GLOBAL NONLINEAR STABILITY OF RAREFACTION WAVES FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH TEMPERATURE AND DENSITY DEPENDENT TRANSPORT COEFFICIENTS

. We study the nonlinear stability of rarefaction waves to the Cauchy problem of one-dimensional compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas when the transport coeﬃcients depend on both temperature and density. When the strength of the rarefaction waves is small or the rarefaction waves of diﬀerent families are separated far enough initially, we show that rarefaction waves are nonlinear stable provided that ( γ − 1) · H 3 ( R )-norm of the initial perturbation is suitably small with γ > 1 being the adiabatic gas constant.

Here v ± > 0, u ± , θ ± > 0 are constants to be specified later, the unknowns v > 0, u, θ > 0, p > 0, and e represent, respectively, the specific volume, the velocity, the absolute temperature, the pressure, and the specific internal energy of the gas. The transport coefficients µ(v, θ) (viscosity) and κ(v, θ) (heat-conductivity) are prescribed through constitutive relations as functions of v and θ which are assumed to satisfy µ(v, θ) > 0, κ(v, θ) > 0 for all v > 0 and θ > 0. Throughout this paper, we focus on the gases for which the constitutive relations between the five thermodynamic variables v, p, e, θ, and s satisfies the relations provided by the kinetic theory of gases (cf. [2]), and we consider only ideal, polytropic gases: where s denotes the specific entropy of the gas and the specific gas constants A, R, and the specific heat at constant volume C v are positive constants and γ > 1 is the adiabatic constant. To simplify the presentation, we can assume without loss of generality that A = R = 1 in the rest of this paper. Our study of the compressible Navier-Stokes equations (1) for a viscous and heat conducting ideal polytropic gas satisfying the constitutive relations (4) with temperature and density dependent transport coefficients µ(v, θ) and κ(v, θ) verifying (3) are motivated by the following considerations: • If one derives the compressible Navier-Stokes equations (1) from the Boltzmann equation with slab symmetry by employing the celebrated Chapman-Enskog expansion, cf. [2,5,15,24,43,46], the corresponding five thermodynamic variables v, p, e, θ, and s satisfy the constitutive relations (4) for ideal, polytropic gases and the transport coefficients µ and κ depend on temperature. In particular, if the inter-molecular potential is proportional to r −α with α > 1, where r represents the intermolecular distance, then µ and κ satisfy µ, κ ∝ θ α+4 2α = θ, for the Maxwellian molecule (α = 4), √ θ, for the elastic spheres (α → +∞); (5) • For certain class of solid-like materials, cf. [3,6,13,20], the viscosity µ depends on density and the heat-conductivity κ depends on density and temperature; • Experimental results for gases at very high temperatures show that both the viscosity µ and the heat-conductivity κ may depend on density and temperature, cf. [20,48].
There have been many results on the global solvability and the precise descriptions of the large time behaviors of the corresponding global solutions to the initial value problem and the initial-boundary value problems of the one-dimensional compressible Navier-Stokes equations (1)-(3)- (4). For the purpose of this introduction we consider only a small selection of the very extensive literature on the corresponding results concerning the Cauchy problem (1)-(3)-(4), (2) as in the following (For the corresponding results on the initial-boundary value problems of (1)-(3)-(4), those interested are referred to the survey paper [31] and the references cited therein): For the case with small initial perturbation, the results is quite complete for general transport coefficients satisfying (3). In fact, by employing the energy method developed by Matsumura and Nishida in [32], if the initial data is small perturbation of certain profiles such as the constant states and some elementary waves like diffusion waves, rarefaction waves, viscous shock waves, viscous contact discontinuities, and some of their superpositions, the corresponding results on the global solvability and the nonlinear stability of these elementary waves to the Cauchy problem (1)-(3)-(4), (2) are well-understood provided that both the initial data and these profiles are assumed to be without vacuum, mass and temperature concentrations, or vanishing temperatures and both the initial perturbation and the strengths of these elementary waves are sufficiently small, cf. [7,8,9,11,17,26,27,28,29,30,33,34] and the references cited therein. Note that although all these results are concerned with the case when the transport coefficients µ and κ are positive constants, the generalizations of these results to the case of temperature and density dependent viscosity µ and heat-conductivity κ satisfying (3) are straightforward for small initial perturbation.
