initial density is suitably small. Before stating the main results, we explain the notations and conventions used throughout this paper. For simplicity we set 

 f dx =

f dx. R3

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

For k ∈ Z and 1 < r < ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows: 

Lr = Lr (R3 ), D k,r = {u ∈ L1loc (R3 )| ∇ k uLr < ∞}, uD k,r := ∇ k uLr , W k,r = Lr ∩ D k,r , H k = W k,2 , D k = D k,2 , D 1 = {u ∈ L6 | ∇uL2 < ∞}.

(1.5)

For the initial data, we define  C0 =

1 ( ρ0 |u0 |2 + G(ρ0 ))dx, 2

(1.6)

where G denotes the potential energy density given by ρ G(ρ) := ρ

P (s) ds. s2

0

It is easy to see G(ρ) =

1 P. γ −1

(1.7)

Then the main results in this paper can be stated as follows: Theorem 1.1. Assume that (1.2) holds. For given positive numbers M0 and M1 , suppose that the initial data (ρ0 , u0 ) satisfy ⎧ γ 2 1 ⎪ ⎨ρ0 |u0 | + ρ0 ∈ L 0 ≤ inf ρ0 ≤ sup ρ0 ≤ M0 , ∇u0 2L2 ≤ M1 , ⎪ ⎩ u0 ∈ D 1 ∩ D 3 , (ρ0 , P (ρ0 )) ∈ H 3 ,

(1.8)

and the compatibility condition −μu0 − (μ + λ)∇divu0 + ∇P (ρ0 ) = ρ0 g,

(1.9)

for some g ∈ D 1 with ρ0 |g|2 ∈ L1 . Then there exists a positive constant  depending on μ, λ, C0 , and M1 such that if M0 ≤ ,

(1.10)

the Cauchy problem (1.1)–(1.4) has a unique global classical solution (ρ, u) satisfying for any 0 < τ < T < ∞, 0 ≤ ρ(x, t) ≤ 2M0 , x ∈ R3 , t ≥ 0, and

(1.11)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(ρ, P (ρ)) ∈ C([0, T ] : H 3 ), u ∈ C([0, T ]; D 1 ∩ D 3 ) ∩ L2 (0, T ; D 4 ) ∩ L∞ (τ, T ; D 4 ), ut ∈ L∞ (0, T ; D 1 ) ∩ L2 (0, T ; D 2 ) ∩ L∞ (τ, T ; D 2 ) ∩ H 1 (τ, T ; D 1 ), √ ρut ∈ L∞ (0, T ; L2 ),

6833

(1.12)

and the following large-time behavior:  lim

t→∞

(|ρ|q + ρ 1/2 |u|4 + |∇u|2 )(x, t)dx = 0,

(1.13)

for all q ∈ (γ , ∞). Remark. It is easy to show that the solution obtained in Theorem 1.1 is a classical solution for positive time. Moreover, in Theorem 1.1 although the upper bound of the initial density is small, yet the initial energy C0 can be arbitrary large and the solution may contain vacuum states. Theorem 1.1 will be proved by combining the local existence result (see Lemma 2.4) with the global a priori estimates derived in Section 3. The key issue in the proof of Theorem 1.1 is to derive the time-independent upper bound of the density. This will be done by modifying the arguments in [16]. However, unlike that in [16] where the smallness of initial energy plays an important role, a key observation lying in our proofs is that the Lp -norm of the gradient of the velocity with 2 < p ≤ 6 is well controlled by the upper bound of the density ρ, and is as small as desired provided the upper bound of the density ρ is small enough. By making a full use of the observation mentioned above, we can then carry out the similar procedure as the one in [16] to prove the time-independent upper bound of the density. The rest of the paper is organized as follows: In Section 2, we state some elementary facts and inequalities which will be needed in later analysis. Section 3 is devoted to derive the necessary a priori estimates on classical solutions which are needed to extend the local existence of solution to all the time. 2. Preliminaries In this section, we will recall some known facts and elementary inequalities which will be used frequently later. First, the following well-known Gagliardo–Nirenberg inequality will be used. Lemma 2.1. For p ∈ [2, 6], q ∈ (1, ∞) and r ∈ (3, ∞), there exists some generic constant C > 0 which may depend on q, r such that for f ∈ H 1 (R3 ) and g ∈ Lq (R3 ) ∩ D 1,r (R3 ), we have p

(6−p)/2p

f Lp ≤ Cf L2

(3p−6)/2p

∇f L2

,

q(r−3)/(3r+q(r−3))

∇gLr

gL∞ ≤ CgLq

3r/(3r+q(r−3))

(2.1) (2.2)

Next, the following Zlotnik inequality will be used to get the uniform (in time) upper bound of the density ρ.

