Yulin YESchool of Mathematics and Statistics,Henan University,Kaifeng 475004,China E-mail:nkyelin@163.com

GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY∗

In this paper,we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ) provided that the initial energy is small in some sense.In our result,we give a relation between the initial energy and the viscosity coefficientµ,and it shows that the initial energy can be large if the coefficient of the viscosityµis taken to be large,which implies that large viscosityµmeans large solution.

global existence;classical solution;compressible Navier-Stokes equations; density-dependent viscosity;vacuum

2010 MR Subject Classification35A09;35Q30

1　Introduction and Main Results

In this paper,we consider the following compressible isentropic Navier-Stokes equations with density-dependent viscosity coefficient in ℝ3:

here,we denote by ρ and u the density and velocity of the fluid,respectively.The pressure p satisfies

The shear and bulk viscosity coefficientsµand λ are enforced to satisfy

In the sequel,we set A=1 without loss of generality.The initial conditions are assumed to bewhere e ρ is a fixed nonnegative constant.

There were huge literatures on the global well-posedness of solutions to Compressible Navier-Stokes equations.When both of the shear and bulk viscosity coefficients are positive constants,the first result of global classical solution was obtained by Matsumura and Nishida [19],in which they required that the initial data was close to a non-vacuum equilibrium in some Sobolev space Hsand the initial density was strictly away from vacuum.In the presence of vacuum,an important breakthrough was due to Lions[17],who first developed an existence theory of global(in time)weak solutions to the Cauchy problem(1.1),(1.3)–(1.4),when the exponent γ is suitably large.Recently,Huang et al.[9]established the well-posedness of global classical solution with small initial energy but possibly large oscillations.Inspired by this work, Deng et al.[2]obtained the similar result in which they required the viscosity coefficientµwas large enough and the small initial energy condition was not necessary.Furthermore,in[3o], Zhang et al.improved the above two results[2,9]and gave a relation between the initial energy and the viscosity coefficientµand showed that when the initial data was of small energy or the viscosity coefficientµwas large enough,there existed a global classical solution.For the cases of full compressible Navier-Stokes equations,Huang et al.[7]and Wen et al.[23]proved the global existence of classical solution when the initial density has non-vacuum or vacuum state at infinity,respectively,in particular,the authors in[23]gave a specific relation between the initial mass and the viscosityµand the heat conduction κ,and showed that there exists a global classical solution provided that either the initial mass is small or the viscosity and heat conduction are large enough,our paper is inspired by this work.

When deriving the compressible Navier-Stokes equations from the Boltzmann equations by the Chapman-Enskog expansions,the viscosity depends on the temperature,and correspondingly depends on the density for isentropic cases.There were a large number of literatures on mathematical studies on the Navier-Stokes equations with density-dependent viscosities. One-dimensional case is referred to[3,4,1o–14,2o,21,24,25]and references therein.For multi-dimensional cases,Vaigant and Kazhikhov[22]considered a special model in which they required the viscosity coefficientµ=const.>o and λ(ρ)=ρβand obtained a remarkable result that the two-dimensional system(1.1)–(1.4)in a torus admits a unique global strong solution for large initial data away from vacuum provided β>3.Recently,under natural compatibility conditions on the initial data,Jiu,Wang and Xin[15,16]considered classical solutions permitting vacuum and improved the index β>3 to β>4/3.Similar results were independently obtained by Huang and Li in[8].Zhang and Fang[29]proved the existence of global weak solution with small energy to the Vaigant-Kazhikhov model and presented the vanishing vacuum behavior.Besides these,for the cases of general density-dependent viscosity coefficient λ(ρ), Zhang[28]established the global well-posedness of classical solution with small initial energy for two-dimensional system but requiring the lower bound of the initial density.Moreover,when the viscosity coefficientµis large or the initial data is of small energy,Zhang[27]and Liu et al.[18]obtained the global existence of classical solution for the three-dimensional system with presence of vacuum,respectively.

