heat conduction and far field vacuum compressible Navier-Stokes equations with zero Entropy bounded solutions to the one-dimensional Advances in Mathematics

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Advances in Mathematics

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Entropy bounded solutions to the one-dimensional compressible Navier-Stokes equations with zero heat conduction and far field vacuum

Jinkai Li^a,∗, Zhouping Xin^b

aSouthChinaResearchCenterforAppliedMathematicsandInterdisciplinary Studies,SouthChinaNormalUniversity,Guangzhou510631,China

bTheInstituteof MathematicalSciences,TheChineseUniversityofHongKong, HongKong,China

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received17December2017 Receivedinrevisedform6 September2019

Accepted13November2019 Availableonlinexxxx

CommunicatedbyC.Fefferman

MSC:

35Q30 76N10

Keywords:

FullcompressibleNavier-Stokes equations

Globalexistenceanduniqueness Strongsolutionswithbounded entropy

Farfieldvacuum

InhomogeneousSobolevspaces

Theentropyisoneofthefundamental statesofafluidand, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importanceinthegasdynamics,themathematicalanalyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at someisolated interior points, it was unknown whether the entropy remains its boundedness. The results obtained in this paper indicate thattheidealgases retaintheiruniformboundednessofthe entropy,locally orgloballyintime, ifthevacuumoccursat thefar field only and thedensity decays slowly enoughat thefar field. Precisely, weconsider the Cauchy problem to theone-dimensionalfullcompressibleNavier-Stokesequations withoutheat conduction,andestablish thelocalandglobal existenceanduniquenessofentropy-boundedsolutions,inthe presence of vacuum at the far field only. It is also shown that,differentfromthecasethatwithcompactlysupported initialdensity,thecompressibleNavier-Stokesequations,with

* Correspondingauthor.

E-mailaddresses:[email protected],[email protected](J. Li),[email protected] (Z. Xin).

https://doi.org/10.1016/j.aim.2019.106923 0001-8708/©2019ElsevierInc. Allrightsreserved.

slowlydecayinginitialdensity,canpropagatetheregularities ininhomogeneousSobolevspaces.

©2019ElsevierInc.Allrightsreserved.

1. Introduction

1.1. Thecompressible Navier-StokesequationsinEuler coordinates

Let ρ,u, and θ, respectively, be the density, velocity, and temperature of a fluid.

Denote by t and x the time and spatial variables. Then, the full compressible Navier- Stokesequationsread as

∂_tρ+ div (ρu) = 0, (1.1)

∂t(ρu) + div (ρu⊗u) = divT+ρf, (1.2)

∂t(ρE) + div (ρuE) + divq= div (Tu) +ρQ, (1.3) where E=^|u₂^|2+eisthespecifictotalenergy,e=e(ρ,θ) isthespecificinternalenergy, T isthestresstensor,qistheinternalenergyfluxdirectlyrelatedtothetransferofheat, f is theexternalforce, andQistheexternalheatsource.

By(1.1)–(1.3),onecanobtainthefollowingequationfore:

∂t(ρe) + div (ρue) +pdivu+ divq=S:∇u+ρQ. (1.4) Thestresstensor T isgivenby

T =S−pI, S= 2μDu+λdivuI, Du= 1

2(∇u+ (∇u)^T),

whereIisthe3×3 identitymatrix,pisthepressure,andμandλareviscositycoefficients, satisfying μ>0 and 2μ+ 3λ≥0.Inthispaper, we consider theidealgases,and state equations are

p=Rρθ, e=c_vθ,

fortwo positiveconstants Randc_v.Then,itfollows from(1.4) that

c_v[∂_t(ρθ) + div (ρuθ)] +pdivu+ divq=S:∇u+ρQ. (1.5) Recallingthestateequationsforpande,bytheGibbsequationθDs=De+pD(¹_ρ), where sisthespecificentropy,onehasthefollowingrelationshipbetweenpands:

p=Ae^cvs ρ^γ, forsomepositive constantA, whereγ−1= _c^R

v.Since R,cv >0, itisclearthatγ >1.

Thanks to this, and using the state equations for pand e again, one can derive from (1.1),(1.2),and(1.5) thefollowingequationfortheentropys:

∂t(ρs) + div (ρsu) + divq θ

=1 θ

S:∇u−q· ∇θ θ

+ρQ

θ.

Fortheinternalenergyfluxq,bytheFourier’s lawofheat conduction,weassumethat q=−κ∇θ,where κ≥0 istheheatconductioncoefficient.

