University of Science and Technology of China, Hefei , P. R. China. Academia Sinica, Beijing , P. R. China

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1 Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations Xiangdi HUANG a,c,jingli b,c, Zhouping XIN c a Department of Mathematics, University of Science and Technology of China, Hefei 36, P. R. China b Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 119, P. R. China c The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong Abstract We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions which may have large oscillations and can contain vacuum states. 1 Introduction The time evolution of the density and the velocity of a general viscous isentropic compressible fluid occupying a domain Ω R 3 is governed by the compressible Navier- Stokes equations: { ρ t +divρu) =, 1.1) ρu) t +divρu u) μδu μ + λ) divu)+ P ρ) =, where ρ, u = u 1,u,u 3 )andp = aρ γ a >,γ > 1) are the fluid density, velocity and pressure, respectively. The constant viscosity coefficients μ and λ satisfy the physical restrictions: μ>, μ+ 3 λ. 1.) This research is supported in part by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK44/6P and CUHK44/8P, and a Focus Area Grant from The Chinese University of Hong Kong. The research of J. Li is partially supported by NSFC Grant No xdhuang@ustc.edu.cn X. Huang), ajingli@gmail.com J. Li), zpxin@ims.cuhk.edu.hk Z. Xin). 1

2 Let Ω = R 3 and ρ be a fixed nonnegative constant. We look for the solutions, ρx,t),ux,t)), to the Cauchy problem for 1.1) with the far field behavior: ux, t), ρx, t) ρ, as x, 1.3) and initial data, ρ, u) t= =ρ,u ), x R ) There are huge literatures on the large time existence and behavior of solutions to 1.1). The one-dimensional problem has been studied extensively by many people, see [7, 18, 6, 7] and the references therein. For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in [3, 8] in the absence of vacuum and recently, for strong solutions also, in [, 4, 5, 5] for the case that the initial density need not be positive and may vanish in open sets. The global classical solutions were first obtained by Matsumura-Nishida [] for initial data close to a nonvacuum equilibrium in some Sobolev space H s. In particular, the theory requires that the solution has small oscillations from a uniform non-vacuum state so that the density is strictly away from the vacuum and the gradient of the density remains bounded uniformly in time. Later, Hoff [8, 9] studied the problem for discontinuous initial data. For the existence of solutions for arbitrary data the far field density is vacuum, that is, ρ = ), the major breakthrough is due to Lions [1] see also Feireisl [6]), where he obtains global existence of weak solutions - defined as solutions with finite energy - when the exponent γ is suitably large. The main restriction on initial data is that the initial energy is finite, so that the density vanishes at far fields, or even has compact support. However, little is known on the structure of such weak solutions. Recently, under the additional assumptions that the viscosity coefficients μ and λ satisfy μ>max{4λ, λ}, 1.5) and for the far field density away from vacuum ρ >), Hoff [1 1]) obtained a new type of global weak solutions with small energy, which have extra regularity information compared with those large weak ones constructed by Lions [1]) and Feireisl [6]). Note that here the weak solutions may contain vacuum though the spatial measure of the set of vacuum has to be small. Moreover, under some additional conditions which prevent the appearance of vacuum states in the data, Hoff [1, 1]) obtained also classical solutions. It should be noted that in the presence of vacuum, the global well-posedness of classical solutions and the regularity and uniqueness of those weak solutions [6,1,1]) remains completely open. Indeed, this is a subtle issue since, in general, one would not expect such general results due to Xin s blow-up results in [9], where it is shown that in the case that the initial density has compact support, any smooth solution to the Cauchy problem of the non-barotropic compressible Navier-Stokes system without heat conduction blows up in finite time for any space dimension, and the same holds for the isentropic case 1.1), at least in one-dimension, and the symmetric two-dimensional case [15]). See also the recent generalizations to the cases for the non-barotropic compressible Navier-Stokes system with heat conduction [3]) and for non-compact but rapidly decreasing at far field initial densities [4]). In this paper, we will study the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations, 1.1), in three-dimensional space with smooth initial data which are of small energy but possibly

3 large oscillations with constant state as far field which could be either vacuum ρ =) or non-vacuum ρ >); in particular, the initial density is allowed to vanish, even has compact support. Before stating the main results, we explain the notations and conventions used throughout this paper. We denote fdx = fdx. R 3 For 1 <r<, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows: { L r = L r R 3 ), D k,r = { u L 1 loc R3 ) k u L r < }, u D k,r k u L r, W k,r = L r D k,r, H k = W k,, D k = D k,, D 1 = { u L 6 u L < }. The initial energy is defined as: C = where G denotes the potential energy density given by Gρ) ρ ) 1 ρ u + Gρ ) dx, 1.6) ρ ρ P ρ) P ρ) s ds. It is clear that { Gρ) = 1 γ 1P, if ρ =, c 1 ρ, ρ)ρ ρ) Gρ) c ρ, ρ)ρ ρ), if ρ >, ρ ρ, for positive constants c 1 ρ, ρ) andc ρ, ρ). Then the main results in this paper can be stated as follows: Theorem 1.1 Assume that 1.) holds. For given numbers M > not necessarily small) and ρ ρ +1, suppose that the initial data ρ,u ) satisfy and the compatibility condition inf ρ sup ρ ρ, u L M, 1.7) u D 1 D 3, ρ ρ, P ρ ) P ρ)) H 3, 1.8) μ u μ + λ) divu + P ρ )=ρ g, 1.9) for some g D 1 with ρ 1/ g L. Then there exists a positive constant ε depending on μ, λ, ρ, a, γ ρ and M such that if C ε, 1.1) the Cauchy problem 1.1) 1.3) 1.4) has a unique global classical solution ρ, u) satisfying for any <τ<t <, ρx, t) ρ, x R 3,t, 1.11) 3

