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)
inR^{3}×(0, T), where the densityρ≥0 and the velocity fieldu∈R^{3}are 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} = (ρ_{0}, ρ_{0}u_{0}),

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 L^{2} 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

ρ_{0}−ρ∞∈H^{1}∩W^{1,q}, u_{0} ∈D^{1}_{0}∩D^{2}, (1.4)
for some constant ρ∞∈[0,∞), and the compatibility condition

−µ∆u_{0}−(µ+λ)∇divu_{0}+∇P(ρ_{0}) =√

ρ_{0}g, (1.5)

for some g ∈ L^{2}, 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}(R^{3}) andW^{k,r} =W^{k,r}(R^{3}) to denote, respectively, the standard
Lebesgue and Sobolev spaces inR^{3}, 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∈L^{1}_{loc}(R^{3})

k∇^{k}ukr <∞o

, D^{k} =D^{k,2},
D_{0}^{1} =n

u∈L^{6}

k∇uk_{2} <∞o
.
For simplicity of notations, we adopt the notation

Z

f dx= Z

R^{3}

f dx.

Our main result is the following:

Theorem 1.1. Suppose that the initial data (ρ_{0}, u_{0}) satisfies

ρ_{0} ≥0, ρ−ρ∞ ∈H^{1}∩W^{1,q}, u_{0} ∈D^{1}_{0}, ρ_{0}u_{0} ∈L^{2},
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 ψ_{0} :=kρ_{0}k∞+kρ_{0}−ρ∞k_{2}+k∇ρ_{0}k_{L}^{2}_{∩L}^{q} +k∇u_{0}k_{2}, such that system
(1.1)–(1.2), subjects to (1.3), admits a unique solution(ρ, u)onR^{3}×(0,T),satisfying

ρ−ρ∞ ∈C([0,T];L^{2})∩L^{∞}(0,T;H^{1}∩W^{1,q}), ρt∈L^{∞}(0,T;L^{2}),
ρu∈C([0,T];L^{2}), u∈L^{∞}(0,T;D^{1}_{0})∩L^{2}(0,T;D^{2}), √

ρu_{t}∈L^{2}(0,T;L^{2}),

√tu ∈L^{∞}(0,T;D^{2})∩L^{2}(0,T;D^{2,q}), √

tu_{t}∈L^{2}(0,T;D_{0}^{1}).

Remark 1.1. (i) Compared with the local well-posedness results established in [7, 8],
in Theorem 1.1, u_{0} is not required to be in D^{2} 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 u_{0} ∈ D^{1}_{0} ∩D^{2} and some compatibility
condition; however, it should be emphasized that all these estimates depend neither
on k∇^{2}u_{0}k_{2} 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 ρ_{0} and u_{0} satisfy the regularity condition

ρ_{0} ≥0, ρ_{0}−ρ∞∈H^{1}∩W^{1,q}, u_{0} ∈D_{0}^{1}∩D^{2},
and the compatibility condition

−µ∆u_{0}−(µ+λ)∇divu+∇P_{0} =√
ρ_{0}g,

for some g ∈L^{2}, where P0 =aρ^{γ}_{0}.

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∗];H^{1}∩W^{1,q}), u∈C([0, T∗];D^{1}_{0}∩D^{2})∩L^{2}(0, T∗;D^{2,q}),
ρ_{t}∈C([0, T∗];L^{2}∩L^{q}), u_{t}∈L^{2}(0, T∗;D_{0}^{1}), √

ρu_{t} ∈L^{∞}(0, T∗;L^{2}).

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

Ψ_{0} :=kρ_{0}k∞+kP_{0}−P∞k_{2}+kP_{0}k∞+k∇P_{0}k_{2}+k∇P_{0}k_{q}+k∇u_{0}k_{2},

with P∞=aρ^{γ}_{∞}, and, in particular, independent ofk∇^{2}u_{0}k_{2} andkgk_{2}. The following
quantity plays the key role in this section

Ψ(t) := (kρk∞+kP −P∞k_{2}+kPk∞+k∇Pk_{2}+k∇Pk_{q}
+k∇uk_{2}+k√

t√

ρu_{t}k_{2})(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 R^{3} ×(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 Ψ_{0}.