In the present paper,we study the global well-posedness of classical solution to the Cauchy problem of the three-dimensional compressible Navier-Stokes equations(1.1)–(1.4)with densitydependent viscosity and vacuum.In our result,we give a relation between the initial energyand the viscosity coefficientµ,and show that there exists a global classical solution if the initial energy is small or the viscosity coefficientµis large enough,besides this,we also obtain the large time behavior of the solution.Our results improve ones of[18,27]and include the results of the constant viscosity cases in[3o].Since the density-dependent viscosity λ(ρ),the non-linearity of the equation is much stronger than the constant viscosity cases,to overcome this difficulty,we require λ(ρ)∈C3[o,∞).And in our Rprocess of the proof,to derive the un R iform upper bound of the density,we use the smallness ofdt in Zlotnik inequality,as a result,we just let the defined Variables A1(T),A2(T)are bounded and do not need to do the time-weighted estimates.Therefore,compared with the previous results[18,3o], our proof is much simpler.

Throughout this paper,we adopt the following simplified notations for the standard homogeneous and inhomogeneous Sobolev spaces(see[5]for more details).

The initial energy of the compressible flows is defined as

It is easy to obtain for some positve constants

Moreover,we assume

where λois a constant.

Now we are ready to state our main result as follows.

Theorem 1.1For any given Ki>o(i=1,2)and ρ>e ρ+1,assume that the initial data (ρo,uo)satifies

and the compatibility condition

for some g∈D1withThen for any T>o,there exists a unique global classical solution(ρ,u)in ℝ3×[o,T]satisfying,

for some constant C>o depending on γ,ρ and some other known constants but independent ofµ,t.

Furthermore,we have the following large time behavior

Remark 1.2Using the Sobolev Embedding theorem,it is easy to show that the solution obtained in Theorem 1.1 is a classical one.

Remark 1.3In our result,the initial energy depends on the viscosity coefficientµand λ,so it is easy to see that the initial energy will be large if the viscosity coefficientµis taken to be large.Hence,our result shows that,for the Cauchy problem of the isentropic compressible Navier-Stokes equations(1.1)–(1.4),there exists a small global classical solution with limited initial energy whenµis a fixed constant or there exists a large global classical solution with general initial data when the viscosity coefficientµis large enough.We give a uniform theorem to show our result,which is more general and include the previous results[18,27].

Remark 1.4It should be noted that our result is also hold even for the constant viscosity, just replacing the λoby λ in(1.9).

The rest of the paper is organized as follows:In Section 2,we collect some elementary facts and inequalities,which will be needed in later analysis.In Section 3,we derive the necessary a priori estimates of the solution which are needed to extend the local solution to a global one.

2　Preliminaries

In this section,we state some known facts and elementary inequalities,which will be used frequently later.First,we recall the following Zlotnik inequality which will be used to obtain the uniform upper bound of the density ρ.

Lemma 2.1Assume that the function y∈W1,1(o,T)solves the ODE system:

where b∈W1,1(o,T)and g∈C(ℝ).If g(∞)=−∞and

for all o≤t1≤t2≤T with some positive constants Noand N1,then one has

where ξ∗∈ℝ is a constant such that

Motivated by[6],we introduce the following Material derivative,the effective viscous flux and vorticity by

Then the momentum equation(1.1)2can be rewritten as

Lemma 2.2Assume that(ρ,u)is a smooth solution of(1.1),(1.3)–(1.5).Then there exists a generic positive constant C such that for any p∈[2,6],we have

Proof(2.2)–(2.4)can be directly obtained by using standard Lpestimate of the elliptic system(2.1)and Sobolev inequality.

Note that−△u=−∇divu+∇×w,which implies that

then we apply Calderon-Zygmund inequality to get

Then(2.5)is hold.

This finishes the proof of Lemma 2.2.

Finally,to extend the local solutions globally in time,we need the following local existence theorem of the classical solutions to(1.1)–(1.3).

Lemma 2.3(see[1,26])Assume that the initial data(ρo,uo)satisfy(1.5)–(1.6)exceptThen there exist a small-time T∗and a unique classical solution(ρ,u)to the Cauchy problem(1.1),(1.3),(1.4)on ℝ3×(o,T∗]such that

3　A Priori Estimates

In this section,we will establish a global priori estimates for the smooth solution to the Cauchy problem(1.1),(1.3),(1.4)to extend the local classical solution guaranteed by Lemma 2.3.Assume that(ρ,u)is a smooth solution of(1.1),(1.3),(1.4)on ℝ3×[o,T]with some fixed time T.To derive the desired estimates,we define

We have the following key a priori estimates on(ρ,u).