4 ρ ρ, P P ρ)) C[,T]; H 3 ), u C[,T]; D 1 D 3 ) L,T; D 4 ) L τ,t; D 4 ), u t L,T; D 1 ) L,T; D ) L τ,t; D ) H 1 τ,t; D 1 ), ρut L,T; L ), 1.1) and the following large-time behavior: ρ ρ q + ρ 1/ u 4 + u )x, t)dx =, 1.13) for all lim t q {, ), for ρ >, γ, ), for ρ =. 1.14) Similar to our previous studies on the Stokes approximation equations in [], we can obtain from 1.13) the following large time behavior of the gradient of the density when vacuum states appear initially and the far field density is away from vacuum, which is completely in contrast to the classical theory [1, ]). Theorem 1. In addition to the conditions of Theorem 1.1, assume further that there exists some point x R 3 such that ρ x )=. Then if ρ >, the unique global classical solution ρ, u) to the Cauchy problem 1.1) 1.3) 1.4) obtained in Theorem 1.1 has to blow up as t, in the sense that for any r>3, A few remarks are in order: lim ρ,t) L t r =. Remark 1.1 The solution obtained in Theorem 1.1 becomes a classical one for positive time. Although it has small energy, yet whose oscillations could be arbitrarily large. In particular, both interior and far field vacuum states are allowed. Remark 1. In the case that the far field density is away from vacuum, i.e., ρ >, the conclusions in Theorem 1.1 generalize the classical theory of Matsumura-Nishida []) to the case of large oscillations since in this case, the requirement of small energy, 1.1), is equivalent to smallness of the mean-square norm of ρ ρ, u ). However, though the large-time asymptotic behavior 1.13) is similar to that in [], yet our solution may contain vacuum states, whose appearance leads to the large time blowup behavior stated in Theorem 1., this is in sharp contrast to that in [1, ] where the gradients of the density are suitably small uniformly for all time. Remark 1.3 When the far field density is vacuum, i.e., ρ =, the small energy assumption, 1.1), is equivalent to that both the kinetic energy and the total pressure are suitably small. There is no requirement on the size of the set of vacuum states. In particular, the initial density may have compact support. Thus, Theorem 1.1 can be regarded as uniqueness and regularity theory of Lions-Feireisl s weak solutions in [6, 1] with small initial energy. It should be noted that the conclusions in Theorem 1.1 for the case of ρ =are somewhat surprising since for the isentropic compressible Navier- Stokes equations 1.1), any non-trivial one-dimensional smooth solution with initial compact supported density blows up in finite time [9]), and the same holds true for two-dimensional smooth spherically symmetric solutions [15]). 4

5 Remark 1.4 It should be emphasized that in Theorem 1.1, the viscosity coefficients are only assumed to satisfy the physical conditions 1.). While the theory on weak small energy solutions, developed in [1, 1], requires the additional assumption 1.5) which is crucial in establishing the time-independent upper bound for the density in the arguments in [1, 1]. Remark 1.5 It is interesting to study the multi-dimensional full compressible Navier- Stokes system when the oscillations are large. This is under further investigation [16]. We now comment on the analysis of this paper. Note that for initial data in the class satisfying 1.7)-1.9), the local existence and uniqueness of classical solutions to the Cauchy problem, 1.1)-1.4), have been established recently in [4]. Thus, to extend the classical solution globally in time, one needs global a priori estimates on smooth solutions to 1.1)-1.4) in suitable higher norms. Some of the main new difficulties are due to the appearance of vacuum and that there are no other constraints on the viscosity coefficients beyond the physical conditions 1.). It turns out that the key issue in this paper is to derive both the time-independent upper bound for the density and the time-depending higher norm estimates of the smooth solution ρ, u). We start with the basic energy estimate and the initial layer analysis, and succeed in deriving weighted spatial mean estimates on the gradient and the material derivatives of the velocity. This is achieved by modifying the basic elegant estimates on the material derivatives of the velocity developed by Hoff [8, 1]) in the theory of small energy weak solutions with non-vacuum far fields. Then we are able to obtain the desired estimates on L 1, min{1,t}; L R 3 ))-norm and the time-independent ones on L 8/3 min{1,t},t; L R 3 ))-norm of the effective viscous flux see.5) for the definition). It follows from these key estimates and Zlotnik s inequality see Lemma.4) that the density admits a time-uniform upper bound which is the key for global estimates of classical solutions. This approach to estimate a uniform upper bound for the density is motivated by our previous analysis on the two-dimensional Stokes approximation equations in []. The next main step is to bound the gradients of the density and the velocity. Motivated by our recent studies [13, 14, 17]) on the blow-up criteria of classical or strong) solutions to 1.1), such bounds can be obtained by solving a logarithm Gronwall inequality based on a Beal-Kato-Majda type inequality see Lemma.5) and the a priori estimates we have just derived, and moreover, such a derivation yields simultaneously also the bound for L 1,T; L R 3 ))-norm of the gradient of the velocity, see Lemma 3.6 and its proof. It should be noted here that we do not require smallness of the gradient of the initial density which prevents the appearance of vacuum [1, ]). Finally, with these a priori estimates on the gradients of the density and the velocity at hand, one can estimate the higher order derivatives by using the same arguments as in [17] to obtain the desired results. The rest of the paper is organized as follows: In Section, we collect some elementary facts and inequalities which will be needed in later analysis. Section 3 is devoted to deriving the necessary a priori estimates on classical solutions which are needed to extend the local solution to all time. Then finally, the main results, Theorem 1.1 and Theorem 1., are proved in Section 4. 5

6 Preliminaries In this section, we will recall some known facts and elementary inequalities which will be used frequently later. We begin with the local existence and uniqueness of classical solutions when the initial density may not be positive and may vanish in an open set. Lemma.1 [4]) For ρ, assume that the initial data ρ,u ) satisfy 1.8)- 1.9). Then there exist a small time T > and a unique classical solution ρ, u) to the Cauchy problem 1.1) 1.3) 1.4) such that ρ ρ, P P ρ)) C[,T ]; H 3 ), u C[,T ]; D 1 D 3 ) L,T ; D 4 ), u t L,T ; D 1 ) L,T ; D ), ρut L,T ; L ), ρutt L,T ; L ), t 1/ u L,T ; D 4.1) ), t 1/ ρu tt L,T ; L ), tu t L,T ; D 3 ), tu tt L,T ; D 1 ) L,T ; D ). Next, the following well-known Gagliardo-Nirenberg inequality will be used later frequently see [19]). Lemma. Gagliardo-Nirenberg) For p [, 6],q 1, ), and r 3, ), there exists some generic constant C> which may depend on q,r such that for f H 1 R 3 ) and g L q R 3 ) D 1,r R 3 ), we have f p L C f 6 p)/ f 3p 6)/,.) p L L g CR3 ) C g qr 3)/3r+qr 3)) L g 3r/3r+qr 3)) q L..3) r We now state some elementary estimates which follow from.) and the standard L p -estimate for the following elliptic system derived from the momentum equations in 1.1): F =divρ u), μ ω = ρ u),.4) where f f t + u f, F μ + λ)divu P ρ)+p ρ), ω u,.5) are the material derivative of f, the effective viscous flux and the vorticity respectively. Lemma.3 Let ρ, u) be a smooth solution of 1.1) 1.3). Then there exists a generic positive constant C depending only on μ and λ such that for any p [, 6] F L p + ω L p C ρ u L p,.6) F L p + ω L p C ρ u 3p 6)/p) u L L + P P ρ) L ) 6 p)/p),.7) u L p C F L p + ω L p)+c P P ρ) L p,.8) u L p C u 6 p)/p) ρ u L L + P P ρ) L 6) 3p 6)/p)..9) 6