For the corresponding results for large data, the story is quite different and the forms of the dependence of transport coefficients µ and κ on the density ρ ≡ v −1 and/or the temperature θ have strong influence on the analysis and the corresponding results obtained up to now can be classified as follows: • For positive constant transport coefficients, a general global existence result is obtained in [1,14,21,22,49], but since the upper and lower bounds on v and θ obtained there are not uniform with respect to the time variable, the corresponding results on the precise description of the large time behaviors of the global solutions constructed there are unknown. For such a problem, for the isentropic case, the global nonlinear stability of rarefaction waves were well-established in [35,36] and for the nonlinear stability of viscous shock waves with large initial perturbation, only partial progress was made in [47]. As for the full compressible Navier-Stokes equations (1), some Nishida-Smoller type results are obtained in [4,18,19,38,39] (For the corresponding original Nishida-Smoller type global existence result for one-dimensional ideal polytropic isentropic compressible Euler system, please refer to [37], whereas the non-isentropic case was analyzed by Liu [25] and also Temple [42]), while for the case of general adiabatic exponent γ > 1, the nonlinear stability of the non-vacuum constant states was solved only recently in [23] and the nonlinear stability of the superposition of rarefaction waves and viscous contact waves was obtained in [10], cf. also [44,45] for the corresponding results for the outflow problem. A key ingredient in all of these analysis is the explicit expression of the specific volume v(t, x) given in [22] and its decent localized version obtained in [12], which leads to the desired uniform pointwise a priori estimates on the specific volume v(t, x) which guarantee that no vacuum nor concentration of mass occur; • Much effort has been invested in generalizing the above approach to other cases and in particular to models satisfying (5). However, as pointed out in [15], this has proved to be challenging and temperature dependence of the viscosity µ has turned out to be especially problematic, but one has been able to incorporate various forms of density dependence in µ and also temperature dependence in κ, cf. [3,15,20,40,41] and the references cited therein. Even so, it is worth to pointing out that since the upper and lower bounds on v(t, x) and θ(t, x) obtained in [3,15,20,40,41] depends on time, the problem on the large time behaviors of the global solutions constructed there is unknown; • For the case when the transport coefficients depend on both density and temperature and satisfy (3), the only results available now are obtained in [24,46], where the nonlinear stability of non-vacuum constant states with large data are obtained.
Thus a natural question is: For the Cauchy problem (1)-(4), (2) with density and temperature dependent transport coefficients satisfying (3), can we obtain the global nonlinear stability of some elementary waves such as rarefaction waves, viscous shock waves, viscous contact waves, and some of their superpositions? Here "global nonlinear stability" means "the corresponding nonlinear stability result with large initial perturbation". And our main purpose here is to deduce the global nonlinear stability of rarefaction waves with large initial perturbation. Since our interest is to show the nonlinear stability of the expansion waves for (1), it is convenient to work with the equations for the entropy s and the absolute temperature θ. To this end, we get from (1) and (4) that and Recall that the gas constants A and R are normalized to be unity. In fact, for smooth solutions, equations (1 . In what follows, we will consider (1) 1 -(1) 2 -(6) with the initial data Here . Since we will focus on the expansion waves to (1), we assume that s + = s − =s in the rest of this paper. For expansion waves, the right hand side of (1) decays faster than each individual term on the left hand side. Therefore, the compressible Navier-Stokes equations (1) may be approximated, time-asymptotically, by the Riemann problem of the compressible Euler equations: with Riemann data We consider the case when the Riemann problem (9), (10) admits a unique global weak (rarefaction wave) solution ,s , and another of the third family, denoted by In other words, the unique weak solution V R ( x t ), U R ( x t ), S R ( x t ) to the Riemann problem (9), (10) is given by are determined by the following equations: To study the nonlinear stability of the expansion waves, as in [34], we first construct a smooth approximation to the above Riemann solution (11). For this purpose, let ω i (t, x) (i = 1, 3) be the unique global smooth solution to the following 474 BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO Cauchy problem: where q > 3 2 , K q = ( +∞ 0 (1 + y 2 ) −q dy) −1 , σ > 0 is a positive constant which will be specified later, and Then, by setting σ = δ = |v − − v + | + |u − − u + |, the smooth approximation of the rarefaction wave profile (V (t, x), U (t, x), S(t, x)) is constructed as follows: where t 0 > 0 is a sufficiently large but fixed positive constant which will be determined later, (V i (t, x), U i (t, x))(i = 1, 3) are defined by the following equations: and Θ(t, x) is defined by We now turn to state our main results in this paper, to this end, if we denote • The constants v ± , u ± , and θ ± do not depend on γ − 1; • The transport coefficients µ(v, θ) and κ(v, θ) satisfy (3) and there exist con- • There exists a sufficiently small positive constant ε 0 which depends only on N 03 , V , V , Θ, and Θ such that Then the Cauchy problem (1) 1 -(1) 2 -(6), (8) admits a unique global solution (v(t, x), u(t, x), s(t, x)) which satisfies hold for all (t, x) ∈ R + × R and some positive constant C ≥ 1 which depends only on N 03 , V , V , Θ, and Θ. Moreover, it holds that Remark 1. Some remarks concerning Theorem 1.1 are listed below: • The introduction of the parameter l is motivated by [4] and the main purpose is to control the possible growth of the solutions to the Cauchy problem (1) 1 -(1) 2 -(6), (8) generated by both the interactions of rarefaction waves from different families and the interactions between the rarefaction waves and the solution itself. It is worth to pointing out that when one considers the nonlinear stability of single rarefaction waves, it is unnecessary to introduce such a parameter and in such a case, the difficulty induced by the interactions of rarefaction waves from different families will not occur and due to the structure of the equations (1) 1 -(1) 2 -(6) and noticing that the nice properties of the smooth approximation (V (t, x), U (t, x), S(t, x)) of the rarefaction waves, especially the fact that (V xx (t, x), U xx (t, x)) L p (R) ∈ L 1 (R + ) for each p > 1 which is due to the way to construct the smooth approximation of the rarefaction waves, cf. (13), introduced first in [35], the difficulty involving the interactions between the rarefaction waves and the solution itself can be controlled suitably even for large rarefaction waves. Thus one can deduce a similar Nishida-Smoller type nonlinear stability result for single strong rarefaction waves.
and noticing that the assumptions we imposed in Theorem 1.1 tell us that -Both v 0 (x) and V (0, x) are bounded by some positive constants independent of γ − 1 from below and above, -(γ − 1)|s| is bounded by some constant independent of γ − 1, it is easy to see that there exists some positive constant C depending only on hold for i = 1, 2, 3. Thus a sufficient condition to guarantee that N 0i (i = 1, 2, 3) is bounded by some positive constant independent of γ − 1 is is bounded by some positive constant independent of γ − 1.
