GLOBAL SMOOTH SOLUTIONS TO THE 1-D COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM WITH LARGE INITIAL DATA

CHEN Ting-ting,CHEN Zhi-chun,CHEN Zheng-zheng

(School of Mathematical Sciences,Anhui University,Hefei 230601,China )

GLOBAL SMOOTH SOLUTIONS TO THE 1-D COMPRESSIBLE NAVIER-STOKES-KORTEWEG SYSTEM WITH LARGE INITIAL DATA

CHEN Ting-ting,CHEN Zhi-chun,CHEN Zheng-zheng

(School of Mathematical Sciences,Anhui University,Hefei 230601,China )

This paper is concerned with the Cauchy problem of the one-dimensional isothermal compressible Navier-Stokes-Korteweg system when the viscosity coefficient and capillarity coefficient are general smooth functions of the density.By using the elementary energy method and Kanel’s technique[25],we obtain the global existence and time-asymptotic behavior of smooth non-vacuum solutions with large initial data,which improves the previous ones in the literature.

compressible Navier-Stokes-Korteweg system;global existence;time-asymptotic behavior;large initial data

1 Introduction

This paper is concerned with the Cauchy problem of the one-dimensional isothermal compressible Navier-Stokes-Korteweg system with density-dependent viscosity coefficient and capillarity coefficient in the Eulerian coordinates

with the initial data

here t and x represent the time variable and the spatial variable,respectively,K is the Korteweg tensor given by

The unknown functions ρ＞0,u,P=P(ρ)denote the density,the velocity,and the pressure of the fluids respectively.μ=μ(ρ)＞0 and κ=κ(ρ)＞0 are the viscosity coefficient and the capillarity coefficient,respectively,and＞0 is a given constant.Throughout this paper, we assume that

System(1.1)can be used to model the motions of compressible isothermal viscous fluids with internal capillarity,see[1–3]for its derivations.Notice that when κ=0,system(1.1) is reduced to the compressible Navier-Stokes system.

There were extensive studies on the mathematical aspects on the compressible Navier-Stokes-Korteweg system.For small initial data,we refer to[8,9,13–15,19–23]for the global existence and large time behavior of smooth solutions in Sobolev space,[5,7,11]for the global existence and uniqueness of strong solutions in Besov space,and[5,6]for the global existence of weak solutions near constant states in the whole space R2.

For large initial data,Kotschote[12],Hattori and Li[10]proved the local existence of strong solutions.Bresch et al.[4]investigated the global existence of weak solutions for an isothermal fluid with the viscosity coefficientsμ(ρ)=ρ,λ(ρ)=0 and the capillarity coefficient κ(ρ)≡in a periodic domain Td(d=2,3),where＞0 are positive constants. Later,such a result was improved by Haspot[6]to some more general density-dependent viscosity coefficients.Tsyganov[16]studied the global existence and time-asymptotic convergence of weak solutions for an isothermal compressible Navier-Stokes-Korteweg system with the viscosity coefficientμ(ρ)≡1 and the capillarity coefficient κ(ρ)=ρ-5on the interval[0,1].Charve and Haspot[17]showed the global existence of strong solutions to system (1.1)withμ(ρ)=ερ and κ(ρ)=ε2ρ-1.Recently,Germain and LeFloch[18]studied the global existence of weak solutions to the Cauchy problem(1.1)–(1.2)with general densitydependent viscosity and capillarity coefficients.Both the vacuum and non-vacuum weak solutions were obtained in[18].Moreover,Chen et al.[23,24]discussed the global existence and large time behavior of smooth and non-vacuum solutions to the Cauchy problem of system(1.1)with the viscosity and capillarity coefficients being some power functions of the density.

However,few results were obtained for the global smooth,large solutions of the isothermal compressible Navier-Stokes-Korteweg system with general density-dependent viscosity coefficient and capillarity coefficient up to now.This paper is devoted to this problem,and we are concerned with the global existence and large time behavior of smooth,non-vacuum solutions to the Cauchy problems(1.1)–(1.2)when the the viscosity coefficientμ(ρ)and the capillarity coefficient κ(ρ)are general smooth functions of the density ρ.

The main result of this paper is stated as follows.

