{'fraction_non_alphanumeric': 0.05007122507122507, 'fraction_numerical': 0.019444444444444445, 'mean_word_length': 4.348952380952381, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The limit to rarefaction wave with vacuum for 1D compressible fluids with temperature-dependent viscosities

In this paper we study the zero dissipation limit of the one-dimensional full compressible Navier-Stokes(CNS) equations with temperature-dependent viscosity and heat-conduction coefficient. It is proved that given a rarefaction wave with one-side vacuum state to the full compressible Euler equations, we can construct a sequence of solutions to the full CNS equations which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence rate is obtained. The main difficulty in our proof lies in the degeneracies of the density, the temperature and the temperature-dependent viscosities at the vacuum region in the zero dissipation limit.

Thus the two i-Riemann invariants (i = 1, 3) can be defined by (cf. [20]) such that ∇ (ρ,u,S) Σ (j) i (ρ, u, S) · r i (ρ, u, S) ≡ 0, i = 1, 3, j = 1, 2. The study of the limiting process of viscous flows when the viscosity tends to zero is one of the important problems in the theory of the compressible fluid. When the solution of the inviscid flow is smooth, the zero dissipation limit can be solved by classical scaling method. However, the inviscid compressible flow contains singularities such as shock, contact discontinuity and the vacuum in general. Therefore, determining how to justify the zero dissipation limit to the Euler equations with basic wave patterns and the vacuum states is a natural and difficult problem.
There have been many results on the zero dissipation limit of the compressible fluid with basic wave patterns without vacuum. For the system of the hyperbolic conservation laws with artificial viscosity Goodman-Xin [4] first verified the viscous limit for piecewise smooth solutions separated by non-interacting shock waves using a matched asymptotic expansion method. Later Yu [25] proved it for the corresponding hyperbolic conservation laws with both shock and initial layers. In 2005, important progress made by Bianchini-Bressan [1] justifies the vanishing viscosity limit in BV space even though the problem is still unsolved for the physical system such as the compressible Navier-Stokes equations.
For the isentropic compressible Navier-Stokes equations with constant viscosity where the conservation of energy in (1.1) is neglected, Hoff-Liu [5] first proved the vanishing viscosity limit for piecewise constant shock even with initial layer. Later Xin [23] obtained the zero dissipation limit for rarefaction waves without vacuum for both rarefaction wave data and well-prepared smooth data. Then Wang [21] generalized the result of Goodmann-Xin [4] to the isentropic Navier-Stokes equations. Recently, Chen-Perepelitsa [3] proved the vanishing viscosity to the compressible Euler equations for the isentropic compressible Navier-Stokes equations (1.1) with constant viscosity by compensated compactness method for the case that the far field of the initial values of Euler system has no vacuums.
For the full Navier-Stokes equations (1.1) with constant viscosity, there are also many results on the zero dissipation limit to the corresponding full Euler system with basic wave patterns without vacuum. We refer to Jiang-Ni-Sun [11] and Xin-Zeng [24] for the rarefaction wave, Wang [22] for the shock wave, Ma [17] for the contact discontinuity, Huang-Wang-Yang [9] and Huang-Jiang-Wang [7] for the superposition of two rarefaction waves and a contact discontinuity, Huang-Wang-Yang [10] for the superposition of one shock and one rarefaction wave and Zhang-Pan-Wang-Tan [26] for the superposition of two shock waves with the initial layer. Recently, Huang-Wang-Wang-Yang [8] succeed in justifies the vanishing viscosity limit of compressible Navier-Stokes equations in the setting of Riemann solutions for the superposition of shock wave, rarefaction wave and contact discontinuity.