There is an extensive literature on the mathematical analysis of the compressible Navier-Stokes equations. In the absence of vacuum, that is the density is bounded from below by some positive constant, the local well-posedness results were proved by Nash [33], Itaya [16], Vol’pert–Hudjaev [39], Tani [36], Valli [37], and Lukaszewicz [28].Thefirstglobalwell-posednessresultwasestablishedbyKazhikhov–Shelukhin[21], where theyprovedtheglobal well-posednessof strongsolutionsof theinitialboundary value problem to the one-dimensional compressible Navier-Stokes equations, for arbi- trary H¹ initial data, and the corresponding result for the Cauchy problem was later provedbyKazhikhov[20];globalwell-posednessofweaksolutionstotheone-dimensional compressible Navier-Stokes equations was proved by Zlotnik–Amosov [43,44] and by Chen–Hoff–Trivisa[1] fortheinitialboundaryvalueproblems,andbyJiang–Zlotnik[19]

forthe Cauchyproblem.Largetime behavior ofsolutionsto theonedimensionalcom- pressibleNavier-StokesequationswithlargeinitialdatawasrecentlyprovedbyLi–Liang [24]. Forthe multi-dimensionalcase, theglobal well-posedness of strongsolutionswere establishedonly for smallperturbed initial data around somenon-vacuumequilibrium or for spherically symmetric large initial data, see Matsumura–Nishida [29–32], Ponce [34], Valli–Zajaczkowski [38], Deckelnick [8], Jiang [17], Hoff [12], Kobayashi–Shibata [22], Danchin [7], Chen–Miao–Zhang [2], and Chikami–Danchin [3]. One of the major differences between one dimensionalcase from the multi-dimensional oneis thatif no vacuum is containedinitially, then novacuum will form later on infinite time, forthe one dimensionalcompressible Navier-Stokes equations, as shownby Hoff-Smoller [13], whilethesimilarresultremainsunknownforthemulti-dimensionalcase.

Inthe presence of vacuum,that isthe density mayvanish on someset, or tends to zeroat thefarfield,thebreakthrough wasmadebyLions [26,27], wherehe provedthe globalexistenceofweaksolutionstotheisentropiccompressibleNavier-Stokesequations, with adiabatic constant γ ≥ ⁹₅; the requirement on γ was later relaxed by Feireisl–

Novotný–Petzeltová [9] to γ > ³₂, and further by Jiang–Zhang [18] toγ > 1 but only fortheaxisymmetricsolutions.ForthefullcompressibleNavier-Stokesequations,global existenceof thevariationalweaksolutionswas provedby Feireisl[10,11]; however,due to the assumptions onthe constitutiveequations made in[10,11], theideal gases were notincludedthere.Localwell-posednessofstrongsolutions,inthepresenceofvacuum, wasprovedfirstfortheisentropiccasebySalvi–Straškraba[35],Cho–Choe–Kim[4],and

Cho–Kim [5], and later for the polytropic case by Cho–Kim [6]. It should be noticed that,in[35,4–6],thesolutionswereestablishedinthehomogeneousSobolevspaces,that is, it is √ρu rather than u itself that has the L^∞(0,T;L²) regularity. Generally, one can not expect that the strong solutions to the compressible Navier-Stokes equations lie in the inhomogeneous Sobolev spaces, if the initial density has compact support.

Actually, it was proved recently by Li–Wang–Xin [23] that: neither isentropic nor the full compressible Navier-Stokes equations on R, with κ = 0 forthe full case, has any solution (ρ,u,θ) inthe inhomogeneous Sobolevspaces C¹([0,T];H^m(R)), with m>2, if ρ0 is compactly supported and some appropriate conditions on the initial data are satisfied;theN-dimensionalfullcompressibleNavier-Stokesequations,withpositiveheat conduction,havenosolution(ρ,u,θ),withfiniteentropy,intheinhomogeneousSobolev spaces C¹([0,T];H^m(R^N)), with m > [^N₂]+ 2, if ρ₀ is compactly supported. Global existence ofstrong andclassical solutionstothe compressibleNavier-Stokesequations, in the presence of initial vacuum,was first proved by Huang–Li–Xin [15], where they establishedtheglobalwell-posednessofstrongandclassicalsolutions,withsmallinitial basic energy,to thethree-dimensionalisentropic compressibleNavier-Stokesequations, see Li–Xin [25] for further developments. However, due to the finite in time blow-up results by Xin [41] and Xin–Yan [42], one can not expect the global well-posedness of classical solutions, ineither inhomogeneous or homogeneous Sobolev spaces, to the full compressible Navier-Stokes equations in the presence of vacuum. In particular, it was proved in [42] that, for the full compressible Navier-Stokes equations, if initially there is an isolated mass group surrounded by the vacuum region, then for the case κ = 0, any classical solutionmust blow-up in finite time, and for the case κ>0, any classicalsolutions,withfiniteentropyinthevacuumregion,mustblow-upinfinitetime.

Global existence of strong solutions to the heat conducting full compressible Navier- StokesequationswereobtainedbyHuang–Li[14] forthecasethatwithnon-vacuumfar field, and by Wen–Zhu [40] for the case thatwith vacuum far field. The spaces of the solutions obtainedin[14,40] cannot excludethepossibility thattheentropyis infinite somewhere inthevacuum region, even ifitis initially finite;infact,due to theresults in[42], thecorresponding entropyin[14,40] mustbe infinitesomewhereinthevacuum region, ifinitially thereisanisolatedmassgroupsurroundedbythevacuumregion.