7 Proof. The standard L p -estimate for the elliptic system.4) yields directly.6), which, together with.) and.5), gives.7). Note that Δu = divu + ω, which implies that u = Δ) 1 divu + Δ) 1 ω. Thus the standard L p estimate shows that u L p C divu L p + ω L p), for p [, 6], which, together with.5), gives.8). Now.9) follows from.),.7) and.8). Next, the following Zlotnik inequality will be used to get the uniform in time) upper bound of the density ρ. Lemma.4 [3]) Let the function y satisfy y t) =gy)+b t) on [,T], y) = y, with g CR) and y,b W 1,1,T). If g ) = and bt ) bt 1 ) N + N 1 t t 1 ).1) for all t 1 <t T with some N and N 1, then where ζ is a constant such that yt) max { y, ζ } + N < on [,T], gζ) N 1 for ζ ζ..11) Finally, we state the following Beal-Kato-Majda type inequality which was proved in [1] when divu and will be used later to estimate u L and ρ L L 6. Lemma.5 For 3 <q<, there is a constant Cq) such that the following estimate holds for all u L R 3 ) D 1,q R 3 ), u L R 3 ) C divu L R 3 ) + ω L R )) 3 loge + u L q R 3 )) + C u L R 3 ) + C..1) Proof. The proof is similar to that of 15) in [1] and is sketched here for completeness. It follows from the Poisson s formula that ux) = 1 Δuy) 4π x y dy.13) divuy)kx y)dy Kx y) ωy)dy v + w, where satisfies Kx y) x y 4π x y 3, Kx y) C x y, Kx y) C x y 3..14) 7

8 It suffices to estimate the term v since w can be handled similarly see [1]). Let δ, 1] be a constant to be chosen and introduce a cut-off function η δ x) satisfying η δ x) =1for x <δ,η δ x) =for x > δ, and η δ x) Cδ 1. Then v can be rewritten as v = η δ y)ky) divux y)dy η δ x y)kx y)divuy)dy.15) + 1 η δ x y)) Kx y)divuy)dy. Each term on the righthand side of.15) can be estimated by.14) as follows: η δ y)ky) divux y)dy δ C η δ y)ky) L q/q 1) u L q C δ r q/q 1) r dr) q 1)/q u L q Cδ q 3)/q u L q, η δ x y)kx y)divuy)dy η δ z) Kz) dz divu L C δ δ δ 1 r r dr divu L C divu L, 1 η δ x y)) Kx y)divuy)dy ) C + Kx y) divuy) dy δ x y 1 x y >1 1 1/ C r 3 r dr divu L + C r 6 r dr) divu L C ln δ divu L + C u L. It follows from.15)-.18) that 1.16).17).18) ) v L C δ q 3)/q u L q +1 ln δ) divu L + u L..19) { Set δ =min 1, u q/q 3) L q }. Then.19) becomes v L Cq) 1+lne + u L q) divu L + u L ). Therefore.1) holds. 3 A priori estimates In this section, we will establish some necessary a priori bounds for smooth solutions to the Cauchy problem 1.1) 1.3) 1.4) to extend the local classical solution guaranteed 8

9 by Lemma.1. Thus, let T>beafixedtimeandρ, u) be the smooth solution to 1.1), 1.3)-1.4), on R 3,T] in the class.1) with smooth initial data ρ,u ) satisfying 1.7)-1.9). To estimate this solution, we set σt) min{1,t} and define T A 1 T ) sup σ u L ) + σρ u dxdt, t [,T ] and A T ) sup σ 3 t [,T ] T ρ u dx + σ 3 u dxdt, A 3 T ) sup u L. t [,T ] We have the following key a priori estimates on ρ, u). Proposition 3.1 For given numbers M > not necessarily small) and ρ ρ + 1, assume that ρ,u ) satisfy 1.7)-1.9). Then there exist positive constants ε and K both depending on μ, λ, ρ, a, γ, ρ and M such that if ρ, u) is a smooth solution of 1.1) 1.3) 1.4) on R 3,T] satisfying sup ρ ρ, R 3 [,T ] A 1 T )+A T ) C 1/, 3.1) A 3 σt )) 3K, the following estimates hold sup ρ 7 R 3 [,T ] 4 ρ, A 1T )+A T ) C 1/, A 3 σt )) K, 3.) provided C ε. Proof. Proposition 3.1 is an easy consequence of the following Lemmas 3., 3.3 and 3.5. In the following, we will use the convention that C denotes a generic positive constant depending on μ, λ, ρ, a, γ, ρ and M, andwewritecα) to emphasize that C depends on α. We start with the following standard energy estimate for ρ, u) and preliminary L bounds for u and ρ u. Lemma 3.1 Let ρ, u) be a smooth solution of 1.1) 1.3) 1.4) with ρx, t) ρ. Then there is a constant C = C ρ) such that ) 1 T μ u ρ u + Gρ) dx + +λ + μ)divu) ) dxdt C, 3.3) and sup t T A 1 T ) CC + C T A T ) CC + CA 1 T )+C T σ u 3 dxdt, 3.4) σ 3 u 4 dxdt. 3.5) 9

