Classical large solution for the viscous compressible fluids with initial vacuum in 1D

R E S E A R C H

Open Access

Classical large solution for the viscous

compressible fluids with initial vacuum in 1D

Zhilei Liang

_{and Yunzhao Lu}

*_{Correspondence:} [email protected]

2_{School of International Trade and} Economics, University of International Business and Economics, Beijing, 100029, P.R. China

Full list of author information is available at the end of the article

Abstract

This paper is concerned with the Cauchy problem for a compressible viscous fluid in one-dimensional (1D) space. By means of the weighted initial density, we obtain the global-in-time existence of a unique classical solution with large initial data. The initial density can be compactly supported or decays to zero not too slowly at infinity.

Keywords: compressible Navier-Stokes; global classical solution; vacuum; Cauchy problem

1 Introduction

The viscous isentropic compressible fluid in one-dimensional (D) space is governed by the compressible Navier-Stokes equations,

ρt+ (ρu)x= ,

(ρu)t+ (ρu)x+Px=μuxx, x∈R,t> ,

(.)

where the unknown functionsρ(x,t),u(x,t), andP=Kργ _{(K> ,γ > ) are the density,} velocity, and pressure, respectively. The viscosity coefficientμ>  is a given constant.

We are interested in the existence of a classical solution to (.) with the far field behavior

(ρ,u)→(, ) as|x| → ∞,t≥, (.)

and initial data

(ρ,u)(x, ) = (ρ≥,u)(x), x∈R. (.)

We first review briefly the well-posedness of solutions for (.). Non-vacuum small per-turbations around a constant have shown that the solutions are classical and globally de-fined in time if the initial data are sufficiently regular; see [–]. If the initial vacuum is allowed, the global weak solutions were first obtained by Lions [] for the isentropic fluids (see also Feireisl []) for large initial data. Later, some regularity information was obtained in [, ]. When it comes to the strong/classical solutions, Kimet al.[–] proved the lo-cal existence and uniqueness for both bounded and unbounded domains⊆R(this also holds for bounded domains inRorR_{). Based on thea prioriestimates developed [–],}

Dinget al.considered the initial-boundary-value (IBV) problem in D space, and mak-ing use of D properties, they [] obtained the global existence of classical solution with

large initial data and vacuum. Similar results was obtained for the annular (or exterior) domain inRandRunder the spherically symmetric assumption; see [, ]. Huanget al.[] showed that for the D Cauchy problem there exists a unique classical solution in the case when the initial total energy is small, but there possible are large oscillations and a vacuum.

However, the existence of a classical solution for the Cauchy problem in D (or D) is another issue. In particular, the argument in [–] could not be directly applied if the domain (N= , ) becomes unbounded since theLp_{-norm (p≥) of the velocity cannot be}

controlled just in terms of theL_{-norm of the gradient of it. Li and Liang [] obtained the}

local existence and uniqueness of strong and classical solutions to the D Cauchy problem with the vacuum as a far field density. The key idea in [] is to controlρuLp, instead

ofuLp, in terms ofρ/u_L and∇u_L by introducing a weight to the initial density.

Recently, Li and Xin [] showed that the classical solution for D Cauchy problem exists globally in time in the case of small initial energy; furthermore, some large-time decay rates of solutions are first presented.

In this paper, for ≤r≤ ∞and integerk≥, we adopt the simplified notations for the standard homogeneous and inhomogeneous Sobolev spaces

Lr=Lr(R), Dk=f ∈L_loc (R) :∂_xkf_L_(R)<∞

, Hk=L∩Dk.

The concern of this paper is the existence, uniqueness, and large-time behavior of clas-sical large solutions to the Cauchy problem (.)-(.), with the vacuum as a far field state, even for the compactly supported density.

Theorem . Define

¯

x=e+x/lne+x. (.)

Assume that the initial functions(ρ,u)satisfy for a∈[,p)

¯

xaρ∈L∩H, ρ/ u∈L, u∈D∩D,

ρxx,Pxx∈L, ρ/pu∈Lp, up/∈D,

(.)

and the following compatibility condition:

–μuxx+Px=ρ/ g, for some g∈L. (.)