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

Lemma 2.2. Assume that y = y(t) ∈ W 1,1 solves the following ODE problem y  (t) = g(y) + b (t) on [0, T ], y(0) = y 0 , with g ∈ C(R) and y, b ∈ W 1,1 (0, T ). If g(∞) = −∞ and b(t2 ) − b(t1 ) ≤ N0 + N1 (t2 − t1 ),

(2.3)

for all 0 ≤ t1 < t2 ≤ T with some N0 ≥ 0 and N1 ≥ 0, then y(t) ≤ max{y 0 , ζ¯ } + N0 < ∞ on [0, T ], where ζ¯ is a constant such that g(ζ ) ≤ N1 , for ζ ≥ ζ¯ .

(2.4)

The following lemma is the local existence and uniqueness of classical solutions when the initial density may not be positive and may vanish in an open set. Lemma 2.3. (See [6].) Assume that the initial data (ρ0 , u0 ) with ρ0 ≥ 0 satisfy (1.8)–(1.9). Then there exist a small time T∗ > 0 and a unique classical solution (ρ, u) to the Cauchy problem (1.1)–(1.4) such that ⎧ (ρ, P (ρ)) ∈ C([0, T∗ ]; H 3 ), ⎪ ⎪ ⎪ ⎪ ⎪ u ∈ C([0, T∗ ]; D 1 ∩ D 3 ) ∩ L2 (0, T∗ ; D 4 ), ⎪ ⎪ ⎪ ⎨u ∈ L∞ (0, T ; D 1 ) ∩ L2 (0, T ; D 2 ), √ρu ∈ L∞ (0, T ; L2 ), t ∗ ∗ t ∗ √ 2 (0, T ; L2 ), t 1/2 u ∈ L∞ (0, T ; D 4 ), ⎪ ρu ∈ L tt ∗ ∗ ⎪ ⎪ ⎪ 1/2 √ρu ∈ L∞ (0, T ; L2 ), tu ∈ L∞ (0, T ; D 3 ), ⎪ ⎪ t tt ∗ t ∗ ⎪ ⎪ ⎩ ∞ 1 2 2 tutt ∈ L (0, T∗ ; D ) ∩ L (0, T∗ ; D ).

(2.5)

We now state some elementary estimates which follow from Gagliardo–Nirenberg inequalities and the standard Lp -estimate for the following elliptic system derived from the momentum equations in (1.1): F = div(ρ u), ˙ μw = ∇ × (ρ u), ˙

(2.6)

f˙ := ft + u · ∇f, F := (2μ + λ)divu − P (ρ), w := ∇ × u

(2.7)

where

are the material derivative of f , the effective viscous flux and the vorticity respectively. Lemma 2.4. Let (ρ, u) be a smooth solutions of (1.1)–(1.4). Then there exists a generic positive constant C such that for any p ∈ [2, 6]

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

∇F Lp ≤ Cρ u ˙ Lp , ∇wLp ≤ Cρ u ˙ Lp , F Lp ≤ Cρ u ˙

3p−6 2p L2

(∇uL2 + P (ρ)L2 )

3p−6

6−p 2p

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(2.8) (2.9)

,

6−p

wLp ≤ Cρ u ˙ L2p ∇uL2p2 , 2

(2.10)

∇uLp ≤ C(F Lp + wLp + P (ρ)Lp ),

(2.11)

∇uLp ≤ C∇u

6−p 2p L2

(ρ u ˙ L2 + P (ρ)L6 )

3p−6 2p

(2.12)

.

Proof. The standard Lp -estimate for the elliptic system (2.6) yields directly (2.8), which together with (2.1) and (2.7) gives (2.9) and (2.10). Note that −u = −∇divu + ∇ × w, which implies that ∇u = −∇(−)−1 ∇divu + ∇(−)−1 ∇ × w. Thus the standard Lp -estimates shows that ∇uLp ≤ C(divuLp + wLp ) ≤ C(F Lp + wLp + P (ρ)Lp ).

(2.13)

That is, (2.11) holds. By Hölder inequality, (2.1) and the second inequality of (2.8), one has (6−p)/2p

∇uLp ≤ ∇uL2

(3p−6)/2p

∇uL6

(6−p)/2p

≤ C∇uL2

(3p−6)/2p

(F L6 + wL6 + P (ρ)L6 )L6

6−p

≤ C∇uL2p2 (ρ u ˙ L2 + P (ρ)L6 )

3p−6 2p

.

(2.14)

This finishes the proof of the lemma. 2 Finally, we state the following logarithmic Sobolev inequality, see [16,18]. Lemma 2.5. For 3 < q < ∞, there is a constant C(q) such that the following estimate holds for all ∇u ∈ L2 (R3 ) ∩ D 1,q (R3 ), ∇uL∞ (R3 ) ≤ C(divuL∞ (R3 ) + wL∞ (R3 ) ) log(e + ∇ 2 uLq (R3 ) ) +C∇uL2 (R3 ) + C.

