Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum

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1 Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum Jing LI a,b, Zhouping XIN b a. Institute of Applied Mathematics, AMSS, & Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 9, P. R. China b. The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong Abstract This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For strong and classical solutions, some a priori decay with rates in large time for both the pressure and the spatial gradient of the velocity field are obtained provided that the initial total energy is suitably small. Moreover, by using these key decay rates and some analysis on the expansion rates of the essential support of the density, we establish the global existence and uniqueness of classical solutions which may be of possibly large oscillations in two spatial dimensions, provided the smooth initial data are of small total energy. In addition, the initial density can even have compact support. This, in particular, yields the global regularity and uniqueness of the re-normalized weak solutions of Lions-Feireisl to the two-dimensional compressible barotropic flows for all adiabatic number γ > provided that the initial total energy is small. Introduction We consider the Navier-Stokes equations { ρ t + divρu =, ρu t + divρu u µ u µ + λ divu + P ρ =,. for viscous compressible barotropic flows. Here, t is time, x Ω R N N =, 3 is the spatial coordinate, and ρ = ρx, t, u = u,, u N x, t, and P ρ = Rρ γ R >, γ >. This research is supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK-44/P, CUHK-448/3P, a Focus Area Grant from The Chinese University of Hong Kong, and a grant from Croucher Foundation. The research of J. Li was partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, and NNSFC Grant No ajingli@gmail.com J. Li, zpxin@ims.cuhk.edu.hk Z. Xin.

2 are the fluid density, velocity and pressure, respectively. Without loss of generality, it is assumed that R =. The constant viscosity coefficients µ and λ satisfy the physical restrictions: µ >, µ + Nλ..3 Let Ω = R N and we consider the Cauchy problem for. with ρ, u vanishing at infinity in some weak sense with given initial data ρ and u, as ρx, = ρ x, ρux, = ρ u x, x Ω = R N..4 There are huge literatures on the large time existence and behavior of solutions to.. The one-dimensional problem has been studied extensively, see [9, 8, 9, 3] and the references therein. For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in [5, 3] in the absence of vacuum and recently, for strong solutions also, in [3 5,9,8] 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 [4] for initial data close to a non-vacuum 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 vacuum. Later, Hoff [,, 3] studied the problem for discontinuous initial data. For the existence of solutions for arbitrary data, the major breakthrough is due to Lions [] see also Feireisl [6, 7], where the global existence of weak solutions when the exponent γ is suitably large are achieved. The main restriction on initial data is that the initial total 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, in particular, the regularity and the uniqueness of such weak solutions remain open. This is a subtle issue, as Xin [3] showed that in the case that the initial density has compact support, any smooth solution in C [, T ] : H s R d s > [d/] + to the Cauchy problem of the full compressible Navier-Stokes system without heat conduction blows up in finite time for any space dimension d, and the same holds for the isentropic case., at least in one-dimension. The assumptions of [3] that the initial density has compact support and that the smooth solution has finite energy are removed recently by Xin-Yan [33] for a large class of initial data containing vacuum. However, this blow-up theory does not apply to the isentropic flows in general, at least in the case of R 3. Indeed, very recently, for the case that the initial density is allowed to vanish and even has compact support, Huang-Li-Xin [6] established the quite surprising global existence and uniqueness of classical solutions with constant state as far field which could be either vacuum or nonvacuum to.-.4 in three-dimensional space with smooth initial data which are of small total energy but possibly large oscillations. Moreover, it was also showed in [6] that for any p >, lim P ρ P ρ L t p R 3 + u L R 3 =,.5 where ρ is the constant far field density. This not only generalizes the classical results of Matsumura-Nishida [4], but also yields the regularity and uniqueness of the weak solutions of Lions and Feireisl [6, 7, ] with initial data of small total energy. Then a natural question arises whether the theory of Huang-Li-Xin [6] remains valid for the case of R. This is interesting partially due to the following reasons: First, a positive answer would yield immediately the regularity and uniqueness of weak solutions of Lions-Feireisl with small initial total energy whose existence has been proved for all

3 3 γ >, see [6, 7]. Second, this question may be subtle due to the recent blow-up result in [3] where it is shown that non-trivial two-dimensional spherically symmetric solution in C [, T ]; H s R s > with initial compactly supported density blows up in finite time. Technically, it is not easy to modify the three-dimensional analysis of [6] to the two-dimensional case with initial density containing vacuum since the analysis of [6] depends crucially on the a priori L 6 -bound on the velocity. For twodimensional problems, only in the case that the far field density is away from vacuum, the techniques of [6] can be modified directly since at this case, for any p [,, the L p -norm of a function u can be bounded by ρ / u L and u L, and the similar results can be obtained [3]. However, when the far field density is vacuum, it seems difficult to bound the L p -norm of u by ρ / u L and u L for any p, so the global existence and large time behavior of strong or classical solutions to the Cauchy problem are much subtle and remain open. Therefore, the main aim of this paper is to study the global existence and large time behavior of strong or classical solutions to.-.4 in some homogeneous Sobolev spaces in two-dimensional space with vacuum as far field density, and at the same time to investigate the decay rates of the pressure and the gradient of velocity in both two and three dimensional spaces provided the initial energy is suitably small, which turn out to be one of the keys for the two-dimensional global well-posedness theory. Before stating the main results, we first explain the notations and conventions used throughout this paper. For R > and Ω = R N N =, 3, set B R {x Ω x < R}, fdx fdx. Moreover, for r, k, and β >, the standard homogeneous and inhomogeneous Sobolev spaces are defined as follows: L r = L r Ω, D k,r = D k,r Ω = {v L loc Ω k v L r Ω}, D = D {,, W k,r = W k,r Ω, H k = W k,, } Ḣ β = f : Ω R f = ξ β ˆfξ dξ <, Ḣ β where ˆf is the Fourier transform of f. Next, we also give the definition of strong solutions as follows: Definition. If all derivatives involved in. for ρ, u are regular distributions, and equations. hold almost everywhere in Ω, T, then ρ, u is called a strong solution to.. For Ω = R N N =, 3, the initial total energy is defined as: C = ρ u + γ P ρ dx. Ω We consider first the two-dimensional case, that is, Ω = R. Without loss of generality, assume that the initial density ρ satisfies R ρ dx =,.6 Ω