By thestateequations for theidealgases, theentropycanbe expressed interms of thedensityandtemperatureas

s=cv

logR

A+ logθ−(γ−1) logρ

,

from which it follows that the entropy may develop singularities or even is not well defined in the vacuum region and, consequently,it is impossible to obtain thedesired regularities of s merely from those of θ and ρ, in the presence of vacuum. Therefore, though the vacuums are allowed for the solutions established in [6,14,40] by choosing (ρ,u,θ) as theunknowns,noregularitiesoftheentropyscanbe impliedinthevacuum region there and, due to the result in [42], the entropy of the solutions obtained in

[14,40] must be infinite in the vacuum region. To the best of our knowledge, in the existingliteratures,therewerenosuchresultsthatprovidedtheuniformlowerorupper boundsoftheentropynearthevacuum.

As stated in the previous paragraph, since the entropy can not be even defined at theplaceswhere thedensityvanishes, itmaybeunreasonable tostudy theentropyfor the full compressible Navier-Stokes equations of the ideal gases, if the vacuum region is anopen set;however,when the vacuum occurs onlyat someisolated interior points or at the far fieldand if, moreover, the entropy behaves well when the fluid tends to these vacuum points or to the far field, it is still possible to define the entropy there.

Therefore,anaturalquestioniswhatkindofbehavioroftheentropy,atthevacuumfar fieldor near the isolated interior vacuum points, can be preserved by the ideal gases, when the flow evolves. The aim of this paper is to give someanswers to this question and, in particular, as indicated in our main results, the ideal gases canpreserve their boundednessoftheentropy,locallyorgloballyintime,ifthevacuumhappensatthefar fieldonly.

Another question thatwe want to addressin this paper is: under whatkind of as- sumptions on the initial density,beyond the case thatthe initial density is uniformly awayfromthevacuum,thecompressibleNavier-Stokesequationsadmitsolutionsinthe inhomogeneous Sobolev spaces. On one hand, recalling the result in [23], for the case thattheinitialdensityhasacompactsupport,thecompressibleNavier-Stokesequations areill-posedintheinhomogeneousSobolevspaces;ontheotherhand,forthecasethat theinitial density is uniformly awayfrom the vacuum,the compressibleNavier-Stokes equations are well-posed in the inhomogeneous Sobolev spaces. Comparing these two cases, and regarding the case that the density has compact support as that the den- sity has supper fast decay at the far field, one may ask whether the fast decay of the density can cause the ill-posedness of the compressible Navier-Stokes equations in the inhomogeneousspaces,orwhetherthecompressibleNavier-Stokesequationswillbewell- posedintheinhomogeneousspaceswhentheinitialdensitydecaysslowlyatthefarfield.

Wewill show inthis paper thatif theinitial density decaysslower than _|^K_x|⁰2,for some positive constant K₀, at the far field, then the compressible Navier-Stokes equations are indeed well-posedin the inhomogeneous Sobolev spaces, where K₀ is an arbitrary positiveconstant.Notethatthisisconsistentwiththewell-posednessresultforthecom- pressible Navier-Stokesequations in theinhomogeneous Sobolev spaces inthe absence ofvacuum.

In this paper, we consider the onedimensional case, and assume thatthere are no externalforcesandheatingsource,i.e.f ≡ Q≡0,andthatthereisnoheatconduction inthefluids,thatisκ= 0,whilethemulti-dimensionalcaseandthecasesthatwithheat conductionwill be studied inthe furtherworks. Under these assumptions, the system considered in this paper is the following one-dimensional compressible Navier-Stokes equations:

ρt+ (ρu)x= 0, (1.6)

ρ(ut+uux)−μuxx+px= 0, (1.7) c_v[(ρθ)_t+ (ρuθ)_x] +pu_x=μ(u_x)². (1.8) Due to p = Rρθ and cv = _γ^R₋₁, equation (1.8) can be rewritten equivalently as a equation forthepressurep,thatis

p_t+up_x+γu_xp=μ(γ−1)(u_x)². (1.9) Itismoreconvenienttouse(1.9),insteadof(1.8),tostateandprovetheresults,inother words, we will use the pressure, instead of the temperature, as oneof the unknowns, throughout this paper; however, it shouldbe mentioned that, as we consider the case thatthevacuumappearsonlyatthefarfield,(1.9) isequivalent to(1.8).

Themain resultsofthis paperare statedandprovedintheLagrangiancoordinates, see Section1.2; however,sincethe solutionsbeing establishedare Lipschitzcontinuous andthedensityvanishesonlyatthefarfields,allresultscanbetransformedbackinthe Euler coordinatesaccordingly.

1.2. Reformulationin Lagrangiancoordinates andmainresults

Let y be the Lagrangiancoordinate,and define thecoordinate transformbetween y and theEulercoordinatexas x=η(y,t),where

⎧⎨

⎩

∂tη(y, t) =u(η(y, t), t), η(y,0) =y.

Denote

(y, t) :=ρ(η(y, t), t), v(y, t) :=u(η(y, t), t), π(y, t) :=p(η(y, t), t), and define

J(y, t) =ηy(y, t).

