Analytical solutions to the 3-D compressible Navier-Stokes equations with free boundaries

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1 Wang et al. Boundary Value Problems 215) 215:94 DOI /s RESEARCH Open Access Analytical solutions to the -D compressible Navier-Stokes equations with free boundaries Mei Wang1*, Zilai Li2 and Wei Li1 * Correspondence: wangmei49@16.com 1 School of Mathematics, Northwest University, Xi an, 71127, China Full list of author information is available at the end of the article Abstract In this paper, we study a class of analytical solutions to the -D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity hρ ) = μ, and the bulk viscosity gρ ) = ρ β β > ). By constructing a class of radial symmetric and self-similar analytical solutions in RN N 2) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity field. Keywords: analytical solution; compressible Navier-Stokes equations; density-dependent 1 Introduction The compressible Navier-Stokes equations with density-dependent viscosity coefficients can be written as ρ + divρu) =, t. ) t ρu) + divρu U) divhρ)du)) gρ) div U) + Pρ) =, where ρx, t), Ux, t), and Pρ) = ρ γ γ > ) stand for the fluid density, velocity, and pressure, respectively. The strain tensor is DU) = U + Ut, and hρ) and gρ) are the Lamé viscosity coefficients satisfying hρ) >, hρ) + Ngρ).. ) During the last several decades, there has been significant progress on the system. ) since it was introduced by Liu et al. in [ ]. There is a huge literature on the global exis 215 Wang et al. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wang et al. Boundary Value Problems 215) 215:94 Page 2 of 15 tence and behaviors of solutions to 1.1) in the case that the viscosity coefficients are both constants. The important progress on the global existence of strong or weak solutions in one spatial dimension or multi-dimension has been made by many authors; refer to [2 8] and references therein. However, the studies in Hoff and Serre [9], Xin [1], and Liu et al. [1] prove that the compressible Navier-Stokes equations with constant viscosity coefficients behave singularly in thepresenceofavacuum. In[1] Liu, Xin and Yang introduced the modified compressible Navier-Stokes equations with density-dependent viscosity coefficients for isentropic fluids. In deriving by Chapman-Enskog expansions from the Boltzmann equation, the viscosity of the compressible Navier-Stokes equations depends on the temperature and thus on the density for isentropic flows. Moreover, it should be emphasized that the viscous Saint-Venant system in the description of the motion for shallow water was also derived recently [11, 12], which is expressed exactly as 1.1) withn = 2, hρ) = ρ, gρ) =, and γ =2. The system 1.1) was first proposed and studied by Vaigant-Kazhikhov in [1]. There is much literature on mathematical studies of the Navier-Stokes equations with densitydependent viscosities. For the one-dimensional case one is referred to [1, 14 24]andreferences therein. For the periodic problem on the torus T 2 and under assumptions that the initial density is uniform away from vacuum and β >wherehρ) is a positive constant and gρ)=ρ β, Vaigant-Kazhikhov found the well-posedness of the classical solution to 1.1), and the global existence and large time behavior of weak solutions was studied by Perepelitsa in [25]. Later on, Jiu-Wang-Xin [26] improved the result and obtained the global well-posedness of the classical solution to the periodic problem with large initial data permitting vacuum. Then Huang-Li [27] relaxed the index β to β > 4 and studied the large time behavior of the solutions. For a 2-D Cauchy problem with vacuum states at far fields, Jiu-Wang-Xin [28] andhuang-li[29] independently considered the global well-posedness of classical solutions in different weighted spaces. Later on, Jiu-Wang-Xin [] studied the global well-posedness of the classical solution to the Cauchy problem 1.1) with hρ) being a positive constant and gρ)=ρ β and large data which keeps non-vacuum states ρ >atfarfields. The first successfulexampleis due to Guoet al. in [1] with spherically symmetric initial data and fixed boundary conditions, and later Guo et al. extended it to the free boundary conditions with discontinuously symmetric initial data [2]. Chen and Zhang in [] showed a local existence result with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. In [2], Guo had discussed the global existence, the Lagrangian structure and large time behavior of weak solutions to the compressible spherically symmetric Navier-Stokes equations 1.1) with the following stress free boundary conditions: ρ γ = ρu r, onr = at), N with hρ)=ρ, gρ)=,andγ [2, ). Notice that the results in [2] can also be generalized to multi-dimensional compressible spherically symmetric Navier-Stokes equation N 2 1.1) with general viscosity coefficients hρ)=ρ θ and gρ)=θ 1)ρ θ with N 1 N < θ 1with