Then the Cauchy problem(.)-(.)admits a unique classical solution(ρ,u),which satis-fies for any T∈(,∞)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ρ∈C([,T];L_∩H_{), P∈L}∞_(,T;H_),

¯

xaρ∈L∞(,T;L∩H), ρu∈L∞(,T;L∩H),

ux∈L∞(,T;L∞∩H),

ρ/p_u∈L∞_(,T;L_{), ∂}

x(up/)∈L(,T;L),

¯

x–_u∈L∞_(,T;L_∩L∞_{), x¯}–/p_u∈Lp_(,T;Lp_∩L∞_),

ρ/u˙∈L∞(,T;L_{), u˙}

x∈L(,T;L), t/uxxx,t/u˙x,tu˙xx∈L∞(,T;L), tρ/_u¨∈L∞_(,T;L_{), tu¨}

x∈L(,T;L),

where

˙

u=ut+uux and u¨=u˙t+uu˙x (.)

denote the material derivatives of u andu˙,respectively.

Remark . We can check that the solution (ρ,u) in Theorem . satisfying (.) is in fact a classical one.

Remark . We mention [] by Xin, where the author shows the solution (ρ,u)∈

C([,T];H_{(R)) must occur with blowup phenomena in the case that the non-trivial initial}

density has a compact support.

Remark . The hypothesisPxx∈Lin (.) can be removed whenγ≥.

Following some ideas developed by Li and Xin [] for the multi-dimensional case,N= , , we give the followingLr_{-norm (r> ) estimate on the pressure.}

Corollary . Let(ρ,u)be one of the solutions described in Theorem..Then it satisfies +∞



P(·,t)r_Lrdt≤C, ∀r∈[ + /γ, +∞). (.)

This work was initially motivated by Dinget al.[], where the authors considered the global existence of classical solution for the D IBV problem. In contrast with [], the

problem under our consideration lies in an unbounded domain, and thus theLp-norm

of the velocityucould not be dominated just in terms of theL_{-norm of the gradient of}

it. In this connection, we follow some ideas in [] and introduce a weight function to the initial density. However, the usage of a Sobolev embedding inequality inRis very different fromR_{. For this, we use some special properties in one-dimensional space (see (.)).}

With these preparations, as well as some ideas in [, , , , ], we establish the global existence and uniqueness of the classical solution to the Cauchy problem (.)-(.).

In the rest of this paper, we first derive some globala prioriestimates in Sections  and . This is necessary when we extend the local solution to all positive time in Section , and thus complete Theorem .. Finally, Corollary . is proven in Section .

2 A priori estimates (I)

We suppose that (ρ,u) is a classical solution to (.)-(.) over the interval [,T] withT∈

(,∞). For simplicity reasons, we may chooseρL= , which implies N

–N

ρ(x)dx≥

 

R

ρ(x)dx=



 (.)

for some large numberN.

First of all, multiplying (.)byuand integrating by parts, we get

Lemma . We have

sup

≤t≤T

ρ/u_L+PL

+μ

T



where as below C denotes a generic constant depending onμ,γ,K,a,and the initial data;

particularly,the expression C(α)emphasizes that C depends onα.

The next lemma derives the bound on the densityρ(x,t) from above.

Lemma . We have

ρ(x,t)≤C, (x,t)∈R×[, +∞). (.)

Proof The proof is almost exactly the same as that in [], Lemma .. Here we state the details for completeness. By (.), one has fort∈[,∞)

ρ(t)_L=ρL= . (.)

Put

ξ(x,t) =

x

–∞

ρu(y,t)dy. (.)

We express (.)in the form

ξxt+

ρu_x= (μux–P)x,

which, along with (.), provides us after integration in variablexwith

ξt+ρu=μux–P= –μ



ρ(ρt+uρx) –P.

Hence

d dtξ

X(t,x),t= –μd

dtlnρ

X(t,x),t–PX(t,x),t

≤–μd

dtlnρ

X(t,x),t, (.)

whereX(t,x) is the particle trajectory satisfying

d

dtX(t,x) =u(X(t,x),t), t> , X(,x) =x.

Integration of (.) in time gives

μlnρX(t,x),t+ξX(t,x),t≤μlnρ(x, ) +ξ(x, ).