Onecanverifyeasily that

J_t=v_y, J|t=0= 1, J = ₀

with 0 beingtheinitialvalueofthedensity.Then,system(1.6),(1.7),and(1.9) trans- forms tothefollowingoneintheLagrangiancoordinate:

J_t=v_y, (1.10)

0vt−μ vy

J

y+πy= 0, (1.11)

πt+γvy

J π= (γ−1)μ vy

J 2

, (1.12)

whereμ>0 andγ >1 areconstants.

We will consider the Cauchy problem and, thus, complement system (1.10)–(1.12) withtheinitialcondition

(J, v, π)|t=0= (J₀, v₀, π₀), (1.13) where J₀ has uniformpositive lower and upper bounds.It shouldbe pointed outthat, bythe definitionof J, theinitial J0 shouldbe identicallyone;however, forthe aim of extending a local solution (J,v,π) to be a global one, we need the local existence of solutionstosystem(1.10)–(1.12),withinitialJ₀notbeingidenticallyone.Therefore,in thispaper,inthestudyofthelocalsolutions,theinitialJ₀isallowedtobenotidentically one,butfortheglobalsolutions,wealwaysassumethatJ0isidenticallyone.

Aswillbe shownlater,theeffectiveviscous fluxGdefinedas G:=μvy

J −π,

whichwasfirstintroducedbyD.Hoff,playsacrucialroleinprovingtheglobalexistence of solutions to system (1.10)–(1.12). By straightforward calculations, it follows from (1.11) and(1.12) that

Gt− μ J

Gy 0

y

=−γvy

JG. (1.14)

Beforestatingthemainresults,wefirstgivesomeconventionsonnecessarynotations to be used throughout this paper and define strong solutions. For 1 ≤ q ≤ ∞, the Lebesgue space L^q(R) consists of all measurable functions f on R with finite norm

f _L^q,where

f _L^q =

⎧⎨

⎩

R|f|^qdx¹_q

, if 1≤q <∞; ess sup_Rf, ifq=∞.

For positive integer m and for 1 ≤ q ≤ ∞, W^m,q = W^m,q(R) is the Sobolev space consisting of allfunctionson R whose generalizedderivatives upto order m belongto L^q.H^mstandsforW^m,2.Forsimplicity,wealsousethenotationsL^qandH^mtodenote the N product spaces (L^q)^N and (H^m)^N, respectively. Wealways use u _q to denote theL^q norm of u. Forsimplicity ofpresentation, wesometimesuse (f₁,f₂,· · ·,f_n) _X todenote thesumN

i=1 fi X or itsequivalentnormN i=1 fi 2

X

¹₂ .

Local and global strong solutions to the problem (1.10)–(1.13), are defined in the followingtwo definitions.

Definition 1.1.Given a positive time T ∈ (0,∞). A triple (J,v,π) is called a strong solutionto theproblem (1.10)–(1.13),onR×(0,T),ifithastheproperties

y∈R,t∈(0,T)inf J(y, t)>0, π≥0 onR×(0, T), J−J₀∈C([0, T];L²), Jy

√ 0 ∈L^∞(0, T;L²), J_t∈L^∞(0, T;L²),

√

0v∈C([0, T];L²), v_y∈L^∞(0, T;L²), √

0v_t, vyy

√ 0

∈L²(0, T;L²), π∈C([0, T];L²), πy

√ 0 ∈L^∞(0, T;L²), π_t∈L⁴(0, T;L²),

satisfiesequations(1.10)–(1.12),a.e.inR×(0,T),andfulfillstheinitialcondition(1.13).

Definition 1.2.A triple (J,v,π) is called a global strong solution to the problem (1.10)–(1.13),ifitisastrongsolutiontothesamesystemonR×(0,T),foranypositive time T ∈(0,∞).

The main results of this paper are the following two theorems concerning the local and globalexistenceofstrongsolutionsto theproblem(1.10)–(1.13).

Theorem 1.1 (Local well-posedness).Let μ > 0 and γ > 1 be constants. Assume that ( ₀,J₀,v₀,π₀)satisfies

y∈(−r,r)inf 0(y)>0, ∀r∈(0,∞), 0≤¯onR, (H1) √

0v₀, v₀, π₀, π₀

√ 0

∈L², π₀≥0on R, (H2) J ≤J0≤J¯onR, J₀

√ 0 ∈L², forpositive constants ,¯J,andJ,¯ and denoteG0:=μv₀−π0.

The followingtwohold:

(i) ThereisapositivetimeT depending only onγ,μ, ,¯J ,J ,¯ v_{0 2}, π_{0 2},and π_{0 ∞}, suchthatsystem(1.10)–(1.12),subjecttotheinitialcondition(1.13),has aunique strongsolution(J,v,π),on R×(0,T).

(ii) Assumeinaddition that

1

√ 0

(y) ≤K₀

2 , ∀y∈R, ⁻₀ ^δ2G₀∈L², (H3) fortwopositiveconstantsδandK0.

