(1.3) There are extensive literatures on the studies of the compressible Navier-Stokes equations

NAVIER-STOKES EQUATIONS WITH VACUUM

HUAJUN GONG, JINKAI LI, XIAN-GAO LIU, AND XIAOTAO ZHANG

Abstract. In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier-Stokes equations is proved with the initial date being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many literatures concerning the well-posedness of compressible Navier-Stokes equations in the presence of vacuum.

1. Introduction

The isentropic compressible Navier-Stokes equations read as

ρ(u_t+ (u· ∇)u)−µ∆u−(λ+µ)∇divu+∇P = 0, (1.1) ρ_t+ div (ρu) = 0, (1.2) inR³×(0, T), where the densityρ≥0 and the velocity fieldu∈R³are the unknowns.

Here P is the scalar pressure given as P =aρ^γ, for two constants a > 0 and γ > 1.

The viscosity constants λ, µ satisfy the physical requirements:

µ >0, 2µ+ 3λ≥0.

System (1.1)–(1.2) is complemented with the following initial-boundary conditions (ρ, ρu)|_t=0 = (ρ₀, ρ₀u₀),

u(x, t)→0, as|x| → ∞. (1.3) There are extensive literatures on the studies of the compressible Navier-Stokes equations. In the absence of vacuum, that is the density has positive lower bound, the system is locally well-posed for large initial data, see, e.g., [23, 41, 46, 48, 49, 51];

however, the global well-posedness is still unknown. It has been known that system in one dimension is globally well-posed for large initial data, see, e.g., [2, 26–29, 57, 58]

and the references therein, see [35] for the large time behavior of the solutions, and also [34, 37, 38] for the global well-posedness for the case that with nonnegative density. For the multi-dimensional case, the global well-posedness holds for small

Date: May 20, 2019.

2010Mathematics Subject Classification. 35D35, 35Q30, 35Q35, 76N10.

Key words and phrases. Isentropic Navier-Stokes equations; local well-posedness; vacuum; with- out compatibility condition.

1

initial data, see, e.g., [6, 10–13, 19, 30, 42–45, 47, 50]. In the presence of vacuum, that is the density may vanish on some set, global existence (but without uniqueness) of weak solutions has been known, see [1, 14–16, 25, 39, 40]. Local well-posedness of strong solutions was proved for suitably regular initial data under some extra compatibility conditions (being mentioned in some details below) in [7–9]. In general, when the vacuum is involved, one can only expect solutions in the homogeneous Sobolev spaces, that is, the L² integrability of u is not expectable, see [31]. Global well-posedness holds if the initial basic energy is sufficiently small, see [20, 21, 36, 54];

however, due to the blow-up results in [55, 56], the corresponding entropy of the global solutions in [20, 54] must be infinite somewhere in the vacuum region, if the initial density is compactly supported.

In this paper, we focus on the well-posedness of the Cauchy problem to system (1.1)–(1.2) in the presence of vacuum. As mentioned in the previous paragraph, local well-posedness of strong solutions to the compressible Navier-Stokes in the presence of vacuum has already been studied in [7–9], where, among some other conditions, the regularity assumption

ρ₀−ρ∞∈H¹∩W^1,q, u₀ ∈D¹₀∩D², (1.4) for some constant ρ∞∈[0,∞), and the compatibility condition

−µ∆u₀−(µ+λ)∇divu₀+∇P(ρ₀) =√

ρ₀g, (1.5)

for some g ∈ L², were used. Similar assumptions as (1.4) and (1.5) were also widely used in studying many other fluid dynamical systems when the vacuum is involved, see, e.g., [3–5, 17, 18, 20–22, 32, 36, 52–54].

Assumptions (1.4) and (1.5) are so widely used when the initial vacuum is taken into consideration, one may ask if the regularities on the initial data stated in (1.4) can be relaxed and if the compatibility condition (1.5) is necessary for the local well- posedness of strong solutions to the corresponding system. In a previous work [33], the second author of this paper considered these questions for the inhomogeneous incompressible Navier-Stokes equations, and found that the compatibility condition is not necessary for the local well-posedness. The aim of the current paper is to give the same answer for the isentropic compressible Navier-Stokes equations. As will be shown in this paper that we can indeed reduce the regularities of the initial velocity in (1.4) and remove the compatibility condition (1.5), without loosing the existence and uniqueness, but the prices that we need to pay are the following: (i) the corresponding strong solutions do not have as high regularities as those in [7–9]

where both (1.4) and (1.5) were assumed; (ii) one can only ask for the continuity, at the initial time, of the momentum ρu, instead of the velocityu itself.

Before stating our main results, let us introduce some notations. Throughout this paper, we useL^r =L^r(R³) andW^k,r =W^k,r(R³) to denote, respectively, the standard Lebesgue and Sobolev spaces inR³, wherekis a positive integer andr∈[1,∞]. When

r = 2, we use H^k instead of W^k,2. For simplicity, we usek · kr =k · kL^r. We denote D^k,r =

n

u∈L¹_loc(R³)

k∇^kukr <∞o

, D^k =D^k,2, D₀¹ =n

u∈L⁶

k∇uk₂ <∞o . For simplicity of notations, we adopt the notation

Z

f dx= Z

R³

f dx.

