THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS*

Haibo YU (于海波)

School of Mathematical Sciences，Huaqiao University，Quanzhou 362021，China E-mail:yuhaibo2049@126.com

Abstract This paper concerns the global existence of strong solutions to the 3D compressible isothermal Navier-Stokes equations with a vacuum at in finity.Based on the special structure of the Zlotnik inequality，the time uniform upper bounds for density are established through some time-dependant a priori estimates under the assumption that the total mass is suitably small.

Key words global strong solution;compressible isothermal Navier-Stokes equations;vacuum

1 Introduction

We consider the 3D compressible Navier-Stokes equations for isothermal flows，which reads as:

where ρ≥0，u=(u1，u2，u3)trand P (ρ)=aρ(a＞0) represent the fluid density，velocity and pressure，respectively.The constants μ and λ are the shear and bulk viscosity coefficients satisfying the physical restrictions:

The main purpose of this paper is to look for global strong solutions to (1.1) with initial data

and the far field behavior

For a polytropic perfect gas，P (ρ)=aργwith γ being the adiabatic exponent.Equation (1.1) is called isentropic gas dynamics for γ＞1，while it is referred to as isothermal gas dynamics for γ=1.The isentropic compressible Navier-Stokes equations has been extensively studied;see[5，16，25，26]for the one-dimensional problem.For the multi-dimensional case，the local existence and uniqueness of classical solutions was proved in[15，22，27，28]in the absence of vacuum and in[1-3，24]for the case in which the initial density need not be positive and may vanish in open sets.The global smooth solutions were first obtained by Matsumura-Nishida[19-21]for initial data close to a non-vacuum equilibrium in Hs.Later，Hoff[7，8]studied the global weak solutions for discontinuous initial data.For the global existence of weak solutions with arbitrary initial data，the major breakthrough was due to Lions in[18](see also Feireisl et al.[4])，which proved the global existence of weak solutions when the adiabatic exponent is suitably large.However，little is known on the structure of such weak solutions.Under the additional assumptions that the viscosity coefficients μ and λ satisfy μ＞max{4λ，-λ}，and that the far field density is away from vacuum，Hoff[9-11]obtained a new type of global weak solution with small energy;this has extra regularity information compared with the solutions in[4，18].Huang et al.[12]established the global existence and uniqueness of classical solutions with constant state as far field which could be either vacuum or non-vacuum to the 3D isentropic compressible Navier-Stokes equations with small total energy but possibly large oscillations.For the case that the L∞-norm of initial density is small，Zhang et al.[29]obtained global existence of classical solutions in the framework of[12].

For the isothermal flow away from vacuum，Nishida[23]established the existence of global BV solutions in dimension one.For three-dimensional space，Hoff[6，7]studied the global weak solutions.For global smooth solutions，see also[19].When initial vacuum is allowed，Huang-Wang[14]obtained one dimensional global weak entropy solutions.Recently，Huang-Li[13]studied global weak and classical solutions for a non-vacuum far field density in R3with small data.A natural question to ask is whether or not smooth solutions exist globally when vacuum appears at infinity.

In this paper，we will study the global existence of strong solutions to the 3D compressible isothermal Navier-Stokes equations with vacuum in the far field.

Before stating our main result，we explain the notations and conventions used throughout this paper.We denote that

For 0≤r≤∞and integer k≥0，we denote the standard homogenous and inhomogenous Sobolev spaces as follows:

Our main result is

Theorem 1.1For any given positive numbersand M，we suppose that the initial data (ρ0，u0) satisfies

and

m≤ε，

then the problem (1.1)-(1.4) has a global strong solution (ρ，u) satisfying，for any 0＜T＜∞，that

Now we briefly outline the main ideas of the proof of Theorem 1.1.The major difficulty of our problem is to establish the time-independent upper bounds for the density ρ.It is worth mentioning that the issue here is much different to those in previous works，such as[7-13]，in which the a priori estimates are based，crucially，on the time uniform bounds of the term.However，for the isothermal flow case，one can hardly obtain uniform upper bounds for this term due to the vacuum at infinity.To establish time uniform a priori estimates，we deal with the local-in-time integralfor any 0≤t1＜t2≤T instead of.Under some necessary a priori assumptions，we build a kind of time-dependant estimate，i.e.，

This kind of estimate turns out to be the key to solving our problem.First of all，by the skill of the piecewise-estimate，we find thatcan be uniformly bounded (see Lemmas 3.5 and 3.6) provided that the initial mass m is suitably small.Our next mission is to establish the timedependant estimates for，which is actually determined by the structure of the Zlotnik inequality (Lemma 2.2).Utilizing a piecewise-estimate method for，we analogously get upper bounds for.In what follows，we have

which is achieved by modifying the basic elegant estimates on the material derivatives of the velocity developed by Hoff[7].Finally，using the time-dependant estimates above，the Zlotnik inequality，together with the smallness of m，implies the pointwise bounds of density.

2 Preliminaries

In this section，we recall some known facts and elementary inequalities which will be used frequently later.