(2.15)

3. A priori estimates To extend the local classical solution guaranteed by Lemma 2.3, we will establish some necessary a priori estimates of smooth solutions. To do this, we assume that (ρ, u) is a smooth solution of (1.1)–(1.4) on R3 × (0, T ] with some T > 0. To estimate this solution, we set σ (t) = min{1, t} and define

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

T  A1 (T ) := sup

t∈[0,T ]

σ ∇u2L2

σρ|u| ˙ 2 dxdt,

+ 0

 A2 (T ) := sup

σ ρ|u| ˙ dx +

t∈[0,T ]

A3 (T ) := sup

t∈[0,T ]

T  3

σ 3 |∇ u| ˙ 2 dxdt,

2

0

∇u2L2 .

We have the following key a priori estimates on (ρ, u). Proposition 3.1. Assume that (ρ0 , u0 ) satisfy (1.8)–(1.9). Then there exist positive constants K1 , K2 and  depending only on C0 , μ, λ, M1 , such that if (ρ, u) is a smooth solution of (1.1)–(1.4) on R3 × (0, T ] satisfying ⎧ ⎪ ⎨0 ≤ ρ(x, t) ≤ 2M0 , (3.1) A3 (σ (T )) ≤ 2K1 , ⎪ ⎩ γ A1 (T ) + A2 (T ) ≤ 2K2 M0 , the following estimates hold 7 γ 0 ≤ ρ(x, t) ≤ M0 , A3 (σ (T )) ≤ K1 , A1 (T ) + A2 (T ) ≤ K2 M0 , 4

(3.2)

provided M0 ≤ . Proof of Proposition 3.1 is an easy consequence of the following Lemmas 3.2–3.4. In the following, we will use the convention that C denotes a generic positive constant depending on a, γ , C0 , μ, λ, M0 , and we write C(α) to emphasize that C depends on α. We start with the following standard energy estimate for (ρ, u) and preliminary L2 bounds for ∇u and ρ u. ˙ Lemma 3.2. Let (ρ, u) be a smooth solution of (1.1)–(1.4). Then there is a constant C such that  sup 0≤t≤T

1 ( ρ|u|2 + G(ρ))dx + 2

T  (μ|∇u|2 + (λ + μ)(divu)2 )dxdt ≤ C0 ,

(3.3)

0 γ A1 (T ) ≤ CM0

T  σ |∇u|3 dxdt,

+C

(3.4)

0

and 2γ A2 (T ) ≤ CA1 (T ) + CM0

T  +C

σ 3 |∇u|4 dxdt. 0

(3.5)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

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Proof. Multiplying the first equation of (1.1) by G (ρ) and the second by uj and integrating, applying the far field condition (1.3), one shows easily the energy inequality (3.3). To prove (3.4), multiplying (1.1)2 by σ m u˙ and then integrating the resulting equality over R3 leads to 

 σ m ρ|u| ˙ 2 dx = − :=

3 

 σ m u˙ · ∇P (ρ)dx + μ

 σ m u˙ · udx + (μ + λ)

σ m u˙ · ∇(divu)dx (3.6)

Ii .

i=1

To deal with the first term, we first notice that P (ρ)t + u · P (ρ) + ρP  (ρ)(divu) = 0 so that, integrating by parts gives  I1 = −

σ m u˙ · ∇P (ρ)dx

 σ m ((divu)t P (ρ) − (u · ∇u) · ∇P )dx

=

  m m−1  P (ρ)(divu)dx σ = ( σ divuP (ρ)dx)t − mσ  +

σ m (P  ρ(divu)2 − P (divu)2 + P ∂i uj ∂j ui )dx

 ≤ ( σ m divuP (ρ)dx)t + mσ m−1 σ  ||divu||2L2 +C()mσ m−1 σ  ||P (ρ)||2L2 + CM0 ∇u2L2 , γ

(3.7)

where we have also used (3.3) and the fact that 0 ≤ ρ ≤ 2M0 . Integration by parts implies  I2 = μ

σ m u˙ · udx

μm m−1  μ σ ||∇u||2L2 σ = − (σ m ∇u2L2 )t + 2 2  −μσ m ∂i uj ∂i (uk ∂k uj )dx μm m−1  μ σ ||∇u||2L2 + C σ ≤ − (σ m ∇u2L2 )t + 2 2 and similarly,

 σ m |∇u|3 dx,

(3.8)

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

 I3 = (μ + λ)

σ m u˙ · ∇(divu)dx

μ+λ m m(μ + λ) m−1  σ ||divu||2L2 (σ divu2L2 )t + σ 2 2  − (μ + λ)σ m divudiv(u · ∇u)dx

=−

μ+λ m m(μ + λ) m−1  ≤− σ ||divu||2L2 + C (σ divu2L2 )t + σ 2 2

 σ m |∇u|3 dx.