3 Wang et al. Boundary Value Problems 215) 215:94 Page of 15 N 2 the space dimension or supplying the following free boundary: ρ γ = θρ θ u r + 2u r ), onr = at). On the other hand, there are many references considering the analytical solutions or blowup solutions to the Navier-Stokes equations, Navier-Stokes-Poisson equations, or Euler-Poisson equations see [4 6]). In [7], Guo has studied the analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients where hρ)=ρ θ and gρ)=θ 1)ρ θ, N =,andwithfreeboundaries. Inthepresentpaper,westudymorepreciselythestructureandasymptoticbehaviorof solutions to the free boundary value problem with both the continuous density condition and the stress free condition. Let hρ)=μ andgρ)=ρ β β >).Atfirst,weconstruct aself-similarsolutionoftheform ρr, t)= f r at) ) a t), ur, t)=a t) at) r, to the continuity equation with special spherically symmetric initial data and given symmetric sphere separating the initial fluids from vacuum, and we derive that at), f z) C 1 are two functions satisfying some ordinary differential equations. At the same time, the free boundary is shown to expand outward in the radial direction at an algebraic rate in time and the density decays to zero asymptotically in time almost everywhere away from the symmetry center, which leads to the formation of vacuum states as the time goes to infinity. One of our main steps in the analysis is to show that at)andf z) C 1 exist globally, which yields the desired analytical solutions to the free boundary value problem with suitable initial data. The plan of this paper is as follows. In Section 2 we will give the main results and the self-similar solution to 1.1). Theorems 2.1 and 2.2 will be proved in Sections.1 and.2, respectively. 2 Notations and main results Set hρ)=μ andgρ)=ρ β, N =,in1.1). Then the isentropic compressible Navier- Stokes system with density-dependent viscosity becomes t ρ + divρu)=, 2.1) t ρu)+divρu U)+ Pρ)=μ U + μ + ρ β ) divu)), for t, + ) andx R.Hereρx, t), Ux, t), and Pρ)=ρ γ γ > 4 ). The initial conditions of 2.1)areimposedthus: ρ, ρu) t= x)=ρ, m )x), x a. 2.2) In this paper, we are concerned with the following spherically symmetric solutions. To this end, we denote x = r, ρx, t)=ρr, t), Ux, t)=ur, t) x r. 2.)

4 Wang et al. Boundary Value Problems 215) 215:94 Page 4 of 15 This leads to the following system of equations: ρ t +ρu) r + 2ρu r ρu) t + ρu 2 + ρ γ ) r + 2ρu2 r for r, t) with =, 2.4) = 2μ + ρ β ) u r + 2u )), 2.5) r r = { r, t) r at), t }, with the initial condition ρ, u) t= = ρ r), u r) ), r a, 2.6) where a > is a constant, and we have the free boundary condition ρ at), t ) =, 2.7) or, when μ =, ρ γ = ρ β u r + 2u r ), onat), 2.8) where a t)=u at), t ), a) = a, t. 2.9) It is easy to get thefollowing usualapriorienergy estimate for smooth solutions to 2.4), 2.5), and the boundary conditions 2.7)or2.8): d dt 1 2 ρu2 + 1 ) γ 1 ργ r 2 dr + Our main result is the following theorem. 2μ + ρ β ) u r r +2u) 2 dr. 2.1) Theorem 2.1 Assume that γ = β > 4. For the radial symmetry compressible Navier-Stokes equations 2.4)-2.5), with the continuous density condition 2.7), there exists a solution of the form ρr, t)= [ γ 1 r2 1 2γ )] a 2 γ 1 1 t) a t), ur, t)= a t) r. 2.11) at) Here at) C 1 [, )) is the free boundary satisfying 2.9) and it exists for all t. Furthermore, at) tends to + as t + with the following rates: 1 C t) γ 1) at) C t). 2.12)