Our main result is the following:

Theorem 1.1. Suppose that the initial data (ρ₀, u₀) satisfies

ρ₀ ≥0, ρ−ρ∞ ∈H¹∩W^1,q, u₀ ∈D¹₀, ρ₀u₀ ∈L², for some ρ_∞ ∈[0,∞) and some q∈(3,6).

Then, there exists a positive time T, depending only on µ, λ, a, γ, q, and the upper bound of ψ₀ :=kρ₀k∞+kρ₀−ρ∞k₂+k∇ρ₀k_L²_∩L^q +k∇u₀k₂, such that system (1.1)–(1.2), subjects to (1.3), admits a unique solution(ρ, u)onR³×(0,T),satisfying

ρ−ρ∞ ∈C([0,T];L²)∩L^∞(0,T;H¹∩W^1,q), ρt∈L^∞(0,T;L²), ρu∈C([0,T];L²), u∈L^∞(0,T;D¹₀)∩L²(0,T;D²), √

ρu_t∈L²(0,T;L²),

√tu ∈L^∞(0,T;D²)∩L²(0,T;D^2,q), √

tu_t∈L²(0,T;D₀¹).

Remark 1.1. (i) Compared with the local well-posedness results established in [7, 8], in Theorem 1.1, u₀ is not required to be in D² and we do not need any compatibility conditions on the initial data.

(ii) The same result as in 1.1 also holds for the initial boundary value problem if imposing suitable boundary conditions on the velocity.

2. Lifespan estimate and some a priori estimates

As preparations of proving the main result being carried out in the next section, the aim of this section is to give the lifespan estimate and some a priori estimates, under the condition that the initial velocity u₀ ∈ D¹₀ ∩D² and some compatibility condition; however, it should be emphasized that all these estimates depend neither on k∇²u₀k₂ nor on the compatibility condition.

We start with the following local existence and uniqueness result, which has been essentially proved in [7–9].

Proposition 2.1. Let ρ∞ ∈ [0,∞) and q ∈ (3,6) be fixed constants. Assume that the data ρ₀ and u₀ satisfy the regularity condition

ρ₀ ≥0, ρ₀−ρ∞∈H¹∩W^1,q, u₀ ∈D₀¹∩D², and the compatibility condition

−µ∆u₀−(µ+λ)∇divu+∇P₀ =√ ρ₀g,

for some g ∈L², where P0 =aρ^γ₀.

Then, there exists a small time T_∗ > 0 and a unique strong solution (ρ, u) to (1.1)–(1.2), subject to (1.3), such that

ρ−ρ∞∈C([0, T∗];H¹∩W^1,q), u∈C([0, T∗];D¹₀∩D²)∩L²(0, T∗;D^2,q), ρ_t∈C([0, T∗];L²∩L^q), u_t∈L²(0, T∗;D₀¹), √

ρu_t ∈L^∞(0, T∗;L²).

As will be shown in this section, the existence time T∗ in the above proposition can be chosen depending only on µ, λ, a, γ, q, and the upper bound of

Ψ₀ :=kρ₀k∞+kP₀−P∞k₂+kP₀k∞+k∇P₀k₂+k∇P₀k_q+k∇u₀k₂,

with P∞=aρ^γ_∞, and, in particular, independent ofk∇²u₀k₂ andkgk₂. The following quantity plays the key role in this section

Ψ(t) := (kρk∞+kP −P∞k₂+kPk∞+k∇Pk₂+k∇Pk_q +k∇uk₂+k√

t√

ρu_tk₂)(t) + 1.

In the rest of this section, until the last proposition, we always assume (ρ, u) is a solution to system (1.1)–(1.2), subject to (1.3), on R³ ×(0, T), satisfying the regularities stated in Proposition 2.1, with T∗ there replaced with T.

Throughout this section, except otherwise explicitly mentioned, we denote by C a generic constant depending only on µ, λ, a, γ, and the upper bound of Ψ₀.

Proposition 2.2. The following estimates hold k∇²uk²₂ ≤ C(Ψ¹⁰+ Ψk√

ρu_tk²₂), k√

ρ(u· ∇)uk²₂ ≤ C(Ψ⁹+ Ψ⁵k√

ρu_tk₂), k∇²uk_q ≤ C(k√

t∇u_tk²₂ +k√

ρu_tk²₂ +t^−5q−64q + Ψ^α1), q∈(3,6), with α₁ := maxn

12,^(5q−6)_2q(6−q)²o .

Proof. Applying the elliptic estimates to (1.1) yields k∇²uk²₂ ≤Ckρk_∞(k√

ρu_tk²₂+k√

ρ(u· ∇)uk²₂) +Ck∇Pk²₂. By the H¨older and Sobolev inequality, one has

k√

ρ(u· ∇)uk²₂ ≤ kρk∞kuk²₆k∇uk2k∇uk6 ≤CΨ⁴k∇²uk2.

Substituting the above inequality into the previous one and using the Cauchy in- equality, one gets

k∇²uk²₂ ≤ C(Ψk√

ρutk²₂+ Ψ⁵k∇²uk2 + Ψ²)

≤ 1

2k∇²uk²₂+C(Ψk√

ρu_tk²₂+ Ψ¹⁰), that is

k∇²uk²₂ ≤C(Ψ¹⁰+ Ψk√

ρu_tk²₂), (2.1)

and, consequently,

k√

ρ(u· ∇)uk²₂ ≤C(Ψ⁹+ Ψ⁵k√

ρu_tk₂), proving the first two conclusions.