(3.9)

Combining (3.6)–(3.9) leads to (σ m B(t)) +

 σ m ρ|u| ˙ 2 dx

≤ Cmσ m−1 σ  [||∇u||2L2 + ||divu||2L2 + ||divu||2L2 ] + C()mσ

m−1 

σ

||P (ρ)||2L2

γ + CM0 ∇u2L2

+C

 σ m |∇u|3 dx,

(3.10)

where B(t) =

μ μ+λ ∇u2L2 + divu2L2 − 2 2

 divuP (ρ))dx

μ+λ μ+λ μ γ ∇u2L2 + divu2L2 − divu2L2 − CM0 2 2 4 μ μ+λ γ ≥ ∇u2L2 + divu2L2 − CM0 . 2 4 ≥

(3.11)

Note that σ  = 0 if t ≥ 1. Integrating (3.10) over (0, T ), choosing m = 1, and using (3.3), one gets (3.4). j Next, in order to prove (3.5), operating σ m u˙j (∂/∂t + div(u·)) to (1.1)2 , summing with respect 3 to j , and integrating the resulting equation over R , one obtains after integration by parts (

1 2

 σ m ρ|u| ˙ 2 dx)t −  =−

m m−1  σ σ 2

 ρ|u| ˙ 2 dx

σ m u˙ j (∂j Pt + div(∂j P u))dx 

+μ

j

σ m u˙ j (ut + div(uuj ))dx 

+ (μ + λ) :=

3  i=1

Ji .

σ m u˙ j (∂t ∂j divu + div(u∂j divu))dx (3.12)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

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It follows from integration by parts and using the equation (1.1)1 that  J1 = −  =

σ m u˙ j (∂j Pt + div(∂j P u))dx σ m (−P  ρdivu∂j u˙ j + P ∂k (∂j u˙ j uk ) − P ∂j (∂k u˙ j uk ))dx γ

˙ L2 ≤ CM0 σ m ∇uL2 ∇ u 2γ

≤ δσ m ∇ u ˙ 2L2 + CM0 σ m ∇u2L2 .

(3.13)

Integration by parts leads to  J2 = μ

j

σ m u˙ j (ut + div(uuj ))dx 

= −μ ≤−

σ m (|∇ u˙ j |2 + ∂i u˙ j ∂k uk ∂i uj − ∂i u˙ j ∂i uk ∂k uj − ∂k u˙ j ∂i uk ∂i uj )dx

3μ m ˙ 2L2 + C σ ∇ u 4

 σ m |∇u|4 dx.

(3.14)

Similarly, μ+λ J3 ≤ − 2



 σ (divu) ˙ dx + C m

2

σ m |∇u|4 dx.

(3.15)

Substituting (3.13)–(3.15) into (3.12) shows that for δ suitably small, it holds that  ( σ m ρ|u| ˙ 2 dx)t + μσ m ∇ u ˙ 2L2 + (μ + λ)σ m divu ˙ 2L2 m ≤ σ m−1 σ  2



2γ + CM0 σ m ∇u2L2

ρ|u| ˙ dx 2

 +C

σ m |∇u|4 dx.

(3.16)

Taking m = 3 in (3.16) and noticing that T 3

σ 2σ 

 ρ|u| ˙ 2 dxdt ≤ CA1 (T ).

0

Integrating (3.16) over (0, T ), we immediately obtain (3.5). The proof of Lemma 3.2 is completed. 2 Next, we estimate A3 (σ (T )), which plays an important role in the proof of uniform upper bound of density. Lemma 3.3. Let (ρ, u) be a smooth solution of (1.1)–(1.4) on R3 × (0, T ] satisfying 0 ≤ ρ ≤ 2M0 . Then there exist positive constants K1 , 1 depending on C0 , μ, λ, M1 , such that

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850 σ(T )

ρ|u| ˙ 2 dx ≤ K1

A3 (σ (T )) +

(3.17)

0

provided A3 (σ (T )) ≤ 2K1 and M0 < 1 . Proof. Choosing m = 0 in (3.10) and integrating it over (0, σ (T )), we find σ(T )

ρ|u| ˙ 2 dx

A3 (σ (T )) + 0 γ ≤ CM1 + CM0

σ(T )

+C

||∇u||3L3 dt

(3.18)

0

To deal with the last term on the right-hand side of (3.18), we make use of (2.12), (3.3) and the fact that 0 ≤ ρ ≤ 2M0 to get that 3/2

||∇u||3L3 ≤ C||∇u||L2 (||ρ u|| ˙ L2 + ||P (ρ)||L6 )3/2 3/4

3/2

3/2

≤ CM0 ||∇u||L2 ||ρ 1/2 u|| ˙ L2 5γ /4

+ CM0

3/2

||∇u||L2

5γ /4

≤ δ||ρ 1/2 u|| ˙ 2L2 + C(δ)M03 ||∇u||6L2 + CM0

3/2

||∇u||L2

(3.19)

Thus, putting (3.19) into (3.18) and choosing δ > 0 sufficiently small, by (3.3) we have σ(T )

ρ|u| ˙ 2 dx

A3 (σ (T )) + 0

σ(T )