The following local existence result is due to[2]:

Proposition 2.1Assume that the initial data (ρ0，u0) satisfies (1.5)-(1.6).Then there exist a time T*＞0 and a strong solution (ρ，u) to (1.1)-(1.4) in R3×[0，T*].

The next lemma arises from Zlotnik[30]，and will be used to prove the uniform (in time) upper bounds for density.

Lemma 2.2Let y∈W1，1(0，T) satisfy the ODE system

y′=g (y)+b′(t) on[0，T]，y (0)=y0，

where b∈W1，1(0，T)，g∈C (R) and g (+∞)=-∞.Assume that there are two constants N0≥0 and N1≥0 such that，for all 0≤t1≤t2≤T，

Then

y (t)≤max{y0，ξ*}+N0＜+∞on[0，T]，

where ξ*∈R is a constant such that

The following well-known Gagliardo-Nirenberg inequality will also be used (see[17]):

Lemma 2.3For p∈[2，6]，q∈(1，∞)，and r∈(3，∞)，there exists some generic constant C＞0 which may depend on q，r such that for f∈H1and g∈Lq∩D1，r，we have

Let F?(2μ+λ) divu-P and ω??×u denote the effective viscous flux and the vorticity，respectively，satisfying that

ΔF=div (ρ)，μΔω=?×(ρ).

Then，the following lemma can be deduced from the standard Lp-estimate for an elliptic system:

Lemma 2.4([12]) If (ρ，u) is a smooth solution of (1.1)-(1.4)，then there exists a generic positive constant C depending only on μ and λ such that，for any p∈[2，6]，

3 A Priori Estimates

Throughout this section，we use the letters C，∈，Ciand∈i(i=1，2，3，···) to denote generic positive constants which may depend on μ，λ，M，initial data and some other constants，but not on T，and we write C (α) to emphasize that C depends on α.

Let (ρ，u) be a strong solution to (1.1)-(1.4) with smooth initial data (ρ0，u0) satisfying (1.5)-(1.6) on R3×(0，T]for any fixed time T＞0.We set σ(t)?min{1，t}and define

Then，we have the following a priori estimates:

Proposition 3.1For any given positive numbersand M，assume that (ρ0，u0) satisfies (1.5)-(1.6).Then there exists a positive constant ε3such that if (ρ，u) is a strong solution of (1.1)-(1.4) on R3×(0，T]，satisfying that

then provided that m≤ε3，the following estimates hold:

ProofProposition 3.1 is a consequence of Lemmas 3.2-3.6. □

3.1Uniform upper bounds for

The next lemma concerns the time-dependant estimate for the local-in-time integral forinstead of the standard energy inequality;this is different to previous results (such as[7-13]，and the references therein).This kind of estimate plays an essential role in this paper.

Lemma 3.2Let the assumptions of Proposition 3.1 hold.Then

for any 0≤t1＜t2≤T.

ProofMultiplying (1.1)2by u in L2，we have，from Young’s inequality，that

Integrating the above inequality over[t1，t2]for 0≤t1＜t2≤T leads to

where we have used (3.1).The proof of Lemma 3.2 is finished. □

Remark 3.3Under the condition of (1.4)，one could not expect to obtain time-uniform upper bounds for the termdt，which plays a key role in previous results.To illuminate this，multiplying (1.1)2by u in L2，we have that

which gives that

In view of (1.4)，the term aρlogρdx is hard to control.

To establish the upper bounds of，we need the following result，which，together with Lemma 3.8，is due to[7]:

Lemma 3.4Let (ρ，u) be a strong solution of (1.1)-(1.4) on R3×(0，T]with 0≤ρ≤.Then it holds that

where η=η(t)≥0 is a piecewise smooth function.

ProofMultiplying (1.1)2by η(t)and integrating over R3yields that

Using (1.1)1，and integrating by parts，we have that

Further，integration by parts then implies that

and similarly，

We can deduces from (2.6)，(2.7) and Young’s inequality that

Substituting J1-J3and (3.6) into (3.5) leads to (3.4)，which finishes the proof of Lemma 3.4. □

We now establish our a priori estimate of A2(σ(T)).

Lemma 3.5Let the assumptions of Proposition 3.1 hold.Then there exists a positive constant ε2such that

provided that m≤ε2.

ProofTaking η=1 and integrating (3.4) over[0，t]for 0＜t≤σ(T)，we have，from Young’s inequality and (3.3)，that

provided that m≤ε1?.Choosing m≤ε2?{ε1，(16C2M)-3/2}，we immediately obtain (3.7) from (3.1).This completes the proof of Lemma 3.5. □

In the next lemma，we proceed to give the upper bounds of A1(T) by using the method of a piecewise-estimate.

Lemma 3.6Let the assumptions of Proposition 3.1 hold.Then there exists a positive constant ε3such that

provided that m≤ε3，where σi(t)?σ(t+1-i) and i is an integer satisfying 1≤i≤[T]-1.

Remark 3.7For simplicity，we only prove the case T＞2.Otherwise，things can be done by choosing a suitably small step size.