It follows from the H¨older and Gagliardo-Nirenberg inequalities that kρ(u· ∇)uk_q ≤ kρk∞kuk ^6q

6−qk∇uk₆

≤ Ckρk_∞kuk

3 q

6k∇uk²⁻

3 q

6 ≤Ckρk_∞k∇uk

3 q

2k∇²uk

2q−3 q

2 , (2.2)

from which, by the Young inequality and using (2.1), one has kρ(u· ∇)uk_q ≤ CΨ^q+3q (Ψ¹⁰+ Ψk√

ρu_tk²₂)^2q−32q

≤ CΨ³(Ψ⁹+k√

ρu_tk²₂)^2q−32q

≤ C(Ψ^2q+ Ψ⁹+k√

ρu_tk²₂)≤C(Ψ¹²+k√

ρu_tk²₂).

It follows from the H¨older and Sobolev inequalities that kρu_tk_q ≤ kρk

5q−6

∞4q k√ ρu_tk

6−q 2q

2 ku_tk

3q−6 2q

6 ≤Ckρk

5q−6

∞4q k√ ρu_tk

6−q 2q

2 k∇u_tk

3q−6 2q

2 , (2.3)

and further by the Young inequality that kρu_tk_q ≤ CΨ^5q−64q t^−3q−64q k√

t∇u_tk

3q−6 2q

2 k√

ρu_tk

6−q 2q

2

≤ C(k√

t∇utk²₂+k√

ρutk²₂+t⁻

5q−6

4q + Ψ

(5q−6)2 2q(6−q)).

Thanks to the above two, and applying the elliptic estimates to (1.1), one obtains k∇²uk_q ≤ C(kρu_tk_q+kρ(u· ∇)uk_q+k∇Pk_q)

≤ C(k√

t∇u_tk²₂+k√

ρu_tk²₂+t^−5q−64q + Ψ^α1),

proving the conclusion.

Proposition 2.3. The following estimate holds sup

0≤t≤T

(k∇uk²₂+kP −P∞k²₂) + Z T

0

k√

ρu_tk²₂dt≤C+C Z T

0

Ψ¹⁰dt.

Proof. Multiplying (1.1) withut, it follows from integration by parts that 1

2 d

dt(µk∇uk²₂ + (µ+λ)kdivuk²₂) +k√

ρu_tk²₂ =− Z

(ρ(u· ∇)u+∇P)·u_tdx.

Integration by parts and noticing that

P_t+u· ∇P +γdivuP = 0, (2.4)

one deduces

− Z

∇P ·u_tdx = Z

(P −P∞)divu_tdx

= d dt

Z

(P −P∞)divudx− Z

P_tdivudx

= d

dt Z

(P −P∞)divudx+ Z

(u· ∇P +γdivuP)divudx.

Therefore d dt

µk∇uk²₂+ (µ+λ)kdivuk²₂−2 Z

(P −P∞)divudx

+k√ ρu_tk²₂

= Z

(u· ∇P +γdivuP)divudx− Z

ρ(u· ∇)u·u_tdx =:I₁+I₂.

By the H¨older, Sobolev, and Young inequalities, and applying Proposition 2.2, one has

I₁ ≤ kuk₆k∇Pk₂kdivuk₃+γkdivuk²₂kPk∞

≤ Ck∇uk2k∇Pk2k∇uk

1 2

2k∇²uk

1 2

2 +CΨ³

≤ CΨ⁵²(Ψ⁵² + Ψ¹⁴k√

ρu_tk₂¹²) +CΨ³

≤ 1 4k√

ρu_tk²₂+CΨ⁵, and

I₂ ≤ k√

ρu_tk₂k√

ρ(u· ∇)uk₂

≤ C(Ψ⁹² + Ψ⁵²k√

ρu_tk₂¹²)k√ ρu_tk₂

≤ 1 4k√

ρu_tk²₂+CΨ¹⁰. Therefore

1 2

d dt

µk∇uk²₂+ (µ+λ)kdivuk²₂−2 Z

(P −P∞)divudx

+k√

ρutk²₂ ≤CΨ¹⁰, from which, one obtains by the Cauchy inequality that

sup

0≤t≤T

k∇uk²₂+ Z T

0

k√

ρu_tk²₂dt≤C

1 + sup

0≤t≤T

kP −P∞k²₂+ Z T

0

Ψ¹⁰dt

. (2.5) Multiplying (2.4) withP−P∞, it follows from integration by parts and the Sobolev inequality that

d

dtkP −P∞k²₂ = −(γ− 1 2)

Z

divu(P −P∞)²dx−γP∞

Z

divu(P −P∞)dx

≤ Ck∇uk2kP −P∞k

1 2

2k∇Pk

3 2

2 +Ck∇uk2kP −P∞k2 ≤CΨ³, which gives

sup

0≤t≤T

kP −P∞k²₂ ≤C+C Z T

0

Ψ³dt.

This, combined with (2.5), leads to the conclusion.

The t-weighted estimate in the next proposition is the key to remove the compat- ibility condition on the initial data.

Proposition 2.4. The following estimate holds sup

0≤t≤T

k√ t√

ρutk²₂+ Z T

0

k√

t∇utk²₂dt≤C+C Z T

0

Ψ¹⁶dt.