10 Proof. Multiplying the first equation in 1.1) by G ρ) and the second by u j and integrating, applying the far field condition 1.3), one shows easily the energy inequality 3.3). The proof of 3.4) and 3.5) is due to Hoff [8]. For integer m, multiplying 1.1) by σ m u then integrating the resulting equality over R 3 leads to σ m ρ u dx = σ m u P + μσ m u u +λ + μ)σ m divu u)dx 3 M i. i=1 Using 1.1) 1 and integrating by parts give M 1 = σ m u Pdx = σ m divu) t P P ρ)) σ m u u) P)dx ) = σ m divup P ρ))dx mσ m 1 σ divup P ρ))dx t + σ m P ρdivu) P divu) + P i u j j u i) dx ) σ m divup P ρ))dx + mσ m 1 σ P P ρ) L u L t 3.6) 3.7) + C ρ) u L ) σ m divup P ρ))dx + C ρ) u L + C ρ)m σ m 1) σ C. t Integration by parts implies M = μσ m u udx = μ σ m u ) L t + μm σm 1 σ u L μσ m i u j i u k k u j )dx μ σ m u ) L t + Cmσm 1 u L + C σ m u 3 dx, 3.8) and similarly, M 3 = λ + μ σ m divu L )t λ + μ)σ m λ + μ mλ + μ) + σ m 1 divu L divudivu u)dx σ m divu ) L t + Cmσm 1 u L + C σ m u 3 dx. 3.9) Combining 3.6)-3.9) leads to B t)+ σ m ρ u dx Cmσ m 1 + C ρ)) u L + C ρ)m σ m 1) σ C + C σ m u 3 dx, 3.1) 1

11 where Bt) μσm u L + μσm u L + μσm 4 u L + divu L + λ + μ)σm λ + μ)σm λ + μ)σm divu L Cσ m C. σ m divup P ρ))dx divu L Cσ m C 1/ divu L Integrating 3.1) over,t), choosing m = 1, and using 3.11), one gets 3.4). 3.11) Next, for integer m, operating σ m u j [ / t +divu )] to 1.1) j, summing with respect to j, and integrating the resulting equation over R 3, one obtains after integration by parts σ m ) ρ u dx m t σm 1 σ ρ u dx = σ m u j [ j P t +div j Pu)]dx + μ σ m u j [ u j t +divu uj )]dx +λ + μ) σ m u j [ t j divu +divu j divu)]dx 3 N i. i=1 3.1) It follows from integration by parts and using the equation 1.1) 1 that N 1 = σ m u j [ j P t +div j Pu)]dx = σ m [ P ρdivu j u j + k j u j u k )P P j k u j u k )]dx 3.13) C ρ)σ m u L u L δσ m u L + C ρ, δ)σ m u L. Integration by parts leads to N = μ σ m u j [ u j t +divu uj )]dx = μ σ m [ u + i u j k u k i u j i u j i u k k u j i u j i u k k u j ]dx 3μ σ m u dx + C σ m u 4 dx ) Similarly, N 3 μ + λ σ m div u) dx + C σ m u 4 dx. 3.15) Substituting 3.13)-3.15) into 3.1) shows that for δ suitably small, it holds that ) σ m ρ u dx + μ σ m u dx +μ + λ) σ m div u) dx t 3.16) mσ m 1 σ ρ u dx + Cσ m u 4 L + C ρ)σ m u 4 L. 11

12 Taking m = 3 in 3.16) and noticing that T 3 σ σ ρ u dxdt CA 1 T ), we immediately obtain 3.5) after integrating 3.16) over,t). Next, the following lemma will give an estimate on A 3 σt )). Lemma 3. Let ρ, u) be a smooth solution of 1.1) 1.3) 1.4) with ρx, t) ρ. Then there exist positive constants K and ε both depending only on μ, λ, ρ, a, γ, ρ and M such that σt ) A 3 σt )) + ρ u dxdt K, 3.17) provided A 3 σt )) 3K and C ε. Proof. Integrating 3.1) over,σt)), choosing m =, and using 3.11), one has σt ) σt ) A 3 σt )) + ρ u dxdt C ρ)c + M)+C ρ) u 3 L dt. 3.18) 3 Note that.9) and 3.3) imply σt ) u 3 L 3 dt C ρ) δ σt ) σt ) ) u 3/ ρ u 3/ + C 1/4 L L dt ρ u dxdt + C ρ, δ) which, together with 3.18), yields easily that σt ) A 3 σt )) + ρ u dxdt C ρ)c + M)+C ρ) K + C ρ)c [A 3 σt ))], σt ) σt ) u 6 L dt u 6 L dt + C ρ)c, for some positive constant K depending only on μ, λ, ρ, a, γ, ρ and M. One thus finishes the proof of 3.17) by choosing ε 9C ρ)k) 1. Lemma 3.3 There exists a positive constant ε 1 μ, λ, ρ, a, γ, ρ, M) ε such that, if ρ, u) is a smooth solution of 1.1) 1.3) 1.4) satisfying 3.1) for K as in Lemma 3., then A 1 T )+A T ) C 1/, 3.19) provided C ε 1. Proof. Lemma 3.1 shows that T T A 1 T )+A T ) C ρ)c + C ρ) σ 3 u 4 L ds + C ρ) σ u 3 4 L ds. 3.) 3 1

13 Due to.8), T σ 3 u 4 L 4 ds C It follows from.7) that T σ 3 F 4 L + ω 4 ) T 4 L ds + C σ 3 P P ρ) 4 4 L ds. 3.1) 4 T σ 3 F 4 L 4 + ω 4 L 4 ) ds T C σ 3 u L + P P ρ) L ) ρ u 3 L ds C ρ) sup σ 3/ )) ρ u L σ 1/ u L + C 1/ T t,t ] ) C ρ) A 1/ 1 T )+C 1/ A 1/ T )A 1 T ) C ρ)c. σρ u dxds 3.) To estimate the second term on the right hand side of 3.1), one deduces from 1.1) 1 that P P ρ) satisfies P P ρ)) t + u P P ρ)) + γp P ρ))divu + γp ρ)divu =. 3.3) Multiplying 3.3) by 3P P ρ)) and integrating the resulting equality over R 3, one gets after using divu = 1 μ+λ F + P P ρ)) that 3γ 1 μ + λ P P ρ) 4 L 4 ) = P P ρ)) 3 dx 3γ 1 P P ρ)) 3 Fdx t μ + λ 3γP ρ) P P ρ)) divudx ) P P ρ)) 3 dx + δ P P ρ) 4 L + C 4 δ F 4 L + C 4 δ u L. 3.4) t Multiplying 3.4) by σ 3, integrating the resulting inequality over,t), and choosing δ suitably small, one may arrive at T σ 3 P P ρ) 4 L 4 dt C sup + C ρ) t T T C ρ)c, P P ρ) 3 L 3 + C σt ) σ 3 F 4 L 4 ds + C ρ)c P P ρ) 3 L 3 dt 3.5) where 3.) has been used. Therefore, collecting 3.1), 3.) and 3.5) shows that T σ 3 u 4 L 4 + P P ρ) 4 L 4 ) ds C ρ)c. 3.6) 13