Proposition 2.2. The following estimates hold
k∇^{2}uk^{2}_{2} ≤ C(Ψ^{10}+ Ψk√

ρu_{t}k^{2}_{2}),
k√

ρ(u· ∇)uk^{2}_{2} ≤ C(Ψ^{9}+ Ψ^{5}k√

ρu_{t}k_{2}),
k∇^{2}uk_{q} ≤ C(k√

t∇u_{t}k^{2}_{2} +k√

ρu_{t}k^{2}_{2} +t^{−}^{5q−6}^{4q} + Ψ^{α}^{1}), q∈(3,6),
with α_{1} := maxn

12,^{(5q−6)}_{2q(6−q)}^{2}o
.

Proof. Applying the elliptic estimates to (1.1) yields
k∇^{2}uk^{2}_{2} ≤Ckρk_{∞}(k√

ρu_{t}k^{2}_{2}+k√

ρ(u· ∇)uk^{2}_{2}) +Ck∇Pk^{2}_{2}.
By the H¨older and Sobolev inequality, one has

k√

ρ(u· ∇)uk^{2}_{2} ≤ kρk∞kuk^{2}_{6}k∇uk2k∇uk6 ≤CΨ^{4}k∇^{2}uk2.

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

k∇^{2}uk^{2}_{2} ≤ C(Ψk√

ρutk^{2}_{2}+ Ψ^{5}k∇^{2}uk2 + Ψ^{2})

≤ 1

2k∇^{2}uk^{2}_{2}+C(Ψk√

ρu_{t}k^{2}_{2}+ Ψ^{10}),
that is

k∇^{2}uk^{2}_{2} ≤C(Ψ^{10}+ Ψk√

ρu_{t}k^{2}_{2}), (2.1)

and, consequently,

k√

ρ(u· ∇)uk^{2}_{2} ≤C(Ψ^{9}+ Ψ^{5}k√

ρu_{t}k_{2}),
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∇uk_{6}

≤ Ckρk_{∞}kuk

3 q

6k∇uk^{2−}

3 q

6 ≤Ckρk_{∞}k∇uk

3 q

2k∇^{2}uk

2q−3 q

2 , (2.2)

from which, by the Young inequality and using (2.1), one has
kρ(u· ∇)uk_{q} ≤ CΨ^{q+3}^{q} (Ψ^{10}+ Ψk√

ρu_{t}k^{2}_{2})^{2q−3}^{2q}

≤ CΨ^{3}(Ψ^{9}+k√

ρu_{t}k^{2}_{2})^{2q−3}^{2q}

≤ C(Ψ^{2q}+ Ψ^{9}+k√

ρu_{t}k^{2}_{2})≤C(Ψ^{12}+k√

ρu_{t}k^{2}_{2}).

It follows from the H¨older and Sobolev inequalities that
kρu_{t}k_{q} ≤ kρk

5q−6

∞4q k√
ρu_{t}k

6−q 2q

2 ku_{t}k

3q−6 2q

6 ≤Ckρk

5q−6

∞4q k√
ρu_{t}k

6−q 2q

2 k∇u_{t}k

3q−6 2q

2 , (2.3)

and further by the Young inequality that
kρu_{t}k_{q} ≤ CΨ^{5q−6}^{4q} t^{−}^{3q−6}^{4q} k√

t∇u_{t}k

3q−6 2q

2 k√

ρu_{t}k

6−q 2q

2

≤ C(k√

t∇utk^{2}_{2}+k√

ρutk^{2}_{2}+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∇^{2}uk_{q} ≤ C(kρu_{t}k_{q}+kρ(u· ∇)uk_{q}+k∇Pk_{q})

≤ C(k√

t∇u_{t}k^{2}_{2}+k√

ρu_{t}k^{2}_{2}+t^{−}^{5q−6}^{4q} + Ψ^{α}^{1}),

proving the conclusion.

