Fluid Dynamic Limit to the Riemann Solutions of Euler Equations: I. Superposition of rarefaction waves and contact discontinuity

Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.

limit of the compressible Navier-Stokes system for viscous and heat conductive fluid in the Lagrangian coordinates: where the functions v(t, x) > 0, u(t, x), θ(t, x) > 0 represent the specific volume, velocity and the absolute temperature of the gas respectively. And p = p(v, θ) is the pressure, e = e(v, θ) is the internal energy, ε > 0 is the viscosity coefficient, κ > 0 is the coefficient of the heat conductivity. Here, both ε and κ are taken as positive constants. And we consider the perfect gas where with s denoting the entropy of the gas and A, R > 0 , γ > 1 being the gas parameters.
Formally, as the coefficients κ and ε tend to zero, the limiting system of (1.1) is the compressible Euler equations (1. 3) The study of this limiting process of viscous flows when the viscosity and heat conductivity coefficients tend 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 usually contains discontinuities, such as shock waves and contact discontinuities. Therefore, how to justify the zero dissipation limit to the Euler equations with basic wave patterns is a natural and difficult problem.
Keeping in mind that the Navier-Stokes equations can be derived from the Boltzmann equation through the Chapman-Enskog expansion when the Knudsen number is close to zero, we assume the following condition on the viscosity constant ε and the heat conductivity coefficient κ in the system (1.1), cf. also [17]: Now we briefly review some recent results on the zero dissipation limit of the compressible fluid with basic wave patterns. For the hyperbolic conservation laws with artificial viscosity u t + f (u) x = εu xx , Goodman-Xin [9] verified the viscous limit for piecewise smooth solutions separated by non-interacting shock waves using a matched asymptotic expansion method. For the compressible isentropic Navier-Stokes equations, Hoff-Liu [12] first proved the vanishing viscosity limit for piecewise constant solutions separated by noninteracting shocks even with initial layer. Later Xin [30] obtained the zero dissipation limit for rarefaction waves and Wang [28] generalized the result of Goodmann-Xin [9] to the isentropic Navier-Stokes equations.
For the inviscid limit of the full compressible Navier-Stokes equations (1.1), Jiang-Ni-Sun [17] justified the zero dissipation limit of the system (1.1) for centered rarefaction waves. Wang [29] proved the zero dissipation limit of the system (1.1) for piecewise smooth solutions separated by shocks using the matched asymptotic expansion method introduced in [9]. Recently, Xin-Zeng [31] considered the zero dissipation limit of the system (1.1) for single rarefaction wave with well prepared initial data and obtained a uniform decay rate in terms of the dissipation coefficients. And Ma [22] obtained the zero dissipation limit of a single strong contact discontinuity in any fixed time interval with a decay rate.
However, to our knowledge, so far there is no result on the zero dissipation limit of the system (1.1) for superposition of different types of basic wave patterns. In the first part of this paper, we investigate the fluid dynamic limit of the compressible Navier-Stokes equations when the corresponding Euler equations have the Riemann solution as a superposition of two rarefaction waves and a contact discontinuity. For this, we need to study the interaction between the rarefaction waves and contact discontinuity.
In the second part of the paper, we study the hydrodynamic limit of the Boltzmann equation [2] with slab symmetry where ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ R 3 , f (t, x, ξ) is the density distribution function of particles at time t with location x and velocity ξ, and ε > 0 is called the Knudsen number which is proportional to the mean free path. Remark that the notation ε here is same as the viscosity of the compressible Navier-Stokes equations (1.1), but it has different physical meanings from (1.1) in different equations and related contexts. For monatomic gas, the rotational invariance of the particles leads to the following bilinear form for the collision operator where ξ ′ , ξ ′ * are the velocities after an elastic collision of two particles with velocities ξ, ξ * before the collision. Here,θ is the angle between the relative velocity ξ − ξ * and the unit vector Γ in S 2 + = {Γ ∈ S 2 : (ξ − ξ * ) · Γ ≥ 0}. The conservation of momentum and energy gives the following relation between the velocities before and after collision: In this paper, we consider the Boltzmann equation for two basic models, that is, the hard sphere model and the hard potential including Maxwellian molecules under the assumption of angular cut-off. For this, we assume that the collision kernel B(|ξ − ξ * |,θ) takes one of the following two forms, B(|ξ − ξ * |,θ) = |(ξ − ξ * , Γ)| = |ξ − ξ * | cosθ, and B(|ξ − ξ * |,θ) = |ξ − ξ * | n−5 n−1 b(θ), b(θ) ∈ L 1 ([0, π]), n ≥ 5. Here, n is the index in the potential of inverse power law which is proportional to r 1−n with r being the distance between two concerned particles.