Then,(J,v,π)has theadditionalregularities

−^δ2

0 G∈L^∞(0, T;L²)∩L⁴(0, T;L^∞), ⁻

δ+1 2

0 G_y∈L²(0, T;L²), (1.15) whereG:=μ^v_J^y −πistheeffectiveviscousflux, and

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

v∈L^∞(0, T;H¹), if v0∈H¹ andδ≥1, ϑ∈L^∞(0, T;H¹), if ϑ₀∈H¹,J₀

0 ∈L²,andδ≥1, s∈L^∞(0, T;L^∞), if s₀∈L^∞ andδ≥γ,

(1.16)

where ϑ := _R^π and s := cvlog π

A^γ

, respectively, are the corresponding temperature and entropy, with := _J⁰ being the density, and ϑ0 := _R^π0

0 and s0 := cvlog

π0

A^γ₀

, respectivelyare theinitialtemperatureandentropy.

Remark1.1. Basically,the condition √¹0

≤ ^K₂⁰ or equivalently | 0| ≤K0

3 2

0 onR meansthat 0 decaysnofaster than _y^K2 at thefarfield.Indeed,if

0(y) = K

y, 0< K<∞,0≤<∞, wherey= (1 +y²)¹², then

1

√ 0

≤K0

2 onR ⇐⇒ ≤2.

Remark1.2. Choose

0(y) = K

y, J0≡1, v0∈C_c^∞, π0=Ae^{cv1 γ}₀, whereK and arepositive numbers.

(i) If _2γ¹₋₁ < ≤2,then ( ₀,v₀,π₀) satisfies conditions(H1), (H2),and (H3), with δ= 1.Therefore, byTheorem 1.1, there isauniquelocal strongsolution (J,v,π), with v being in the inhomogeneous Sobolevspace L^∞(0,T;H¹); if moreoverthat γ > ⁵₄ and _2(γ−1)¹ < ≤2,then ϑ0 ∈H¹, and, consequently,the temperatureϑ alsoliesintheinhomogeneousSobolevspaceL^∞(0,T;H¹).Notethatthisdoesnot contradicttotheill-posednessresultsforthecompressibleNavier-Stokesequations in[23],as theinitialdensitythereisassumedto becompactlysupported.

(ii) If _γ¹ < ≤2, then ( ₀,v₀,π₀) satisfies conditions (H1),(H2), (H3),with δ =γ, and s0 ≡ 1. Therefore, by Theorem 1.1, there is a unique local strong solution

(J,v,π), and thecorresponding entropys isuniformly bounded onR×(0,T).To thebestofourknowledge,thisisthefirsttimethattheboundednessoftheentropy isachieved,inthepresenceofvacuumatthefarfield,forthecompressibleNavier- Stokesequations.

(iii) Combining(i)with(ii)showsthatifγ > ⁵₄ andmax

1

γ,_2(γ¹₋₁₎

< ≤2,thereis auniquelocal strongsolution(J,v,π),with thepropertiesthatthecorresponding entropyisuniformlybounded,andthevelocityandthecorrespondingtemperature lieintheinhomogeneous spaceL^∞(0,T;H¹).

Remark1.3.

(i) The assumption inf_{y∈(−R,R) 0}(y) >0, for all R ∈ (0,∞), is used for the bound- edness of the entropy and the regularities of the velocity and temperaturein the inhomogeneousSobolevspaces,butitisnotneededforthelocalwell-posedness(in thehomogeneousSobolevspaces).

(ii) Thecompressible Navier-Stokesequationspropagate theregularitiesinthe homo- geneous Sobolev spaces, see [4–6], but not in the inhomogeneous Sobolev spaces (in particular,theL² regularity ofv cannotbe propagated),see [23],iftheinitial density has a compact support. While (ii) of Theorem 1.1 shows that, if the ini- tial density decays slowly to the vacuum at the far field, then the regularities in thehomogeneous Sobolevspaces,inparticular, theL² regularity ofv, canbealso propagated bythecompressibleNavier-Stokesequations.

(iii) The result in (ii) of Theorem 1.1 also indicates that the uniform boundedness of theentropycanbe propagatedbythecompressibleNavier-Stokesequations,ifthe initialdensitydecaysslowlytothevacuumat thefarfield.

Remark1.4.It followsfromthedefinitionofstrongsolutionsthat

√ 0v∈L^∞(0, T;L²), vy∈L²(0, T;H¹), whichimpliesv∈L²(0,T;W^1,∞(R)).DefinetheEulercoordinateas

x=η(y, t), ∂_tη(y, t) =v(y, t), η(y,0) =y.

Note that

∂tηy=vy =∂tJ, ηy(y,0) = 1 =J(y,0).

Thus, ηy ≡J on R×(0,T).Since J has uniform positive lower and upper bounds on R×(0,T), for any fixed t ∈ (0,T), η is reversible in y. Therefore, one candefine the densityρ,velocity u,andpressurep,intheEulercoordinateas

ρ(x, t) = (y, t), u(x, t) =v(y, t), p(x, t) =π(y, t),

where (y,t) := _J(y,t)^0(y). It can be checked that (ρ,u,p) hasappropriate regularities, in particularu∈L¹(0,T;W^1,∞(R)),and itisasolutionto system(1.6), (1.7),and (1.9), subjecttotheinitial data( 0,v0,π0);whiletheuniquenessintheEulercoordinatecan be provenby transformingit to theLagrangiancoordinate, as u∈L¹(0,T;W^1,∞(R)), andapplyingtheuniquenessresultstatedinProposition3.1.