Proposition 2.3. The following estimate holds sup

0≤t≤T

(k∇uk^{2}_{2}+kP −P∞k^{2}_{2}) +
Z T

0

k√

ρu_{t}k^{2}_{2}dt≤C+C
Z T

0

Ψ^{10}dt.

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

2 d

dt(µk∇uk^{2}_{2} + (µ+λ)kdivuk^{2}_{2}) +k√

ρu_{t}k^{2}_{2} =−
Z

(ρ(u· ∇)u+∇P)·u_{t}dx.

Integration by parts and noticing that

P_{t}+u· ∇P +γdivuP = 0, (2.4)

one deduces

− Z

∇P ·u_{t}dx =
Z

(P −P∞)divu_{t}dx

= d dt

Z

(P −P∞)divudx− Z

P_{t}divudx

= d

dt Z

(P −P∞)divudx+ Z

(u· ∇P +γdivuP)divudx.

Therefore d dt

µk∇uk^{2}_{2}+ (µ+λ)kdivuk^{2}_{2}−2
Z

(P −P∞)divudx

+k√
ρu_{t}k^{2}_{2}

= Z

(u· ∇P +γdivuP)divudx− Z

ρ(u· ∇)u·u_{t}dx =:I_{1}+I_{2}.

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

I_{1} ≤ kuk_{6}k∇Pk_{2}kdivuk_{3}+γkdivuk^{2}_{2}kPk∞

≤ Ck∇uk2k∇Pk2k∇uk

1 2

2k∇^{2}uk

1 2

2 +CΨ^{3}

≤ CΨ^{5}^{2}(Ψ^{5}^{2} + Ψ^{1}^{4}k√

ρu_{t}k_{2}^{1}^{2}) +CΨ^{3}

≤ 1 4k√

ρu_{t}k^{2}_{2}+CΨ^{5},
and

I_{2} ≤ k√

ρu_{t}k_{2}k√

ρ(u· ∇)uk_{2}

≤ C(Ψ^{9}^{2} + Ψ^{5}^{2}k√

ρu_{t}k_{2}^{1}^{2})k√
ρu_{t}k_{2}

≤ 1 4k√

ρu_{t}k^{2}_{2}+CΨ^{10}.
Therefore

1 2

d dt

µk∇uk^{2}_{2}+ (µ+λ)kdivuk^{2}_{2}−2
Z

(P −P∞)divudx

+k√

ρutk^{2}_{2} ≤CΨ^{10},
from which, one obtains by the Cauchy inequality that

sup

0≤t≤T

k∇uk^{2}_{2}+
Z T

0

k√

ρu_{t}k^{2}_{2}dt≤C

1 + sup

0≤t≤T

kP −P∞k^{2}_{2}+
Z T

0

Ψ^{10}dt

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

d

dtkP −P∞k^{2}_{2} = −(γ− 1
2)

Z

divu(P −P∞)^{2}dx−γP∞

Z

divu(P −P∞)dx

≤ Ck∇uk2kP −P∞k

1 2

2k∇Pk

3 2

2 +Ck∇uk2kP −P∞k2 ≤CΨ^{3},
which gives

sup

0≤t≤T

kP −P∞k^{2}_{2} ≤C+C
Z T

0

Ψ^{3}dt.

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√

ρutk^{2}_{2}+
Z T

0

k√

t∇utk^{2}_{2}dt≤C+C
Z T

0

Ψ^{16}dt.

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

ρ(∂_{t}u_{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 R^{3} and then using the continuity
equation (1.2), one has

1 2

d dtk√

ρu_{t}k^{2}_{2}+µk∇u_{t}k^{2}_{2}+ (λ+µ)kdivu_{t}k^{2}_{2}

= Z

P_{t}divu_{t}dx+
Z

div (ρu)|u_{t}|^{2}dx+
Z

div (ρu)(u· ∇)u·u_{t}dx

− Z

ρ(u_{t}· ∇u)·u_{t}dx=:II_{1}+II_{2}+II_{3}+II_{4}.