4 4 which implies that there exists a positive constant N such that ρ dx ρ dx = B N..7 We can now state our first main result, Theorem., concerning the global existence of strong solutions. Theorem. Let Ω = R. In addition to.6 and.7, suppose that the initial data ρ, u satisfy for any given numbers M >, ρ, a >, q >, and β, ], and where ρ ρ, x a ρ L H W,q, u Ḣβ D, ρ / u L,.8 u Ḣβ + ρ x a L M,.9 x e + x / log e + x.. Then there exists a positive constant ε depending on µ, λ, γ, a, ρ, β, N, and M such that if C ε,. the problem.-.4 has a unique global strong solution ρ, u satisfying for any < T <, ρx, t ρ, x, t R [, T ],. ρ C[, T ]; L H W,q, and x a ρ L, T ; L H W,q, ρu, u, x u, t ρu t L, T ; L, u L, T ; H L q+/q, T ; W,q, t u L, T ; W,q, ρut, t u t, t x u t L R, T,.3 inf ρx, tdx B N 4,.4 +t log α e+t for any α > and some positive constant N depending only on α, N, and M. Moreover, ρ, u has the following decay rates, that is, for t, where u, t L p Cpt +/p, for p [,, P, t L r Crt +/r, for r,, ω, t L + F, t L Ct,.5 ω u u, F µ + λdivu P,.6 are respectively the vorticity and the effective viscous flux, and Cα depends on α besides µ, λ, γ, a, ρ, β, N, and M.

5 5 Remark. It should be noted here that the decay rate estimates.5 combined with the estimate on upper bound of the expansion rate of the essential support of the density.4 play a crucial role in deriving the global existence of strong and classical solutions to the two-dimensional problem.-.4. This is in contrast to the threedimensional case [6] where the global existence of classical solutions to.-.4 was achieved without any bounds on the decay rates of the solutions partially due to the a priori L 6 -bounds on the velocity field. As will be seen in the proof, the key observation is the decay with a rate for the mean-square norm of the pressure in.5. If the initial data ρ, u satisfy some additional regularity and compatibility conditions, the global strong solutions obtained by Theorem. become classical ones, that is, Theorem. Let Ω = R. In addition to the assumptions in Theorem., assume further that ρ, u satisfy ρ, P ρ L L q, x δ ρ, x δ P ρ, u L,.7 for some constant δ,, and the following compatibility condition: µ u µ + λ divu + P ρ = ρ / g,.8 with some g L. Then, in addition to.-.5, the strong solution ρ, u obtained by Theorem. satisfies for any < T <, ρ, P ρ C[, T ]; L L q, x δ ρ, x δ P ρ, u L, T ; L, ρut, t u t, t x u t, t ρu tt, t u t L, T ; L, t 3 u L, T ; L L q, u t, x u t, t u tt, t x u tt L, T ; L, t ρu L, T ; L q+/..9 Remark. The solution obtained in Theorem. becomes a classical one for positive time [9]. Although it has small energy, yet whose oscillations could be arbitrarily large. In particular, both interior and far field vacuum are allowed. There is no requirement on the size of the set of vacuum states. Therefore, the initial density may have compact support. Moreover, by the strong-weak uniqueness theorem of Lions [], Theorem. and Theorem. can be regarded as uniqueness and regularity theory of Lions-Feireisl s weak solutions with small initial energy, whose existence has been proved for all γ > in [6, ]. Remark.3 It is worth noting that the conclusions in Theorem. and Theorem. are somewhat surprising since for the isentropic compressible Navier-Stokes equations., any non-trivial two-dimensional spherically symmetric solution ρ, u C [, T ], H s s > with initial compact supported density blows up in finite time [3]. Indeed, as in [3], the key point of [3] to prove the blowup phenomena is based on the fact that the support of the density will not grow in time in the space C [, T ]; H m. However, in the current case, though the density ρ C[, T ]; H, yet the velocity u satisfies only u C, T ]; H k. Note that the function u { u H k }

6 6 decays much slower for large values of the spatial variable x than u H k+. Therefore, it seems that it is the slow decay of the velocity field for large values of the spatial variable x that leads to the global existence of smooth solutions. Unfortunately, such an argument cannot be valid for the full compressible Navier-Stokes system since the blowup results of Xin-Yan in [33] work for any classical solutions with compactly supported initial density. For the three-dimensional case, that is, Ω = R 3, we have the following results concerning the decay properties of the global classical solutions whose existence is essentially due to [6]. Theorem.3 Let Ω = R 3. For given numbers M >, ρ, β /, ], and q 3, 6, suppose that the initial data ρ, u satisfy ρ, P ρ H W,q, P ρ, ρ u L, u Ḣβ, u H,. and the compatibility condition ρ ρ, u Ḣβ M,. µ u µ + λ divu + P ρ = ρ / g,. for some g L. Moreover, if γ > 3/, assume that ρ L..3 Then there exists a positive constant ε depending on µ, λ, γ, ρ, β, and M such that if C ε,.4 the Cauchy problem.-.4 has a unique global classical solution ρ, u in R 3, satisfying for any < τ < T <, ρx, t ρ, x R 3, t,.5 ρ C[, T ]; L 3/ H W,q, P C[, T ]; L H W,q, u L, T ; L 6, u L, T ; H L, T ; H L τ, T ; H W,q.6, u t L, T ; L L τ, T ; H H τ, T ; L. Moreover, for r,, there exist positive constants Cr and C depending on µ, λ, γ, ρ, β, and M such that for t, u, t L p Ct +/p, for p [, 6], P, t L r Crt +/r, for r,,.7 u, t L + µ + λdivu P, t L Ct, where if γ > 3/, Cr and C both depend on ρ L R 3 also.