5 Wang et al. Boundary Value Problems 215) 215:94 Page 5 of 15 Remark 2.1 The solution 2.11) constructed in Theorem 2.1 satisfies the following properties: ρ, t), as t +, 2.1) t) 4 = πa t) +, ast +, 2.14) where the domain of the fluid is defined by t)= { r, t) r at), t }. On the other hand, for the stress free boundary condition 2.8), we have the following. Theorem 2.2 Assume that γ =2β 1, γ > 5. Then the free boundary value problem for the radial symmetry compressible Navier-Stokes system,2.4)-2.5), with the stress free condition 2.8),has a unique solution with the free boundary r = at) given by and at)=f 1 1) ρ β γ a )+γ β)t ) 1 γ β) 2.15) ρr, t)= f r at) ) a t), ur, t)= r ρ β γ a )+γ β)t), 2.16) where f z)>,fz) C[, 1]) C 1, 1]) satisfies 1 γ 2 z + γ f 1) 1 γ f γ 2 z)f z) 1 γ + 1 ) f 1) 6 1 γ γ 11 2 f 6 z)f z)=. 2.17) 2 Remark 2.2 It can be verified easily that the solution 2.15) intheorem2.2, satisfies the following properties: ρ, t), ρ at), t ), as t +, 2.18) t) 4 = πa t) +, ast ) Moreover, for r at), we have ur, t) = a t) at) r a t) 1 γ γ C1 + t) 1, as t +, 2.2) ur r, t) 1 = ρ β γ, as t ) a )+γ β)t) The proofs of the main theorems First, we will give a self-similar solution to the continuity equation 2.4). Lemma.1 For any two C 1 functions f z) and at)>,define ρr, t)= f r at) ) a t), ur, t) t)=a r..1) at)

6 Wang et al. Boundary Value Problems 215) 215:94 Page 6 of 15 Then ρ, u)r, t) solves the continuity equation 2.4), i.e., ρ t +ρ) r u + ρu r + 2ρu r =..2) Here, we can choose at) as a free boundary which satisfies the condition 2.7)or2.8). So in the following we will determine the form of the function f x) andthenprovethe global existence of the free boundary at). To this end, denoting z = r at),onecanobtain from 2.5)that a t) a t) f z)z + γ f γ 1 z)f 1 z) a γ +1 t) a t) a β+2 t) βf β 1 z)f z)=..) Next, we will solve.) according to the free boundary conditions 2.7) or2.8), respectively..1 The continuous boundary condition Assume that γ = β > 4.Werequire z = γ f γ 2 z)f z)..4) Then it follows from.)-.4)that [ f z)= f γ 1 1) + γ 1 2γ 1 z 2 )] 1 γ 1,.5) a t) a 2 β t)+a t)a 1 β t)=..6) Using the boundary condition 2.7)yieldsf z)=[ γ 1 2γ 1 z2 )] 1 γ 1,andthen ρr, t)= [ γ 1 2γ 1 z2 )] 1 γ 1 a t), ur, t)= a t) r..7) at) Clearly, if at) > is a free boundary satisfying the condition 2.7), then it is straightforward to check that ρ, u)definedby.7) is a solution of 2.4)-2.5), where at) can be determined by a t)=a 1 + t a2 β s) ds 2 β a2 β t)+ 2 β a2 β,.8) a) = a, a ) = a 1, with a >anda 1 being the initial location and slope of the free boundary. Thus, it remains to solve the boundary value problem.8). We start with estimates on solutions of.8). Lemma.2 Let at) C 1 [, T] be a solution to.8) for T >.Then there exist two uniform positive constants C 1 and C 2 >,independent of T, such that 1 C t) γ 1) at) C t) 6β 4 β 1)..9)