Next,wehavethefollowingglobal well-posedness.

Theorem1.2(Globalwell-posedness). Assumethat (H1)–(H2)hold, andthat

0∈L¹, π₀∈L¹, ₀(y)≥ A₀

(1 +|y|)², ∀y∈R, (H4) forsomepositiveconstant A0.

Then,thefollowingtwohold:

(i) Thereisauniqueglobal strongsolution(J,v,π)tosystem(1.10)–(1.12),subjectto theinitialcondition (J,v,π)|t=0= (1,v0,π0).Moreover,itholds that

R

1

2 ⁰(y)v²(y, t) + 1

γ−1J(y, t)π(y, t)

dy=E0,

R

0(y)v(y, t)dy=m0, inf

y∈RJ(y, t)≥c0, forany t∈[0,∞),where

E0:=

R

0v₀² 2 + π0

γ−1

dy, m0=

R

0v0dy, c0=e⁻²

√2 μ

√E001.

(ii) Assumefurther that (H3)holds,for twopositiveconstantsδ andK0.Then, (1.15) and(1.16) hold forany T ∈(0,∞).

Remark 1.5.If the initial data has more regularities, then the corresponding solution ( ,v,π) in Theorem 1.2 canbe classical ones and, consequently, we obtain the global existenceofclassicalsolutionstothecompressibleNavier-Stokesequationswithoutheat conduction,inthe presenceof vacuum at far fields.Tothe bestof ourknowledge, this isthefirstresultontheglobalexistenceofstrongsolutionstothecompressibleNavier- Stokesequationswithoutheatconduction,forarbitrarylargeinitialdata,inthepresence offarfieldvacuum.Notethatthisglobalexistenceresultdoesnotcontradicttothefinite timeblow-upresultsin[42],astheassumptionofhavinginitialisolatedmassgroupthere isexcludedinourcase.

Remark1.6.Thefollowingassumption in(H4)

0(y)≥ A₀

(1 +|y|)², ∀y∈R

canbe removed.Infact,noticingthat,essentially,therole thatthis assumptionplayed in theproof of Theorem 1.2is to justify integration by parts ofsome integralsdefined on thewholeline,so thatonecangetthebasicenergyinequalityandtheestimates on G, see Proposition 4.1, Proposition4.4, and Proposition 4.6. Alternatively, to get the desiredbasicenergyinequalityandtheestimatesonG,onecanapproximatetheCauchy problembyasequenceofinitial-boundaryvalueproblems,whilefortheinitial-boundary value problems, theintegration byparts to the corresponding integrals canbe carried outwithouttheaboveassumption.

Remark1.7.Choose

0(y) = K

y, J₀≡1, v₀∈C_c^∞, π₀=Ae^{cv1 γ}₀, where K and arepositive numbers.

(i) If 1 < ≤ 2, then ( 0,v0,π0) satisfies assumptions (H1), (H2), (H3), with δ = 1, and (H4). Therefore, by Theorem 1.2, there is a unique global strong solution (J,v,π),withvbeingintheinhomogeneousSobolevspaceL^∞(0,T;H¹);ifmoreover that γ > ⁵₄ and max

1,_2(γ¹₋₁₎

< ≤2, then ϑ0 ∈H¹, and, consequently, the temperatureϑalsoliesintheinhomogeneousSobolevspaceL^∞(0,T;H¹).

(ii) Ifmax 1,¹_γ

< ≤2,then( 0,v0,π0) satisfiesconditions(H1),(H2),(H3),with δ=γ,(H4),ands₀≡1.Therefore,byTheorem1.2,thereisauniquestrongsolution (J,v,π),and thecorresponding entropysisuniformly boundedonR×(0,T).

(iii) Consequently, if γ > ⁵₄ and max

1,¹_γ,_2(γ¹₋₁₎

< ≤ 2, then there is a unique globalstrong solution(J,v,π),withthepropertiesthatthecorresponding entropy isuniformly bounded andthatthevelocity andthecorresponding temperaturelie intheinhomogeneous spaceL^∞(0,T;H¹).

Remark1.8.Sameas inRemark1.4,onecanobtainthecorrespondingglobal existence of solutionsintheEuler coordinatestothecompressibleNavier-Stokesequations(1.6), (1.7),and (1.9),subjectto theinitialdata( ₀,v₀,π₀).

Therest ofthis paperisarrangedasfollows: inSection2,weconsider thesystemin theabsenceofvacuum,andcarryoutsomeaprioriestimates,whichareindependentof thepositive lower boundofthedensity;theproof ofTheorem1.1 isgiveninSection3, while thatofTheorem 1.2isgiveninthelast section.

2. Localexistenceinthe absenceofvacuum

Inthissection,westudytheCauchyproblem(1.10)–(1.13),intheabsenceofvacuum, thatis, thedensity 0 is assumed to haveapositive lower bound. Wefocus onsomea priori estimates of the solutionswhich are independent of the positive lower bound of thedensity ₀.