7 7 Remark.4 It should be pointed out that the large time asymptotic decay with rates of the global strong or classical solutions,.5 and.7, are completely new for the multi-dimensional compressible Navier-Stokes equations. in the presence of vacuum. They show in particular that the L -norm of both the pressure and the gradient of the velocity decay in time with a rate t /, and the gradient of the vorticity and the effective viscous flux decay faster than themselves. However, whether the second derivatives of the velocity field decay or not remains open. This is an interesting problem and left for the future. We now make some comments on the analysis of this paper. Note that for initial data in the class satisfying.8,.9,.7, and.8 except u Ḣβ, the local existence and uniqueness of classical solutions to the Cauchy problem,.-.4, have been established recently in [9]. Thus, to extend the classical solution globally in time, one needs some global a priori estimates on smooth solutions to.-.4 in suitable higher norms. It turns out that as in the three-dimensional case [6], the key issue here 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, so some basic ideas used in [6] will be adapted here, yet new difficulties arises in the two-dimensional case. Indeed, the analysis in [6] relies heavily on the basic fact that, for the three-dimensional case, the L 6 -norm of v D R 3 can be bounded by the L -norm of the gradient of v which fails for v D R. In fact, for two-dimensional case, some of the main new difficulties are due to the appearance of vacuum at far field and the lack of integrability of the velocity and its material derivatives in the whole two-dimensional space. To overcome these difficulties, first, using the L -integrability of the density, we observe that the L -norm in both space and time of the pressure is time-independent see 3.8. This is crucial to show that the H -norm of the effective viscous flux decays at the rate of t / for large time see 3.6 which plays a key role in obtaining the decay property of the L - norm of the effective viscous flux. Then, after some careful estimates of the expansion rates of the essential support of the density see 3.39, we succeed in obtaining that, for large time, the L p -norm of the gradient of the effective viscous flux see.6 for the definition can be bounded by the product of + t 5 and some function gt whose temporal L -norm is independent of time see Based on these key ingredients, we are able to obtain the desired estimates on L, min{, T }; L R -norm and the time-independent ones on L 4 min{, T }, T ; L R -norm of the effective viscous flux see 3.6. Then, motivated by [], we deduce from these estimates and Zlotnik s inequality see Lemma.6 that the density admits a time-uniform upper bound which is the key for global estimates of classical solutions. The next main step is to bound the gradients of the density and the velocity. Similar to [4 6], such bounds can be obtained by solving a logarithm Gronwall inequality based on a Beale-Kato-Majda type inequality see Lemma.7 and the a priori estimates we have just derived, and moreover, such a derivation yields simultaneously also the bound for L, T ; L R - norm of the gradient of the velocity, see Lemma 4. and its proof. 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 [4, 9] 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. Sections 3 and 4 are 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, Theorems.-.3,

8 8 are proved in Section 5. Preliminaries In this section, for Ω = R, we will recall some known facts and elementary inequalities which will be used frequently later. We begin with the local existence of strong and classical solutions whose proof can be found in [9]. Lemma. Let Ω = R. Assume that ρ, u satisfies.8 except u Ḣβ. Then there exist a small time T > and a unique strong solution ρ, u to the problem.-.4 in R, T satisfying.3 and.4. Moreover, if ρ, u satisfies.7 and.8 besides.8, ρ, u satisfies.9 also. Next, the following well-known Gagliardo-Nirenberg inequality see [6] will be used later. Lemma. Gagliardo-Nirenberg For p [,, q,, and r,, there exists some generic constant C > which may depend on p, q, and r such that for f H R and g L q R D,r R, we have f p L p R C f L R f p L R,. g CR C g qr /r+qr g r/r+qr.. L q R L r R The following weighted L p bounds for elements of the Hilbert space D R can be found in [, Theorem B.]. Lemma.3 For m [, and θ + m/,, there exists a positive constant C such that we have for all v D, R, R v m /m e + x loge + x θ dx C v L B + C v L R..3 The combination of Lemma.3 with the Poincaré inequality yields Lemma.4 For x as in., suppose that ρ L R is a function such that ρ M, M ρdx, ρ x α L R,.4 B N for N and positive constants M, M, and α. Then, for r [,, there exists a positive constant C depending only on M, M, α, and r such that /r ρ v r dx CN 3 + ρ x α L R ρ / v L R + v L R,.5 R for each v { v D R ρ / v L R }.

9 9 Proof. First, for f L B N, denote the average of f over B N by f BN fxdx. B N B N It then follows from.4 that ρ BN v BN = ρbn ρ v v BN dx + ρvdx B N B N B N B N M N v v BN L B N + M / N ρ / v L B N 8M v L B N + M / N ρ / v L B N,.6 where in the last inequality one has used the following Poincaré inequality [8, 7.45] v v BN L B N 4N v L B N..7 Then, it follows from.6 and.4 that v BN CM, M N v L B N + CM, M N ρ / v L B N, which together with.7 leads to v dx v v BN dx + B N v BN B N B N CM, M N 6 v L B N + CM, M N 4 ρ / v L B N..8 Finally, it follows from Holder s inequality,.3,.8, and.4 that for r [, and σ = 4/4 + α,, ρ v r dx ρ x α σ L /σ R v r x ασ L 4/ασ R ρ σ R L R C + ρ x α L R r N 3 ρ / v L R + v L R, which gives.5. This completes the proof of Lemma.4. Next, for,, denoting the material derivative of f by f f t +u f, we state some elementary estimates which follow from. and the standard L p -estimate for the following elliptic system derived from the momentum equations in.: where F and ω are as in.6. F = divρ u, µ ω = ρ u,.9 Lemma.5 Let Ω = R and ρ, u be a smooth solution of.. Then for p there exists a positive constant C depending only on p, µ, and λ such that F L p R + ω L p R C ρ u L p R,. F L p R + ω L p R C ρ u /p L R u L R + P L R /p,. u L p R C ρ u /p L R u L R + P L R /p + C P L p R..

10 Proof. On the one hand, the standard L p -estimate for the elliptic system.9 yields. directly, which, together with. and.6, gives.. On the other hand, since u = divu ω, we have u = divu ω..3 Thus applying the standard L p -estimate to.3 shows u L p R Cp divu L p R + ω L p R Cp F L p R + Cp ω L p R + Cp P L p R, which, along with., gives.. The proof of Lemma.5 is completed. Next, to get the uniform in time upper bound of the density ρ, we need the following Zlotnik inequality. Lemma.6 [34] Let the function y satisfy y t = gy + b t on [, T ], y = y, with g CR and y, b W,, T. If g = and bt bt N + N t t.4 for all t < t T with some N and N, then where ζ is a constant such that yt max { y, ζ } + N < on [, T ], gζ N for ζ ζ..5 Finally, the following Beale-Kato-Majda type inequality, which was proved in [, 7] when divu, will be used later to estimate u L and ρ L Lqq >. Lemma.7 For < q <, there is a constant Cq such that the following estimate holds for all u L R D,q R, u L R C divu L R + ω L R loge + u L q R + C u L R + C. 3 A priori estimatesi: lower order estimates In this section, for Ω = R, we will establish some necessary a priori bounds for smooth solutions to the Cauchy problem.-.4 to extend the local strong and classical solutions guaranteed by Lemma.. Thus, let T > be a fixed time and ρ, u be the smooth solution to.-.4 on R, T ] with smooth initial data ρ, u satisfying.8 and.9. Set σt min{, t}. Define T A T sup σ u L + σ ρ u dxdt, 3. and A T sup σ T ρ u dx + We have the following key a priori estimates on ρ, u. σ u dxdt. 3.