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 Ψ^{4}+||∇u||^{2}_{2}||∇P||^{2}_{L}2∩L^{q}

+ µ

8||∇u_{t}||^{2}_{2}

≤ CΨ^{4}+µ

8||∇u_{t}||^{2}_{2}.

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

II_{2} = −
Z

R^{3}

ρu· ∇|u_{t}|^{2}dx

≤ 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

≤ CΨ^{7}||√

ρu_{t}||^{2}_{2}+ µ

8||∇u_{t}||^{2}_{2},
II_{3} ≤

Z

R^{3}

ρ|u|(|∇u|^{2}|u_{t}|+|u||∇^{2}u||u_{t}|+|u||∇u||∇u_{t}|)dx

≤ C||ρ||∞(||u||_{6}||∇u||^{2}_{3}||u_{t}||_{6}+||u||^{2}_{6}||∇^{2}u||_{2}||u_{t}||_{6}
+||u||^{2}_{6}||∇u||_{6}||∇u_{t}||_{2})

≤ C||ρ||_{∞}||∇u||^{2}_{2}||∇^{2}u||_{2}||∇u_{t}||_{2}

≤ CΨ^{6}||∇^{2}u||^{2}_{2}+µ

8||∇u_{t}||^{2}_{L}2

≤ C(Ψ^{16}+ Ψ^{7}||√

ρu_{t}||^{2}_{2}) + µ

8||∇u_{t}||^{2}_{L}2,
and

II_{4} ≤
Z

R^{3}

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

≤ CΨ^{7}||√

ρu_{t}||^{2}_{2} +µ

8||∇u_{t}||^{2}_{2}.
Therefore, we have

d dtk√

ρu_{t}k^{2}_{2}+µk∇u_{t}k^{2}_{2} ≤C(Ψ^{16}+ Ψ^{7}||√

ρu_{t}||^{2}_{2}),
which, multiplied by t, gives

d dtk√

t√

ρutk^{2}_{2}+µk√

t∇utk^{2}_{2} ≤ C(Ψ^{16}+ Ψ^{7}||√
t√

ρut||^{2}_{2}+k√
ρutk^{2}_{2})

≤ C(Ψ^{16}+k√

ρu_{t}k^{2}_{2}).

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∇^{2}ukq)dt ≤C+C
Z T

0

Ψ^{α}^{2}dt,
with α2 := max{16, α1}= max

n

16,^{(5q−6)}_{2q(6−q)}^{2}
o

.

Proof. Noticing thatt^{−}^{5q−6}^{4q} ∈(0,1), forq ∈(3,6), and recalling the following estimate
by Proposition 2.2

k∇^{2}uk_{q} ≤C(k√

t∇u_{t}k^{2}_{2}+k√

ρu_{t}k^{2}_{2}+t^{−}^{5q−6}^{4q} + Ψ^{α}^{1}),

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−θ}_{2} k∇^{2}uk^{θ}_{q}dt

≤ C Z T

0

(k∇uk_{2}+k∇^{2}uk_{q})dt

≤ C Z T

0

(k√

t∇u_{t}k^{2}_{2}+k√

ρu_{t}k^{2}_{2}+t^{−}^{5q−6}^{4q} + Ψ^{α}^{1})dt

≤ C+C Z T

0

Ψ^{α}^{2}dt,

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^{α}^{2}dt

, 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 ∈R^{3}, there is a unique x ∈R^{3},
such that X(t;x) = y, and, in particular, X(t;R^{3}) = R^{3}; in fact, to show this, it
suffices to consider the backward problem _{dt}^{d}Z(t) = u(Z(t), t), X(T;x) = y.Then, by
(1.2), it has

d

dtρ(X(t;x), t) = ∂_{t}ρ(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) =ρ_{0}(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∇Pk_{2}+||∇P||_{q})≤Cexp

C Z T

0

Ψ^{α}^{2}dt

, q∈(3,6),
where α_{2} 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 overR^{3}, one has
d

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

d

dt||∇P||_{p} ≤C(||∇u||∞||∇P||_{p}+||P||∞||∇^{2}u||_{p}).