Theorem 1.1Suppose the following conditions hold:

(i)The initial data(ρ0(x)-,u0(x))∈H4(R)×H3(R),and there exist two positive constants m0,m1such that m0≤ρ0(x)≤m1for all x∈R.

(ii)The smooth functionsμ(ρ)and κ(ρ)satisfyμ(ρ),κ(ρ)＞0 for ρ＞0,and one of the following two conditions hold:

and the time-asymptotic behavior

here C1is a positive constant depending only on m0,m1,and C2is a positive constant depending only on m0,m1,

When the viscosity coefficientμ(ρ)and the capillarity coefficient κ(ρ)are given by

where α,β∈R are some constants,condition(ii)of Theorem 1.1 corresponds to

while condition(iii)of Theorem 1.1 is equivalent to

or

Thus from Theorem 1.1,we have the following corollary.

Corollary 1.1Let condition(i)of Theorem 1.1 holds.Suppose that the viscosity coefficientμ(ρ)and the capillarity coefficient κ(ρ)are given by(1.7)and the constants α,β satisfy one of the following conditions:

then the same conclusions of Theorem 1.1 hold.

Remark 1.1Some remarks on Theorem 1.1 and Corollary 1.1 are given as follows:

(1)Conditions(ii)and(iii)of Theorem 1.1 are used to deduce the positive lower and upper bounds of the density ρ(t,x),see Lemmas 2.3–2.5 for details.

(2)In Theorem 1.1,the viscosity coefficientμ(ρ)and the capillarity coefficient κ(ρ)are general smooth functions of ρ satisfying conditions(ii)and(iii)of Theorem 1.1,which are more general than those in[23,24],where only some power like density-dependent viscosity and capillarity coefficients are studied.

On the other hand,Germain and LeFloch[18]also discussed the global existence of weak solutions away from vacuum for problems(1.1)–(1.2)withμ(ρ)=ραand κ(ρ)=ρβunder the condition that

or

which means that 0≤α＜1.From condition(A)of Corollary 1.1,we see that α∈thus Corollary 1.1 also improves the results of[18]to the case α∈Moreover,case (B)of Corollary 1.1 is completely new compared to the results in[18,23,24].Thus in these sense,our main result Theorem 1.1 can be viewed as an extension of the works[18,23,24].

Now we make some comments on the analysis of this paper.The proof of Theorem 1.1 is motivated by the previous works[18,23,24].When the viscosity coefficientμ(ρ) and the capillarity coefficient κ(ρ)are some power functions of the density,the authors in [23,24]studied the global existence and large time behavior of smooth solutions away from vacuum to the Cauchy problem of system(1.1)with large initial data in the Lagrangian coordinates.However,for the viscosity coefficientμ(ρ)and the capillarity coefficient κ(ρ) being some general smooth functions of the density,it is much more easier for us to study such a problem in the Eulerian coordinates rather than the Lagrangian coordinates.To prove Theorem 1.1,we mainly use the method of Kanel[25]and the energy estimates.The key step is to derive the positive lower and upper bounds for the density ρ(t,x).First,due to effect of the Korteweg tensor,an estimate ofappears in the basic energy estimate(see Lemma 2.1).Based on this and a new inequality for the renormalized internal energy(see Lemma 2.2),the lower and upper bounds of ρ(t,x)for cases(ii)(a)of Theorem 1.1 can be derived easily by applying Kanel’s method[25](see Lemma 2.3).Second,we perform an uniform-in-time estimate onunder condition(iii)of Theorem 1.1(see Lemma 2.4).We remark that Lemma 2.4 is proved by using the approach of Kanel[25],rather than introducing the effective velocity as[4,17,18].Then by employing Kanel’s method[25]againand Lemmas 2.1,2.2 and 2.4,the lower and upper bounds of ρ(t,x)for the cases(ii)(b)of Theorem 1.1 follows immediately(see Lemma 2.5).Having obtained the lower and upper bounds on ρ(t,x),the higher order energy estimates of solutions to the Cauchy problem (1.1)–(1.2)can be deduced by using the lower order estimates and Gronwall’s inequality, and then Theorem 1.1 follows by the standard continuation argument.In the next section, we will give the proof of Theorem 1.1.