11 Proposition 3. Under the conditions of Theorem., there exists some positive constant ε depending on µ, λ, γ, a, ρ, β, N, and M such that if ρ, u is a smooth solution of.-.4 on R, T ] satisfying the following estimates hold provided C ε. sup ρ ρ, A T + A T C /, 3.3 R [,T ] sup ρ 7 ρ/4, A T + A T + R [,T ] T σ P L dt C /, 3.4 The proof of Proposition 3. will be postponed to the end of this section. In the following, we will use the convention that C denotes a generic positive constant depending on µ, λ, γ, a, ρ, β, N, and M, and use Cα to emphasize that C depends on α. We begin with the following standard energy estimate for ρ, u and preliminary L bounds for u and ρ u. Lemma 3. Let ρ, u be a smooth solution of.-.4 on R, T ]. Then there is a positive constant C depending only on µ, λ, and γ such that ρ u + T γ P dx + µ u dxdt C, 3.5 and sup sup A T CC + C sup P L + C A T CA T + C T T σ u 3 + P u dxdt, 3.6 σ u 4 L 4 + P 4 L 4 dt. 3.7 Proof. First, the standard energy inequality reads: ρ u + P T µ u dx + + µ + λdivu dxdt C, γ which together with.3 shows 3.5. Next, multiplying. by u and then integrating the resulting equality over R lead to ρ u dx = u P dx + µ u udx + µ + λ divu udx. 3.8 Since P satisfies P t + u P + γp divu =, 3.9 integration by parts yields that u P dx = divu t P u u P dx γ = divup dx + P divu + P i u j j u i dx t divup dx + C P u dx. t 3.

12 Integration by parts also implies that µ u udx = µ µ u L t µ u L t + C i u j i u k k u j dx u 3 dx, 3. and that µ + λ divu udx = λ + µ λ + µ divu L t λ + µ divu L t + C u 3 dx. divudivu udx 3. Putting into 3.8 leads to B t + ρ u dx C P u dx + C u 3 L 3, 3.3 where satisfies Bt µ u L + λ + µ divu L divup dx 3.4 µ 4 u L C P L Bt C u L + C P L. 3.5 Then, integrating 3.3 multiplied by σ over, T and using 3.5 and 3.5 yield 3.6 directly. Finally, to prove 3.7, we will use the basic estimates of u due to Hoff []. Operating / t + divu to. j, one gets by some simple calculations that ρ u j t + ρu u j µ u j µ + λ j div u = µ i i u u j + divu i u j µdiv i u i u j µ + λ j i u u i divu µ + λdiv j udivu + γ j P divu + divp j u. 3.6 Multiplying 3.6 by u and integrating the resulting equation over R lead to ρ u dx + µ u dx C u 4 L + C P 4 4 L, t which multiplied by σ gives 3.7 and completes the proof of Lemma 3.. Remark 3. It is easy to check that the estimates 3.3 and 3.7 also hold for Ω = R 3. Next, we give a key observation that pressure decays in time. Lemma 3.3 Let ρ, u be a smooth solution of.-.4 on R, T ] satisfying 3.3. Then there exists a positive constant C ρ depending only on µ, λ, γ, and ρ such that T A T + A T + σ P L dt C ρc. 3.8

13 3 Proof. First, it follows from., 3.5, and 3.3 that T C σ u 4 L 4 + P 4 L 4 dt T σ ρ u L σ u L + σ P T L dt + C σ P 4 L dt 4 C ρ A T + C T σ ρ / u L dt + C ρ T σ P L dt. To estimate the last term on the right-hand side of 3.9, noticing that. gives 3.9 P = divρ u + µ + λdivu, 3. we obtain from Hölder s and Sobolev s inequalities that P dx C divρ u L 4γ P L 4γ/4γ + C u L P L C ρ u L 4γ/γ+ ρ / L ρ γ / L γ + C u L P L C ρ / L 4γ ρ / u L ρ / L ρ γ / L γ + C u L P L C P L ρ / u L + C u L P L, where in the last inequality, one has used ρdx = ρ dx =, 3. due to the mass conservation equation.. Thus, we arrive at P L C ρ / u L + C u L, 3. which, along with 3.6, 3.7, 3.9, 3.5, and 3.3 gives T A T + A T C ρc + C ρ σ u 3 L dt Then, on the one hand, one deduces from., 3.5, and 3.3 that σt σ u 3 L 3 dt C σt CA / σt C ρc. σ ρ / u L u L + P L dt + C ρc σt u L + P L dt + C ρc 3.4 On the other hand, Hölder s inequality, 3.9, 3.3, and 3. imply T σt T T σ u 3 L dt δ u 4 3 L dt + Cδ u 4 L dt σt σt δc ρa T + CδC ρc. 3.5 Finally, putting 3.4 and 3.5 into 3.3 and choosing δ suitably small lead to A T + A T C ρc,