By the Gronwall inequality, one has sup

0≤t≤T

||∇P||p ≤C

k∇P0kp+ Z T

0

kPk∞k∇^{2}ukpdt

exp

C Z T

0

k∇uk∞dt

≤C

1 + Z T

0

k∇^{2}ukpdt

exp

C Z T

0

Ψ^{α}^{2}dt

.

(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^{α}^{2}dt

exp

C Z T

0

Ψ^{α}^{2}dt

≤ Cexp

C Z T

0

Ψ^{α}^{2}dt

,

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

(Ψ^{5}+ Ψ^{1}^{2}k√

ρu_{t}k_{2})dt

exp

C Z T

0

Ψ^{α}^{2}dt

≤ C

1 + Z T

0

(Ψ^{5}+k√

ρu_{t}k^{2}_{2})dt

exp

C Z T

0

Ψ^{α}^{2}dt

≤ C

1 + Z T

0

Ψ^{10}dt

exp

C Z T

0

Ψ^{α}^{2}dt

≤ Cexp

C Z T

0

Ψ^{α}^{2}dt

.

This proves the conclusion.

Proposition 2.8. The following estimates hold sup

0≤t≤T

(kρ−ρ∞k_{2}+k∇ρk_{2}+k∇ρk_{q})≤Cexp

C Z T

0

Ψ^{α}^{2}dt

, q ∈(3,6),

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

0≤t≤T

k√

t∇^{2}uk^{2}_{2} ≤C sup

0≤t≤T

(Ψ^{10}+ Ψk√
t√

ρu_{t}k^{2}_{2}),
where α_{2} is the number in Proposition 2.5.

Proof. The estimate of kρ−ρ∞k_{2} follows in the same way as that for kP −P∞k_{2}
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∇^{2}uk^{2}_{2} 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 ψ_{0} := kρ_{0}k∞+kρ_{0} −ρ∞k_{2} +k∇ρ_{0}k_{L}^{2}∩L^{q} +k∇u_{0}k_{2}, and, in
particular, independent of ρ and k∇^{2}u_{0}k_{2}, such that system (1.1)–(1.2), subject to
(1.3), has a unique solution (ρ, u) on R^{3}×(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∞k_{2}+k∇ρk_{2} +k∇ρk_{q}+kρ_{t}k_{2}) ≤ C,
sup

0≤t≤T

k∇uk^{2}_{2}+
Z T

0

(k∇^{2}uk^{2}_{2} +k√

ρu_{t}k^{2}_{2})dt ≤ C,
sup

0≤t≤T

(k√ t√

ρu_{t}k^{2}_{2}+k√

t∇^{2}uk^{2}_{2}) +
Z T

0

(k√

t∇u_{t}k^{2}_{2} +k√

t∇^{2}uk^{2}_{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), onR^{3}×(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˜ R^{3}×(0,T¯)
and on R^{3}×(0,T˜), respectively, coincide on R^{3}×(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 R^{3}×(0, T_{k}). Define (ρ, u) on R^{3}×(0, T^{max}) as

(ρ, u)(x, t) = (ρ_{k}, u_{k})(x, t), x∈R^{3}, 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), onR^{3}×(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_{m}exp

C_{m}

Z t 0

Ψ^{α}^{2}dτ

, t ∈(0, T^{max}),

whereC_{m}is a positive constant depending only onµ,λ,a,γ,q, and the upper bound
of ψ_{0}. Here we have used the fact that Ψ_{0} can be controlled by ψ_{0}. One can easily
derive from the above inequality that

Ψ(t)≤2^{α}^{1}^{2}C_{m}, ∀t ∈ 0,min

T^{max},(2^{α}^{2}C_{m}^{α}^{2}^{+1})^{−1} . (2.8)
Thanks to the above estimate, one can get by applying Propositions 2.5 and 2.8 that