γ ≤ CM1 + CM0

+ CM03

γ ≤ CM1 + CM0

5γ /4 + CM0

||∇u||6L2 dt

5γ /4 + CM0

0

≤

σ(T )

3/2

||∇u||L2 dt 0

+ CM03 [A3 (σ (T ))]2

K1 + CM03 [A3 (σ (T ))]2 2

(3.20)

where the positive constant K1 is defined as follows: γ

5γ /4

K1 := 2C(M1 + M0 + M0

)

1 Thus, if M0 ≤ min{1, ( 8CK )1/3 } := 1 and A3 (σ (T )) ≤ 2K1 , (3.17) holds. 1

Now we can conclude the estimates of A1 (T ) and A2 (T ).

2

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

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Lemma 3.4. There exist positive constants K2 , 2 depending on C0 , μ, λ, M1 such that if (ρ, u) is a smooth solution of (1.1)–(1.4) satisfying (3.1), then γ

A1 (T ) + A2 (T ) ≤ K2 M0

(3.21)

γ

provided M0 ≤ 2 and A1 (T ) + A2 (T ) ≤ 2K2 M0 . Proof. It follows from (3.4) and (3.5) that A1 (T ) + A2 (T ) γ ≤ C(M0

2γ + M0 ) + C

T

T σ ||∇u||3L3 dt

+C

0

σ 3 ||∇u||4L4 dt.

(3.22)

0

Due to (2.11), T

T σ

3

||∇u||4L4 dt

≤C

0

T σ

3

||F ||4L4 dt

+C

0

σ 3 ||ω||4L4 dt 0

T +C

σ 3 ||P (ρ)||4L4 dt.

(3.23)

0

Recalling the definitions of A1 (T ), A2 (T ), and using the fact that 0 ≤ ρ ≤ 2M0 , we infer from (2.9) that T

T σ

0

3

F 4L4 dt

≤C

σ 3 (∇uL2 + P (ρ)L2 )ρ u ˙ 3L2 dt 0 3/2

T

≤ CM0

γ +3

0 3/2 ≤ CM0

σ 3 ρ 1/2 u ˙ 3L2 dt 0

T sup {(σ 0≤t≤T

γ +3

+ CM0 2 3

T

σ 3 ∇uL2 ρ 1/2 u ˙ 3L2 dt + CM0 2

3

3/2

ρ

1/2

u ˙ L2 )(σ

1/2

||∇u||L2 )}

σ ρ 1/2 u ˙ 2L2 dt 0

T sup (σ 3/2 ρ 1/2 u ˙ L2 )

0≤t≤T 1

σ ρ 1/2 u ˙ 2L2 dt 0

γ +3 2

≤ CM02 A12 (T )A22 (T ) + CM0

1

A1 (T )A22 (T ).

(3.24)

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

Similarly, it follows from (2.10) that T

T σ

3

w4L4 dt

≤C

0

3

3

1

σ 3 ∇uL2 ρ u ˙ 3L2 dt ≤ CM02 A12 (T )A22 (T ).

(3.25)

0

To estimate the third term on the right side of (3.23), one deduces from (1.1) that P (ρ) satisfies (P (ρ))t + u · ∇(P (ρ)) + γ P (ρ)divu = 0.

(3.26)

Multiplying (3.26) by 3(P (ρ))2 and integrating the resulting equality over R3 , one gets after 1 (F + P (ρ)) that (keeping in mind that γ > 1) using divu = 2μ+λ   3γ − 1 3γ − 1 P (ρ)4L4 = −( (P (ρ))3 dx)t − (P (ρ))3 F dx 2μ + λ 2μ + λ  ≤ −( (P (ρ))3 dx)t + ηP (ρ)4L4 + C(η)F 4L4 ,

(3.27)

which, integrated over (0, T ) and combined with (3.3), yields T

T

2γ

σ 3 ||P (ρ)||4L4 dt ≤ CM0 + C 0

σ 3 ||F ||4L4 dt.

(3.28)

0

So, substituting (3.24)–(3.28) into (3.23) gives T

2γ

3

3

γ +3

1

1

σ 3 ||∇u||4L4 dt ≤ CM0 + CM02 A12 (T )A22 (T ) + CM0 2 A1 (T )A22 (T ).

(3.29)

0

We still need to deal with ||∇u||L3 . To do so, we first utilize (3.3) to get that if T > σ (T ) T

σ(T )

σ ||∇u||3L3 dt

≤

0

T

σ ||∇u||3L3 dt + 0

||∇u||3L3 dt

σ (T )

σ(T )

σ ||∇u||3L3 dt

≤ 0

T + C( σ (T )

σ(T )

σ ||∇u||3L3 dt

≤ 0

T ||∇u||2L2 dt)1/2 (

||∇u||4L4 dt)1/2

σ (T )

T + C( σ (T )

||∇u||4L4 dt)1/2 .