Proposition 3.1Assume that the initial data satisfies(1.5),(1.6).If the solution(ρ,u) satisfies

for some constant C>o depending onγ and some other known constants but independent ofµand t.

Proposition 3.1 is an easy consequence of Lemmas 3.2–3.4.

First,we state the following standard energy estimate for(ρ,u).

Lemma 3.2Under the conditions of Proposition 3.1,it holds that

ProofMultiplying the mass and the momentum equations by G′(ρ)and u,respectively, then integrating over ℝ3×[o,T],we can easily obtain(3.5),and we omit the details here.

Lemma 3.3Under the conditions of Proposition 3.1,it holds that

ProofMultiplying(1.1)2by˙u,and integrating the resulting equality by parts over ℝ3, we have

where we have used the following constructer of equations

Integrating(3.7)over(o,T),we have

Then together with(3.3),we obtain that

which implies that

so,we can also obtain that

Next,multiplying(3.8)by 3(p−p(e ρ))2and integrating the resulting equality over ℝ3,and using the effective viscous flux,we can obtain that

then integrating the resulting inequality over(o,T),and choosing ε sufficiently small,one obtains that

Therefore,substituting(3.16)and(3.14)into(3.13),we have

which together with(3.11)and(3.17),gives

Then we complete the proof of Lemma 3.3.

Lemma 3.4Under the conditions of Proposition 3.1,it holds that

ProofFirst,applying∂t+∂k(uk·)to momentum equation(1.1)2,gives

Multiplying on both sides of(3.21)by˙ujand integrating by parts over ℝ3,we have

Integrating(3.22)over(o,T)and substituting(3.17)into the resulting inequality gives

Thus the proof of Lemma 3.4 is completed.

Lemma 3.5Under the conditions of Proposition 3.1,it holds that

for any(x,t)∈ℝ3×[o,T],provided

ProofFirst,we rewrite the mass eq.(1.1)1as

where Dtρ=ρt+u·∇ρ and

Then to estimate‖b(t)‖L∞,using Lemma 2.2 and(3.14),we deduce that for all o≤t1

So,one can choose Noand N1as

Note that

then we deduce from Lemma 2.1 and(3.25)by choosing

provided that

Thus the proof of Lemma 3.5 is completed.

Next,by standard energy estimates,we can easily obtain the spatial gradient and high order estimates of the smooth solution(ρ,u)as follows,we refer readers to[18],and we omit the details.

Lemma 3.6Let(ρ,u)be a smooth solution of(1.1)–(1.3)on ℝ3×[o,T].Then

Lemma 3.7Let(ρ,u)be a smooth solution of(1.1)–(1.3)on ℝ3×[o,T].Then

Lemma 3.8Let(ρ,u)be a smooth solution of(1.1)–(1.3)on ℝ3×[o,T].Then

Lemma 3.9Let(ρ,u)be a smooth solution of(1.1)–(1.3)on ℝ3×[o,T].Then

Lemma 3.10Let(ρ,u)be a smooth solution of(1.1)–(1.3)on ℝ3×[o,T].Then for τ∈(o,T],there exists a positive constant C(τ,T),depending on τ and T,such that

With all the estimates above and using the standard arguments based on the local existence theorem,one can easily extend the classical solution of(ρ,u)to a global one,and the large time behavior is similar obtained too,we omit the details and the readers can be referred to[18]. Then the proof of Theorem 1.1 is completed.