Thenthefollowinglocal existenceresultholds:

Proposition 2.1.Given afunction ₀ satisfying ≤ 0≤ ¯on R, fortwo positivecon- stants and .¯ LetN0beapositiveconstantsuchthat _J¹+ ¯J+ J₀ 2+ (v0,π0) H¹ ≤N0. Assumethat theinitialdata(J0,v0,π0)satisfies

J ≤J0≤J¯onR, J₀ ∈L², v0∈H¹, 0≤π0∈H¹, fortwopositiveconstantsJ andJ¯.

Then,there isaunique localstrongsolution (J,v,π)tosystem(1.10)–(1.12),subject totheinitialcondition(1.13),on R×(0,T),satisfying

3

4J ≤J ≤5

4J ,¯ π≥0, onR×[0, T], J−J0∈C([0, T];H¹), Jt∈L^∞(0, T;L²),

v∈C([0, T];H¹)∩L²(0, T;H²), vt∈L²(0, T;L²), π∈C([0, T];H¹), πt∈L²(0, T;L²),

whereT is apositiveconstantdepending only onμ,γ, , ,¯ and N0.

Proof. This will be provedbythe fixed pointargument. Let M and T be two positive constantswithTbeingsuitablysmall,tobedeterminedbythequantity_N0.SetX_M,T :=

v v _L∞(0,T;H¹)∩L²(0,T;H²)≤M

.Givenv∈X_M,T,defineJ andπ,successively,asthe uniquesolutionsto thefollowing twoordinarydifferentialequations:

Jt=vy, and

πt+γvy

J π= (γ−1)μvy

J 2

, withinitialdataJ|t=0=J₀ andπ|t=0=π₀,respectively.

WeclaimthatJ andπdefinedasabovehavethepropertiesstatedintheproposition.

Notethat

π=e^−γ0tvyJdsπ0+μ(γ−1) t 0

e^{−γstvyJdτ}vy

J 2

ds. (2.1)

Thus,theassumptionsμ>0,γ >1,andπ0≥0 implyπ≥0.Sincevy∈L^∞(0,T;L²)∩ L²(0,T;H¹), so J −J0 = t

0vyds ∈ C([0,T];H¹) and Jt = vy ∈ L^∞(0,T;L²). By Gagliardo-Nirenberg andHölderinequalities,

J−J0 ∞(t) = t 0

vyds ∞

≤C_∗ t 0

vy

1 2

2 vyy

1 2

2ds

≤C_∗

sup

0≤s≤t

v_{y 2} ¹₂⎛

⎝ t 0

v_yy ²₂ds

⎞

⎠

1 4

t³⁴ ≤C_∗M T³⁴ ≤J 4, as long as 0≤ t ≤ T ≤(_4C^J

∗M)⁴³, and, thus, ³₄J ≤ J ≤ ⁵₄J¯on R×[0,T]. Thanks to this and that v_y ∈ L²(0,T;H¹) → L²(0,T;L^∞) and J_y ∈ L^∞(0,T;L²), one obtains that ^v_J^y ∈L²(0,T;L²) and,bytheHölderinequality, ^v_J^y

y= ^v_J^yy −^vy_J^J2y ∈L²(0,T;L²).

Therefore, ^v_J^y ∈L²(0,T;H¹) and furtherL²(0,T;L^∞) by theembedding. Using these, recalling (2.1),andnoticingthat

πy=μ(γ−1) t 0

e^{−γstvyJdτ}

⎡

⎣2vy

J vy

J

y−γvy

J 2t

s

vy

J

ydτ

⎤

⎦ds

+e^−γ0tvyJds

⎛

⎝π₀ −γ t

0

vy

J

yds

⎞

⎠,

one can get easily by the Hölder inequality that π ∈ L^∞(0,T;H¹) and further that π ∈ L^∞(0,T;L^∞) by theembedding. Recalling that ^v_J^y ∈L²(0,T;L^∞∩H¹) and π∈ L^∞(0,T;H¹∩L^∞),oneobtainsthatπt= (γ−1)μ ^v_J^y2

−γ^v_J^yπ∈L²(0,T;L²) and πyt= 2μ(γ−1)vy

J vy

J

y−γ

πy

vy

J +πvy

J

y

∈L¹(0, T;L²),

inotherwords,π_t∈L²(0,T;L²)∩L¹(0,T;H¹).Thisandπ∈L^∞(0,T;H¹) leadtothat π∈C([0,T];H¹) andπt∈L²(0,T;L²).

Take T ≤(_4C^J

∗M)⁴³. Then, as statedin theprevious paragraph, itholds that ³₄J ≤ J ≤ ⁵₄J¯ on R×(0,T). Let V be unique solution to the following uniform parabolic equation

V_t− μ

0

Vy

J

y

=−πy 0

, inR×(0, T),

subject to theinitial dataV|t=0 =v₀.Then, the classictheory forparabolic equations shows thatV ∈ L^∞(0,T;H¹)∩L²(0,T;H¹) and Vt ∈ L²(0,T;L²). Define a mapping M:v→V,withV definedasabove.Bystandardenergyestimates,onecanshowthat

Mis acontractingmappingonXM,T, forsomeM andT depending onlyonμ,γ, , ,¯ and N0. Consequently, by the contracting mapping principle, there is a unique fixed point, denoted by v, to M in XM,T. Then (J,v,π), with (J,π) defined in the way as above,isadesiredsolutiontosystem(1.10)–(1.12),subjectto(1.13),onR×(0,T).Since theproofislengthybutstandard,thedetailsareomittedhere.