14 4 which together with 3. and 3.5 gives 3.8 and completes the proof of Lemma 3.3. Next, we derive the rates of decay for u and P, which are essential to obtain the uniform in time upper bound of the density for large time. Lemma 3.4 For p [,, there exists a positive constant Cp, ρ depending only on p, µ, λ, γ, and ρ such that, if ρ, u is a smooth solution of.-.4 on R, T ] satisfying 3.3, then sup σt t T t p u p L + P p p L + t ρ / u p L Cp, ρc. 3.6 Proof. First, for p, multiplying 3.9 by pp p and integrating the resulting equality over R, one gets after using divu = µ+λ F + P that P p L p t + pγ P p+ = pγ P p F dx µ + λ L p+ µ + λ pγ 3.7 P p+ + Cp F p+, µ + λ L p+ L p+ which together with. gives µ + λ pγ P p L + P p p+ Cp F p+ t L p+ L p+ In particular, choosing p = in 3.8 shows Cp u L + P L ρ u p L. 3.8 P L t + γ µ + λ P 3 L 3 δ ρ / u L + Cδ u 4 L + P 4 L. 3.9 Next, it follows from 3.3 and. that B t + ρ u dx C P 3 L + C u 3 3 L 3 C P 3 L + C ρ u 3 L u L + P L C P 3 L + δ ρ / u 3 L + C ρ, δ u 4 L + P 4 L. 3.3 Choosing C + µ + λc + /γ suitably large such that µ 4 u L + P L Bt + C P L C u L + C P L, 3.3 adding 3.9 multiplied by C to 3.3, and choosing δ suitably small lead to Bt + C P L + ρ u + P 3 dx C P 4 L + C u 4 L, 3.3 which multiplied by t, together with Gronwall s inequality, 3.3, 3.8, 3.5, and 3.3 yields sup t u L + P T ρ L + t u + P 3 dxdt C ρc σt t T σt

15 5 Next, multiplying 3.7 by t together with. gives t ρ u dx + µt u dx t t ρ u dx + C ρt ρ u L u L + P L + C ρt P 4 L Choosing p = 3 in 3.8 and adding 3.8 multiplied by C + t to 3.34 lead to t ρ u dx + µ + λ C + t P 3 L + µt u 3γ 3 L + t P 4 L 4 t ρ Ct u + P 3 dx + C ρt ρ / u L u L + P L, which combined with Gronwall s inequality, 3.33, and 3.3 yields ρ sup t u + P 3 T dx + t u L + P 4 L dt 4 C ρc σt t T σt Finally, we claim that for m =,,, T sup t m P m+ + t m P m+ dt Cm, ρc L m+ L m+, 3.36 σt t T σt which together with., 3.33, and 3.35 gives 3.6. We shall prove 3.36 by induction. In fact, 3.33 shows that 3.36 holds for m =. Assume that 3.36 holds for m = n, that is, T sup t n P n+ + t n P n+ dt Cn, ρc L n+ L n σt t T σt Multiplying 3.8 where p = n + by t n+, one obtains after using 3.35 µ + λ t n+ P n+ n + γ L n+ t + tn+ P n+3 L n Cn, ρt n P n+ + Cn, ρc L n+ u L + P L. Integrating 3.38 over [σt, T ] together with 3.37 and 3.8 shows that 3.36 holds for m = n +. By induction, we obtain 3.36 and finish the proof of Lemma 3.4. Next, the following Lemma 3.5 combined with Lemma.4 will be useful to estimate the L p -norm of ρ u and obtain the uniform in time upper bound of the density for large time. Lemma 3.5 Let ρ, u be a smooth solution of.-.4 on R, T ] satisfying the assumptions in Theorem. and 3.3. Then for any α >, there exists a positive constant N depending only on α, N, and M such that for all t, T ], ρx, tdx B N +t log α +t

16 6 Proof. First, multiplying. by + x / and integrating the resulting equality over R, we obtain after integration by parts and using both 3.5 and 3. that d ρ + x / dx C ρ u dx dt C C. / ρdx / ρ u dx This gives sup s t ρ + x / dx CM + t. 3.4 Next, for ϕy C R such that ϕy, ϕy = { if y, if y, ϕ, multiplying. by ϕy with y = δx + t log α e + t for small δ > which will be determined later, we obtain d dt δ ρϕydx = ρ y ϕ y t dx + + t log α ρu y ϕdx e + t C δ C δ + t log α ρ x dx e + t + t log α e + t CM δ + t log α e + t, where in the last inequality we have used 3.4. Since α >, this yields ρϕ y dx ρ xϕx δdx Cα, M δ 4, 3.4 where we choose δ = N + 4Cα, M. Finally, it follows from 3.4 that for N δ = N + 4Cα, M, ρdx ρϕ δx + t log α e + t dx 4, B N +t log α e+t which finishes the proof of Lemma 3.5. Next, to obtain the upper bound of the density for small time, we still need the following lemma. Lemma 3.6 Let ρ, u be a smooth solution of.-.4 on R, T ] satisfying 3.3 and the assumptions in Theorem.. Then there exists a positive constant K depending only on µ, λ, γ, a, ρ, β, N, and M such that sup t β u L + t σt σt t β ρ u dxdt K ρ, M. 3.4

17 7 Proof. First, set { } µ / ν min + µ + λ /, β, /]. β If β,, Sobolev s inequality implies ρ u +ν dx ρ u dx + C ρ + C ρ u / β Ḣ β For the case that β =, one obtains from.5 that ρ u +ν dx C ρ ρ u dx + ρ u / β dx C ρ, M u dx +ν/ C ρ, M Then, multiplying. by + ν u ν u and integrating the resulting equation over R lead to d ρ u +ν dx + + ν u ν µ u + µ + λdivu dx dt + νν µ + λ divu u ν u dx + C ρ γ u ν u dx + ν µ + λdivu u ν + νµ dx + u ν u dx 4 + C ρ u +ν dx + C ρ +νγ ν/ dx, which together with Gronwall s inequality, 3.43, and 3.44 thus gives ρ u +ν dx C ρ, M sup t σt Next, as in [], for the linear differential operator L defined by let w and w be the solution to: and Lw j ρw j t + ρu wj µ w j + µ + λ j divw = ρẇ j µ w j + µ + λ j divw, j =,, Lw =, w x, = w x, 3.46 Lw = P ρ, w x, =, 3.47 respectively. A straightforward energy estimate of 3.46 shows that: sup t σt ρ w dx + σt w dxdt Cρ w dx. 3.48