Proof. Differentiating (1.1) in t and using (1.2) yield

ρ(∂_tu_t+ (u· ∇)u_t)−µ∆u_t−(λ+µ)∇divu_t

=−∇Pt+ div (ρu)(ut+ (u· ∇)u)−ρ(ut· ∇)u.

Multiplying it by u_t, integrating by parts over R³ and then using the continuity equation (1.2), one has

1 2

d dtk√

ρu_tk²₂+µk∇u_tk²₂+ (λ+µ)kdivu_tk²₂

= Z

P_tdivu_tdx+ Z

div (ρu)|u_t|²dx+ Z

div (ρu)(u· ∇)u·u_tdx

− Z

ρ(u_t· ∇u)·u_tdx=:II₁+II₂+II₃+II₄.

Recalling (2.4) and using the Sobolev and Young inequalities, one deduces II1 = −

Z

(γdivuP +u· ∇P)divutdx

≤ C(kPk∞||∇u||₂||∇u_t||₂+||u||₆||∇P||₃||∇u_t||₂)

≤ C Ψ⁴+||∇u||²₂||∇P||²_L2∩L^q

+ µ

8||∇u_t||²₂

≤ CΨ⁴+µ

8||∇u_t||²₂.

Integrating by parts, using the H¨older, Sobolev and Young inequalities, and applying Proposition 2.2, we have

II₂ = − Z

R³

ρu· ∇|u_t|²dx

≤ C||ρ||

1

∞2 ||u||6||√ ρut||

1 2

2||√ ρut||

1 2

6||∇ut||2

≤ C||ρ||

3

∞4 ||∇u||₂||√ ρu_t||

1 2

2||∇u_t||

3 2

2

≤ CΨ⁷||√

ρu_t||²₂+ µ

8||∇u_t||²₂, II₃ ≤

Z

R³

ρ|u|(|∇u|²|u_t|+|u||∇²u||u_t|+|u||∇u||∇u_t|)dx

≤ C||ρ||∞(||u||₆||∇u||²₃||u_t||₆+||u||²₆||∇²u||₂||u_t||₆ +||u||²₆||∇u||₆||∇u_t||₂)

≤ C||ρ||_∞||∇u||²₂||∇²u||₂||∇u_t||₂

≤ CΨ⁶||∇²u||²₂+µ

8||∇u_t||²_L2

≤ C(Ψ¹⁶+ Ψ⁷||√

ρu_t||²₂) + µ

8||∇u_t||²_L2, and

II₄ ≤ Z

R³

ρ|u_t|²|∇u|dx

≤ C||ρ||

1

∞2 ||∇u||2||√ ρut||

1 2

2||√ ρut||

1 2

6||ut||6

≤ C||ρ||

3

∞4 ||∇u||₂||√ ρu_t||

1 2

2||∇u_t||

3 2

2

≤ CΨ⁷||√

ρu_t||²₂ +µ

8||∇u_t||²₂. Therefore, we have

d dtk√

ρu_tk²₂+µk∇u_tk²₂ ≤C(Ψ¹⁶+ Ψ⁷||√

ρu_t||²₂), which, multiplied by t, gives

d dtk√

t√

ρutk²₂+µk√

t∇utk²₂ ≤ C(Ψ¹⁶+ Ψ⁷||√ t√

ρut||²₂+k√ ρutk²₂)

≤ C(Ψ¹⁶+k√

ρu_tk²₂).

Integrating this in t and applying Proposition 2.3, the conclusion follows.

Proposition 2.5. The following estimate holds Z T

0

(k∇uk∞+k∇²ukq)dt ≤C+C Z T

0

Ψ^α2dt, with α2 := max{16, α1}= max

n

16,^(5q−6)_2q(6−q)² o

.

Proof. Noticing thatt^−5q−64q ∈(0,1), forq ∈(3,6), and recalling the following estimate by Proposition 2.2

k∇²uk_q ≤C(k√

t∇u_tk²₂+k√

ρu_tk²₂+t^−5q−64q + Ψ^α1),

it follows from the Gagliardo-Nirenberg and Young inequalities and Propositions 2.3 and 2.4 that

Z T 0

k∇uk∞dt ≤ C Z T

0

k∇uk^1−θ₂ k∇²uk^θ_qdt

≤ C Z T

0

(k∇uk₂+k∇²uk_q)dt

≤ C Z T

0

(k√

t∇u_tk²₂+k√

ρu_tk²₂+t^−5q−64q + Ψ^α1)dt

≤ C+C Z T

0

Ψ^α2dt,

where θ = _5q−6^3q ∈(0,1), proving the conclusion.

Proposition 2.6. The following estimate holds sup

0≤t≤T

(kρk∞+kPk∞)≤Cexp

C Z T

0

Ψ^α2dt

, where α2 is the number in Proposition 2.5.

Proof. DefineX(t;x) as

_d

dtX(t;x) =u(X(t;x), t), X(0;x) = x.