(3.30)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

6843

For the first term on the right-hand side, we infer from the second inequality of (3.19) that σ(T )

σ ||∇u||3L3 dt 0 3/4 ≤ CM0

σ(T )

3/2 3/2 σ ||∇u||L2 ||ρ 1/2 u|| ˙ L2 dt

5γ 4

σ(T )

+ CM0

3/2

σ ||∇u||L2 dt

0

0

3/4

5γ 4

1/4

≤ CM0 A1 (T )A3 (σ (T )) + CM0 ,

(3.31)

where we have used (3.3) and following estimates: σ(T )

3/2

3/2

σ ||∇u||L2 ||ρ 1/2 u|| ˙ L2 dt 0 σ(T )

≤

sup

(σ

1/4

t∈[0,σ (T )]

||∇u||L2 )

1/2

3/2

σ 3/4 ||∇u||L2 ||ρ 1/2 u|| ˙ L2 dt 0

3/4 ≤ CA1 (T )

sup t∈[0,σ (T )]

1/2

1/2

{(||∇u||L2 )(σ 1/4 ||∇u||L2 )}

1/4 ≤ CA1 (T )A3 (σ (T ))

(3.32)

due to (3.3), Hölder inequality and the definitions of A1 (T ), A3 (σ (T )). Thus, combining (3.31), (3.32) with (3.29) leads to T σ ||∇u||3L3 dt 0 5γ

γ

3

3

1

≤ C(M0 + M04 ) + CM04 A14 (T )A24 (T ) γ +3

1

1

3/4

1/4

3/4

1/4

+ CM0 4 A12 (T )A24 (T ) + CM0 A1 (T )A3 (σ (T )). Hence, we deduce from (3.29) and (3.33) that A1 (T ) + A2 (T ) 5γ

γ

2γ

≤ C(M0 + M04 + M0 ) + CM0 A1 (T )A3 (σ (T )) 3

3

1

3

3

1

+ CM04 A14 (T )A24 (T ) + CM02 A12 (T )A22 (T ) γ +3

1

1

γ +3

1

+ CM0 4 A12 (T )A24 (T ) + CM0 2 A1 (T )A22 (T )

(3.33)

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

≤

K2 γ 3/4 1/4 M + CM0 A1 (T )A3 (σ (T )) 6 0 3

3

1

3

3

1

+ CM04 A14 (T )A24 (T ) + CM02 A12 (T )A22 (T ) γ +3

1

γ +3

1

1

+ CM0 4 A12 (T )A24 (T ) + CM0 2 A1 (T )A22 (T )

(3.34)

where K2 := 18C, M0 ≤ 1. γ Note that γ > 1 and A1 (T ) + A2 (T ) ≤ 2K2 M0 . (3.34) implies that there is a constant 2 such that if M0 ≤ 2 , (3.21) holds. 2 To prove the uniform upper bound of density, we still need the following estimate. Lemma 3.5. Assume that (ρ, u) is a smooth solution of (1.1)–(1.4) on R3 × (0, T ) satisfying 0 ≤ ρ ≤ 2M0 . Then there is a constant K3 depending on μ, λ, C0 , M1 such that 

σ(T )

σρ|u| ˙ dx +

σ |∇ u| ˙ 2 dxdt ≤ K3 .

2

sup t∈[0,σ (T )]

(3.35)

0

Proof. Choosing m = 1 in (3.16) and integrating the resulting over R3 × (0, σ (T )), by (3.3) and (3.17), we obtain 

σ(T )

σρ|u| ˙ dx +

σ |∇ u| ˙ 2 dxdt

2

sup t∈[0,σ (T )]

2γ ≤ CK1 + CM0

0 σ(T )

+C

σ ||∇u||4L4 dt. 0

Similar to the derivation of (3.24), we deduce from (2.12) that σ(T )

σ ||∇u||4L4 dt 0 σ(T )

≤C

σ ||∇u||L2 (||P (ρ)||L6 + ||ρ u˙ L2 )3 dt 0 5γ 2

≤ CM0

3/2 + CM0

σ(T )

σ ||∇u||L2 ||ρ 1/2 u|| ˙ 3L2 dt 0

(3.36)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

5γ 2

≤ CM0

3/2 + CM0

5γ 2

σ(T )

{(σ

sup

1/2

t∈[0,σ (T )]

γ +3 2

≤ CM0 + 2CM0

6845

||∇u||L2 )(σ

1/2

||ρ

1/2

u|| ˙ L2 )}

||ρ 1/2 u|| ˙ 2L2 dt 0

1/2

K1 K2

sup t∈[0,σ (T )]

(σ 1/2 ||ρ 1/2 u|| ˙ L2 ).