FEIMIN HUANG, YI WANG AND TONG YANG
Formally, when the Knudsen number ε tends to zero, the limit of the Boltzmann equation (1.5) is the classical system of Euler equations x, ξ)dξ, i = 1, 2, 3, (1.7) Here, ρ is the density, u = (u 1 , u 2 , u 3 ) is the macroscopic velocity, E is the internal energy of the gas, and p = Rρθ with R being the gas constant is the pressure. Note that the temperature θ is related to the internal energy by E = 3 2 Rθ, and ϕ i (ξ)(i = 0, 1, 2, 3, 4) are the collision invariants given by How to justify the above limit, that is, the Euler equation (1.6) from Boltzmann equation (1.5) when Knudsen number ε tends to zero is an open problem going way back to the time of Maxwell. For this, Hilbert introduced the famous Hilbert expansion to show formally that the first order approximation of the Boltzmann equation gives the Euler equations. On the other hand, it is important to verify this limit process rigorously in mathematics. For the case when the Euler equation has smooth solutions, the zero Knudsen number limit of the Boltzmann equation has been studied even in the case with an initial layer, cf. Ukai-Asano [26], Caflish [3], Lachowicz [18] and Nishida [24] etc. However, as is well-known, solutions of the Euler equations (1.6) in general develop singularities, such as shock waves and contact discontinuities. Therefore, how to verify the fluid limit from Boltzmann equation to the Euler equations with basic wave patterns becomes an natural problem. In this direction, Yu [32] showed that when the solution of the Euler equations (1.6) contains only non-interacting shocks, there exists a sequence of solutions to the Boltzmann equation that converge to a local Maxwellian defined by the solution of the Euler equations (1.6) uniformly away from the shock in any fixed time interval. In this work, the inner and outer expansions developed by Goodman-Xin [9] for conservation laws and the Hilbert expansion were skillfully and cleverly used. Recently, Huang-Wang-Yang [15] proved the fluid dynamic limit of the Boltzmann equation to the Euler equations for a single contact discontinuity where the uniform decay rate was also obtained. And Xin-Zeng [31] proved the fluid dynamic limit of the compressible Navier-Stokes equations and Boltzmann equation to the Euler equations with non-interacting rarefaction waves. About the detailed introductions of the Boltzmann equation and its hydrodynamic limit, see the books [4], [7] etc.
In this paper, we will study the hydrodynamic limit of the Boltzmann equation when the corresponding Euler equations have a Riemann solution as a superposition of two rarefaction waves and a contact discontinuity. More precisely, given a Riemann solution of the Euler equations (1.6) with superposition of two rarefaction waves and a contact discontinuity, we will show that there exists a family of solutions to the Boltzmann equation that converge to a local Maxwellian defined by the Euler solution uniformly away from the contact discontinuity for strictly positive time as ε → 0. Moreover, a uniform convergence rate in ε is also given.
As mentioned above for the compressible Navier-Stokes equations, we also need to study the detailed wave interactions through this limiting process.
For later use, we now briefly present the micro-macro decomposition around the local Maxwellian defined by the solution to the Boltzmann equation, cf. [19] and [21]. For a solution f (t, x, ξ) of the Boltzmann equation (1.5), set where the local Maxwellian M(t, x, ξ) = M [ρ,u,θ] (ξ) represents the macroscopic (fluid) component of the solution, which is naturally defined by the five conserved quantities, i.e., the mass density ρ(t, x), the momentum ρu(t, x), and the total energy And G(t, x, ξ) being the difference between the solution and the above local Maxwellian represents the microscopic (non-fluid) component. For convenience, we denote the inner product of h and g in L 2 ξ (R 3 ) with respect to a given MaxwellianM by: IfM is the local Maxwellian M defined in (1.8), with respect to the corresponding inner product, the macroscopic space is spanned by the following five pairwise orthogonal base In the following, ifM is the local Maxwellian M, we just use the simplified notation ·, · to denote the inner product ·, · M . The macroscopic projection P 0 and microscopic projection P 1 can be defined as follows h, χ j χ j , The projections P 0 and P 1 are orthogonal and satisfy P 0 P 0 = P 0 , P 1 P 1 = P 1 , P 0 P 1 = P 1 P 0 = 0.