One can show that for any t ∈(0, T), and for anyy ∈R³, there is a unique x ∈R³, such that X(t;x) = y, and, in particular, X(t;R³) = R³; in fact, to show this, it suffices to consider the backward problem _dt^dZ(t) = u(Z(t), t), X(T;x) = y.Then, by (1.2), it has

d

dtρ(X(t;x), t) = ∂_tρ(X(t;x), t) +u(X(t;x), t)· ∇ρ(X(t;x), t)

= −divu(X(t;x), t)ρ(X(t;x), t), and, thus,

ρ(X(t;x), t) =ρ₀(x) exp

− Z t

0

divu(X(τ;x), τ)dτ

. (2.6)

Therefore,

kρk∞(t) = kρ(X(t;x), t)k∞(t)

≤ kρ0k∞exp Z T

0

k∇uk∞dt

,

and the conclusion follows by applying Proposition 2.4.

Proposition 2.7. The following estimate holds sup

0≤t≤T

(k∇Pk₂+||∇P||_q)≤Cexp

C Z T

0

Ψ^α2dt

, q∈(3,6), where α₂ is the number in Proposition 2.5.

Proof. From (2.4), one has

∂_t∇P +γdivu∇P +γP∇divu+ (u· ∇)∇P +∇P∇u= 0.

Multiplying the above by |∇P|^p−2∇P, integrating overR³, one has d

dtk∇Pk^p_p ≤C(||∇u||∞||∇P||^p_p+||P||∞||∇²u||_p||∇P||^p−1_p ), which gives

d

dt||∇P||_p ≤C(||∇u||∞||∇P||_p+||P||∞||∇²u||_p).

By the Gronwall inequality, one has sup

0≤t≤T

||∇P||p ≤C

k∇P0kp+ Z T

0

kPk∞k∇²ukpdt

exp

C Z T

0

k∇uk∞dt

≤C

1 + Z T

0

k∇²ukpdt

exp

C Z T

0

Ψ^α2dt

.

(2.7)

Thanks to the above, it follows from Proposition 2.5 and Proposition 2.6 that sup

0≤t≤T

||∇P||_q ≤ C

1 + Z T

0

Ψ^α2dt

exp

C Z T

0

Ψ^α2dt

≤ Cexp

C Z T

0

Ψ^α2dt

,

where we have used the fact that e^z ≥ 1 +z for z ≥ 0. By Proposition 2.2 and Proposition 2.3, it follows from (2.7) and the Cauchy inequality that

sup

0≤t≤T

||∇P||₂ ≤ C

1 + Z T

0

(Ψ⁵+ Ψ¹²k√

ρu_tk₂)dt

exp

C Z T

0

Ψ^α2dt

≤ C

1 + Z T

0

(Ψ⁵+k√

ρu_tk²₂)dt

exp

C Z T

0

Ψ^α2dt

≤ C

1 + Z T

0

Ψ¹⁰dt

exp

C Z T

0

Ψ^α2dt

≤ Cexp

C Z T

0

Ψ^α2dt

.

This proves the conclusion.

Proposition 2.8. The following estimates hold sup

0≤t≤T

(kρ−ρ∞k₂+k∇ρk₂+k∇ρk_q)≤Cexp

C Z T

0

Ψ^α2dt

, q ∈(3,6),

with constant C depending also on kρ0−ρ∞k2 +k∇ρ0k2+k∇ρ0kq, and sup

0≤t≤T

k√

t∇²uk²₂ ≤C sup

0≤t≤T

(Ψ¹⁰+ Ψk√ t√

ρu_tk²₂), where α₂ is the number in Proposition 2.5.

Proof. The estimate of kρ−ρ∞k₂ follows in the same way as that for kP −P∞k₂ in Proposition 2.3, while those for k∇ρk2 and k∇ρkq can be proved similarly as in Proposition 2.7. The conclusion for k√

t∇²uk²₂ follows from combining Propositions

2.2 and 2.4.

We end up this section with the following proposition on the lifespan estimate and a priori estimates.

Proposition 2.9. Assume in addition to the conditions in Proposition 2.1 thatρ≥ρ for some positive constant ρ.

Then, there are two positive constants T and C depending only on µ, λ, a, γ, q, and the upper bound of ψ₀ := kρ₀k∞+kρ₀ −ρ∞k₂ +k∇ρ₀k_L²∩L^q +k∇u₀k₂, and, in particular, independent of ρ and k∇²u₀k₂, such that system (1.1)–(1.2), subject to (1.3), has a unique solution (ρ, u) on R³×(0,T), enjoying the regularities stated in Proposition 2.1, with T∗ there replaced by T, and the following a priori estimates

sup

0≤t≤T

(kρk∞+kρ−ρ∞k₂+k∇ρk₂ +k∇ρk_q+kρ_tk₂) ≤ C, sup

0≤t≤T

k∇uk²₂+ Z T

0

(k∇²uk²₂ +k√

ρu_tk²₂)dt ≤ C, sup

0≤t≤T

(k√ t√

ρu_tk²₂+k√

t∇²uk²₂) + Z T

0

(k√

t∇u_tk²₂ +k√

t∇²uk²_q)dt ≤ C.