References

[1]Cho Y,Kim H.On classical solutions of the compressible Navier-Stokes equation with nonnegative initial densities.Manuscripta Math,2006,120(1):91–129

[2]Deng X M,Zhang P X,Zhao J N.Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data.Sci China Math,2012,55(12):2457–2468

[3]Ding S J,Wen H Y,Zhu C J.Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum.J Differ Equ,2011,251(6):1696–1725

[4]Dou C S,Jiu Q S.A remark on free boundary problem of 1-D compressible Navier-Stokes equations with density-dependent viscosity.Math Meth Appl Sci,2010,33:103–116

[5]Galdi G P.An Introduction to the Mathematical Theory of the Navier-Stokes Equations.New York: Springer-Verlag,1994

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

[7]Huang X D,Li J.Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes System with vacuum and large oscillations.Kyoto Univ Res Inform Repository,2013,1830:98–118

[8]Huang X D,Li J.Global well-posedness of classical solutions to the Cauchy problem of two-dimensional baratropic compressible Navier-Stokes system with vacuum and large initial data.Preprint,arXiv:1207. 3746,2012

[9]Huang X D,Li J,Xin Z P.Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations.Comm Pure Appl Math,2012, 65(4):549–585

[10]Jiang S.Global smooth solutions of the equations of a viscous,heat-conducting one-dimensional gas with density-dependent viscosity.Math Nachr,1998,190:169-183

[11]Jiang S,Xin Z P,Zhang P.Global weak solutions to 1D compressible isentropy Navier-Stokes with densitydependent viscosity.Methods Appl Anal,2005,12(3):239–252

[12]Jiu Q S,Li M J,Ye Y L.Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data.J Differ Equ,2014,257(2):311–350

[13]Jiu Q S,Xin Z P.The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients.Kinet Relat Models,2008,1(2):313–330

[14]Jiu Q S,Wang Y,Xin Z P.Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with Density-dependent viscosity.Comm Partial Differential Equations,2011,36:602–634

[15]Jiu Q S,Wang Y,Xin Z P.Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum.J Math Fluid Mech,2014,16(3):483–521

[16]Jiu Q S,Wang Y,Xin Z P.Global well-posedness of the Cauchy problem of 2D compressible Navier-Stokes equations in weighted spaces.J Differ Equ,2013,255(3):351–404

[17]Lions P L.Mathematical Topics in Fluid Dynamics 2,Compressible Models.Oxford:Oxford Science Publication,1998

[18]Liu S Q,Zhang J W,Zhao J N.Global classical solutions for 3D compressible Navier-Stokes equations with vacuum and a density-dependent viscosity coefficient.J Math Anal Appl,2013,401(2):795–810

[19]Matsumura A,Nishida T.The initial value problem for the equations of motion of viscous and heatconductive gases.J Math Kyoto Univ,1980,20(1):67–104

[20]Mellet A,Vasseur A.Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations.SIAM J Math Anal,2008,39(4):1344–1365

[21]Okada M,Matusu N S,Makino T.Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity.Ann Univ Ferrara Sez VII(NS),2002,48:1–20

[22]Vaigant V A,Kazhikhov V A.On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid.Sibirsk Mat Zh,1995,36(6):1283–1316

[23]Wen H Y,Zhu C J.Global classical solution to 3D full compressible Navier-Stokes equations with vacuum at infinity.arXiv:1208.1199

[24]Yang T,Yao Z A,Zhu C J.Compressible Navier-Stokes equations with density-dependent viscosity and vacuum.Comm Partial Differ Equ,2001,26(5/6):965–981

[25]Ye Y L.Global classical solution to 1D compressible Navier-Stokes equations with no vacuum at infinity. Math Meth Appl Sci,2016,39(4):776–795

[26]Ye Y L,Dou C S,Jiu Q S.Local well-posedness to the Cauchy problem of the 3-D compressible Navier-Stokes equations with density-dependent viscosity.Acta Math Sci,2014,34B(3):851–871

[27]Zhang P X.Global classical solution to the 3D isentropic compressible Navier-Stokes equations with general initial data and a density-dependent viscosity coefficient.Math Methods Appl Sci,2015,38(2):1158–1177

[28]Zhang T.Global solutions of compressible barotropic Navier-Stokes equations with a density-dependent viscosity coefficient.J Math Phys,2011,52(4):043510

[29]Zhang T,Fang D Y.Compressible flows with a density-dependent viscosity coefficient.SIAM J Math Anal, 2010,41(6):2453–2488

[30]Zhang P X,Zhao J N.Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data.Sci China Math,2014,57(7):1463–1478

∗January 29,2015;revised May 30,2015.