(kρ−ρ_{∞}k_{2}+k∇ρk_{2}+k∇ρk_{q}+k√

t∇^{2}uk^{2}_{2})(t) +
Z t

0

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

kρ_{t}k_{2} = ku· ∇ρ+ divuρk_{2}

≤ kuk_{6}k∇ρk_{3} +k∇uk_{2}kρk∞

≤ C(1 +k∇ρk_{L}^{2}_{∩L}^{q})k∇uk_{2} ≤C, (2.10)
for any 0 < t < min{T^{max},(2^{α}^{2}C_{m}^{α}^{2}^{+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∈R^{3}, 0< t <min

T^{max},(2^{α}^{2}C_{m}^{α}^{2}^{+1})^{−1} . (2.11)
We claim that T^{max} > (2^{α}^{2}C_{m}^{α}^{2}^{+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})∈H^{1}∩W^{1,q}, u(·, T^{max})∈D_{0}^{1}∩D^{2}.

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^{α}^{2}C_{m}^{α}^{2}^{+1})^{−1}.

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

0 k√

t∇^{2}uk^{2}_{q}dt ≤ C, on the same
time interval.

It remains to verify RT 0 k√

t∇^{2}uk^{2}_{q}dt ≤ 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∇^{2}ukq ≤ C(kρutkq+kρ(u· ∇)ukq+k∇Pkq)

≤ C(kρk

5q−6

∞4q k√
ρu_{t}k

6−q 2q

2 k∇u_{t}k

3q−6 2q

2 +kρk∞k∇uk

3 q

2k∇^{2}uk

2q−3 q

2 +k∇ρk_{q})

≤ C(1 +k∇^{2}uk^{2}_{2}+k√

ρu_{t}k_{2}+k∇u_{t}k_{2}),
and further that

Z T 0

k√

t∇^{2}uk^{2}_{q}dt ≤C
Z T

0

(1 +k∇^{2}uk^{2}_{2}k√

t∇^{2}uk^{2}_{2}+k√

ρu_{t}k^{2}_{2}+k√

t∇u_{t}k^{2}_{2})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)f^{2}(t),
f(0) = 0,

where α, β and δ are three nonnegative functions, with α, δ, tβ ∈L^{1}((0, T)).

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

tg(0) exp 1

2 Z t

0

(α(s) +A^{2}B^{2}sβ(s))ds

, g(t) +

Z t 0

G(s)ds ≤ g(0) exp Z t

0

(α(s) +A^{2}B^{2}sβ(s))ds

,

where B = 1 +e^{R}^{0}^{T}^{δ(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), onR^{3}×(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−θ}_{2} k∇^{2}uk^{θ}_{q}dt ≤C
Z T

0

k√

t∇^{2}uk^{θ}_{q}t^{−}^{θ}^{2}dt

≤ C Z T

0

k√

t∇^{2}uk^{2}_{q}dt

^{θ}2 Z T
0

t^{−}^{2−θ}^{θ} dt
^{1−}^{θ}2

<∞,
where θ = _{5q−6}^{3q} ∈(0,1). Therefore, ∇ˆu,∇ˇu∈L^{1}(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|^{2}dx ≤ C
Z

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

≤ C(||∇ρ||ˆ _{3}||σ||_{2}||v||_{6}+||∇ˇu||∞||σ||^{2}_{2}) +C||ρ||ˆ ∞||∇v||_{2}||σ||_{2}

≤ C||∇ˇu||_{∞}||σ||^{2}_{2}+C(kˆρk_{∞}+k∇ˆρk_{3})||∇v||_{2}||σ||_{2},
and, thus,

d

dtkσk_{2} ≤C||∇ˇu||_{∞}||σ||_{2}+C(kρkˆ _{∞}+k∇ˆρk_{3})||∇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 ^{2}dx+
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 ≤ kWk_{2}k∇vk_{2}+kσk^{3}