Making use of (.), (.), (.), and (.), we find

μlnρX(t,x),t≤μlnρ(x, ) +ξ(x, ) +ξX(t,x),t

≤μlnρ(x) +

R

ρ|u|dx+

≤μρ+ρ/u_Lρ/_L +ρ/u_Lρ/_L

≤C.

This finishes the proof.

Lemma . We have

uxL+ T



ρ/u˙_Ldt≤C(T). (.)

Proof Multiplying (.)byu˙and integrating by parts give rise to

d dt

R μ |ux|

_dx+

Rρ|˙u|

_dx

=

RPu˙xdx– μ

Rux(uux)xdx

=

RPu˙xdx– μ 

Ru



xdx

= d

dt

RPuxdx–

R

Pt+ (Pu)x

uxdx+

RPu



xdx–

μ 

Ru



xdx

= d

dt

RPuxdx+γ

RPu



xdx–

μ 

Ru



xdx, (.)

where the last equality comes from

Pt+Pxu+γPux= , (.)

owing to (.). By virtue of (.), (.), and the Sobolev inequality, it satisfies

uxL∞+PL∞

≤Cμux–PL∞+CPL∞

≤CuxL+C(μu_x–P)_x

L+CPL+CP_L∞

≤CuxL+ρ/u˙ L

+C. (.)

So, it follows from (.) that

d

dtB(t) +ρ

/_u˙

L≤C

 +uxL∞+PL∞

ux_L

≤Cux_L+Cux_L+

 ρ

/_u˙

L, (.)

where

B(t) =

R

μ |ux|

_–Pu

x

dx (.)

satisfies

μ ux



Noting (.), (.), and (.), we integrate (.) to get

ux_L_(R)+ T



ρ/u˙_L_(R)dt≤C(T) +C T



ux_L_(R)dt,

which completes the proof in view of the Gronwall inequality and (.).

Lemma . For all t∈[, +∞),

A

–A

ρ(x,t)dx≥ 

, (.)

where A=N( +t)ln(e+t),and Nis taken from(.).

Proof The idea is borrowed from []. Using (.) and (.), we multiply (.)by|x|to

obtain

d dt

Rρ|x|dx≤C

Rρ|u|dx≤Cρ

/

Lρ/u_L≤C.

This together with (.) brings about

R

ρ(x,t)|x|dx≤C( +t). (.)

Define a cut-off functionϕ(x)∈C(R) satisfying

≤ϕ(x)≤, ϕ(x) =

, |x|< , , |x|> , ϕ

≤C. (.)

Multiplied byϕ(y) withy=ηx[( +t)ln(e+t)]–, from (.)

d dt

Rρϕ(y)dx=

Rρϕ

ytdx+

Rρuϕ

yxdx

≥– η

( +t)_ln(e+t)

Rρ|x|dx–

η ( +t)ln(e+t)

Rρ|u|dx

≥– Cη

( +t)ln(e+t),

where the last inequality is due to (.) and (.). Integrating the above inequality in time we conclude that

Rρϕ(y)dx≥

Rρϕ(ηx)dx–Cη≥  .

This proves (.) by choosingη= (N+ C)–.

Our next object is to derive some weightedLp_{estimate onu, which plays a key role in}

Lemma . We have for all p∈[,∞)

sup

≤t≤T

ux¯–/pp_Lp≤C(p,T)ρ/pu

p

Lp+up/

x  L . (.)

Proof Multiplying (.)byp|u|p–uwithp≥, we obtain

d dt

Rρ|u|

p_dx+μ(p– ) p

R

up/_xdx

= (p– )

RPu

(p–)/_up/

xdx

≤μ(p– )

p

R

up/_xdx+C(p)

RP

_|u|p–_dx

≤μ(p– )

p

R

up/_xdx+C(p)

Rρ|u|

p_dx+C,

where in the last inequality we have used (.) and (.). Integrating in time leads to

ρ/pu_Lp+

T



_up/

x



Ldt≤C(T,p). (.)

Let

up₌ 

N

N

–N updx.