(3.37)

Combining (3.36), (3.37) and applying Cauchy–Swcharz inequality, one gets (3.35). 2 We now proceed to derive a uniform (in time) upper bound for the density. Lemma 3.6. If (ρ, u) is a smooth solution of (1.1)–(1.4) on R3 × (0, T ) satisfying 0 ≤ ρ ≤ 2M0 . Then there exists a positive constant 3 depending on μ, λ, C0 , M1 such that 7M0 , 4

sup ρL∞ ≤ 0≤t≤T

(3.38)

provided M0 ≤ 3 . Proof. Rewrite the equation of the mass conservation (1.1)1 as Dt ρ = g(ρ) + b (t),

(3.39)

where γ +1

1 Dt ρ := ρt + u · ∇ρ, g(ρ) := − aρ 2μ+λ , b(t) := − 2μ+λ

t 0

(3.40)

ρF ds.

To apply Lemma 2.2 to (3.40), we need to deal with b(t). To do this, we first utilize (2.2), (2.8) and (2.9) to deduce that for all 0 ≤ t1 ≤ t2 ≤ σ (T ), σ(T )

M0 |b(t2 ) − b(t1 )| ≤ C 2μ + λ

σ(T )

F (·, t)

L∞

dt ≤ CM0

0

3/4

0

σ(T )

≤ CM0

1/4

F L2 ∇F L6 dt

3/4

(∇uL2 + P (ρ)L2 )1/4 ρ u ˙ L6 dt 0

14+γ 8

σ(T )

σ −3/8 (σ ∇ u ˙ 2L2 )3/8 dt

≤ CM0

0 7/4 + CM0

σ(T )

σ −3/8 (∇u2L2 )1/8 (σ ∇ u ˙ 2L2 )3/8 dt

0 14+γ 8

≤ C(M0

7/4

3/8

+ M0 )K3

(3.41)

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

where (3.1), (3.35) and Hölder inequality are used. Thus, for 0 ≤ t ≤ σ (T ) we can choose N0 , N1 and ζ¯ in Lemma 2.2 as follows: 14+γ 8

N0 = C(M0

7/4 3/8 + M0 )K3 , N1 = 0, ζ¯ = 0.

Then, it is clear that g(ζ ) ≤ −

aζ γ +1 ≤ N1 = 0, for all ζ ≥ ζ¯ = 0 2μ + λ

and consequently, we can make use of Lemma 2.2 to deduce ||ρ||L∞ ≤ max{M0 , 0} + N0 ≤

sup t∈[0,σ (T )]

3M0 2

(3.42)

provided that 14+γ 8

N0 = C(M0

7/4

3/8

+ M0 )K3

≤

M0 2

i.e. 6+γ

3/4

3/8

C(M0 8 + M0 )K3

1 ≤ . 2

(3.43)

For σ (T ) ≤ t ≤ T , using Lemma 2.1, (2.8), (3.1), (3.21) and the fact that 0 ≤ ρ ≤ 2M0 , we see that T

T ||∇F ||2L2 dt

T

≤C

σ (T )

||ρ u|| ˙ 2L2 dt

γ +1

≤ CM0

σ (T )

||ρ 1/2 u|| ˙ 2L2 dt ≤ CK2 M0

σ (T )

and T

T ||∇F ||2L6 dt

σ (T )

≤C

T ||ρ u|| ˙ 2L6 dt

σ (T )

γ +2

≤ CM02

||∇ u|| ˙ 2L2 dt ≤ CK2 M0

σ (T )

so that, it holds for σ (T ) ≤ t1 ≤ t2 ≤ T that M |b(t2 ) − b(t1 )| ≤ C 2μ + λ γ +1

t2 F (·, t)L∞ dt t1

aM0 1−γ (t2 − t1 ) + CM0 ≤ 2μ + λ

T ||F ||2L∞ dt σ (T )

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

T

γ +1

aM0 1−γ ≤ (t2 − t1 ) + CM0 2μ + λ ≤

γ +1 aM0

2μ + λ

6847

||∇F ||L2 ||∇F ||L6 dt σ (T )

(t2 − t1 ) + C(M02 + M03 ).

(3.44)

Therefore, one can choose N1 , N0 and ζ¯ in Lemma 2.2 as follows: γ +1

N1 =

aM0 , N0 = CK2 (M02 + M03 ), ζ¯ = M0 2μ + λ

Since γ +1

aM0 aζ γ +1 g(ζ ) ≤ − ≤ −N1 = − , for all ζ ≥ ζ¯ = M0 2μ + λ 2μ + λ we thus infer from Lemma 2.2 that sup

t∈[σ (T ),T ]

||ρ||L∞ ≤ max{M0 , 3M0 /2} + N0 ≤

7M0 4

(3.45)

provided such that CK2 (M03 + M03 ) ≤

M0 4

i.e. 1 CK2 (M0 + M02 ) ≤ . 4 Combining (3.43) and (3.46), the proof of this lemma is completed.