2kˇu_{t}k_{6}kvk_{6}
+kσk_{2}kˇuk_{6}k∇ˇuk_{6}kvk_{6}+k∇ˇuk∞kp

ρvkˆ ^{2}_{2}

≤ C(kWk_{2}k∇vk_{2}+kσk^{3}

2k∇ˇu_{t}k_{2}k∇vk_{2}

+kσk_{2}k∇ˇuk_{2}k∇^{2}ukˇ _{2}k∇vk_{2}+k∇ˇuk∞kp
ρvkˆ ^{2}_{2})

≤ µ

2k∇vk^{2}_{2}+C(kWk^{2}_{2} +k∇ˇu_{t}k^{2}_{2}kσk^{2}3
2

+k∇ˇuk^{2}_{2}k∇^{2}ukˇ ^{2}_{2}kσk^{2}_{2}+k∇ˇuk∞kp
ρvkˆ ^{2}_{2}),
which plugged into (3.6) leads to

d dtkp

ρvkˆ ^{2}_{2}+µk∇vk^{2}_{2} ≤ Ck∇ˇuk_{∞}kp

ρvkˆ ^{2}_{2}+C(1 +k∇ˇutk^{2}_{2}+k∇ˇuk^{2}_{2}|∇^{2}ukˇ ^{2}_{2})

×(kWk^{2}_{2}+kσk^{2}_{2}+kσk^{2}3
2

). (3.7)

The appearance ofkσk^{3}

2 in the above inequality requires the energy estimate forkσk^{3}
given in the below. 2

Testing (3.1) with sign(σ)|σ|^{1}^{2} and using the H¨older and Sobolev inequalities that
d

dt Z

|σ|^{3}^{2}dx ≤ C
Z

(|v· ∇ˆρ| |σ|^{1}^{2} +|divˇu||σ|^{3}^{2} +|divvρ| |σ|ˆ ^{1}^{2})

≤ 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||^{3}^{2}3
2

+C||∇ρ||ˆ _{2}||∇v||_{2}||σ||^{1}^{2}3
2

, which gives

d
dtkσk^{3}

2 ≤C||∇ˇu||_{∞}||σ||^{3}

2 +C||∇ρ||ˆ _{2}||∇v||_{2}. (3.8)
Denote

f1(t) = (kσk^{3}

2 +kσk2+kWk2)(t), g1(t) =kp

ρvkˆ ^{2}_{2}(t), G1(t) =µk∇vk^{2}_{2}(t),
δ1(t) = Ck∇ˇuk∞(t), A1 =C sup

0≤t≤T

(kˆρk∞+kPˆk∞+k∇ˆρk_{L}^{2}∩L^{3} +k∇Pˆk3)(t),
α_{1}(t) =Ck∇ˇuk∞(t), β_{1}(t) =C(1 +k∇ˇu_{t}k^{2}_{2}+k∇ˇuk^{2}_{2}k∇^{2}ukˇ ^{2}_{2})(t),

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

d

dtf_{1}(t)≤δ_{1}(t)f_{1}(t) +A_{1}p
G_{1}(t),

d

dtg_{1}(t) +G_{1}(t)≤α_{1}(t)g_{1}(t) +β_{1}(t)f_{1}^{2}(t),
f_{1}(0) = 0.

By the regularities of ( ˆρ,u) and ( ˇˆ ρ,u), and recallingˇ ∇ˇu ∈ L^{1}(0, T;L^{∞}), one can
easily verify that α_{1}, δ_{1}, tβ_{1} ∈L^{1}((0, T)). Therefore, one can apply Lemma 3.1 to get
f_{1}(t) = g_{1}(t) = G_{1}(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
ρ^{4}_{∞}

Z

|u_{t}|^{2}dx =
Z

|ρ∞−ρ+ρ|^{4}|u_{t}|^{2}dx

≤ C Z

(|ρ−ρ∞|^{4}+ρ^{4})|u_{t}|^{2}dx

≤ C(kρ−ρ∞k^{4}_{6}ku_{t}k^{2}_{6}+kρk^{3}_{∞}k√
ρu_{t}k^{2}_{2})