NotationsThroughout this paper,C denotes some generic constant which may vary in different estimates.If the dependence needs to be explicitly pointed out,the notation C(·,···,·)or Ci(·,···,·)(i∈N)is used.f′(ρ)denotes the derivative of the function f(ρ) with respect to ρ.For function spaces,Lp(R)(1≤p≤+∞)is the standard Lebesgue space with the norm‖·‖Lp,and Hl(R)stands for the usual l-th order Sobolev space with its norm

2 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1.To do this,we seek the solutions of the Cauchy problems(1.1)–(1.2)in the following set of functions

where M≥m＞0 and T＞0 are some positive constants.

Under the assumptions of Theorem 1.1,we have the following local existence result.

Proposition 2.1(Local existence)Under the assumptions of Theorem 1.1,there exists a sufficiently small positive constant t1depending only on m0,m1,such that the Cauchy problems(1.1)–(1.2)admits a unique smooth solution(ρ,u)(t,x)∈

where b＞1 is a positive constant depending only on m0,m1.

The proof of Proposition 2.1 can be done by using the dual argument and iteration technique,which is similar to that of Theorem 1.1 in[10]and thus omitted here for brevity. Suppose that the local solution(ρ,u)(t,x)obtained in Proposition 2.1 has been extended to the time step t=T≥t1for some positive constant T＞0.To prove Theorem 1.1,one needs only to show the following a priori estimates.

Proposition 2.2(A priori estimates)Under the assumptions of Theorem 1.1,suppose that(ρ,u)(t,x)∈X(0,T;M0,M1)is a solution of the Cauchy problem(1.1)–(1.2)for somepositive constants T and M0,M1＞0.Then there exist two positive constants C1and C2which are independent of T,M0,M1such that the following estimates hold:

Proposition 2.2 can be obtained by a series of lemmas below.We first give the following key lemma.

Lemma 2.1(Basic energy estimates)Under the assumptions of Proposition 2.2,it holds that

for all t∈[0,T],where the functionis defined by

ProofIn view of the continuity equation(1.1)1,we have

On the other hand,by using(1.1)1again,the movement equation(1.1)2can be rewritten as

Substituting(2.6)into(2.5),we get

Here and hereafter,{···}xdenotes the terms which will disappear after integrating with respect to x.

Moreover,it follows from(1.1)1that

Combining(2.7)and(2.8),and integrating the resultant equation with respect to t and x over[0,t]×R,we can get(2.3).This completes the proof of Lemma 2.1.

In order to apply Kanel’s method[25]to show the lower and upper bound of the density ρ(t,x),we need to establish the following lemma.

Lemma 2.2There exists a uniform positive constant c0such that

ProofUsing the L’Hospital rule,we obtain

Consequently,there exist a sufficiently small constant δ and a large constantsuch that

and c0=minwe have(2.9)holds.This completes the proof of Lemma 2.2.

Based on Lemmas 2.1–2.2,we now show the lower and upper bounds of ρ(t,x)by using Kanel’s method[25].

Lemma 2.3(Lower and upper bounds of ρ(t,x)for the cases(ii)(a)of Theorem 1.1)Under the assumptions of Proposition 2.2,if the capillarity coefficient κ(ρ)satisfies the condition(ii)(a)of Theorem 1.1,then there exists a positive constant C3depending only

for all(t,x)∈[0,T]×R.

ProofLet

then under the condition(ii)(a)of Theorem 1.1,we have

On the other hand,we deduce from Lemmas 2.1–2.2 that

(2.13)thus follows from(2.14)and(2.15)immediately.This completes the proof of Lemma 2.3.

Next,we give the estimate on

Lemma 2.4Let condition(i)of Theorem 1.1 holds and

Then if f(ρ)≤0,there exists a positive constant C4depending only on m0,m1and‖(ρ0-

ProofFirst,by the continuity equation(1.1)1,we have

where we have used the fact that

Putting(2.17)into(2.6),and multiplying the resultant equation by

A direct calculation yields that

Combining(2.19)and(2.20),and integrating the resultant equation in t and x over[0,t]×R, we have

where we have used the fact that

By employing integrations by parts,we obtain

Inserting(2.22)–(2.23)into(2.21),and using(2.3),we arrive at

(2.24)together with the assumption that f(ρ)≤0 implies(2.16)immediately.This completes the proof of Lemma 2.4.

Lemma 2.5Let conditions(i)and(ii)(b)of Theorem 1.1 hold and f(ρ)≤0,then there exists a positive constant C5depending only on m0,m1andthat

for all(t,x)∈[0,T]×R.