(3.46) 2

Proof of Proposition 3.1. Taking K1 , K2 , K3 as the ones in Lemma 3.3, 3.4 and 3.5 respectively, and choosing  = min{1 , 2 , 3 }, one immediately arrives the desired result of Proposition 3.1. 2 With the help of Proposition 3.1, one can easily derive the higher-order estimates of solution (ρ, u) in a similar manner as those obtained in [16, Lemmas 3.7–3.11]. We write only the results. In the following lemmas, we will always assume that M0 appropriate small and the constant C may depend on 1

T , ρ02 gL2 , ∇gL2 , ∇u0 H 2 , ρ0 H 3 , P (ρ0 )H 3 , besides μ, λ, C0 , γ and M0 where g is the function in (1.9).

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

Lemma 3.7. The following estimates hold 

T  |∇ut |2 dxdt ≤ C,

ρ|ut | dx + 2

sup 0≤t≤T

(3.47)

0

sup (ρH 2 + P (ρ)H 2 ) ≤ C.

t∈[0,T ]

(3.48)

Lemma 3.8. The following estimates hold: T sup (ρt H 1 + Pt H 1 ) +

t∈[0,T ]

(ρtt 2H 1 + Ptt 2H 1 )dt ≤ C,

(3.49)

0

 sup

T  |∇ut | dx +

ρu2tt dxdt ≤ C.

2

t∈[0,T ]

(3.50)

0

Lemma 3.9. It holds that sup (ρH 3 + P (ρ)H 3 ) ≤ C,

t∈[0,T ]

(3.51)

T sup (∇ut L2 + ∇uH 2 ) +

t∈[0,T ]

(∇u2H 3 + ∇ut 2H 1 )dt ≤ C.

(3.52)

0

Lemma 3.10. For any τ ∈ (0, T ), there exists some positive constant C(τ ) such that T  |∇utt |2 dxdt ≤ C(τ ).

sup (∇ut H 1 + ∇ uL2 ) + 4

τ ≤t≤T

(3.53)

τ

Now, combining all the priori estimates with the local existence theorem (i.e. Lemma 2.3), we can prove Theorem 1.1 in a similar manner as in [8]. Proof of Theorem 1.1. By Lemma 2.3, there exists a T∗ > 0 such that the Cauchy problem (1.1)–(1.4) has a unique classical solution (ρ, u) on (0, T∗ ]. We will use the a priori estimates, Proposition 3.1 and Lemmas 3.9 and 3.10, to extend the local classical solution (ρ, u) to all the time. First, since A1 (0) + A2 (0) = 0, A3 (0) ≤ K1 , ρ0 ≤ M0 , there exists a T1 ∈ (0, T∗ ] such that (3.1) holds for T = T1 . Set T ∗ = sup{T | (3.1) holds}.

(3.54)

J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

6849

Then T ∗ ≥ T1 > 0. Hence, for any 0 < τ < T ≤ T ∗ with T finite, it follows from Lemma 3.9 and Lemma 3.10 that ∇ut , ∇ 3 u ∈ C([τ, T ]; L2 ∩ L4 ), ∇u, ∇ 2 u ∈ C([τ, T ]; L2 ∩ C(R3 )),

(3.55)

where we have used the standard embedding L∞ (τ, T ; H 1 ) ∩ H 1 (τ, T ; H −1 ) → C([τ, T ]; Lq ), for any q ∈ [2, 6). Due to (3.47), (3.50) and (3.53), one can get T (ρ|ut |2 )t L1 dt τ

T (ρt |ut |2 L1 + 2ρut · utt L1 )dt

≤ τ

T ≤C

(ρ|divu||ut |2 L1 + |u||∇ρ||ut |2 L1 + ρ 1/2 ut L2 ρ 1/2 utt L2 )dt τ

T ≤C

(ρ|ut |2 L1 ∇uL∞ + uL6 ∇ρL2 ut 2L6 + ρ 1/2 utt L2 )dt τ

≤ C, which yields ρ 1/2 ut ∈ C([τ, T ]; L2 ). This, together with (3.55), gives ρ 1/2 u, ˙ ∇ u˙ ∈ C([τ, T ]; L2 ).

(3.56)

T ∗ = ∞.

(3.57)

Next, we claim that

Otherwise, T ∗ < ∞. Then by Proposition 3.1, (3.2) holds for T = T ∗ . It follows from Lemma 3.9, Lemma 3.10 and (3.56) that (ρ(x, T ∗ ), u(x, T ∗ )) satisfies (1.8) and (1.9) with g(x) = u(x, ˙ T ∗ ), x ∈ R3 . Lemma 2.3 implies that there exists T ∗ ∗ > T ∗ , such that (3.1) holds for T = T ∗ ∗ , which contradicts (3.54). Hence, (3.57) holds. Lemmas 2.3, 3.9–3.10 and (3.55) thus show that (ρ, u) is in fact the unique classical solution defined on (0, T ] for any 0 < T < T ∗ = ∞. The proof of (1.13) is similar to that in [16]. 2

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J. Qian, J. Zhao / J. Differential Equations 259 (2015) 6830–6850

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