18 8 Then, multiplying 3.46 by w t and integrating the resulting equality over R yield that for t, σt ], µ w L + µ + λ divw L t + ρ w dx = ρw u w dx C ρ ρ / ẇ L ρ /+ν u L +ν w /+ν w L ν/+ν L ρ w dx + C ρ, M w L, where in the last inequality we have used 3.45 and the following simple fact: w L C ρẇ L, 3.49 due to the standard L -estimate of the elliptic system Gronwall s inequality together with 3.49 and 3.48 gives σt sup w L + ρ w dxdt C ρ, M w L, 3.5 t σt and sup t w L + t σt σt t ρ w dxdt C ρ, M w L. 3.5 Since the solution operator w w, t is linear, by the standard Stein-Weiss interpolation argument [], one can deduce from 3.5 and 3.5 that for any θ [β, ], sup t θ w L + t σt σt t θ ρ w dxdt C ρ, M w, 3.5 Ḣ θ with a uniform constant C independent of θ. Finally, we estimate w. It follows from a similar way as for the proof of. and. that and that for p, µ + λdivw P L + w L C ρẇ L, 3.53 w L p C µ + λdivw P L p + C P L p + w L p δ ρẇ L + C ρ, p, δ w L + C ρ, p, δc /p Multiplying 3.47 by w t and integrating the resulting equation over R yield that for t, σt ], µ w L + µ + λ divw L P divw dx + ρ w dx t = ρw u w dx P t divw dx 3.55 C ρ ρ / ẇ L ρ /+ν u L +ν w L +ν/ν P t divw dx C ρ, Mδ ρ / ẇ L + Cδ, ρ, M w L + u L +,

19 9 where in the last inequality we have used 3.54, 3.45, and the following simple fact: P t divw dx = P u µ + λdivw P dx µ + λ + P divudx µ + λ C P u L ρẇ L + C P L u L δ ρ / ẇ L + Cδ, ρ u L +, due to 3.9 and Gronwall s inequality together with 3.55 gives sup w L + t σt σt ρ w dxdt C ρ, M Taking w = u so that w + w = u, we then derive 3.4 from 3.5 and 3.56 directly. Thus, we finish the proof of Lemma 3.6. 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 [], see also [6]. Lemma 3.7 There exists a positive constant ε = ε ρ, M depending on µ, λ, γ, a, ρ, β, N, and M such that, if ρ, u is a smooth solution of.-.4 on R, T ] satisfying 3.3 and the assumptions in Theorem., then provided C ε. sup ρt L 7 ρ 4, 3.57 Proof. First, we rewrite the equation of the mass conservation. as where D t ρ = gρ + b t, 3.58 D t ρ ρ t + u ρ, gρ ργ+, bt µ + λ µ + λ t ρf dt. Next, it follows from., 3.4, 3.39, and.5 that for t > and p [,, F, t L p Cp ρ u, t L p Cp, ρ, M + t 5 ρ / u, t L + u, t L, 3.59 which, together with the Gagliardo-Nirenberg inequality. for q =, yields that for

20 r 4 + 4/β and δ r + βr /3r 4,, bσt C ρ σt C ρ, M σt r /4r σ β r /4r σ β F r/r L F L dt r σ r+ βr /4r σ F L r r/4r dt 3r 4/4r σt σ r/4r C ρ, M σ δ T dt σ F L rdt C ρ, M σt C ρ, MC r/4r, σ ρ / u L + σ u L dt r/4r where in the second, fourth, and last inequalities one has used respectively 3.4, 3.59, and 3.8. This combined with 3.58 yields that sup ρ L ρ + C ρ, MC /4 3 ρ t [,σt ], 3.6 provided C ε min{, ρ/c ρ, M 4 }. Next, it follows from. and 3.6 that for t [σt, T ], F, t H C u, t L + P, t L + ρ u, t L C ρc / t /, 3.6 which together with. and 3.59 shows T σt T C F, t 4 L dt σt C ρ, MC 35/8 C ρ, MC 35/8, F, t 35/9 F, t /9 dt L 7 L 7 T σt t 5/8 ρ / u L + u L /9 dt 3.6 where in the last inequality, one has used 3.3. This shows that for all σt t t T, t bt bt C ρ F, t L dt t µ + λ t t + C ρ, M T σt µ + λ t t + C ρ, MC 35/8, F, t 4 L dt

21 which implies that one can choose N and N in.4 as: N = µ + λ, N = C ρ, MC 35/8. Hence, we set ζ = in.5 since for all ζ, Lemma.6 and 3.6 thus lead to provided gζ = ζγ+ µ + λ N = µ + λ. sup ρ L 3 ρ t [σt,t ] + N 7 ρ 4, 3.63 C ε min{ε, ε }, ρ 8/35 for ε. 4C ρ, M The combination of 3.6 with 3.63 completes the proof of Lemma 3.7. With Lemmas 3.3 and 3.7 at hand, we are now in a position to prove Proposition 3.. Proof of Proposition 3.. It follows from 3.8 that provided T A T + A T + σ P L dt C /, 3.64 C ε 3 C ρ. Letting ε min{ε, ε 3 }, we obtain 3.4 directly from 3.57 and 3.64 and finish the proof of Proposition A priori estimates II: higher order estimates Form now on, for smooth initial data ρ, u satisfying.8 and.9, assume that ρ, u is a smooth solution of.-.4 on R, T ] satisfying 3.3. Then, we derive some necessary uniform estimates on the spatial gradient of the smooth solution ρ, u. Lemma 4. There is a positive constant C depending only on T, µ, λ, γ, a, ρ, β, N, M, and ρ H W,q such that sup ρ H W,q + u L + t u L T + u L + u q+/q L q + t u L q dt C. Proof. First, it follows from 3.3, 3.3, Gronwall s inequality, and 3.5 that T sup u L + t [,T ] 4. ρ u dxdt C, 4.

22 which together with. shows T u 4 L 4 dt C. 4.3 Multiplying 3.7 by t and integrating the resulting inequality over, T combined with 4. and 4.3 lead to T sup t ρ u dx + t u L dt C. 4.4 Next, we prove 4. by using Lemma.7 as in [5]. For p [, q], ρ p satisfies Thus, due to ρ p t + div ρ p u + p ρ p divu + p ρ p ρ t u ρ + pρ ρ p ρ divu =. d dt ρ L p C + u L ρ L p + C u L p C + u L ρ L p + C ρ u L p, 4.5 u L p C ρ u L p + P L p, 4.6 which follows from the standard L p -estimate for the following elliptic system: µ u + µ + λ divu = ρ u + P, u as x. Next, it follows from the Gargliardo-Nirenberg inequality, 4., and. that divu L + ω L C F L + C P L + C ω L Cq + Cq F q/q L q Cq + Cq ρ u q/q L q, which, together with Lemma.7, yields that + Cq ω q/q L q u L C divu L + ω L loge + u L q + C u L + C C + ρ u q/q L loge + ρ u q L q + ρ L q + C C + ρ u L q loge + ρ L q Next, it follows from the Hölder inequality and 3.59 that ρ u L q ρ u q /q L ρ u qq /q L q C ρ u q /q L ρ / u L + u L qq /q C ρ / u L + C ρ / u q /q L u qq /q L, which combined with 4. and 4.4 implies that T C C. ρ u +/q L + t ρ u q L q dt T ρ / u L + t u L + t q3 q q /q 3 q q dt 4.9