14 Finally, we estimate the last term on the right hand side of 3.). First, 3.6) implies that T σt ) σ u 3 dxds T σt ) Next, one deduces from.9) and 3.17) that u 4 + u )dxds CC. 3.7) σt ) σ u 3 L ds 3 σt ) ) C ρ) σ u 3/ ρ u 3/ + C 1/4 L L ds σt ) ) 3/4 C ρ) σ 1/4 u 3/ σ ρ u dx) ds + C ρ)c L σ u ) ) σt ) 1/4 1/ C ρ) sup L u u σ 1/ L L t,σt )] ρ u dx) 3/4 ds + C ρ)c C ρ, M)A 1 C 1/4 + C ρ)c C ρ, M)C 3/4, 3.8) provided C ε. It thus follows from 3.) and 3.6)-3.8) that the left hand side of 3.19) is bounded by C ρ, M)C 3/4 C 1/ provided { C ε 1 min ε, C ρ, M)) 4}. The proof of Lemma 3.3 is completed. Lemma 3.4 There exists a positive constant C depending on μ, λ, ρ, a, γ ρ and M such that the following estimates hold for a smooth solution ρ, u) of 1.1) 1.3) 1.4) as in Lemma 3.3: provided C ε 1. T sup u L + t [,T ] ρ u dxdt C ρ, M), 3.9) σ u dxdt C ρ, M). 3.3) sup t,t ] T σρ u dx + Proof. 3.9) is a direct consequence of 3.17) and 3.19). To prove 3.3), we integrate 14

15 3.16) over,t)andchoosem =1toget T sup σρ u dx + σ u L dt t,t ] σt ) T ρ u dxdt + C σ u 4 L dt + C ρ)c 4 C ρ, M)+C C ρ, M)+C C ρ, M)+C T σt ) σt ) sup t,σt )] σ 3 u 4 L 4 dt + C σt ) σ u 4 L 4 dt σ u L ρ u 3 L + P P ρ) 3 ) L dt 6 σ 1/ u L )σ 1/ ρ u L ) C ρ, M)+C ρ, M) sup σ 1 ρ 1/ u L, t,t ] ) σt ) ρ u L dt 3.31) where in the third inequality, 3.6) and.9) have been used. Then 3.3) follows from 3.31) and Young s inequality. We now proceed to derive a uniform in time) upper bound for the density, which turns out to be the key to obtain all the higher order estimates and thus to extend the classical solution globally. We will use an approach motivated by our previous study on the two-dimensional Stokes approximation equations []). Lemma 3.5 There exists a positive constant ε = ε ρ, M) as described in Theorem 1.1 such that, if ρ, u) is a smooth solution of 1.1) 1.3) 1.4) as in Lemma 3.3, then provided C ε. sup ρt) L 7 ρ t T 4, Proof. Rewrite the equation of the mass conservation 1.1) 1 as where D t ρ = gρ)+b t), D t ρ ρ t + u ρ, gρ) aρ μ + λ ργ ρ γ ), bt) 1 μ + λ t ρf dt. For t [,σt )], one deduces from Lemma.,.6) and 3.3) that for all t 1 < 15

16 t σt ), bt ) bt 1 ) C C ρ) C ρ) C ρ) σt ) σt ) σt ) σt ) C ρ)c 1/16 C ρ)c 1/16 ρf ),t) L dt F,t) 1/4 L F,t) 3/4 L 6 dt u 1/4 L + P P ρ) 1/4 L ) u 3/4 L dt ) ) 1/4 σ 1/ σ 1/ 1/8 σ u L + C σ 3/8 u 3/8 L ) dt σt ) 1+ C ρ, M)C 1/16, σ σ +1) 1/ u 3/8 L ) dt ) 5/8 ) 3/8 σt ) σ 4/5 dt σ u L dt 1 provided C ε 1. Therefore, for t [,σt )], one can choose N and N 1 in.1) as follows: and ζ = ρ in.11). Then N 1 =, N = C ρ, M)C 1/16, gζ) = aζ μ + λ ζγ ρ γ ) N 1 =, for all ζ ζ = ρ. Lemma.4 thus yields that sup ρ L max{ ρ, ρ} + N ρ + C ρ, M)C 1/16 3 ρ t [,σt )], 3.3) provided C min{ε 1,ε }, ) ρ 16 for ε. C ρ, M) On the other hand, for t [σt ),T], one deduces from Lemma., 3.19), 3.3), and.6) that for all σt ) t 1 t T, t bt ) bt 1 ) C ρ) F,t) L dt t 1 a μ + λ t t 1 )+C ρ) a μ + λ t t 1 )+C ρ) T σt ) T σt ) T a μ + λ t t 1 )+C ρ)c 1/6 a μ + λ t t 1 )+C ρ)c /3, 16 F,t) 8/3 L dt F,t) /3 L F,t) L 6 dt σt ) u,t) L dt