Before concluding this section, we outline the main idea used to yield the above result. As is usual in constructing global solutions with large-amplitude to nonlinear partial differential equations, the main difficulty lies in how to control the possible growth of the solutions induced by the nonlinearity of the equations under considerations. For our problem, in addition to the above difficulty, we need also to control the possible growth of the solutions to the Cauchy problem (1) 1 -(1) 2 -(6), (8) generated by the interactions of rarefaction waves from different families and the interaction between the solutions and the rarefaction waves. The key point is to get the desired uniform positive lower and upper bounds on the specific volume v(t, x) and the absolute temperature θ(t, x). Our main observations are the following: • Firstly, the constitutive relations (4) together with the equation (7) suggest that one can perform the desired energy type estimates based on the a priori assumption sup for some sufficiently small > 0 and some T > 0 which implies that Θ ≤ θ(t, x) ≤ Θ for all (t, x) ∈ [0, T ] × R; • Secondly, with the above estimate on θ(t, x) in hand, we then perform some energy type estimates by using the smallness of both such an and γ − 1 to control the possible growth of the solutions of to the Cauchy problem (1) 1 -(1) 2 -(6), (8) induced by the nonlinearity of the equations (1) 1 , (1) 2 , and (6) and by exploiting the largeness of the parameter l to control the possible growth generated by both the interactions of rarefaction waves from different families and the interactions between the rarefaction waves and the solution itself. These estimates together with the argument developed by Kanel' in [16] can lead to an estimate on the uniform lower and upper bounds on v(t, x) in terms of N 03 , V , V , Θ, and Θ; • Thirdly, a further energy type estimate can yield the following estimate on θ(t, x) − Θ(t, x) H 3 (R) in terms of the factor γ − 1, N 03 , V , V , Θ, and Θ for some positive constant C N 03 , V , V , Θ, Θ depending only on N 03 , V , V , Θ, and Θ. Such an estimate implies that is of the order √ γ − 1 and then a carefully designed continuation argument can close the whole analysis so that the local solution can be extended step by step to a global one provided that both γ − 1 and l −1 are assumed to be sufficiently small. This paper is arranged as follows: We will give some properties of the smooth approximation of the rarefaction wave solutions in Section 2. In Section 3, by using the largeness of l and the smallness of both θ(t, x) − Θ(t, x) H 3 (R) and γ − 1 > 0 to control the possible growth of the solution (v(t, x), u(t, x), s(t, x)) to the Cauchy problem (1) 1 -(1) 2 -(6), (8) caused by the nonlinearity of the Navier-Stokes equations (1) 1 -(1) 2 -(6) under our consideration and by both the interactions of rarefaction waves from different families and the interactions between the rarefaction waves and the solution itself, we deduce the desired energy type estimates. Finally, we extend the local solution step by step to a global one by combining the a priori estimates obtained in Section 3 with the continuation argument and give the proof of Theorem 1.1 in Section 4.
Notations. Throughout the rest of this paper, C or O(1) will be used to denote a generic positive constant which is independent of t, t 0 , δ, γ − 1, and x but may depends on N 03 , v ± , u ± , θ ± , V , V , Θ, and Θ and C i (·, ·)(i ∈ Z + ) stands for some generic constants depending only on the quantities listed in the parentheses. Note that all these constants may vary from line to line.
For two functions f (x) and g(x), f (x) ∼ g(x) as x → a means that there exists a generic positive constant C > 0 which is independent of t, t 0 , δ, γ −1, and x but may depend on N 03 , v ± , u ± , θ ± , V , V , Θ, and Θ such that B means that there is a generic positive constant C > 0 independent of t, t 0 , δ, γ − 1, and x such that B ≤ CB , while B ∼ B means that B B and B B. H l (R)(l ≥ 0) denotes the usual Sobolev space with standard norm · l , and · 0 = · will be used to denote the usual L 2 -norm.

2.
Preliminaries. In this section, we give some basic estimates and identities which will be used later. First, we list some properties of the global smooth functions (V (t, x), U (t, x), S(t, x), Θ(t, x)) constructed in (13) and (14).
To present some estimates on (V (t, x), U (t, x), S(t, x)), we first state some results on the Cauchy problem (13) in the following lemma Lemma 2.1. For each i ∈ {1, 3}, the Cauchy problem (13) admits a unique global smooth solution ω i (t, x) which satisfies the following properties: x t is the unique rarefaction wave solution of the corresponding Riemann problem of (1.19) 1 , i.e., Based on the results obtained in Lemma 2.1 and from (14) and (15), we can deduce that x)) constructed in (14) and (15) have the following properties: .