ByProposition2.1, there is apositive time T1,such thatthe problem (1.10)–(1.13) hasauniquesolution(J,v,π),onthetime interval(0,T1),satisfying

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

3

4J ≤J ≤ ⁵₄J ,¯ π≥0, onR×[0, T1], J−J₀∈C([0, T₁];H¹), J_t∈L^∞(0, T₁;L²),

v∈C([0, T1];H¹)∩L²(0, T1;H²), vt∈L²(0, T1;L²), π∈C([0, T1];H¹), πt∈L²(0, T1;L²).

Starting from the time T₁, noticing that (J,v,π)|t=T1 satisfies the conditions on the initial data stated in Proposition 2.1, one can extend the solution (J,v,π) forward in timetoanothertimeT2=T1+t1,forsomepositivetimet1dependingonlyonμ,γ, , ,¯ andtheupperboundof(T1),where,forsimplicity ofnotations,wehavedenoted

(t) :=

yinf∈RJ −1

+ J _∞+ Jy 2+ v H¹+ π H¹

(t), (2.2) suchthat (J,v,π) isthe uniquesolutionto the problem (1.10)–(1.13) on time internal (0,T₂),andthatitenjoysthesameregularitiesasaboveinthetimeinterval(0,T₂),and (³₄)²J ≤J≤(⁵₄)²J¯onR×[0,T₂].Continuingthisprocedure,oneobtainstwosequences ofpositive numbers {tj}^∞j=1 and {Tj}^∞J=1,with tj depending onlyonμ,γ, , ,¯ and the upperboundof(Tj),andTj+1=Tj+tj,suchthatthesolution(J,v,π) canbeextended totimeintervals(0,Tj),satisfying

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

(³₄)^jJ ≤J ≤(⁵₄)^jJ ,¯ π≥0, onR×[0, T_j], J −J0∈C([0, Tj];H¹), Jt∈L^∞(0, Tj;L²),

v∈C([0, Tj];H¹)∩L²(0, Tj;H²), vt∈L²(0, Tj;L²), π∈C([0, T_j];H¹), π_t∈L²(0, T_j;L²),

forj= 1,2,· · ·.SetthemaximalexistingtimeT_∞as T_∞=T1+

∞ j=1

tj.

Then,thesolution(J,v,π) canbe extendedto thetimeinterval(0,T_∞),suchthat

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0< inf

y∈R,t∈[0,T]J(y, t)≤ sup

y∈R,t∈[0,T]

J(y, t)<∞, π≥0 onR×[0, T],

J −J0∈C([0, T];H¹), Jt∈L^∞(0, T;L²),

v∈C([0, T];H¹)∩L²(0, T;H²), v_t∈L²(0, T;L²), π∈C([0, T];H¹), π_t∈L²(0, T;L²),

(2.3)

foranyT ∈(0,T_∞).Moreover,ifT_∞<∞,itmusthavelim_j→∞tj= 0 and,consequently, onehas

j→∞lim (Tj) =∞, (2.4)

where (t) is defined by (2.2). Otherwise, if (2.4) is not true, then (T_j) ≤ N0 for somepositive constant_N0;inthis case,noticingthattheexistencetime t_j providedin Proposition2.1dependsonlyonμ,γ, , ,¯ andtheupperboundof(T_j),onecanchoose t_j>0 tobeindependent ofj,contradictingto thefactthatlim_j→∞t_j = 0.

Thanks to the statements in the above paragraph, in the rest of this section, we alwaysassumethat(J,v,π) istheuniquesolutionto(1.10)–(1.13) andthatithasbeen extended,inthesamewayasabove,tothemaximalexistingtimeinterval(0,T_∞),where themaximal timeT_∞ isconstructedinthesameway asabove.

To obtaintheaprioriestimateson(J,v,π),we defineapositive time Ts:= sup T ∈(0, T_∞)

J

2 ≤J ≤2 ¯J onR×[0, T]

!

. (2.5)

Westartwiththefollowing estimateonG:

Proposition 2.2. Thereisapositiveconstant t¹_∗=t¹_∗(γ,μ, ,¯ J , √

J₀G_{0 2}),suchthat

sup

0≤t≤T_∗¹

√J G ²₂+μ

T_∗¹

0

Gy

√ 0

²

2

dt≤3(1 + "

J₀G₀ ²₂),

T_∗¹

0

G ⁴_∞dt≤C(μ, , J ,¯ "

J0G0 2),

where G0:=μv₀−π0,T_∗¹:= min{1,t¹_∗,Ts},andTs isdefined by(2.5).

Proof. Multiplying (1.14) by J G, and integrating the resultant over R, one gets by integration bypartsthat

R

J GG_tdy+μ

R

(Gy)²

0

dy=−γ

R

v_yG²dy. (2.6)