17 provided C ε 1. Therefore, one can choose N 1 and N in.1) as: a N 1 = μ + λ, N = C ρ)c /3. Note that gζ) = aζ μ + λ ζγ ρ γ ) N 1 = a, for all ζ ρ +1. μ + λ So one can set ζ = ρ + 1 in.11). Lemma.4 and 3.3) thus yield that { } 3 ρ sup ρ L max, ρ +1 + N 3 ρ t [σt ),T ] + C ρ)c/3 7 ρ 4, 3.33) provided ) ρ 3/ C ε min{ε 1,ε,ε 3 }, for ε ) 4C ρ) The combination of 3.3) with 3.33) completes the proof of Lemma 3.5. From now on, we will always assume that the initial energy C satisfies 3.34) and the constant C may depend on T, ρ 1/ g L, g L, u H, ρ ρ H 3, P ρ ) P ρ) H 3, besides μ, λ, ρ, a, γ, ρ and M, where g is as in 1.9). Next, we will derive important estimates on the spatial gradient of the smooth solution ρ, u). Lemma 3.6 The following estimates hold ρ u dx + sup t T T u dxdt C, 3.35) T sup ρ L L 6 + u H 1)+ u L dt C. 3.36) t T Proof. Taking m = in 3.16), one can deduce from Gagliardo-Nirenberg s inequality.),.6), 3.9) and 3.16) that ) ρ u dx + μ u dx +μ + λ) div u) dx t C u 4 L + C ρ) u 4 L C u L u 3 L + C 6 C F 3 L + ω 3 6 L + P P ρ) 3 ) 6 L + C 6 C F 3 L + ω 3 ) L + C C ρ u 3 L + C C ρ 1/ u 4 L + C. 3.37) Taking into account on the compatibility condition 1.9), we can define ρ ux, t =)= ρ g. 3.38) 17

18 Then 3.35) follows from Gronwall s inequality, 3.37), 3.9), and 3.38). Next, we prove 3.36) by using Lemma.5 as in [14]. For p 6, ρ p satisfies ρ p ) t +div ρ p u)+p 1) ρ p divu + p ρ p ρ) t u ρ)+pρ ρ p ρ divu =. Thus, due to t ρ L p C1 + u L ) ρ L p + C u L p C1 + u L ) ρ L p + C ρ u L p, 3.39) u L p C ρ u L p + P L p), 3.4) which follows from the standard L p -estimate for the following elliptic system: μδu +μ + λ) divu = ρ u + P. It follows from Lemma.5 and 3.4) that u L C divu L + ω L )loge + u L 6)+C u L + C C divu L + ω L )loge + u L 6 + P L 6)+C C divu L + ω L )loge + u L ) + C divu L + ω L )loge + ρ L 6)+C. 3.41) Set ft) e + ρ L 6, gt) 1+ divu L + ω L )loge + u L )+ u L. Combining 3.41) with 3.39) and setting p = 6 in 3.39), one gets f t) Cgt)ft)+Cgt)ft)lnft)+Cgt), which yields ln ft)) Cgt)+Cgt)lnft), 3.4) due to ft) > 1. Note that.5), Lemma.,.6), 3.35), and Lemma 3.5 imply T gt)dt C C C C C, T T T T divu L + ω L ) dt + C F L + P P ρ) L + ω L ) dt + C F L + F L 6 + ω L + ω L 6 ) dt + C u L dt + C 3.43) which, together with 3.4) and Gronwall s inequality, shows that sup ft) C. t T 18

19 Consequently, As a consequence of 3.41), 3.43) and 3.44), one obtains sup ρ L 6 C. 3.44) t T T u L dt C. 3.45) Next, taking p = in 3.39), one gets by using 3.45), 3.9) and Gronwall s inequality that sup ρ L C, t T which, together with 3.4), 3.35), 3.9), 3.44), and 3.45), gives 3.36). The proof of Lemma 3.6 is completed. The following Lemmas will deal with the higher order estimates of the solutions which are needed to guarantee the extension of local classical solution to be a global one. The proofs are similar to the ones in [17], and we sketch them here for completeness. Lemma 3.7 The following estimates hold sup t T T ρ u t dx + u t dxdt C, 3.46) sup ρ ρ H + P ρ) P ρ) H ) C. 3.47) t [,T ] Proof. Estimate 3.46) follows directly from the following simple facts: ρ u t dx ρ u dx + ρ u u dx C + C ρ 1/ u L u L 6 u L 6 C, and u t L u L + u u) L u L + C u L u L + C u 4 L 4 u L + C, due to Lemma 3.6. Next, we prove 3.47). Note that P satisfies P t + u P + γpdivu =, 3.48) which, together with 1.1) 1 and a simple computation, yields that d P L + ρ ) dt L C1 + u L ) P L + ρ ) L + C F H + C ω H + C, 3.49) 19

20 where we have used the following simple fact: u H m C divu H m + ω H m) C F H m + ω H m + P P ρ) H m), for m =1,. Noticing that F and ω satisfy.4), we get by the standard L -estimate for elliptic system, 3.35) and 3.36) that F H + ω H C F L + ω L + ρ u L + ρ u) L ) C1 + ρ L 3 u L 6 + u L ) C1 + u L ), which, together with 3.49), Lemma 3.6, and Gronwall s inequality, gives directly P L + ) ρ L C. sup t [,T ] Thus the proof of Lemma 3.7 is completed. Lemma 3.8 The following estimates hold: T sup ρ t H 1 + P t H 1)+ t T sup u t dx + t T T ρtt L + P tt L ) dt C, 3.5) ρu ttdxdt C. 3.51) Proof. We first prove 3.5). One deduces from 3.48) and 3.36) that Differentiating 3.48) yields P t L C u L P L + C u L C. 3.5) P t + u P + u P + γ P divu + γp divu =. Hence, by 3.36) and 3.47), one gets P t L C u L P L + C u L 3 P L 6 + C u L C. 3.53) The combination of 3.5) with 3.53) implies Note that P tt satisfies sup P t H 1 C. 3.54) t T P tt + γp t divu + γpdivu t + u t P + u P t =. 3.55) Thus, one gets from 3.55) 3.54) 3.36) and 3.46) that T C C. P tt L dt T P t L 6 u L 3 + u t L + u t L 6 P L 3 + P t L ) dt