On account of the Poincaré inequality and (.), we infer

up≤_ N

–N

ρup_dx≤_ N

–N

ρup–up_{dx+ } N

–N

ρ|u|pdx

≤C N

–N

up–up_dx+C

R

ρ|u|pdx

≤

u

p_+C(N

)

R

_up/

x



dx+C

Rρ|u|

p_dx,

that is,

N

–N

updx≤C(N)

R

up/_xdx+C(N)

Rρ|u|

p_{dx. (.)}

Next, a straight computation shows for an even numberp= , , . . .



R

( –ϕ)up

x_ln_|x| dx≤

R

–

xln|x|

( –ϕ)updx

= 

R ( –ϕ)

xln|x|u

p/_up/

xdx–

R ϕ_

xln|x|u p_dx

≤

R

( –ϕ)up

x_ln_|x| dx+C

R

up/_xdx+C

{≤|x|≤}

whereϕis defined in (.). This together with (.) gives rise to

R

( –ϕ)up

x_ln_|x| dx≤C

R

up/

x



dx+C

Rρ|u|

p_{dx, p= , , . . . . (.)}

In terms of the Cauchy inequality, (.) holds true for allp≥.

Lemma . We have

sup

≤t≤T

x_¯aρ_L_∩H≤C(T). (.)

Proof It follows from (.)that

ρx¯a_t+uρx¯a_x=auρx¯a(lnx¯)x. (.)

Observe (.) and the following simple facts:

|∂xx¯|+|∂xxx¯| ≤Cln

e+x, (.)

x_¯(a–p)/plne+x_L∞≤C, fora<p,

and integrate (.) to obtain forp≥

d dt

Rρx¯

a_dx≤C

Rρ|u|¯x

(a–)_lne+x_dx

≤Cρ/pu_Lpρ(p–)/px¯(a–)ln

e+x_Lp/(p–)

≤Cρ/pu_Lpx¯(a–p)/pln

e+x_L∞

R ρx¯adx

(p–)/p

≤C

R ρx¯adx

(p–)/p

,

and hence

sup

≤t≤T

ρx¯a_L≤C(T,p). (.)

Differentiating (.) in variablexonce more we get

ρx¯a_xt+uρx¯a_xx+ ux

ρx¯a_x+ρx¯auxx–μ–Px

= –μ–Pxρx¯a+a

uxρx¯a(lnx¯)x+u

ρx¯a_x(lnx¯)x+uρx¯a(lnx¯)xx

. (.)

Notice that

Pxρx¯a=Kγργ

ρx¯a_x–aρx¯a(lnx¯)x

and

it satisfies from (.) after multiplication by (ρx¯a₎ x

d dtρx¯

a

xL≤C

 +uxL∞+u(lnx¯)x_L∞ρx¯a

xL

+Cρx¯a L∞ρ

/_u˙

L

+Cρx¯a_L

 +ux(lnx¯)x_L∞+u(lnx¯)xx_L∞

. (.)

Inequalities (.), (.)-(.), and (.) ensure that forp∈(,∞)

u(lnx¯)x+u(lnx¯)xx_L∞ ≤C(p)ux¯–/p_L∞

≤C(p)ux¯–/p_Lp+ux¯–/p

xL

≤C(p,T)ρ/pu_Lp+up/

x

/p

L +uxL+ux¯– L

≤C(p,T) +up/_x/p L

(.)

and that forp= 

ux¯–_L∞≤C(T)ρ/u_L+uxL

≤C(T). (.)

Thanks to (.), (.), (.), and (.), we conclude from (.) that

d dtρx¯

a

xL≤C

 +ρ/u˙_L+up/

xL

 +ρx¯a_x L

,

which implies by using (.), (.), and the Gronwall inequality

ρx¯a_xL≤C(T).

This together with (.) finishes the proof.

Lemma . We have

ρ/u˙_L+ T



˙ux_Ldt≤C(T) (.)

and

uxL∞+uxxL ≤C(T). (.)

Proof Operating∂t+∂x(u·) to (.), using (.) and (.), we obtain

ρu˙t+ρuu˙x–μu˙xx= –μ

|ux|

x–

Pt+ (Pu)x

x+ (uxP)x

= –μ|ux|

Integration of (.) after multiplication byu˙leads to

 

d dtρ

/_u˙

L+μ˙ux_L=μ

Ru˙x|ux|

_dx–γ

Ru˙xuxP dx,

which combining with (.), (.), (.), and (.) lets us conclude that

d dtρ

/_u˙

L+˙ux_L≤Cux_L+CP_L

≤CuxL∞ux_L+CPL

≤Cρ/u˙_L+C. (.)