RAREFACTION WAVES FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS 479
It is obvious that V x (t) 2 L 2 is not integrable with respect to t. However we can get for any r > 0 and p > 1 that Remark 2. Recall that the quantities g(V, Θ), r(V, Θ), and q(V, Θ) represent the interaction of waves from different families. From the estimates obtained in (iii) of Lemma 2.2, it is easy to see that one can control g(V, Θ), r(V, Θ), and q(V, Θ) suitably if the parameter l = t0 δ introduced in (14) in the construction of a smooth approximation of the rarefaction wave profile is chosen sufficiently large.
we can deduce that (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) solves On the other hand, it is well-known that is a convex entropy to the compressible Navier-Stokes equations (1) around the smooth rarefaction wave profile (V (t, x), U (t, x), Θ(t, x), S(t, x)) which satisfies the following identity Moreover, The identity (22) plays an essential role in our analysis and we'd like to use several sentences here to explain our main idea to deduce our main result. Those terms appeared in J, especially the last two terms, represent the terms induced by the nonlinearity of the compressible Navier-Stokes equations (1), while the term in K measures the interaction of rarefaction waves from different families, and those terms in L reflect the interaction between the rarefaction waves and the solution itself. In the following sections, we will show how to control the corresponding terms related to J by the smallness of γ − 1 and the terms related to K and L by the largeness of the parameter l introduced in (14). 3. Energy estimates. To prove Theorem 1.1, we first define the following set of functions for which the solutions to the Cauchy problem (19), (20) will be sought: For the local solvability of the Cauchy problem (19), (20) in the above set of functions, one has Suppose that such a local solution (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) constructed in Lemma 3.1 has been extended to the time step t = T > t 1 and satisfies the a priori assumption for all x ∈ R, 0 ≤ t ≤ T , some sufficiently small positive constant > 0, and some generic positive constants M 1 , N 1 (without loss of generality, we may assume in the rest of this manuscript that M 1 ≥ 1, N 1 ≥ 1), what we want to do next is to deduce some energy type estimates in terms of (ϕ 0 , ψ 0 , φ0 √ γ−1 ) 3 , V , V , Θ, and Θ, but are independent of , M 1 , and N 1 . Before going to the details of the analysis, we'd like to mention again that our main idea here is to use the largeness of l and the smallness of both and γ − 1 to control the possible growth of the solution (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) constructed in Lemma 3.1 induced by the nonlinearity of the equations (19), the interaction of rarefaction waves from different families, and the interaction between the rarefaction waves and the solution (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) itself. Now we turn to perform the desired energy type estimates based on the a priori assumption (23). Before doing so, we first point out that although the precise expressions of the positive constants C(V , V , Θ, Θ) and C(M 1 ) appeared in the right hand side of the energy type estimates throughout the rest of this paper can indeed be given explicitly, to simplify the presentation, we will simply denote them by C(V , V , Θ, Θ) and C(M 1 ) even though they may vary from line to line.
Before performing the energy type estimates on the solution (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) defined on the strip Π T = [0, T ] × R, we first deduce from the as- and

BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO
With the above estimate in hand, we have by integrating the entropy identity (21) with respect to t and x over [ Here From the a priori assumption (23) and its consequence (25), (24), we have Here R j = C(V , V , Θ, Θ)|R j |(j = 1, 2, 3, 4, 5) and note that the positive constant C(V , V , Θ, Θ) in (26) is independent of M 1 and N 1 . Now we estimate R j (j = 1, 2, 3, 4, 5) term by term. In fact, from the a priori assumption (23), Cauchy-Schwarz's inequality, and Lemma 2.2, we have and Substituting the above estimates into (26), we can get that If we choose l ≥ l 1 > 1 sufficiently large such that then we can deduce by exploiting Gronwall's inequality in (27) that Now we turn to control the term (τ, x)dxdτ appeared in the right hand side of (29). To this end, we get from (22) that Integrating this identity with respect to t and x over [0, t] × R, we obtain Here Now we deal with R 6 , R 7 , R 8 , R 9 , and R 10 term by term. For this purpose, by applying the Cauchy-Schwarz inequality, we can get from Lemma 2.2 and the a priori assumption (23) that

486
BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO and As to R 10 , noticing that where K j (j = 1, 2, 3, 4) denote the corresponding terms in the right hand side of the above indentity.