Proof. Define the maximal time T_max as

T_max := max{T ∈T}, where

T := {T ∈[T∗,∞)| There is a solution (ρ, u) in the class XT

to system (1.1)−(1.2), subject to (1.3), onR³×(0, T)},

where XT is the class of (ρ, u) enjoying the regularities as stated in Proposition 2.1, with T∗ there replaced with T. By Proposition 2.1, it is clear that T^max is well defined and T^max ≥ T∗. Moreover, by the uniqueness result, see the proof of the uniqueness part of Theorem 1.1 in the next section, one can easily show that any two solutions ( ¯ρ,u) and ( ˜¯ ρ,u) to system (1.1)–(1.2), subject to (1.3), on˜ R³×(0,T¯) and on R³×(0,T˜), respectively, coincide on R³×(0,min{T ,¯ T˜}). Choose T_k ∈ T

with Tk ↑ T^max as k ↑ ∞. By definition of T, there is a solution (ρk, uk) to system (1.1)–(1.2), subject to (1.3), on R³×(0, T_k). Define (ρ, u) on R³×(0, T^max) as

(ρ, u)(x, t) = (ρ_k, u_k)(x, t), x∈R³, t∈(0, T_k), k= 1,2,· · · .

Applying the uniqueness result again, the definition of (ρ, u) is independent of the choice of the sequence {T_k}^∞_k=1. By the construction of (ρ, u), one can verify that (ρ, u) is a solution to (1.1)–(1.2), subject to (1.3), onR³×(0, T^max), and (ρ, u)∈XT, for any T ∈(0, T^max).

By Propositions 2.3, 2.4, 2.6, and 2.7, it is clear Ψ(t)≤C_mexp

C_m

Z t 0

Ψ^α2dτ

, t ∈(0, T^max),

whereC_mis a positive constant depending only onµ,λ,a,γ,q, and the upper bound of ψ₀. Here we have used the fact that Ψ₀ can be controlled by ψ₀. One can easily derive from the above inequality that

Ψ(t)≤2^α12C_m, ∀t ∈ 0,min

T^max,(2^α2C_m^α2+1)⁻¹ . (2.8) Thanks to the above estimate, one can get by applying Propositions 2.5 and 2.8 that

(kρ−ρ_∞k₂+k∇ρk₂+k∇ρk_q+k√

t∇²uk²₂)(t) + Z t

0

k∇uk_∞dτ ≤C, (2.9) for any t ∈ (0,min{T^max,(2^α2C_m^α2+1)⁻¹}), and for a positive constant C depending only on µ, λ, a, γ, q, and the upper bound of ψ₀. Thanks to (2.8)–(2.9) and using (1.2) one can further obtain

kρ_tk₂ = ku· ∇ρ+ divuρk₂

≤ kuk₆k∇ρk₃ +k∇uk₂kρk∞

≤ C(1 +k∇ρk_L²_∩L^q)k∇uk₂ ≤C, (2.10) for any 0 < t < min{T^max,(2^α2C_m^α2+1)⁻¹}. Using the estimate Rt

0 k∇uk∞dτ ≤ C in (2.9) and recalling (2.6), it is clear that

ρ(x, t)≥Cρ, x∈R³, 0< t <min

T^max,(2^α2C_m^α2+1)⁻¹ . (2.11) We claim that T^max > (2^α2C_m^α2+1)⁻¹. Assume in contradiction that this does not hold. Then, all the estimates in (2.8)–(2.11) hold for any t ∈ (0, T^max). Estimates (2.8)–(2.11), holding on time interval (0, T^max), guarantee that (ρ, u)(·, t) can be uniquely extended to timeT^max, with (ρ, u)(·, T^max) defined as the limit of (ρ, u)(·, t) as t↑T^max, and that

(ρ−ρ∞)(·, T^max)∈H¹∩W^1,q, u(·, T^max)∈D₀¹∩D².

Thanks to this and recalling (2.11), it is clear that the compatibility condition holds at time T^max. Therefore, by the local well-posedness result, i.e., Proposition 2.1, one can further extend solution (ρ, u) beyond the time T^max, which contradicts to the definition of T^max. This contradiction proves the claim that T^max > (2^α2C_m^α2+1)⁻¹.

As a result, one obtains a solution (ρ, u) on time interval (0,(2^α2C_m^α2+1)⁻¹) satisfying all the a priori estimates in (2.8)–(2.10), except RT

0 k√

t∇²uk²_qdt ≤ C, on the same time interval.

It remains to verify RT 0 k√

t∇²uk²_qdt ≤ C. To this end, recalling (2.2) and (2.3), it follows from the elliptic estimate, the estimates just obtained, and the Young inequality that

k∇²ukq ≤ C(kρutkq+kρ(u· ∇)ukq+k∇Pkq)

≤ C(kρk

5q−6

∞4q k√ ρu_tk

6−q 2q

2 k∇u_tk

3q−6 2q

2 +kρk∞k∇uk

3 q

2k∇²uk

2q−3 q

2 +k∇ρk_q)

≤ C(1 +k∇²uk²₂+k√

ρu_tk₂+k∇u_tk₂), and further that

Z T 0

k√

t∇²uk²_qdt ≤C Z T

0

(1 +k∇²uk²₂k√

t∇²uk²₂+k√

ρu_tk²₂+k√

t∇u_tk²₂)dt≤C,

proving the conclusion.

3. Proof of Theorem 1.1 This section is devoted to proving Theorem 1.1.

The following lemma, proved in [33], will be used in proving the uniqueness.

Lemma 3.1. Given a positive time T and nonnegative functions f, g, G on [0, T], with f and g being absolutely continuous on [0, T]. Suppose that







d

dtf(t)≤δ(t)f(t) +Ap G(t),

d

dtg(t) +G(t)≤α(t)g(t) +β(t)f²(t), f(0) = 0,

where α, β and δ are three nonnegative functions, with α, δ, tβ ∈L¹((0, T)).