Remembering (.), we integrate (.) to obtain (.). Next, by virtue of (.) and (.), it follows from (.)that

uxxL≤C(μ)u_xx–μ–P_x

L+PxL

≤Cρ/u˙_L+Cρx¯a_H≤C,

which together with (.) yields (.).

3 A priori estimates (II)

In this section, we derive the higher-order regularity estimates of the solutions.

Lemma . We have

sup

≤t≤T

t˙ux_L+ T



tρ/u¨_L+˙uxx_L

dt≤C(T). (.)

Proof Multiplying (.) byu¨leads to

 

Rρ|¨u|

_dx=μ

Ru˙xxu dx¨ –μ

R

|ux|

xu dx¨ +γ

R(uxP)xu dx¨ . (.)

We estimate the terms on the right-hand side of (.) as follows. First,

μ

Ru˙xxu dx¨ = μ

Ru˙xx(

∂tu˙+uu˙x)dx= –

μ 

d dt

Ru˙



x–

μ 

Ru˙



xuxdx.

Second,

–μ

R

|ux|

xu dx¨

=μ

R|ux|

_(u˙

t+uu˙x)xdx

=μd

dt

R|ux|

_u˙

xdx– μ

Rux(u˙–uux)xu˙xdx+ μ

R|ux|

_(uu˙

x)xdx

=μd

dt

R|ux|

_u˙

xdx– μ

Ruxu˙



+ μ

Rux(uux)xu˙xdx+ μ

R|ux|

_(uu˙

x)xdx

=μd

dt

R|ux|

_u˙

xdx– μ

Ruxu˙



xdx+ μ

R(ux)

_u˙

xdx.

By (.), the final term becomes

R(uxP)xu dx¨

= –

RuxP(u˙t+uu˙x)xdx

= –d

dt

RuxPu˙xdx+

R

˙

u_xP+u˙xux

Pt+ (uP)x–Pux

dx

+

Ru˙xxPuuxdx–

RPux(uu˙x)xdx

= –d

dt

RuxPu˙xdx+

R

˙

u_xP–γu˙xuxPux

dx–

Ru˙xPu



xdx.

Substitute the last three inequalities into (.), take (.), (.), and (.) into account, and we arrive at

d dtI(t) +

 

R

ρ|¨u|dx

≤C +uxL∞+PL∞

˙ux_L+C

ux_L+P_L

≤C˙ux_L+C, (.)

where

I(t) =

R

μ |˙ux|

_–μ|u

x|u˙x+γPuxu˙x

dx

satisfies

μ ˙ux



L–C≤I(t)≤C˙ux_L+C.

Thanks to (.), we multiply (.) bytand integrate the resulting expression to deduce

sup

≤t≤T

t˙ux_L+ T



tρ/u¨_Ldt≤C(T). (.)

It follows from (.) that

˙uxxL≤Cρu¨_L+Cu_xL∞+P_L∞u_xxL+P_xL

≤Cρ/u¨_L+CuxxL+Cρx¯a H

≤Cρ/u¨_L+ 

in which (.), (.), (.), and (.) have been used. Integrating (.) gives

T



t˙uxx_Ldt≤C(T),

owing to (.).

Lemma . We have

sup

≤t≤T

ρxxL+P_xxL+tu_xxx L

≤C(T). (.)

Proof Differentiating (.)with respect toxtwice we get

ρxxt= –uρxxx– uxρxx– uxxρx–uxxxρ,

which satisfies after multiplication by ρxx,

d

dtρxxL≤CuxL∞ρxxL+CρxL∞uxxL+CρL∞uxxxL

≤CρxxL+Cu_xxxL+C, (.)

where the last inequality follows from (.), (.), and (.). A similar argument to (.) lets us deduce from (.) that

d

dtPxxL ≤CPxxL+CuxxxL+C. (.)