To bound these K j (j = 1, 2, 3, 4), by exploiting Lemma 2.2 and the a priori assumption (23) again, we can get that K 1 and K 2 can be estimated as follows we have BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO and K 3,j (j = 1, 2, 3) can be estimated as in the following Inserting (34), (35), and (36) into (33), one can deduce that K 3 can be bounded as in the following As to K 4 , since we have from the smallness of that

BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO
Plugging (39) and (40) into (38), it yields that Inserting the estimates (31), (32), (37), and (41) into the identity for R 10 , one can get that Having obtained the estimates on R j (j = 6, 7, 8, 9, 10), if we choose γ − 1 > 0 and 0 < < 1 sufficiently small and l ≥ l 2 ≥ l 1 > 1 sufficiently large such that and one can obtain by inserting the estimates on R j (j = 6, 7, 8, 9, 10) into (30) that Now we turn to deal with the term Integrating this identity with respect to t and x over [0, t] × R, we can get that Here R i (i = 11, · · · , 16) denote those terms corresponding to the terms in the right hand side of the above identity.
To estimate |R i |, i = 11, . . . , 16 term by term, we have from the a priori assumption (23), Cauchy-Schwarz's inequality, and Lemma 2.2 that and Having obtained the above estimates, if we choose l ≥ l 3 ≥ l 2 > 1 suitably large such that then we can obtain the following result Now if γ − 1 is further assumed to be small enough such that we can conclude from (29), (44), and (46) that Here and in the rest of this manuscript, without loss of generality, we can assume that ψ 0 , ϕ 0 , φ0 Having obtained the estimate (48), we now turn to use Kanel's method [16] to deduce a uniform positive lower and upper bounds for v(t, x). To do so, we need to deduce the we can get from Lemma 2.2 that Thus the estimate (48) tells us that The above analysis yields the following result Now we use the estimate (49) to yield an estimate on the positive lower and upper bounds on the specific volume v(t, x). To this end, we first get from the a priori assumption (23) together with the estimates (25) and (24) that Secondly, since the constants v ± , u ± , and θ ± do not depend on γ − 1 and µ(v) is assumed to satisfy (16), one can easily deduce that there exist a positive constant C(Θ, Θ, v ± , u ± , θ ± ) depending only on Θ, Θ, v ± , u ± , and θ ± and a function µ(v) satisfying (50), (51), and (52) together with the estimate (49) imply With the estimate (53) in hand, if we setting we can deduce from the assumption (51) that there exist positive constants Moreover, one can get from the estimate (53) that Then we deduce from the above inequalities that there exists a generic positive constant C which may depend only on v ± , u ± , θ ± , V , V , Θ, Θ such that Furthermore, we can get from the above estimate and the estimate (49) that Now we turn to derive certain energy type estimates on (ϕ x (t, x), φ x (t, x), ψ x (t, x)). To this end, we need to yield an estimate on ψ x (t) 2 , which is the main content of the following lemma Proof. From (18) and (19) 2 , we have Integrating this identity with respect to t and x over [0, t] × R, we obtain Letting Then we collect the above estimates in (56) to deduce the desired estimate and this completes the proof of Lemma 3.5.