FEIMIN HUANG, YI WANG AND TONG YANG
Note that a function h(ξ) is called microscopic or non-fluid if where ϕ i (ξ) is the collision invariants. Under the above micro-macro decomposition, the solution f (t, x, ξ) of the Boltzmann equation (1.5) satisfies By multiplying the equation (1.9) by the collision invariants ϕ i (ξ)(i = 0, 1, 2, 3, 4) and integrating the resulting equations with respect to ξ over R 3 , one has the following fluid-type system for the fluid components: (1.10) Note that the above fluid-type system is not closed and one more equation for the non-fluid component G is needed and it can be obtained by applying the projection operator P 1 to the equation (1.9): Here L M is the linearized collision operator of Q(f, f ) with respect to the local Maxwellian M: . Note that the null space N of L M is spanned by the macroscopic variables: Furthermore, there exists a positive constant σ 0 > 0 such that for any function h(ξ) ∈ N ⊥ , cf. [10], is the collision frequency. For the hard sphere model and the hard potential including Maxwellian molecules with angular cut-off, the collision frequency ν(|ξ|) has the following property for some positive constants ν 0 , c and 0 ≤ κ 0 ≤ 1.
Consequently, the linearized collision operator L M is a dissipative operator on L 2 (R 3 ), and its inverse L −1 M exists in N ⊥ . It follows from (1.11) that (1.13) Plugging the equation (1.12) into (1.10) gives (1.14) where the viscosity coefficient µ(θ) > 0 and the heat conductivity coefficient λ(θ) > 0 are smooth functions of the temperature θ. Here, we normalize the gas constant R to be 2 3 so that E = θ and p = 2 3 ρθ. The explicit formula of µ(θ) and λ(θ) can be found for example in [5], we omit it here for brevity.
Since the problem considered in this paper is one dimensional in the space variable x ∈ R, in the macroscopic level, it is more convenient to rewrite the equation (1.5) and the system (1.6) in the Lagrangian coordinates as in the study of conservation laws. That is, set the coordinate transformation: x ⇒ x 0 ρ(t, y)dy, t ⇒ t.
We will still denote the Lagrangian coordinates by (t, x) for simplicity of notation. Then (1.5) and (1.6) in the Lagrangian coordinates become, respectively, and (1.16) Also, (1.10)-(1.14) take the form (1.21) With the above preparation, the main results in this paper for both the compressible Navier-Stokes equations and the Boltzmann equation will be given in the next section. And the proof of the zero dissipation limit for the compressible Navier-Stokes equations will be given in Section 3 while the proof of hydrodynamic limit for the Boltzmann equation will be given in the last section.

Compressible Navier-Stokes equations.
It is well known that for the Euler equations, there are three basic wave patterns, shock, rarefaction wave and contact discontinuity. And the Riemann solution to the Euler equations has a basic wave pattern consisting the superposition of these three waves with the contact discontinuity in the middle. For later use, let us firstly recall the wave curves for the two types of basic waves studied in this paper.
Given the right end state (v + , u + , θ + ), the following wave curves in the phase space (v, u, θ) are defined with v > 0 and θ > 0 for the Euler equations.
then the following Riemann problem of the Euler system (1.3) with Riemann initial data x > 0 admits a single contact discontinuity solution x > 0, t > 0. (2.3) As in [14], the viscous version of the above contact discontinuity, called viscous contact wave (V CD , U CD , Θ CD )(t, x), can be defined as follows. Since we expect that P CD ≈ p + = p − , and |U CD | ≪ 1, the leading order of the energy equation ( Thus, we can get the following nonlinear diffusion equation which has a unique self-similar solutionΘ(t, . Now the viscous contact wave (V CD , U CD , Θ CD )(t, x) can be defined by , Here, it is straightforward to check that the viscous contact wave defined in (2.4) satisfies as |x| → +∞, where δ CD = |θ + − θ − | represents the strength of the viscous contact wave and C 0 is a positive generic constant. Note that in the above definition, the higher order term ε[Rγ−ν(γ−1)] γp+Θ t is used in Θ CD (t, x) so that the viscous contact wave (V CD , U CD , Θ CD )(t, x) satisfies the momentum equation exactly. Precisely, (V CD , U CD , Θ CD )(t, x) satisfies the system where P CD = RΘ CD V CD and the error term Q CD has the property that is different from the one used in [14] and [16]. Here, this ansatz is chosen such that the mass equation and the momentum equation are satisfied exactly while the error term occurs only in the energy equation. However, note that the approximate energy equation that the viscous contact wave satisfies is not in the conservative form.