23 3 Then, substituting 4.8 into 4.5 where p = q, we deduce from Gronwall s inequality and 4.9 that which, along with 4.6 and 4.9, shows T sup ρ L q C, u q+/q L q + t u L q dt C. 4. Finally, taking p = in 4.5, one gets by using 4., 4., and Gronwall s inequality that sup ρ L C, which, together with 4.6, 4.4, and 4., yields 4.. The proof of Lemma 4. is completed. Lemma 4. There is a positive constant C depending only on T, µ, λ, γ, a, ρ, β, N, M, and x a ρ L L q such that sup x a ρ L H W,q C. 4. Proof. First, it follows from.3, 3.4, 3.39, and.8 that for any η, ] and any s >, u x η L s/η Cη, s. 4. Multiplying. by x a and integrating the resulting equality over R lead to d ρ x a dx C ρ u x a log e + x dx dt C ρ x a +8/8+a L 8+a/7+a u x 4/8+a L 8+a C ρ x a dx + C. This gives sup ρ x a dx C. 4.3 Then, one derives from. that v ρ x a satisfies v t + u v avu log x + vdivu =, which, together with some estimates as for 4.5, gives that for any p [, q] v L p t C + u L + u log x L v L p + C v L u log x L p + u log x L p + u L p C + u W,q v L p + C v L u L p + u x /5 L 4p x 3/ L 4p/3 + u L p C + u L p + u W,q + v L p + v L q, 4.4

24 4 where in the second and the last inequalities, one has used 4. and 4.3. Choosing p = q in 4.4, we obtain after using Gronwall s inequality and 4. that sup ρ x a L q C. 4.5 Finally, setting p = in 4.4, we deduce from 4. and 4.5 that sup ρ x a L C, which combined with 4.3 and 4.5 thus gives 4. and finishes the proof of Lemma 4.. Lemma 4.3 There is a positive constant C depending only on T, µ, λ, γ, a, ρ, β, N, M, and x a ρ L L q such that T sup t ρ / u t L + u H + t u t L dt C. 4.6 Proof. Differentiating. with respect to t gives ρu tt + ρu u t µ u t µ + λ divu t = ρ t u t + u u ρu t u P t. 4.7 Multiplying 4.7 by u t and integrating the resulting equation over R, we obtain after using. that d µ ut ρ u t dx + + µ + λdivu t dx dt = ρu u t u t dx ρu u u u t dx ρu t u u t dx + P t divu t dx 4.8 C ρ u u t u t + u + u u dx + C ρ u u u t dx + C ρ u t u dx + Cδ P t L + δ u t L. Each term on the right-hand side of 4.8 can be estimated as follows: First, the combination of 4. with 4. gives that for any η, ] and any s >, ρ η u L s/η + u x η L s/η Cη, s. 4.9 Moreover, it follows from.5, 3.4, and 3.39 that ρ / u t L 6 C ρ / u t L + C u t L, 4. which together with 4.9, 4., and Holder s inequality yields that for δ,, ρ u u t u t + u + u u dx C ρ / u L 6 ρ / u t / L ρ / u t / L 6 ut L + u L 4 + C ρ /4 u L ρ / u t / ρ / u L t / u L 6 L / ut L + u L + C ρ / u t ρ / / u L t L + u t L δ u t L + Cδ u L + ρ / u t L +. 4.

25 5 Next, Holder s inequality, 4.9, and 4. lead to ρ u u u t dx + ρ u t u dx C ρ / u L u 8 L 4 u t L + u L ρ / u t 3/ ρ / u L 6 t / L δ u t L + Cδ u L + ρ / u t L Next, it follows from 4.9, 4., and 4. that P t L C x a u L q/q ρ γ L x a ρ L q + C u L C. 4.3 Finally, putting into 4.8 and choosing δ suitably small, we obtain after using 4.6 and 4. that d ρ u t dx + µ u t dx C ρ u t dx + C ρ u dx dt It follows from 4.6 and 4.9 that u H + ρ / u u L which together with 4. shows C + C ρ / u L + C ρ / u L 6 u /3 u /3 L L C + C ρ / u L + u L, u H + ρ / u t L C ρ / u L + C. 4.5 This combined with 4.4, 4., and Gronwall s inequality gives 4.6 and finishes the proof of Lemma 4.3. From now on, assume that ρ, u is a smooth solution of.-.4 on R, T ] satisfying 3.3 for smooth initial data ρ, u satisfying.8,.9,.7, and.8. Moreover, in addition to T, µ, λ, γ, a, ρ, β, N, M, and x a ρ L Lq, the generic positive constant C may depend on u L, x δ ρ L, x δ P ρ L, and g L, with g as in.8. Lemma 4.4 It holds that sup ρ / u t L + u H T + u t L dt C. 4.6 Proof. Taking into account on the compatibility condition.8, we can define ρ ux, t = = g. 4.7 Integrating 3.7 over, T together with 4.7 and 4.3 yields directly that sup ρ / u L + T u L dt C, which, along with 4.5 and 4.4, gives 4.6 and finishes the proof of Lemma 4.4. The following higher order estimates of the solutions which are needed to guarantee the extension of local classical solution to be a global one are similar to those in [9], so we omit their proofs here.