21 One can handle ρ t and ρ tt similarly. Thus 3.5) is proved. Next, we prove 3.51). Differentiating 1.1) with respect to t, then multiplying the resulting equation by u tt, one gets after integration by parts that d μ dt u t + λ + μ divu t ) dx + ρu tt dx = d 1 ) ρ t u t dx ρ t u u u t dx + P t divu t dx dt + 1 ρ tt u t dx + ρ t u u) t u t dx ρu t u u tt dx 3.56) ρu u t u tt dx P tt divu t dx d dt I + 5 I i. i=1 It follows from 1.1) 1, 3.36), 3.5) and 3.46) that I = 1 ρ t u t dx ρ t u u u t dx + P t divu t dx divρu) u t dx + C ρ t L 3 u u L u t L 6 + C P t L u t L C ρ u u t u t dx + C u t L 3.57) and C u L 6 ρ 1/ u t 1/ L u t 1/ L 6 u t L + C u t L δ u t L + C δ, I 1 = ρ tt u t dx = ρ t u + ρu t ) u t )dx ) C ρ t L 3 u L + ρ 1/ u t 1/ u L t 1/ u L 6 t L 6 u t L C u t L + C u t 5/ L C u t 4 L + C, I = ρ t u u) t u t dx = ρ tt u u u t + ρ t u t u u t + ρ t u u t u t ) dx ρ tt L u u L 3 u t L 6 + ρ t L u t L 3 u L 6 + ρ t L 3 u L u t L u t L 6 C ρ tt L + C u t L. 3.58) 3.59) 1

22 Cauchy s inequality gives I 3 + I 4 = ρu t u u tt dx + ρu u t u tt dx C ρ 1/ u tt L u t L 6 u L 3 + u L u t L ) δ ρ 1/ u tt L + C δ u t L, 3.6) and I 5 = P tt divu t dx P tt L divu t L C P tt L + u t L. 3.61) Due to the regularity of the local solution,.1), t u t C[,T ]; L ). Thus u t,t /) L T t u t L,T ;L ) C, 3.6) where C may also depend on g L. Collecting all the estimates 3.57)-3.6), one deduces from 3.56), 3.5), 3.46) and Gronwall s inequality that T sup u t L + ρ u tt dxdt C. 3.63) T / t T T / On the other hand,.1) gives sup u t L + t T / T / ρ u tt dxdt C. 3.64) The combination of 3.63) with 3.64) yields 3.51) immediately. This completes the proof of Lemma 3.8. Lemma 3.9 It holds that sup ρ ρ H 3 + P P ρ) H 3) C, t [,T ] 3.65) T sup u t L + u H )+ u H 3 + u t ) H dt C ) t [,T ] Proof. It follows from 3.51) and 3.36) that ρ u) L ρ u t L + ρ u t L + ρ u u L + ρ u L + ρ u u L ρ L 3 u t L 6 + C u t L + C ρ L 3 u L u L 6 + C u L 3 u L 6 + C u L u L C,

23 which together with 3.35) gives The standard H 1 -estimate for elliptic system gives sup ρ u H 1 C. 3.67) t T u H 1 C μδu +μ + λ) divu H 1 = C ρ u + P H 1 C ρ u H 1 + P H 1) C, 3.68) due to 1.1), 3.67) and 3.47). As a consequence of 3.36) and 3.68), one has sup u H C. 3.69) t T Therefore, the standard L -estimate for elliptic system, 3.36), and Lemma 3.8 yield that u t L C μδu t +μ + λ) divu t L = ρu tt + ρ t u t + ρ t u u + ρu t u + ρu u t + P t L C ρu tt L + ρ t L 3 u t L 6 + ρ t L 3 u L u L 6) + C u t L 6 u L 3 + u L u t L + P t L ) C ρu tt L + C, 3.7) which, together with 3.51), implies T u t H 1 dt C. 3.71) Applying the standard H -estimate for elliptic system again leads to u H C μδu +μ + λ) divu H C ρ u H + C P H C + C u t H 1 + C 3 P L, 3.7) where one has used 3.67) and the following simple facts: ρu t ) L C ρ u t L + ρ u t L + ) u t L C ρ L u t H 1 + ρ L 3 u t L 6 + ) u t L C + C u t H 1, and ρu u) L C ρu) u L + ρu) u L + 3 ) u L C 1+ ρu) L u H + ρu) L 3 ) u L 6 C 1+ ρ L u L + ρ L 6 u L 3 + ) u L C, 3

24 due to 3.47) and 3.69). By using 3.69), 3.7), and 3.47), one may get that 3 P L ) t C 3 u P L + u P L + u 3 P L + 4 u L ) 3 P L C 3 u L P H + u L 3 P L 6 + u L 3 P L ) 3 P L + C 1+ u t L + 3 P L ) 3 P L C + C u t H 1 + C 3 P L, which, together with Gronwall s inequality and 3.71), yields that sup 3 P L C. 3.73) t T Collecting all these estimates 3.71)-3.73) and 3.47) shows T sup P P ρ) H 3 + u H dt C. 3.74) 3 t T It is easy to check similar arguments work for ρ ρ by using 3.74). Hence, sup ρ ρ H 3 C. 3.75) t T Combing 3.74) with 3.75) shows 3.65). Estimate 3.66) thus follows from 3.51), 3.69), 3.71), and 3.74). Hence the proof of Lemma 3.9 is finished. Lemma 3.1 For any τ,t), there exists some positive constant Cτ) such that T sup ut H u L ) + u tt dxdt Cτ). 3.76) τ t T τ Proof. Differentiate 1.1) with respect to t to get ρu ttt + ρu u tt μδu tt μ + λ) divu tt =divρu)u tt +divρu) t u t ρu) t u t ρ tt u +ρ t u t ) u ρu tt u P tt. 3.77) Multiplying 3.77) by u tt and then integrating the resulting equation over R 3, one gets after integration by parts that 1 d ρ u tt dx + μ u tt +μ + λ)divu tt ) )dx dt = 4 u i ttρu u i ttdx ρu) t [ u t u tt )+ u t u tt ] dx 3.78) ρ tt u +ρ t u t ) u u tt dx ρu tt u u tt dx + P tt divu tt dx 5 J 5. i=1 We estimate each J i i =1,, 5) as follows: 4