Then, then following estimates hold f(t) ≤ ABp

tg(0) exp 1

2 Z t

0

(α(s) +A²B²sβ(s))ds

, g(t) +

Z t 0

G(s)ds ≤ g(0) exp Z t

0

(α(s) +A²B²sβ(s))ds

,

where B = 1 +e^R0Tδ(τ)dτ. In particular, if g(0) = 0, then f ≡g ≡0 on (0, T).

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1. We first prove the uniqueness and then the existence.

Uniqueness: Let ( ˇρ,u), ( ˆˇ ρ,u) be two solutions of system (1.1)-(1.2), subject toˆ (1.3), onR³×(0, T), satisfying the regularities stated in the theorem. Foru∈ {ˆu,u},ˇ

by the Gagliardo-Nirenberg and H¨older inequalities, one has Z T

0

k∇uk∞dt ≤ C Z T

0

k∇uk^1−θ₂ k∇²uk^θ_qdt ≤C Z T

0

k√

t∇²uk^θ_qt^−θ2dt

≤ C Z T

0

k√

t∇²uk²_qdt

^θ2 Z T 0

t^−2−θθ dt ^1−θ2

<∞, where θ = _5q−6^3q ∈(0,1). Therefore, ∇ˆu,∇ˇu∈L¹(0, T;L^∞).

Denote

σ= ˇρ−ρ,ˆ W = ˇP −P ,ˆ v = ˇu−u.ˆ Then, straightforward calculations show

σ_t+v· ∇ˆρ+ ˇu· ∇σ+ div ˇuσ+ divvρˆ= 0, (3.1) ˆ

ρ(v_t+ ˆu· ∇v)−µ∆v −(λ+µ)∇divv +∇W =−σˇu_t−σˇu· ∇ˇu−ρvˆ · ∇ˇu, (3.2) Wt+v· ∇Pˆ+ ˇu· ∇W +γdiv ˇuW +γdivvPˆ = 0. (3.3) Testing (3.1) with σ and using the H¨older inequality, we have

d dt

Z

|σ|²dx ≤ C Z

(|v· ∇ρ| |σ|ˆ +|divˇu||σ|²+|divvρ| |σ|)ˆ

≤ C(||∇ρ||ˆ ₃||σ||₂||v||₆+||∇ˇu||∞||σ||²₂) +C||ρ||ˆ ∞||∇v||₂||σ||₂

≤ C||∇ˇu||_∞||σ||²₂+C(kˆρk_∞+k∇ˆρk₃)||∇v||₂||σ||₂, and, thus,

d

dtkσk₂ ≤C||∇ˇu||_∞||σ||₂+C(kρkˆ _∞+k∇ˆρk₃)||∇v||₂. (3.4) Similarly, by testing (3.3) with W yields

d

dt||W||₂ ≤C||∇ˇu||∞||W||₂+C(||∇Pˆ||₃+||Pˆ||∞)||∇v||₂. (3.5) Testing (3.2) with v and using the H¨older and Young inequalities, we have

1 2

d dt

Z

ρ|v|ˆ ²dx+ Z

[µ|∇v|²+ (λ+µ)(divv)²]dx

≤ Z

(|W||∇v|+|σ||ˇut||v|+|σ||ˇu||∇ˇu||v|+|ρ(vˆ · ∇ˇu)·v|)dx=:RHS. (3.6) We proceed the proof separately for the cases ρ∞ = 0 andρ∞>0.

Case I: ρ∞ = 0.

By the H¨older, Sobolev, and Young inequalities, we can control RHS as RHS ≤ kWk₂k∇vk₂+kσk³

2kˇu_tk₆kvk₆ +kσk₂kˇuk₆k∇ˇuk₆kvk₆+k∇ˇuk∞kp

ρvkˆ ²₂

≤ C(kWk₂k∇vk₂+kσk³

2k∇ˇu_tk₂k∇vk₂

+kσk₂k∇ˇuk₂k∇²ukˇ ₂k∇vk₂+k∇ˇuk∞kp ρvkˆ ²₂)

≤ µ

2k∇vk²₂+C(kWk²₂ +k∇ˇu_tk²₂kσk²3 2

+k∇ˇuk²₂k∇²ukˇ ²₂kσk²₂+k∇ˇuk∞kp ρvkˆ ²₂), which plugged into (3.6) leads to

d dtkp

ρvkˆ ²₂+µk∇vk²₂ ≤ Ck∇ˇuk_∞kp

ρvkˆ ²₂+C(1 +k∇ˇutk²₂+k∇ˇuk²₂|∇²ukˇ ²₂)

×(kWk²₂+kσk²₂+kσk²3 2

). (3.7)

The appearance ofkσk³

2 in the above inequality requires the energy estimate forkσk³ given in the below. 2

Testing (3.1) with sign(σ)|σ|¹² and using the H¨older and Sobolev inequalities that d

dt Z

|σ|³²dx ≤ C Z

(|v· ∇ˆρ| |σ|¹² +|divˇu||σ|³² +|divvρ| |σ|ˆ ¹²)