It is well-known that the vacuum states are generic in inviscid compressible Euler equations (1.5) since the vacuum states may occur in the Riemann solutions instantaneously as t > 0 even if the initial Riemann data are non-vacuum states on both sides at t = 0. Therefore, vacuum states are important physical states in gas dynamics and often yield degeneracies and certain singularities in the physical system, which cause some essential analytical difficulties. For example, the velocity can not even be defined in the vacuum region. In the setting of Riemann solutions, as pointed out by , among the three elementary hyperbolic waves, i.e., shock and rarefaction waves and contact discontinuities, to the one-dimensional isentropic compressible Euler equations (1.5), only the rarefaction wave can be connected to the vacuum states. There are some mathematical results on the time-asymptotic stability and the vanishing viscosity limit to the rarefaction wave with the vacuum. Perepelitsa [19] consider the timeasymptotic stability of solutions to 1-d isentropic compressible Navier-Stokes equations with fixed and constant viscosity toward rarefaction waves connected to vacuum in Lagrangian coordinate. Then Jiu-Wang-Xin [12] study the large time asymptotic behavior toward rarefaction wave with vacuum for solution to the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, which can be viewed as the compressible Navier-Stokes equations (1.1) with fixed ǫ = 1 in isentropic regime. Note that in isentropic regime, there is no energy equation (1.1) 3 and the viscosity coefficient in the momentum equation (1.1) 2 transfer to depend on the density. More recently, Huang-Li-Wang [6] justified the vanishing viscosity limit of one-dimensional isentropic compressible Navier-Stokes equations with constant viscosity to the rarefaction wave with one-side vacuum state to the corresponding compressible Euler equations. However, for the full compressible Navier-Stokes equations (1.1) with temperaturedependent viscosities, as far as we know, there is no any result on the zero dissipation limit to the rarefaction wave with the vacuum due to various difficulties mentioned below. Now we give a description of the rarefaction wave connected to the vacuum to the full compressible Euler equations (1.5); see also the reference [20]. For definiteness, 3-rarefaction wave will be considered. If we investigate the compressible Euler system (1.5) with the Riemann initial data where the left side is the vacuum state and ρ + > 0, u + , θ + > 0 are prescribed constants on the right state, then the Riemann problem (1.5), (1.7) admits a 3−rarefaction wave connected to the vacuum on the left hand side. By the fact that along the 3−rarefaction wave curve, 3−Riemann invariant Σ 3 (ρ + , u + , θ + ) being the speed of the gas coming into the vacuum from the 3-rarefaction wave. On the other hand, in the vacuum region, the absolute temperature θ also becomes zero due to the state equation (1.3), that is, θ = ρ γ−1 e S and the fact that the entropy S keeps constant along the 3-rarefaction wave. Correspondingly, the main difficulty of the present paper lies in how to deal with the degeneracies of the temperature-dependent viscosities in the vacuum region in the zero dissipation limit process.
As described above, the 3−rarefaction wave connecting the vacuum state ρ = 0 to (ρ + , u + , θ + ) is the self-similar solution (ρ r 3 , u r 3 , θ r 3 )(ξ), (ξ = x t ) of (1.5) defined by (1.9) and Σ (2) 3 = S r 3 = S + := −(γ − 1) log ρ + + log θ + . (1.10) Thus one can define the momentum m r 3 and the total internal energy n r 3 of 3-rarefaction wave by and In the present paper, we want to construct a sequence of global-in-time solutions (ρ ǫ , m ǫ := ρ ǫ u ǫ , n ǫ := ρ ǫ θ ǫ )(x, t) to the compressible Navier-Stokes equations (1.1) with temperaturedependent viscosities, which converge to the 3-rarefaction wave (ρ r 3 , m r 3 , n r 3 )(x/t) with vacuum defined above as ǫ tends to zero. The effects of initial layers will be ignored by choosing the well-prepared initial data (3.1) depending on the viscosity for the Navier-Stokes equations. As mentioned before, the main novelty and difficulty here is determining how to control the degeneracies caused by the vacuum in the rarefaction wave. To overcome this difficulty, we first cut off the 3-rarefaction wave with vacuum along the rarefaction wave curve suitably (ν is the cut-off parameter and the details can be seen in Section 2), and use the fact that the viscosity ǫ can control the degeneracies caused by the vacuum in rarefaction waves by choosing suitably ν = ν(ǫ). In fact, we choose ν = ǫ a | ln ǫ| with a defined in (1.14) in the present paper. The other observation is that we can carry out the energy estimates under the a priori assumptions (3.10)-(3.11) such that the perturbation is suitably small in L ∞ (R) norm with some decay rate with respect to ǫ.
On the other hand, compared with the previous works [6] for the isentropic Naiver-Stokes equations with the constant viscosity case, some new difficulties occur for the full Navier-Stokes equations (1.1) with temperature-dependent viscosities considered in the present paper. Firstly, in order to overcome the difficulties caused by the non-isentropic regime, the relative entropyentropy flux pair (η, q) defined in (3.17) is used as in [16]. Secondly, since in the vacuum region, the temperature becomes zero due to the entropy keeps constant by the structure of rarefaction wave. Therefore, not only the density but also the temperature cause the degeneracies in the vacuum region and so the temperature-dependent viscosities do. Thus, to deal with the terms for the temperature-dependent viscosities, such as (3.35), becomes subtle. In fact, the derivative estimates of the perturbation of the density depends on the second order derivative estimates of velocity with some degenerate coefficients (see (3.50)), which is quite different from the constant viscosity case in [6] and [14]. By choosing the convergence rate a suitably as in (1.14) and then the parameters ν, δ as in (3.12) closes the a priori estimates and yields the desired result. Now we state our main result as follows.