From (44), (46), (55), and (54), we deduce the following result Lemma 3.6. Under the assumptions listed in Lemma 3.4, we have that and The above analysis implies that under a priori assumption (23), the smallness assumption of γ − 1, > 0 and the largeness of l, there exists some positive constant C which may depend only on v ± , u ± , θ ± , V , V , Θ, Θ such that . Now we derive the second order energy estimates on (ϕ(t, x), ψ(t, x), φ(t, x)). To do so, for the corresponding estimates on ψ xx (t) 2 , we differentiate (19) 2 with respect to x once and multiply the final result by −ψ xxx (t, x) to get that Integrating this identity with respect to t and x over [0, t] × R, we have Letting I i = C(V , V , Θ, Θ)|I i | (i = 1, 2, 3) and by using Lemma 2.2, (57), and Cauchy-schwarz's inequality, we can bound I i (i = 1, 2, 3) term by term as follows BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO . Inserting the estimates of I i (1 ≤ i ≤ 3) into (58), we can deduce from (54), (25), and (24) that To deduce an estimate on φ xx (t) 2 , we have by differentiating (19) 3 with respect to x once and by multiplying the result by −φ xxx (t, x) to get that that Integrating the above identity with respect to t and x over [0, t] × R, we have and I i (i = 4, . . . , 11) can be controlled term by term as follows: where we have used the following estimate Plugging the above estimates on I i (i = 4, . . . , 11) into (60), one has from (54), (25), and (24) that To get an estimate on ϕ xx (t) 2 , we first differentiate (22) with respect to x once, multiply the result by µ(v, θ) ϕx v x , and then integrate the final identity with respect to t and x over [0, t] × R to deduce that Here To deduce an estimate on I 12 , due to and noticing BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO we can get from Lemma 2.2, (57), and Cauchy-Schwarz inequality that Similarly, I 13 and I 14 can be estimated as in the following As to I 15 , it is easy to see that BINGKANG HUANG, LUSHENG WANG AND QINGHUA XIAO D 1 can be estimated as For D i (i = 2, 3, 4, 5), under the assumption that and γ −1 are chosen sufficiently small, we can obtain and Thus if we assume further that γ − 1 is sufficiently small such that hold for some sufficiently large constant K 1 depending only on v ± , u ± , θ ± , V , V , Θ, and Θ, we can get from the above estimates on D i (i = 1, 2, 3, 4, 5) that I 15 can be estimated as Based on the above estimates, we finally get that where we used the following result . A suitable linear combination of (59), (61), and (63) yields the following result Lemma 3.7. Under the same condition listed in Lemma 3.4, if we further assume that γ − 1 is sufficiently small such that the assumption (62) holds, then we have Proof. In fact, by using the smallness of γ − 1, we can get by multiplying (63) by a sufficiently large positive number λ and by adding the resulting inequality with (59) and (61) that holds for some sufficiently large constant K 2 depending only on v ± , u ± , θ ± , V , V , Θ, and Θ, then we have . As a direct consequence of the results obtained in Lemmas 3.2-3.8, we have the following corollary.  Let (ϕ(t, x), ψ(t, x), φ(t, x)) be the local solution constructed in Lemma 3.1 which has been extended to the time step t = T ≥ t 1 and assume that (ϕ(t, x), ψ(t, x), φ(t, x)) satisfies the a priori assumption in (23), then if l is chosen sufficiently large and > 0 and γ − 1 > 0 are chosen sufficiently small such that 3 ≤ 1 hold for some positive constants C(V , V , Θ, Θ) and C(M 1 ) which depend only on V , V , Θ, Θ and M 1 respectively, 1 N 2 01 e K1N 2 01 ≤ 1 20 , K 2 (γ − 1) 1 2 N 2 02 e N 2 01 exp(K2N 2 01 ) ≤ 1 hold true for some positive constants C(M 1 ) depending only on M 1 and K 1 , K 2 which are sufficiently large but depend only on v ± , u ± , θ ± , V , V , Θ, and Θ, one can deduce that there exists a generic positive constant K 3 which depends only on V , V , Θ, and Θ such that    , such that the Cauchy problem (19), (20) admits a unique smooth solution (ϕ(t, x), ψ(t, x), φ(t, x), ξ(t, x)) ∈ X 3 0, t 1 ; V , V ; Θ, Θ which satisfies for all 0 ≤ t ≤ t 1 , x ∈ R and holds for all 0 ≤ t ≤ t 1 .

5.
Acknowledgments. Bingkang Huang was partially supported by "the Fundamental Research Funds for the Central Universities", Lusheng Wang was partially supported by two grants from the National Natural Science Foundation of China under contracts 11271160 and 11261160485, respectively, Qinghua Xiao was supported by two grants from the National Natural Science Foundation of China under contracts 11171340 and 11501556, respectively.