ProofSet

then it follows from assumption(ii)(b)of Theorem 1.1 that

On the other hand,Lemmas 2.1 and 2.4 imply that

From(2.26)and(2.27),we have(2.25)at once.This completes the proof Lemma 2.5.

As a consequence of Lemmas 2.3–2.5,we have

Corollary 2.1Under the assumptions of Lemmas 2.3–2.5,it holds that for 0≤t≤T,

where C6＞0 is a constant depending only on m0,m1and

The next lemma gives an estimate on

Lemma 2.6There exists a positive constant C7depending only on m0,m1and‖(ρ0-such that for 0≤t≤T,

ProofWe derive from Lemmas 2.3–2.5 that

On the other hand,Lemmas 2.3–2.5 also imply that

From the Cauchy equality and(2.30),we infer that

Then(2.29)follows from(2.31)and(2.32)immediately.This completes the proof of Lemma 2.6.

For the estimate on‖ux(t)‖2,we have

Lemma 2.7There exists a positive constant C8depending only on m0,m1and‖(ρ0-such that for 0≤t≤T,

ProofMultiplying(2.6)by-uxx,and using the continuity equation(1.1)1,we have

Integrating(2.34)in t and x over[0,t]×R gives

where

It follows from the Cauchy inequality,the Sobolev inequality,the Young inequality,

Lemmas 2.3 and 2.5,and Corollary 2.1 that

Putting(2.36)–(2.37)into(2.35),and using Growwall’s equality,we obtain(2.33).This completes the proof of Lemma 2.7.

Finally,we estimate the term

Lemma 2.8There exists a positive constant C9depending only on m0,m1,and‖u0‖1such that for t∈[0,T],

ProofDifferentiating(1.1)2once with respect to x,then multiplying the resultant equation by ρxx,and using equation(1.1)1,we have

Integrating(2.39)with respect to t and x over[0,t]×R,using the Cauchy inequality,the Sobolev inequality,Lemmas 2.3–2.7 and Corollary 2.1,we can get Lemma 2.8,the proof is similar to Lemma 2.7 and thus omitted here.This completes the proof of Lemma 2.8.

It follows from Corollary 2.1,and Lemmas 2.6–2.8 that there exists a positive constant C10depending only on m0,m1,and‖u0‖1such that for 0≤t≤T,

Similarly,we can also obtain

where C11is a positive constant depending only on m0,m1,

Proof of Proposition 2.2Proposition 2.2 follows from(2.40)and(2.41)immediately.

Proof of Theorem 1.1By Propositions 2.1–2.2 and the standard continuity argument,we can extend the local-in-time smooth solution to be a global one(i.e.,T=+∞). Thus(1.4)and(1.5)follows from(2.1)and(2.2),respectively.Moreover,estimate(2.2)and system(1.1)imply that

which implies that

Furthermore,we deduce from(2.2),(2.43)and the Sobolev inequality that

From(2.43)and(2.44),we have(1.6)at once.This completes the proof of Theorem 1.1.

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一維可壓縮Navier-Stokes-Korteweg方程組的大初值整體光滑解

陳婷婷,陳志春,陳正爭

(安徽大學(xué)數(shù)學(xué)科學(xué)學(xué)院,安徽合肥230601)

本文研究了當(dāng)粘性系數(shù)和毛細(xì)系數(shù)是密度函數(shù)的一般光滑函數(shù)時,一維等溫的可壓縮Navier-Stokes-Korteweg方程的Cauchy問題.利用基本能量方法和Kanel的技巧,得到了大初值、非真空光滑解的整體存在性與時間漸近行為.本文結(jié)果推廣了已有文獻(xiàn)中的結(jié)論.

可壓縮Navier-Stokes-Korteweg方程;整體存在性;時間漸近行為;大初值

O175.29

tion:35Q35;35L65;35B40

A

0255-7797(2017)01-0091-16

?Received date:2016-04-09Accepted date:2016-04-20

Foundation item:Supported by National Natural Science Foundation of China(11426031)and Undergraduate Scientific Research Training Program of Anhui University(ZLTS2015141).

Biography:Chen Tingting(1995–),female,born at Tongling,Anhui,undergraduate,major in partial differential equation.

Chen Zhengzheng.