(2) As the viscosity ǫ → 0, (ρ ǫ , m ǫ , n ǫ )(x, t) converges to (ρ r 3 , m r 3 , n r 3 )(x/t) pointwisely except the original point (0, 0). Furthermore, for any given positive constant l, there exists a constant C l > 0, independent of ǫ, such that with the positive constant a given by . (1.14) Remark 1.2. From (1.13) and (1.14), one can see that the decay rate ǫ a | ln ǫ| in Theorem 1.1 decreases monotonically with respect to α, which is consistent to the observation that the viscosity becomes weaker as α becomes larger due to the vacuum in the rarefaction wave, although the convergence rates in (1.13) and (1.14) may not be optimal.
The rest of the paper is organized as follows. In section 2, we construct a smooth 3-rarefaction wave profile which approximates the cut-off rarefaction wave for the Euler equations based on the inviscid Burgers equation. Then the global-in-time solution to CNS (1.1) is obtained around the smooth 3-rarefaction wave profile and finally the proof the Theorem 1.1 is shown in Section 3.
Throughout this paper, H k (R), k = 0, 1, 2, ..., denotes the l-th order Sobolev space with its norm while L 2 (dz) means the L 2 integral over R with respect to the Lebesgue measure dz, and z = x or y. For simplicity, we also write C as generic positive constants which are independent of time t or τ and viscosity ǫ unless otherwise stated.

Rarefaction waves
Since there is no exact rarefaction wave profile for the Navier-Stokes equations (1.1), the following approximate rarefaction wave profile satisfying the Euler equations was motivated by Matsumura-Nishihara [18] and Xin [23].
Consider the Riemann problem for the inviscid Burgers equation: Motivated by [18,23], the approximate rarefaction wave to the compressible Navier-Stokes equations (1.1) can be constructed by the Burgers equation where δ > 0 is a small parameter to be determined and the hyperbolic tangent function tanh x = e x −e −x e x +e −x . In fact, we choose δ = ǫ a in (3.12) with a given by (1.14). Note that the solution w r δ (t, x) of the problem (2.3) can be given explicitly by And w r δ (t, x) has the following properties: (2) The following estimates hold for all t > 0, δ > 0 and p∈ [1, ∞]: Note that Lemma 2.1 is a little different from the one in [23] as stated in [6]. For the detailed proof of Lemma 2.1, one can refer to [23] and [6] and we omit it here for brevity.
In what follows, the analysis is always carried out under the a priori assumptions where a is given by (1.14), [0, τ 1 (ǫ)] is the time interval in which the solution exists and τ 1 (ǫ) may depend on ǫ. Take ν = ǫ a | ln ǫ|, δ = ǫ a , (3.12) in the sequel. Then it follows that ν ≥ Cǫ a with C ≥ max{2, 2e −S Under the a priori assumption (3.10), one can get In fact, if ǫ ≪ 1, then one has Similarly, note thatθ =ρ γ−1 eS ≥ ν γ−1 eS by the definition of the rarefaction wave profile defined in (2.10), it holds that Since the proof for the local existence of the solution to (3.4) − (3.6) is standard, we omit it for brevity. Note that in order to get the convergence rate of the local solution with respect to ǫ as in (3.10), the local existence time interval, denoted by [0, τ 0 ] where τ 0 may depend on ǫ, that is, τ 0 = τ 0 (ǫ). The next step for the proof of Theorem 3.1 is to extend the local solution to the global solution in [0, ∞) for small but fixed viscosity coefficient and heat conduction coefficient ǫ. To do so, it is sufficient to show the following a priori estimates for fixed ǫ with 0 < ǫ ≪ 1.

15)
and Proof of Lemma 3.2: The proof of Lemma 3.2 consists of the following steps.
About the convergence of the momentum, we have ≤ C l ǫ a + δ| ln δ| + ν ≤ C l ǫ a | ln ǫ|.
About the convergence of the total energy, we can get where we have used the fact that sup τ ∈[0,+∞) Then the proof of Theorem 1.1 is completed.