7 Wang et al. Boundary Value Problems 215) 215:94 Page 7 of 15 Proof We first verify the fact that at) 1 for all t [, T]. Note that.7)implies ρ at), t ) =, u, t) =..1) We introduce Ht)= = ) 2ρr r 1+t)u 2 dr t)2 γ 1 ρr 4 dr 21+t) t)2 γ 1 ρ γ r 2 dr ρur dr +1+t) 2 ρu 2 r 2 dr ρ γ r 2 dr..11) Due to a t)=uat), t)and.1), a direct computation gives H t)= ρt r 4 2ρur ) ρu dr +1+t) 2 2 ) t + 2 ρ γ γ 1 ) ) r 2 dr t +21+t) ρu 2 r 2 ρu) t r + 2 ) γ 1 ργ r 2 dr = I 1 + I 2 + I..12) Equations 2.4)and.1)yield I 1 = One has ρur 4 ) dr =. r ρu I 2 =1+t) 2 2 ) t + 2 ρ γ γ 1 ) ) r 2 dr t =1+t) 2 2ρu) t ur 2 ρ t u 2 r 2 + 2γ γ 1 ργ 1 ρ t r 2 dr =1+t) 2 2u [ ρu 2) r r2 ρ γ ) r r2] 2γ γ 1 ργ 1 ρu) r r 2 2γ 2μ γ 1 ργ u2γ +ρu) r u 2 r 2 2ρu r +2ur 2 + ρ γ ) u r + 2u )) dr r r =1+t) 2 2γ ρ γ 1 γ ur 2) r ρu r 2) 2μ r +2ur2 + ρ γ ) u r + 2u )) dr r r 2μ =1+t) 2 2ur 2 ) + λρ) u r + 2u )) dr r r =1+t) 2 [ ρ γ ) r 2uur r 2 +4u 2 r ) + 2μ + ρ γ ) 4uu r r 4u 2)] = 1+t) 2 ρ γ 2uu r r 2 +4u 2 r ) r +1+t)2 2μ + ρ γ ) 4uu r r 4u 2) = 1+t) 2 2ρ γ u 2 r r2 +16ρ γ u 2) dr,

8 Wang et al. Boundary Value Problems 215) 215:94 Page 8 of 15 I =21+t) ρu 2 r 2 ρu) t r + 2 γ 1 ργ r 2 dr 5 γ 2μ =21+t) γ 1 ργ r 2 + ρ γ ) u r + 2u r =21+t) 21+t) =21+t) ρ γ u r r +2ur 2) r )) r dr r 2μ + ρ γ ) 2u r r 2 2ur ) + 5 γ γ 1 ργ r 2 dr ρ γ u r r 2 +8ρ γ ur + 5 γ γ 1 ργ r 2 dr. Thus, substituting the above estimates into.12) and using the Cauchy-Schwarz inequality, one deduces H t) = 1+t) 2 2ρ γ u 2 r r2 +16ρ γ u 2) dr +1+t) 6ρ γ u r r ρ γ ur + 5 γ γ 1 ργ r 2 dr where one has used 21 + t) t) ρ γ r 2 dr + 25 γ ) 1 + t) γ 1 ρ γ u r r 2 dr 21 + t) 2 ρ γ u 2 r r2 dr ρ γ ur dr t) 2 ρ γ u 2 dr +4 Note also that the conservation of total mass implies that ρ γ r 2 dr,.1) ρ γ r 2 dr, ρ γ r 2 dr. ρr 2 dr = a ρ r 2 dr =1. In the case γ 5,.1)yields and so H t) Ht) H) + and consequently 17γ 1) E, E = 2 a ) 1 2 ρ u 2 + ργ r 2 dr,.14) γ 1 17γ 1) E t,.15) 2 ρ γ r 2 dr C1 + t) 1..16)