26 6 Lemma 4.5 The following estimates hold: sup T + sup T sup t u t L + x δ ρ L + x δ P L sup C, 4.8 t ρ / u tt L + u t L dt C, 4.9 ρ L q + P L q C, 4.3 t ρ / u tt L + 3 u L L q + u t H + ρu L q+/ t u tt L + u tt x L dt C Proofs of Theorems.-.3 With all the a priori estimates in Sections 3 and 4 at hand, we are ready to prove the main results of this paper in this section. Proof of Theorem.. By Lemma., there exists a T > such that the Cauchy problem.-.4 has a unique strong solution ρ, u on R, T ]. We will use the a priori estimates, Proposition 3. and Lemmas , to extend the local strong solution ρ, u to all time. First, it follows from 3., 3., and.8 that A + A =, ρ ρ. Therefore, there exists a T, T ] such that 3.3 holds for T = T. Next, set T = sup{t 3.3 holds}. 5. Then T T >. Hence, for any < τ < T T with T finite, one deduces from 4.6 that for any q, u C[τ, T ]; L L q, 5. where one has used the standard embedding L τ, T ; H H τ, T ; H C [τ, T ]; L q, for any q [,. Moreover, it follows from 4., 4., and [, Lemma.3] that Finally, we claim that ρ C[, T ]; L H W,q. 5.3 T =. 5.4 Otherwise, T <. Then by Proposition 3., 3.4 holds for T = T. It follows from 3.5, 4., 5. and 5.3 that ρx, T, ux, T satisfies.8 except u, T Ḣβ. Thus, Lemma. implies that there exists some T > T, such that 3.3 holds for T = T, which contradicts 5.. Hence, 5.4 holds. Lemmas. and thus

27 7 show that ρ, u is in fact the unique strong solution defined on R, T ] for any < T < T =. The proof of Theorem. is completed. Proof of Theorem.. Similar to the proof of Theorem., one can prove Theorem. by using Lemma., Proposition 3., and Lemmas To prove Theorem.3, we need the following elementary estimates similar to those of Lemma.5 whose proof can be found in [6, Lemma.3]. Lemma 5. Let Ω = R 3 and ρ, u be a smooth solution of.. 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, 5.5 F L p + ω L p C ρ u 3p 6/p L u L + P L 6 p/p, 5.6 u L p C F L p + ω L p + C P L p, 5.7 where F = µ + λdivu P and ω = u are the effective viscous flux and the vorticity respectively. Proof of Theorem.3. It suffices to prove.7. In fact, it follows from [6, Proposition 3. and 3.6] that there exists some ε depending only on µ, λ, γ, ρ, β, and M such that sup u L + ρ L γ L + ρ/ u L provided C ε. t< + If γ 3/, it then holds that u L + ρ / u L + u L dt C, 5.8 sup ρ L 3/ C sup ρ γ/3 L C. 5.9 γ t< t< If γ > 3/, since ρ L,. yields that for t, ρx, tdx = ρ xdx, which combined with 5.8 implies sup ρ L 3/ C sup ρ /3 C. 5. L t< t< Similar to 3., one deduces from. that P = divρ u + µ + λdivu, which together with the Sobolev inequality gives P L C divρ u L + C u L C ρ u L 6/5 + C u L C ρ / L 3/ ρ / u L + C u L C ρ / u L + C u L,

28 8 where in the last inequality one has used 5.9 and 5.. This combined with 5.8 leads to Next, similar to 3.7, for p, we have P p L p t + pγ µ + λ P L dt C. 5. P p+ L p+ = pγ µ + λ P p F dx, 5. which together with Holder s inequality yields P p L p t + pγ P p+ Cp F p+. µ + λ L p+ L 5.3 p+ Next, for Bt defined as in 3.4, it follows from 3.3 and 5.7 that B t + ρ u dx C P 3 L + C u 3 3 L 3 C P 3 L 3 + C F 3 L 3 + C ω 3 L Choosing C + µ + λc + /γ suitably large such that µ 4 u L + P L Bt + C P L C u L + C P L, 5.5 setting p = in 5.3, and adding 5.3 multiplied by C to 5.4 yield that for t, Bt + C P L + ρ u + P 3 dx C F 3 L 3 + C ω 3 L 3 ρ / u L + C u 4 L + P 4 L, 5.6 where in the second inequality we have used 5.6 and 5.8. Multiplying 5.6 by t, along with Gronwall s inequality, 5.5, 5.8, and 5., gives sup t< t u L + P ρ L + t u + P 3 dxdt C. 5.7 Then, multiplying 3.7 by t together with 5.7 gives t ρ u dx + µt u dx t t ρ u dx + Ct F 4 L + Ct ω 4 4 L + Ct P 4 4 L Setting p = 3 in 5.3 and adding 5.3 multiplied by µ + λ C + t /3γ to 5.8 lead to t ρ u dx + µ + λ C + t P 3 L + µt u 3 L + t P 4 L 4 3γ ρ Ct u + P 3 dx + Ct F 4 L + Ct ω 4 4 L 4 ρ Ct u + P 3 dx + Ct ρ / u 3 L u L + P L ρ Ct u + P 3 dx + Ct ρ / u L ρ / u L + u L + P L, t

29 9 where in the second inequality we have used 5.6. This combined with Gronwall s inequality, 5.7, 5.8, and 5. yields ρ sup t u + P 3 dx + t< t u L + P 4 L 4 dt C. 5.9 This combined with. gives.7 provided we show that for m =,,, sup t m P m+ + L t< m+ t m P m+ L m+ dt Cm, 5. which will be proved by induction. Since 5.7 shows that 5. holds for m =, we assume that 5. holds for m = n, that is, sup t n P n+ + L t< n+ t n P n+ L n+ dt Cn. 5. Setting p = n + in 5. and multiplying 5. by t n+ give µ + λ t n+ P n+ n + γ L n+ t + tn+ P n+3 L n+3 Cnt n P n+ + Cnt n+ P n+ F L n+ L n+ L. It follows from the Gagliardo-Nirenberg inequality, 5.5, and 5.9 that F L dt C C C C, F / L 6 F / L 6 dt ρ u / L ρ u / L 6 dt t / u / L dt 5. which, along with 5., 5., and Gronwall s inequality, thus shows that 5. holds for m = n +. By induction, we obtain 5. and finish the proof of.7. The proof of Theorem.3 is completed. References [] Beale, J. T.; Kato, T.; Majda. A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys , [] Bergh, J.; Lofstrom, J. Interpolation spaces, An introduction. Springer-Verlag, Berlin-Heidelberg-New York, 976. [3] Cho, Y.; Choe, H. J.; Kim, H. Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83 4, [4] Cho, Y.; Kim, H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscript Math. 6, 9-9. [5] Choe, H. J.; Kim, H. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Eqs. 9 3,

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