≤ C(k∇ρk^{4}_{2}k∇u_{t}k^{2}_{2} +kρk^{3}_{∞}k√

ρu_{t}k^{2}_{2}),
and, thus,

Z T 0

tku_{t}k^{2}_{2}dt≤C sup

0≤t≤T

(k∇ρk^{4}_{2}+kρk^{3}_{∞})
Z T

0

(k√

t∇u_{t}k^{2}_{2}+k√

ρu_{t}k^{2}_{2})dt <∞,
that is, √

tu_{t}∈L^{2}(0, T;L^{2}), foru∈ {ˆu,u}.ˇ

By the H¨older, Sobolev, and Cauchy inequalities, we deduce
RHS ≤ kWk_{2}k∇vk_{2}+kσk_{2}kˇu_{t}k_{3}kvk_{6}

+kσk_{2}kˇuk_{6}k∇ˇuk_{6}kvk_{6}+k∇ˇuk∞kp
ρvkˆ ^{2}_{2}

≤ kWk_{2}k∇vk_{2}+k∇ˇuk∞kp
ρvkˆ ^{2}_{2}
+(kˇu_{t}k

1 2

2k∇ˇu_{t}k

1 2

2 +k∇ˇuk_{2}k∇^{2}ukˇ _{2})k∇vk_{2}kσk_{2}

≤ µ

2k∇vk^{2}_{2} +Ck∇ˇuk∞kp

ρvkˆ ^{2}_{2}+C(1 +kˇu_{t}k_{2}k∇ˇu_{t}k_{2}
+k∇ˇuk^{2}_{2}k∇^{2}ukˇ ^{2}_{2})(kσk^{2}_{2}+kWk^{2}_{2}).

Plugging this into (3.6) leads to d

dtkp

ρvkˆ ^{2}_{2}+µk∇vk^{2}_{2} ≤ Ck∇ˇuk∞kp

ρvkˆ ^{2}_{2}+C(1 +kˇu_{t}k_{2}k∇ˇu_{t}k_{2}

+k∇ˇuk^{2}_{2}k∇^{2}ukˇ ^{2}_{2})(kσk^{2}_{2}+kWk^{2}_{2}). (3.9)
Denote

f_{2}(t) = (kσk_{2}+kWk_{2})(t), g_{2}(t) =kp

ρvkˆ ^{2}_{2}(t), G_{2}(t) =µk∇vk^{2}_{2}(t),
δ_{2}(t) = Ck∇ˇuk_{∞}(t), A_{2} =C sup

0≤t≤T

(kˆρk_{∞}+kPˆk_{∞}+k∇ˆρk_{2}+k∇Pˆk_{3})(t),
α_{2}(t) =Ck∇ˇuk∞(t), β_{2}(t) =C(1 +kuˇ_{t}k_{2}k∇ˇu_{t}k_{2}+k∇ˇuk^{2}_{2}k∇^{2}ukˇ ^{2}_{2})(t),
then, it follows from (3.4), (3.5), and (3.9) that

d

dtf2(t)≤δ2(t)f2(t) +A2

pG2(t),

d

dtg_{2}(t) +G_{2}(t)≤α_{2}(t)g_{2}(t) +β_{2}(t)f_{2}^{2}(t),
f_{2}(0) = 0.

By the regularities of ( ˆρ,u) and ( ˇˆ ρ,u), and recallingˇ ∇u∈ L^{1}(0, T;L^{∞}) and √
tut ∈
L^{2}(0, T;L^{2}), for u∈ {ˆu,u}, one can easily verify thatˇ α2, δ2, tβ2 ∈L^{1}((0, T)). There-
fore, one can apply Lemma 3.1 to get f_{2}(t) = g_{2}(t) = G_{2}(t) = 0, on (0, T), which
implies the uniqueness for Case II.