In order to examineuxxxL, we differentiate (.)_and obtain

μuxxx=ρxu˙+ρu˙x+Pxx,

which, along with (.), (.), (.), and (.), leads us to conclude

uxxxL≤Cρ_xu˙_L+ρu˙_xL+P_xxL

≤Cρx¯a_Hx¯–au˙_L∞+˙uxL+P_xxL

≤C +˙uxL+P_xxL

. (.)

Once (.) is obtained, inequalities (.) and (.) imply that

d dt

ρxxL+P_xxL

≤CρxxL+P_xxL

+C˙uxL+C,

which generates by using (.) and the Gronwall inequality

ρxxL+P_xxL≤C(T). (.)

As a result of (.) and (.), we have from (.)

tuxxx_L≤C(T).

Lemma . We have

sup

≤t≤T

tρ/u¨_L+˙uxxL

+

T



t¨ux_Ldt≤C. (.)

Proof Utilizing (.)and (.), operating by∂t+∂x(u·) to (.), it yields after a

straight-forward but tedious calculation

ρu¨t+ρuu¨x–μu¨xx= –μ(uxu˙x)x+ μ

u_x_x+γ(Pu˙x)x–γ(γ + )

u_xP_x, which gives after multiplication byu¨

 

d dt

Rρ|¨u|

_dx+μ

R|¨ux|

_dx

= μ

Ru¨xuxu˙xdx– μ

Ru¨xu



xdx–γ

Ru¨xu˙xP dx+γ(γ+ )

Ru¨xu



xP dx,

from which, together with (.), (.), (.), and the Cauchy inequality, one deduces

d dtρ

/_u¨

L+¨ux_L≤C+C˙ux_L. (.)

Recalling (.), integration of (.) and multiplication byt_yields

tρ/u¨_L+ T



t¨ux_Ldt≤C(T).

Consequently, it follows from (.) that

sup

≤t≤T

t˙uxxL≤C(T).

The proof is finished.

4 Proof of Theorem 1.1

Now we are ready to prove Theorem .. For this we first state the local-in-time existence and uniqueness of the classical solution to (.)-(.).

Lemma . Under the hypotheses in Theorem.,the Cauchy problem(.)-(.)admits a unique classical solution(ρ,u)overR×(,T∗]for some small T∗> ,which satisfies the properties(.).

Proof The existence part is exactly the same procedure as in []. For the sake of brevity, we only prove the uniqueness. Assume that (ρ,u) and (ρ¯,u¯) are two different solutions with the identical initial data. Minus (.)satisfied by (ρ,u) and (ρ¯,u¯) yields

ρUt+ρuUx–μUxx= –ρUu¯x–H(u¯t+u¯u¯x) –

P(ρ) –P(ρ¯)_x, (.)

with

By virtue of (.), we multiply (.) byUand conclude

d dt

RρU

_dx+μ

RU



xdx

≤C

RρU

_dx+C

R|H||U||˙¯u|dx+C

P(ρ) –P(ρ¯)_L, (.)

whereu˙¯=u¯t+u¯u¯x. Hölder’s inequality together with (.) and (.) shows

R|H||U||˙¯u|dx≤C

u_˙¯x¯–_L∞Hx¯_LUx¯–_L

≤C(ε) +˙¯ux



LHx¯

L+ερ/U



L+Ux_L

. (.)

Next, from the mass equation we have

Ht+uH¯ x+Hu¯x+ρUx+Uρx= .

Multiplied byHx¯, it satisfies

d dtHx¯



L≤Cu¯x_L∞+u¯x¯–/_L∞Hx¯_L

+CHx¯_Lρx¯_L∞UxL+Ux¯– L∞x¯

_ρ

x_L

≤C(ε) +u¯_xLHx¯



L+ε

Ux_L+ρ/U



L

, (.)

where the second inequality comes from (.), (.), (.), (.), and (.).

Remark . The restrictiona≥ is from (.),i.e.,

x_¯ρx_L≤Cx¯ρ_H≤Cx¯aρ_H≤C.

Substituting (.) and (.) into (.), choosingεsmall, we discover

d dtHx¯



L+ρ/U



L

+Ux_L

≤C +˙¯uxL+u¯

xLHx¯



L+ρ/U



L

+CP(ρ) –P(ρ¯)_L. (.)

