Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.


Introduction and the Main Result
We study the asymptotic behavior, as the viscosity and heat-conductivity go to zero, respectively, of solutions to the Cauchy problem for the Navier-Stokes equations for a one-dimensional compressible heat-conducting fluid (in Lagrangian coordinates): with (discontinuous) initial data ( )( ) ( )( ) where , u v ± ± and e ± are given constant states.The system (1.1), describing the motion of the fluid, is the conservation laws of mass, momentum and energy.
The asymptotic behavior of viscous flows, as the viscosity vanishes, is one of the important topics in the theory of compressible flows.It is expected that a general weak entropy solution to the Euler equations should be (strong) limit of solutions to the corresponding Navier-Stokes equations with same initial data as the viscosity and heat conductivity tend to zero, respectively.
For the one-dimensional compressible isentropic Navier-Stokes equations and the corresponding inviscid p-system ( ) 0, 0, the vanishing viscosity limit for the Cauchy problem has been studied by several researchers.In [1] Di Perna uses the method of compensated compactness and established almost everywhere convergence of admissible solutions ( ) of (1.4) to an admissible solution of (1.5), provided that ( ) is uniformly L ∞ bounded and v ε is uniform bounded away from zero.However, this uniform boundedness is difficult to verify in general, and the abstract analysis in [1] gets little information on the qualitative nature of the viscous solutions.In [2] Hoff and Liu investigate the inviscid limit problem for (1.4) in the case that the underlying inviscid flow is a single weak shock wave, and they show that solutions of the compressible Navier-Stokes equations with shock data exist and converge to the inviscid shocks, as viscosity vanishes, uniformly away from the shocks.
Based on [2] [3], Xin in [4] shows that the solution to the Cauchy problem for the system (1.4) with weak centered rarefaction wave data exists for all time and converges to the weak centered rarefaction wave solution of the corresponding Euler equations, as the viscosity tends to zero, uniformly away from the initial discontinuity.Moreover, for a given centered rarefaction wave to the Euler equations with finite strength, he constructs a viscous solution to the compressible Navier-Stokes system with initial data depending on the viscosity, such that the viscous solution approaches the centered rarefaction wave as the viscosity goes to zero at the rate 1 4 ln ε ε uniformly for all time away from 0 t = .In the vanishing viscosity limit, the Prandtl boundary layers (characteristic boundaries) are studied for the multidimensional linearized compressible Navier-Stokes equ-ations by using asymptotic analysis in [5] [6] [7], while the boundary layer stability in the case of non-characteristic boundaries and one spatial dimension is discussed in [8] [9].We mention that there is an extensive literature on the vanishing artificial viscosity limit for hyperbolic systems of conservation laws, see, for example, [1] [3] [10]- [19], also cf. the monographs [20] [21] [22] and the references therein.We also mention that the convergence of 1-d Broadwell model and the relaxation limit of a rate-type viscoelastic system to the isentropic Euler equations with centered rarefaction wave initial data are studied in [23] [24], respectively.And, in [25], the solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength had been proved exist globally in time, and moreover, as the viscosity and heat-conductivity coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.
However, in those paper, κ is generally dependent of ε , while in this pa- per, we will show the dissipation limit in the case that κ and ε are indepen- dent of each other.
Our aim in this paper is to study the relation between the solution ( ) with the same constant states ( ) with initial data ) where , u v ± ± and s ± are the constant states.The corresponding inviscid Euler equations read: We assume in this paper that the pressure p is a smooth function of its arguments satisfying Notice that the condition (1.14) assures the system (1.13) has characteristic speeds and there are two family of rarefaction waves for the Euler equations (1.13).For illustration, we describe only the 1-rarefaction waves, and thus assume The case for the 3-rarefaction waves can be dealt with similarly.
Suppose the end states ( ) can be connected by 1-rarefaction waves.