25 Hölder s inequality gives J 1 C ρ 1/ u tt L u tt L u L δ u tt L + C δ ρ 1/ u tt L. 3.79) It follows from 3.46), 3.5), 3.51), and 3.36) that J C ρu t L 3 + ρ t u L 3) u tt L 6 u t L + u tt L u t L 6) ) C ρ 1/ u t 1/ u L t 1/ + ρ L 6 t L 6 u L 6 u tt L δ u tt L + C δ, 3.8) and J 3 C ρ tt L u L u L 3 + ρ t L 6 u t L 6 u L ) u tt L 6 δ u tt L + C δ ρ tt L, J 4 + J 5 C ρu tt L u L 3 u tt L 6 + C P tt L u tt L δ u tt L + C δ ρ 1/ u tt L + C δ P tt L. 3.81) 3.8) For any τ,t ), since t 1/ ρu tt L,T ; L ) by.1), there exists some t τ/,τ) such that ρ u tt dxt ) 1 t t 1/ ρu tt L,T ;L ) Cτ). 3.83) Substituting 3.79)-3.8) into 3.78) and choosing δ suitably small, one obtains by using 3.5) 3.83) and Gronwall s inequality that T sup ρ u tt dx + u tt dxdt Cτ), t t T t which, together with 3.7) and 3.51), yields that T sup u t H 1 + u tt dxdt Cτ), 3.84) τ t T τ due to t <τ.now, 3.76) follows from 3.7), 3.84), and 3.65). We finish the proof of Lemma Proof of Theorems 1.1 and 1. With all the a priori estimates in Section 3 at hand, we are ready to prove the main results of this paper in this section. Proof of Theorem 1.1. By Lemma.1, there exists a T > such that the Cauchy problem 1.1), 1.3), and 1.4) has a unique classical solution ρ, u) on,t ]. We will use the a priori estimates, Proposition 3.1 and Lemmas 3.9 and 3.1, to extend the local classical solution ρ, u) toalltime. First, since A 1 ) + A ) =, A 3 ) M, ρ ρ, 5

26 there exists a T 1,T ] such that 3.1) holds for T = T 1. Set T =sup{t 3.1) holds}. 4.1) Then T T 1 >. Hence, for any <τ<t T with T finite, it follows from Lemmas 3.9 and 3.1 that )) u t, 3 u C[τ,T]; L L 4 ), u, u C [τ,t]; L C R 3, 4.) where we have used the standard embedding L τ,t; H 1 ) H 1 τ,t; H 1 ) C [τ,t]; L q ), for any q [, 6). Due to 3.46), 3.51), and 3.76), one can get T τ T C C C, which yields ρ u t ) t L 1dt τ T τ T τ ρt u t ) L 1 + ρu t u tt L 1 dt This, together with 4.), gives Next, we claim that ) ρ divu u t L 1 + u ρ u t L 1 + ρ 1/ u t L ρ 1/ u tt L dt ) ρ u t L 1 u L + u L 6 ρ L u t L + ρ 1/ u 6 tt L dt ρ 1/ u t C[τ,T]; L ). ρ 1/ u, u C[τ,T]; L ). 4.3) T =. 4.4) Otherwise, T <. Then by Proposition 3.1, 3.) holds for T = T. It follows from Lemmas 3.9 and 3.1 and 4.3) that ρx, T ),ux, T )) satisfies 1.8) and 1.9) with gx) = ux, T ),x R 3. Lemma.1 implies that there exists T >T, such that 3.1) holds for T = T, which contradicts 4.1). Hence, 4.4) holds. Lemmas.1, 3.9 and 3.1 and 4.) thus show that ρ, u) is in fact the unique classical solution defined on,t] for any <T <T =. Finally, to finish the proof of Theorem 1.1, it remains to prove 1.13). Multiplying 3.3) by 4P P ρ)) 3 and integrating the resulting equality over R 3, one has ) P P ρ) 4 L 4 t) = 4γ 1) P P ρ)) 4 divudx γ P ρ)p P ρ)) 3 divudx, 6

27 which yields that P P ρ) 4 ) L t) dt C P P ρ) 4 4 L 4 + u 4 ) L dt C, 4.5) 4 1 due to 3.6). Combining 3.6) with 4.5) leads to 1 lim P P ρ) L t 4 =, which together with 3.3) implies lim t ρ ρ q dx =, for all q satisfying 1.14). Note that 3.3) and.) imply ) 1/ ρ 1/ u 4 dx ρ u dx u 3 L C u 3 6 L. Thus 1.13) follows provided that Setting It) μ u L + λ + μ divu L, choosing m = in 3.6), and using 3.8) and 3.9), one has I t) C where one has used the following simple estimate: M 1 = u Pdx = P P ρ))div udx lim u L =. 4.6) t ρ u dx + C u 3 L 3 + CC 1/ u L, 4.7) CC 1/ u L. We thus deduce from 4.7), 3.19), and 3.6) that ) I t) dt C ρ 1/ u 4 L + u L u 4 L + u 4 L dt 1 1 ) C ρ 1/ u L + u 4 L + u 4 L dt which, together with C, 1 1 It) dt C 1 u L dt C, 7

28 implies 4.6). The proof of Theorem 1.1 is finished. Proof of Theorem 1.. The proof is similar to that of Theorem 1. in []. We just sketch it here. Otherwise, there exist some constant C > and a subsequence { } t nj j=1,t n j such that ) ρ,tnj L r C. Hence, the Gagliardo-Nirenberg inequality.3) yields that there exists some positive constant C independent of t nj such that for a = r/r 3), 1), ρx, tnj ) ρ CR3 ) C ρx, tnj ) a ρx, L r tnj ) ρ 1 a L 3 CC a ρx, tnj ) ρ 1 a. 4.8) L 3 Due to 1.13), the right hand side of 4.8) goes to as t nj. Hence, ρx, tnj ) ρ CR3 ) ast n j. 4.9) On the other hand, since ρ, u) is a classical solution, thus there exists a unique particle path x t) withx ) = x such that ρx t),t) for all t. So, we conclude from this identity that ρx, t nj ) ρ CR3 ) ρx t nj ),t nj ) ρ ρ >, which contradicts 4.9). This completes the proof of Theorem 1.. References [1] Beal, J. T., Kato, T., Majda. A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys 94, ) [] Cho, Y., Choe, H. J., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83, ) [3] Cho, Y., Jin, B.J.: Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl. 3), ) [4] Cho, Y., Kim, H.: On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscript Math. 1, ) [5] Choe, H. J., Kim, H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Eqs. 19, ) [6] Feireisl, E., Novotny, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 34), ) 8

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