One deduces from (.) that

P(ρ) –P(ρ¯)_t+u¯P(ρ) –P(ρ¯)_x+UP(ρ)x+γ

P(ρ) –P(ρ¯)u¯x+γP(ρ)Ux= ,

which gives rise to

d

dtP(ρ) –P(ρ¯)L≤CP(ρ) –P(ρ¯)_L+CUxL+Cρ/U L.

Hence,

P(ρ) –P(ρ¯)_L≤C t



UxL+ρ/U L

Inserting (.) back into (.) yields

d dtG≤C

 +˙¯uxL+u¯

xL

G,

with

G(t) =Hx¯_L+ρ/U_L+ t



ρ/U_L+Ux_L

ds.

In view of (.) and (.), from the Gronwall inequality one concludesG(t) = , which implies thatH(x,t) =U(x,t) =  almost everywhere (x,t)∈R×(,T).

Next we show that the solution exists globally in time. SupposeT∗≥T∗is the maximal

time of existence. We claim that

T∗=∞. (.)

If (.) is violated,i.e.,T∗<∞. Then thea prioriestimates in Sections  and  guarantee that the conditions (.) make sense atT=T∗, and, moreover, the compatibility condition (.) could be replaced by

–μuxx

x,T∗+Px

x,T∗=ρ/x,T∗g∗,

withg∗= –ρ/_u˙(x,T∗_)∈L_{. However, the local existence result Lemma . tells that (ρ,u)}

could be extended to (,T∗+T˜∗] with another smallT˜∗> . The contradiction implies that

(.) is true, and thus Theorem . is completed.

5 Proof of Corollary 1.1

In this final section the generic constantCis independent ofT. Following [], we define

A(T) = sup

t∈[,T]

σux_L+ T



σρ/u˙_Ldt

and

A(T) = sup

t∈[,T]

σρ/u˙_L+ T



σ˙ux_Ldt,

whereσ∈min{t, }. Remember (.); one has

d dtB(t) +

Rρ|˙u|

_dx=γ

RP|ux|

_dx–μ



Ru



xdx, (.)

whereB(t) is taken from (.). Multiplied byσ, it yields from (.) after integration

A(T)≤CσP_L+C T



σB(t)dt+C T



σ

R

P|ux|+|ux|

dx dt

≤CP_L+C T



ux_Ldt+C T



σ

R|ux|

_{dx dt}

≤C+C T



where we have used (.), (.), and (.). If we multiply (.) byσ_{, we obtain}

A(T)≤CA(T) +C

T



σux_L+P_L

dt. (.)

In view of and (.)and (.), the Sobolev inequality ensures that

ux_L≤Cμux–P_L+CP_L

≤Cρu˙_L

ux_L+P_L

+CP_L

≤Cρ/u˙_L

ux_L+P_L

+CP_L.

By this we estimate

C T



σux_Ldt≤C T



σρ/u˙_L

ux_L+P_L

+P_L

dt

≤CA/_ (T)A/_ (T)

T



σ/ux_Ldt

+CA(T) +

T



σP_L+P_L

dt

≤ 

A(T) +CA(T) +

T



σP_L+P_L

dt,

which guarantees that (.) satisfies

A(T)≤CA(T) +

T



σP_L+P_L

dt

≤C+C T



σux_Ldt+ T



σP_L+P_L

dt, (.)

where in the second inequality we have used (.).

In order to examine the last term on the right hand-side of (.), we integrate (.)and

obtain

P(x,t) =μux– x

–∞

ρu˙,

which implies

P(x,t)≤μ|ux|+ρu˙L≤μ|u_x|+ρ/u˙ L.

By this we compute

RP

(γ+)/γ =K/γ

RρP



≤C

Rρ|ux|

_+C

Rρ

ρ/u˙_L

≤Cux_L+Cρ/u˙



Because ofγ > , combination of (.) and (.) gives

P_L≤

RP

(γ+)/γ_P(γ–)/γ

≤C

RP

(γ+)/γ _≤

Cux_L+Cρ/u˙



L,

and thus

P_L+P_L≤CP_L

P_L+ 

≤CP_L≤CP_L≤Cux_L+Cρ/u˙



L. (.)

Inserting (.) into (.), using (.), (.), and the definition ofA(T), we obtain

A(T) +A(T)≤C+C

T



σux_Ldt. (.)