S. F. Cui
which is uniquely determined by the system (1.13) and the rarefaction wave initial data ( ) For the internal energy ( ) , e v θ , we assume ( ) For the sake of convenience, throughout this paper we denote In this paper, we prove that the solution of system (1.
, , and so does the other jumps, where To prove Theorem 1.1 and to overcome the difficulties induced by non-isentropy of the flow, we shall adapt and modify the arguments in [25], but we do not use the natural scaling argument, and we do not assume that We point out here that in view of Theorem 1.1, an initial jump discontinuity at 0 x = can be allowed in (1.2).The evolution of this jump discontinuity is an important aspect in our analysis.It has been shown in [28] that the discontinuity evolution follows a curve

[ ] [ ]
x u v = −  in x-t plane, and the jump discontinuity in , x v u and x θ decays exponentially in time, while the discontinuity in u and θ are smoothed out at positive time, see [28] for details.We shall exploit this fact in the proof of Theorem 1.1.
In Section 2 we reformulate the problem and give the proof of Theorem 1.1, while Section 3 is dedicated to the derivation of a priori estimates used in Section 2.
Throughout this paper, we use the following notation: .

Reformulation and the Proof of Theorem 1.1
In this section, we will reduce the proof of Theorem 1.1 to the nonlinear time-asymptotic stability analysis of rarefaction waves for the system (1.11) under non-smooth perturbations.First, we derive some necessary estimates on the rarefaction waves of the Euler equations (1.13) based on the inviscid Burgers equation, in particularly, we construct an explicit smooth 1-rarefaction wave which well approximates a given centered 1-rarefaction wave.We start with the Riemann problem for the Burgers equation: where To construct a smooth rarefaction wave solution of the Burgers equation which approximates the centered rarefaction wave, we set for 0 and for each 0 δ > , we solve the following initial value problem , and we define ( ) ( ) ( ) ( ) and due to Lemma 2.1, the following lemma holds for LEMMA 2.2.(S.JIANG [25]) The functions 2) For any 1 p ≤ ≤ ∞ , there is a constant ( ) Substituting the above decomposition into (1.1),(1.9) and (1.10), we obtain the system for the functions , , , ϕ ψ φ ξ : where ( ) ϕ ψ φ ξ and its derivatives are sufficiently smooth away from 0 x = but up to 0 x = and ( ) ( ) ϕ ψ φ goes to zero uniformly as 0 t → .This convergence then yields Theorem 1.1 due to Lemmas 2.1 and 2.2.LEMMA 2.3.(Hoff [28]) Suppose that ( ) is suitably small so that there exist two positive constants v and v with ( ) Then, there is a constant 0 T > , such that the Cauchy problem (2.9), (2.10) has a solution ( ) in the same function class as for ( ) Moreover, , , ϕ ψ φ satisfy 1) There exists a positive constant C, such that

Uniform a Priori Estimates
In this section we derive the key a priori estimates given in Proposition 2.4.First, we introduce the normalized entropy: , , , , , , where we have used the fact that An easy computation implies that η satisfies the equation: Employing (3.1), one has LEMMA3.1.Suppose that the assumptions of Proposition 2.4 hold.Then, where Recalling the definition of ( ) 0 , N t t and applying Lemma 2.2, for given , j R α can be estimated as follows.
which, by integrating with respect to x and t, leads to The terms on the right hand side of (3.11) can be bounded as follows (see [25] for detail), Substituting the above estimates into (3.11),we obtain (3.9).□ Similarly, we can bound the derivatives of φ as follows.
LEMMA 3.4Assume that the assumptions of Proposition 2.4 hold.Then, (   where the right hand side can be estimated as follows (see [25] for detail), p e v θ = and e denote the specific volume, the velocity, the S. F. Cui DOI: 10.4236/jamp.2018.6101802143 Journal of Applied Mathematics and Physics temperature, the pressure and the internal energy respectively, and , ε κ are the viscosity and heat conductivity coefficients, respectively.At infinity, the initial data x t of the corresponding inviscid Euler equations: uniformly as time goes to infinity, i.e., shall show that the Cauchy problem (2.9), (2.10) possesses a unique global solution ( t t .
LEMMA3.2.Suppose that the assumptions of Proposition 2.4 hold.Then Suppose that the assumptions of Proposition 2.4 hold.Then, .