9 Wang et al. Boundary Value Problems 215) 215:94 Page 9 of 15 Thus, as a consequence of.16) and the conservation of mass, we have, for any t >, 1= which implies ) 1 γ ) γ 1 ρr 2 dr ρ γ r 2 dr r 2 γ dr Cat) γ 1 + t) γ 1, 1 at) C1 + t) γ 1)..17) For 1 < γ < 5,.1)yields Ht)1 + t) γ 5 ) C1 + t) γ 5, which yields Ht) C1 + t) α..18) Here 1 if 4 < γ < 5, α = 1+σ, σ >small, ifγ = 4,.19) 5 γ, if1<γ < 4. As in.15)-.17), one can show that at) C1 + t) 2 α γ 1)..2) Next, we derive an upper bound for at).it followsfrom.8),.2) that Then a t) a 1 + C t) β 1 β 1) + C t) 2 α)2 β) β 1) + C a 2 β a 1 + C1 + t) β 1 β 1) + C t) 2 β) β 1) + C a 2 β. at) Ct + C1 + t) 6β 4 β 1) C1 + t) 6β 4 β 1). This yields.9) and completes the proof of Lemma.2. We are now ready to give the existence and uniqueness of the solution to the boundary value problem.8). Lemma. There exists a sufficiently small T such that.8) has a solution at),which is unique in C 1 [, T] and satisfied with < 1 2 a < at)<2a. Proof The lemma can be proved by a fixed point argument. In fact, set g at) ) = a 1 + t a 2 β s) ds 2 β a2 β t)+ 2 β a2 β.

10 Wang et al. Boundary Value Problems 215) 215:94 Page 1 of 15 Then.8)canberewrittenas dat) = g at) ), a) = a, g a) ) = a ) = a 1. dt Let T 1 be a positive small constant to be determined. Define { X = at) C 1 [, T 1 ], < 1 } 2 a < at)<2a, t [, T 1 ]. Then, for any a 1 t)anda 2 t) X,sinceβ >1,wehave g a1 t) ) g a 2 t) ) t = a 2 β 1 s) a 2 β 2 s) ) ds + ) 1 4 6β 2 β 2 a a 1 a 2 β 2 t) ) 1 4 6β t + 2 a a1 s) a 2 s) β 2 ds 1 2 a ) 4 6β ) 2 β + T 1 L sup t T 1 a 1 t) a 2 t), 2 β a 2 t) a 2 β 1 t) ) 2 β sup a 1 t) a 2 t) β 2 t T 1 where L = 1 2 a ) 1 β β 1) 2 β + T 1)isaconstant.WenowdefineamappingonX by Tat)=a + t g as) ) ds, t [, T 1 ]. Then Tat) C 1 [, T 1 ], and, for any t < T 1,onecandeducethat Tat) =a + a + 2a t a 1 + a 1 s 2 β a 2 β τ) dτ 2 β a2 β s)+ ) 1 2 β ) ) 1 2 β 2 a t + 2 a t 2 ) 2 β a2 β ds if and t a 1 2 β 1 2 a ) 2 β ) 2 +4a 1 2 a ) 2 β a 1 2 β 1 2 a ) 2 β ) a ) 2 β = T 2, ) Tat) a a 1 2 β a2 β t 1 2 a if t 1 2 a = T a 1. 2 β a2 β