≤ C(||∇ˆρ||₂||σ||

1 2 3 2

||v||₆+||∇ˇu||∞||σ||

3 2 3 2

) +C||ˆρ||₆||∇v||₂||σ||

1 2 3 2

≤ C||∇ˇu||∞||σ||³²3 2

+C||∇ρ||ˆ ₂||∇v||₂||σ||¹²3 2

, which gives

d dtkσk³

2 ≤C||∇ˇu||_∞||σ||³

2 +C||∇ρ||ˆ ₂||∇v||₂. (3.8) Denote

f1(t) = (kσk³

2 +kσk2+kWk2)(t), g1(t) =kp

ρvkˆ ²₂(t), G1(t) =µk∇vk²₂(t), δ1(t) = Ck∇ˇuk∞(t), A1 =C sup

0≤t≤T

(kˆρk∞+kPˆk∞+k∇ˆρk_L²∩L³ +k∇Pˆk3)(t), α₁(t) =Ck∇ˇuk∞(t), β₁(t) =C(1 +k∇ˇu_tk²₂+k∇ˇuk²₂k∇²ukˇ ²₂)(t),

then, it follows from (3.4), (3.5), (3.7), and (3.8) that







d

dtf₁(t)≤δ₁(t)f₁(t) +A₁p G₁(t),

d

dtg₁(t) +G₁(t)≤α₁(t)g₁(t) +β₁(t)f₁²(t), f₁(0) = 0.

By the regularities of ( ˆρ,u) and ( ˇˆ ρ,u), and recallingˇ ∇ˇu ∈ L¹(0, T;L^∞), one can easily verify that α₁, δ₁, tβ₁ ∈L¹((0, T)). Therefore, one can apply Lemma 3.1 to get f₁(t) = g₁(t) = G₁(t) = 0, on (0, T), which implies the uniqueness for Case I.

Case II: ρ∞ >0.

By the H¨older and Sobolev inequalities, it follows for (ρ, u)∈ {( ˆρ,u),ˆ ( ˇρ,u)}ˇ that ρ⁴_∞

Z

|u_t|²dx = Z

|ρ∞−ρ+ρ|⁴|u_t|²dx

≤ C Z

(|ρ−ρ∞|⁴+ρ⁴)|u_t|²dx

≤ C(kρ−ρ∞k⁴₆ku_tk²₆+kρk³_∞k√ ρu_tk²₂)

≤ C(k∇ρk⁴₂k∇u_tk²₂ +kρk³_∞k√

ρu_tk²₂), and, thus,

Z T 0

tku_tk²₂dt≤C sup

0≤t≤T

(k∇ρk⁴₂+kρk³_∞) Z T

0

(k√

t∇u_tk²₂+k√

ρu_tk²₂)dt <∞, that is, √

tu_t∈L²(0, T;L²), foru∈ {ˆu,u}.ˇ

By the H¨older, Sobolev, and Cauchy inequalities, we deduce RHS ≤ kWk₂k∇vk₂+kσk₂kˇu_tk₃kvk₆

+kσk₂kˇuk₆k∇ˇuk₆kvk₆+k∇ˇuk∞kp ρvkˆ ²₂

≤ kWk₂k∇vk₂+k∇ˇuk∞kp ρvkˆ ²₂ +(kˇu_tk

1 2

2k∇ˇu_tk

1 2

2 +k∇ˇuk₂k∇²ukˇ ₂)k∇vk₂kσk₂

≤ µ

2k∇vk²₂ +Ck∇ˇuk∞kp

ρvkˆ ²₂+C(1 +kˇu_tk₂k∇ˇu_tk₂ +k∇ˇuk²₂k∇²ukˇ ²₂)(kσk²₂+kWk²₂).

Plugging this into (3.6) leads to d

dtkp

ρvkˆ ²₂+µk∇vk²₂ ≤ Ck∇ˇuk∞kp

ρvkˆ ²₂+C(1 +kˇu_tk₂k∇ˇu_tk₂

+k∇ˇuk²₂k∇²ukˇ ²₂)(kσk²₂+kWk²₂). (3.9) Denote

f₂(t) = (kσk₂+kWk₂)(t), g₂(t) =kp

ρvkˆ ²₂(t), G₂(t) =µk∇vk²₂(t), δ₂(t) = Ck∇ˇuk_∞(t), A₂ =C sup

0≤t≤T

(kˆρk_∞+kPˆk_∞+k∇ˆρk₂+k∇Pˆk₃)(t), α₂(t) =Ck∇ˇuk∞(t), β₂(t) =C(1 +kuˇ_tk₂k∇ˇu_tk₂+k∇ˇuk²₂k∇²ukˇ ²₂)(t), then, it follows from (3.4), (3.5), and (3.9) that







d

dtf2(t)≤δ2(t)f2(t) +A2

pG2(t),

d

dtg₂(t) +G₂(t)≤α₂(t)g₂(t) +β₂(t)f₂²(t), f₂(0) = 0.

By the regularities of ( ˆρ,u) and ( ˇˆ ρ,u), and recallingˇ ∇u∈ L¹(0, T;L^∞) and √ tut ∈ L²(0, T;L²), for u∈ {ˆu,u}, one can easily verify thatˇ α2, δ2, tβ2 ∈L¹((0, T)). There- fore, one can apply Lemma 3.1 to get f₂(t) = g₂(t) = G₂(t) = 0, on (0, T), which implies the uniqueness for Case II.