ProofFor integer i (1≤i≤[T]-1)，taking η=σi(t) and integrating (3.4) over (i-1，i+1]imply

where we have used (3.1) and (3.3).We deduce from (3.9) that

due to σ1(t)=σ(t) and

provided that m≤ε3?.Note that the constant C3is independent of i.(3.8) follows from (3.10) and (3.11).The proof of Lemma 3.7 is complete. □

3.2 Upper bounds for density

To derive the pointwise upper bounds for density，the key element here is the timedependant estimate for.As such，we need the following lemma:

Lemma 3.8Let (ρ，u) be a strong solution of (1.1)-(1.4) on R3×(0，T]with 0≤ρ≤.Then it holds that

where η=η(t)≥0 is a piecewise smooth function.

ProofApplying，summing with respect to j，and integrating the resulting equation over R3，we obtain，after integration by parts，that

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

Integration by parts then leads to

Similarly，we have

Analogous to (3.6)，we derive that

Collecting all of the above estimates into (3.13)，we finish the proof of this lemma. □

Based on Lemma 3.8，we establish the desired time-dependant estimates for.

Lemma 3.9Let the assumptions of Proposition 3.1 hold.Then we have that

for 0≤t1＜t2≤T，provided that m≤ε3.

ProofFor any integer 1≤i≤[T]-1，integrating (3.12) with η=over (i-1，i+1]，we have，from (3.2)，(3.3)，(3.8) and Young’s inequality，that

According to (3.16)，we reach

Hence，(3.14) follows from (3.17) and (3.18).

We now proceed to give the estimate on?.First，we integrate (3.4) over[t1，t2]?[0，T]and take η=σ to obtain，from (3.2) and (3.3)，that

Second，integrating (3.12) over[t1，t2]and taking η=σ2，we find that

where (3.2)，(3.3)，(3.14) and (3.19) have been used.Then，(3.15) follows from (3.20).The proof of Lemma 3.9 is thus finished. □

We still need the following result before showing the upper bounds of density:

Lemma 3.10Let the assumptions of Proposition 3.1 hold.Then we have that

provided that m≤ε3.

ProofTaking η=σ and integrating (3.12) over[0，σ(T)]，we get，from (3.2)，(3.3)，(3.7) and Young’s inequality，that

which completes the proof of this lemma. □

With Lemmas 3.9 and 3.10 in hand，we derive the uniform upper bounds of the density ρ in the following lemma:

Lemma 3.11There exists a positive constant ε as described in Theorem 1.1 such that if (ρ，u) is a strong solution of (1.1)-(1.4) on R3×(0，T]with 0≤ρ≤+1，then

provided that m≤ε.

ProofWe first rewrite (1.1)1as

For t∈[0，σ(T)]，we deduce，from H?lder’s inequality，(2.3)-(2.6)，(3.2) and (3.21)，that for all 0≤t1＜t2≤σ(T)，

provided that m≤ε3.Therefore，for t∈[0，σ(T)]，we can choose N0and N1in (2.1) as

N1=0，N0=C4m1/16

and ξ*=0 in (2.2).Then，

Thus，we deduce from Lemma 2.2 that

provided that

For t∈[σ(T)，T]，we derive from (2.3)-(2.6)，(3.2) and (3.15)，for all σ(T)≤t1＜t2≤T，that

provided that

Therefore，for t∈[σ(T)，T]，we can choose N0and N1in (2.1) as

and ξ*=1 in (2.2).Then，

Thus，due to Lemma 2.2，we arrive at

provided that

Combining (3.23) with (3.24)，Lemma 3.11 is proved. □

4 Proof of Theorem 1.1

From now on，we always assume that m≤ε，as in Lemma 3.11，and that the constant C may depend on T and g.

Lemma 4.1We have that

ProofTaking η=1 in (3.12) and integrating the resulting equation over (0，σ(T)]，we have，from (3.7) and Young’s inequality，that

which finishes the proof of Lemma 4.1. □

The following result follows directly from Lemmas 3.9 and 4.1:

Lemma 4.2We have that

Similarly to[12，Lemma 3.6]，we have the following:

Lemma 4.3It holds that

Proof of Theorem 1.1By Proposition 2.1，there exists a positive time T*＞0 such that the initial value problem (1.1)-(1.4) has a strong solution (ρ，u) on R3×(0，T*].Next，we prove that the local strong solution (ρ，u) is indeed globally defined for all time.To this end，we assume from now on that m≤ε holds，with ε＞0 being the same as in Lemma 3.11.

Now，we claim that

Otherwise，T*＜∞.It follows from Lemmas 3.6 and 4.3 that (ρ，u)(x，T*) satisfies (1.5).Moreover，Lemma 4.2 yields that∈L2.Thus，Proposition 2.1，together with the continuity arguments，implies that there exists a T**＞T*such that (ρ，u) can be extended to be a strong solution of (1.1)-(1.4) on R3×(0，T**)，which contradicts (4.1).Hence，(4.2) holds.The proof of Theorem 1.1 is therefore complete. □