11 Wang et al. Boundary Value Problems 215) 215:94 Page 11 of 15 Thus, if T 1 min{t 2, T },thentat) X.Furthermore,since Ta 1 s) Ta 2 s) = t g a 1 s) ) ds t g a 2 s) ) ds LT 1 sup a 1 t) a 2 t), t T 1 T will be a contraction mapping if 2 β + T 1)T 1 < 1 2 a ) β 1 1 β), i.e., 2 β ) a ) β 1 1 β) T 1 < 2 2 β = T 4. The above argument shows that T : X X is a contraction with the sup-norm for any T 1 = min{t 2, T, T 4 }. By the contraction mapping theorem, there exists a unique at) C 1 [, T 1 ]suchthattat)=at)andthena t)=gat)), which yields.8). This completes the proof of the lemma. Now, Theorem 2.1followsfromLemma., theaprioriestimates, Lemma.2, andthe standard continuity argument. This completes the proof of Theorem The stress free boundary condition First, it follows from the free boundary 2.8)andfrom2.4)that ρ at) ) = ρ β γ a )+γ β)t ) 1 β γ..21) Using the ansatz in.1)showsthat at)=f 1 1) ρ β γ a )+γ β)t ) 1 γ β)..22) Then.1)becomes ρr, t)= f r at) ) a t), ur, t)= r ρ β γ a )+γ β)t)..2) Set α = f 1), and then.22) tellsusthatα >. Recall the assumption that β = 1 2 γ + 1 ), γ > 5.Then.)becomes 1 γ 2 z + γα1 γ f γ 2 z)f z) 1 γ + 1 ) α 6 1 γ γ 11 2 f 6 z)f z)=..24) 2 Denoting gz)= f z) γ 5 ) 6 for any z [, 1], then the above equality becomes α g z){g g1) = 1. γ 1 γ 5 z) γ +1 } = α 2 γ 1)γ 5)z, 12γ.25) Next, we will prove that.25) can be solved on [, 1]. To this end, we start with apriori estimates and the uniqueness.

12 Wang et al. Boundary Value Problems 215) 215:94 Page 12 of 15 Lemma.4 For any γ > 5, let gz) be a solution to the system.25) in C[, 1]) C1, 1]). Then γ +1 ) γ 5 γ 1 < gz) 1,.26) for all z,1].furthermore, such a solution is unique. γ 1 Proof If g γ 5 γ +1 z) =,then.25) impliesthatz must be zero, i.e. g γ 1 γ 5 ) = γ 1 Namely, if z, then one has g γ 5 z) γ 1 If for any z, 1], g γ 5 z) [, ) γ 5 γ +1 γ 1 1 gz)< <1, γ +1. γ +1 ), then.25)impliesthatg z) andthus γ 1 which is a contradiction. Thus we can deduce that g γ 5 γ +1 z) > together with.25)togetg z) and consequently γ +1 γ +1. for all z, 1] and ) γ 5 γ 1 < gz) 1, z, 1]..27) It remains to prove the uniqueness. To this end, let ḡz) C[, 1]) C 1, 1]) be another solution to.25)withḡ1) = 1 and γ 5 γ +1 ) γ 1 < gz) 1 for all z, 1]. Define wz)=gz) ḡz). Then wz) solves the following problem: d γ γ 5) {wz) dz γ +1)γ 1) {[wz)+ḡz)] 1) 1) γ 5 ḡz) γ 5 }} =, w1) =, ḡ1) = 1..28) Set I = { z [, 1] wξ), z ξ 1 }. Here I because of 1 I.Definez = inf I and then z [, 1]. Obviously, the uniqueness of solutions to the system.25) will be shown by proving that z andusinga continuity argument. If not, then z, 1], and wz )=.Foranyz, z ),.26)tellsusthat γ +1 ) γ 5 γ 1 < gz) 1, z, z )..29) Integrating.28)over[z, z ]shows wz) γ γ 5) {[ ] 1) 1) wz)+ḡz) γ 5 ḡz) } γ 5 =..) γ +1)γ 1)