It is only left for us to deal with_Tσux_Ldt. Notice that

uxLr ≤C_μu_x–P_Lr+CP_Lr

≤Cρu˙(_Lr–)/r

ux/_Lr+P/_Lr

+CPLr

≤Cρ/u˙(_Lr–)/r

ux/_Lr+P/_Lr

+CPLr (r= , ). (.)

This, along with (.) and (.), guarantees that

C σ(T)



σux_Ldt≤C σ(T)



σρ/u˙_L

ux_L+P_L

dt+C

≤CA/_ σ(T) σ(T) 

ux_L+P_L

dt+C

≤CA/_ σ(T)+C≤εA(T) +C (.)

and that

C T

σ(T)

σux_Ldt

≤ε

T

σ(T)

ux_Ldt+C(ε) T

σ(T)

ux_Ldt

≤ε

T

σ(T)

ρ/u˙_Lux_L+ρ/u˙



LP_L+P_L

+C(ε)

≤ε

A(T) +A(T) +

T

σ(T)

P_L

+C(ε)

≤εA(T) +A(T)

+C(ε), (.)

Substituting (.) and (.) back into (.), and selectingεso small as to

A+A≤C. (.)

Having (.) in hand, we use (.), (.), and (.) to get

∞



P(_L(γγ+)/+)/γγdt≤C+ ∞



ux_Ldt+ ∞



ρ/u˙_Ldt≤C.

This together with (.) yields (.) as desired.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by ZL, and YL prepared the manuscript initially and performed all the steps of the proofs in this research. We all have read and approved the final manuscript.

Author details

1_{School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, P.R. China.} 2_{School of International Trade and Economics, University of International Business and Economics, Beijing, 100029,} P.R. China.

Acknowledgements

The work is partially supported by National Natural Sciences Foundation of China No. 11301422.

Received: 29 December 2014 Accepted: 30 May 2015 References

1. Hoff, D: Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ.120, 215-254 (1995)

2. Hoff, D: Discontinuous solutions of the Navier-Stokes equations for multi-dimensional heat-conducting fluids. Arch. Ration. Mech. Anal.193, 303-354 (1997)

3. Matsumura, A, Nishida, T: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.20(1), 67-104 (1980)

4. Kazhikhov, A, Shelukhin, V: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh.41, 282-291 (1977)

5. Lions, P: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Oxford Science Publication, Oxford (1998)

6. Feireisl, E: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

7. Desjardins, B: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Commun. Partial Differ. Equ.22(5-6), 977-1008 (1997)

8. Hoff, D: Compressible flow in a half-space with Navier boundary conditions. J. Math. Fluid Mech.7, 315-338 (2005) 9. Cho, Y, Choe, H, Kim, H: Unique solvability of the initial boundary value problems for compressible viscous fluids.

J. Math. Pures Appl.83(9), 243-275 (2004)

10. Choe, H, Kim, H: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ.190, 504-523 (2003)

11. Cho, Y, Kim, H: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscr. Math.120, 91-129 (2006)

12. Ding, S, Wen, H, Zhu, C: Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. J. Differ. Equ.251, 1696-1725 (2011)

13. Choe, H, Kim, H: Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fluids. Math. Methods Appl. Sci.28, 1-28 (2005)

14. Ding, S, Wen, H, Yao, L, Zhu, C: Global classical spherically symmetric solution of compressible isentropic Navier-Stokes equations with vacuum. SIAM J. Math. Anal.44, 1257-1278 (2012)

15. Huang, X, Li, J, Xin, Z: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun. Pure Appl. Math.65, 549-585 (2012) 16. Li, J, Liang, Z: Local well-posedness of strong and classical solutions to Cauchy problem of the two-dimensional

barotropic compressible Navier-Stokes equations with vacuum. J. Math. Pures Appl.102, 640-671 (2014) 17. Li, J, Xin, Z: Global well-posedness and large time asymptotic behavior of classical solutions to the compressible

Navier-Stokes equations with vacuum. arXiv:1310.1673v1

18. Xin, Z: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math.51, 229-240 (1998)