13 Wang et al. Boundary Value Problems 215) 215:94 Page 1 of 15 Since for any γ > 5 and 1) γ 5 [ wz)+ḡz) ] 1) γ 5 ḡz) ) 1) γ 5 = > 2, a Taylor expansion gives 1)[ḡz) ] γ 1 γ 5 wz)+o1)w 2 z).1) γ 5 for sufficiently small wz). Putting.1)into.)andusingthefactwz )=,onehas { 1 γ +1 } [ḡz) ] γ 1 γ 5 wz) for z close to z.noticethat 1 [ḡz) ] γ 1 γ 5 <, z, z ) γ +1 γ γ 5) γ +1)γ 1) O1)w2 z)=,.2) by virtue of.29). Then one can easily deduce that wz), z z δ, z )forsomeδ >. This contradicts to z = inf I.Thusz and the proof of Lemma.4 is completed. Now we are ready to give an existence result to system.25). Lemma.5 For any γ > 5, there is a positive function y = gz) in C[, 1]) C1, 1]) satisfying.25). Proof We can rewrite.25) as follows: g z)=ggz), z)= g1) = 1. α 2 γ 1)γ 5)z 12γ γ 1 g γ 5 γ +1 z),.) We look for a solution to.)suchthat gz) C [, 1] ) C 1, 1] ), γ +1 ) γ 5 γ 1 < gz) 1, z, 1]..4) γ 5 γ +1 Set R = {z, gz)) 1 z a, 1 gz) b} for small a, 1), b, 1 ) γ 1 ). Then we can easily deduce that G gz), z ) M, z, gz) ) R, where M is a positive constant only depending on γ, a, b. Since Ggz), z) is continuous in R, bychoosingh = min{a, b }, one can show that the M solution to the initial value problem.2) exists in the neighborhood 1 z h. Similarly, we can extend this solution from the left of the neighborhood 1 z h step by step. Let the maximum interval of the existence of solutions be α,1] for some α ; with gz) C[α,1]) C 1 α, 1]), we will show that α. If not, then α, 1). By Lemma.4,onehasgα)> γ 5 γ +1 ) γ 1 and.) 1 is well defined in the small neighborhood of z = α for C 1 function gz). Thus similar arguments to those above show that the solution of this initial value problem.2) in the neighborhood α

14 Wang et al. Boundary Value Problems 215) 215:94 Page 14 of 15 z h for small h >exists.thatistosay,thesolutiongz) can be extended to the interval [α h, 1], which contradicts the fact that α, 1] is the maximum interval of the existenceofsolutions.thenwehaveobtainedac[, 1]) C 1, 1]) solution gz) tothe system.25). Finally, by virtue of.2) and Lemmas.4-.5, we have obtained the global existence of y = f z) C[, 1]) C 1, 1]) to.24) and the proof of Theorem 2.2 is complete. According to Lemma.4, f ) > γ +1 ) γ 6 1 f 1), and f z) =ρr, t)a t), we can deduce from.21)-.22) the following. Corollary.1 The density at the origin of the center has the following estimate: and then γ +1 ) 6 γ 1 [ ρ 1 γ 6 a )+ ] 6 γ 1 t 6 γ 1 < ρ, t)= f ) f 1) [ ρ 1 γ 6 a )+ ] 6 γ 1 γ 1 t, 6 ρ, t), as t +. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics, Northwest University, Xi an, 71127, China. 2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan Province 454, P.R. China. Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions which helped to improve the presentation of the paper. This research is partially supported by National Natural Science Foundation of China 1115, , and also supported by Research Fund for the Doctoral Program of Higher Education of China Received: 5 February 215 Accepted: 19 May 215 References 1. Liu, TP, Xin, ZP, Yang, T: Vacuum states of compressible flow. Discrete Contin. Dyn. Syst. 4, ) 2. Danchin, R: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, ). Feireisl, E, Novotoý, A, Petzeltová, H: On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids. J. Math. Fluid Mech., ) 4. Hoff, D: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ration. Mech. Anal. 12, ) 5. Jiang, S, Zhang, P: Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun. Math. Phys. 215, ) 6. Kazhikhov, AV, Shelukhin, VV: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, ) 7. Lions, PL: Mathematical Topics in Fluid Dynamics 2, Compressible Models. Oxford Science Publication, Oxford 1998) 8. Matsumura, A, Nishida, T: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 2, ) 9. Hoff, D, Serre, D: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51, ) 1. Xin, ZP: Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density. Commun. Pure Appl. Math. 51, ) 11. Gerbeau, JF, Perthame, B: Derivation of viscous Saint-Venant system for laminar shallow water, numerical validation. Discrete Contin. Dyn. Syst., Ser. B B1, ) 12. Marche, F: Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. Eur. J. Mech. B, Fluids 26, ) 1. Vaigant, VA, Kazhikhov, AV: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. Sib. Mat. Zh. 66), ) inrussian). Translation insiberian Math. J. 66), )

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