Rarefaction waves.
We now turn to the rarefaction waves. Since there is no exact rarefaction wave profile for either the Navier-Stokes equations or the Boltzmann equation, the following approximate rarefaction wave profile satisfying the Euler equations was motivated by [23] and [30]. For the completeness of the presentation, we include its definition and the properties in this subsection.
, then there exists a i-rarefaction wave (v ri , u ri , θ ri )(x/t) which is a global solution of the following Riemann problem (2.8)

Consider the following inviscid Burgers equation with Riemann data
If w − < w + , then the above Riemann problem admits a rarefaction wave solution (2.10) Obviously, we have the following Lemma, Lemma 2.1. For any shift t 0 > 0 in the time variable, we have where C is a positive constant depending only on w ± .
Remark that Lemma 2.1 plays an important role in the wave interaction estimates for the rarefaction waves.
As in [30], the approximate rarefaction wave (V R , U R , Θ R )(t, x) to the problem (1.1) can be constructed by the solution of the Burgers equation where σ > 0 is a small parameter to be determined. Note that the solution w r σ (t, x) of the problem (2.11) is given by And w r σ (t, x) has the following properties: Then the smooth approximate rarefaction wave profile denoted by where t 0 is the shift used to control the interaction between waves in different families with the property that t 0 → 0 as ε → 0. In the following, we choose where P Ri = p(V Ri , Θ Ri ). By Lemmas 2.1 and 2.2, the properties on the rarefaction waves can be summarized as follows.
constructed in (2.12) have the following properties: (2) For any 1 ≤ p ≤ +∞, the following estimates holds, where the positive constant C only depends on p and the wave strength; (4) There exist positive constants C and σ 0 such that for σ ∈ (0, σ 0 ) and t, t 0 > 0, 2.1.3. Superposition of rarefaction waves and contact discontinuity. In this subsection, we will define the solution profile that consists of the superposition of two rarefaction waves and a contact discontinuity.
So the wave pattern (V ,Ū ,Θ)(t, x) consisting of 1-rarefaction wave, 2-contact discontinuity and 3-rarefaction wave that solves the corresponding Riemann problem of the Euler system (1.3) can be defined by Correspondingly, the approximate wave pattern (V, U, Θ)(t, x) of the compressible Navier-Stokes equations can be defined by Thus, from the construction of the contact wave and Lemma 2.3, we have the following relation between the approximate wave pattern (V, U, Θ)(t, x) of the compressible Navier-Stokes equations and the exact inviscid wave pattern (V ,Ū ,Θ)(t, x) to the Euler equations with t 0 = ε 1 5 and σ = ε Moreover, (V, U, Θ)(t, x) satisfies the following system where P = p(V, Θ), and (2.19) Similarly, we have (2.20) Here Q 11 and Q 21 represent the interactions coming from different wave patterns, Q 12 and Q 22 represent the error terms coming from the approximate rarefaction wave profiles, and Q CD is the error term defined in (2.7) due to the viscous contact wave.
Firstly, we estimate the interaction terms Q 11 and Q 21 by dividing the whole domain Ω = {(t, x)|(t, x) ∈ R + × R} into three regions: . Now from Lemma 2.3, we have the following estimates in each section: Hence, in summary, we have for some positive constants C. Now we consider the system (1.1) with the initial values Introduce the following scaled variables (2.23) In the following, we will use the notations (v, u, θ)(τ, y) and (V, U, Θ)(τ, y) for the unknown functions and the approximate wave profiles in the scaled variables. Set the perturbation around the composite wave pattern (V, U, Θ)(τ, y) by Then the perturbation (φ, ψ, ζ)(τ, y) satisfies the system (2.24) And this system will be studied in Section 3.

2.1.4.
Main result to the compressible Navier-Stokes equations. We are now ready to state the main result on the compressible Navier-Stokes equations as follows.
, and the positive constant C h depends only on h but is independent of ε.
Remark 2. Theorem 2.4 shows that, away from the initial time t = 0 and the contact discontinuity located at x = 0 with the expansion rate x 2 ε(1+t) , for the viscosity coefficient ε < ε 0 , there exists a unique global solution (v ε , u ε , θ ε )(t, x) of the compressible Navier-Stokes equations (1.1) which tends to the Riemann solution (V ,Ū ,Θ)(t, x) consisting of two rarefaction waves and a contact discontinuity when ε → 0 and κ = O(ε) → 0. Moreover, a uniform convergence rate ε 1 5 holds on the set Σ h for any h > 0.
Remark 3. Theorem 2.4 holds uniformly when (t, x) ∈ Σ h for any fixed h > 0 if the contact wave strength δ CD and the viscosity coefficient ε are suitably small. However, if we restrict the problem to a set Σ h ∩ {t ≤ T } for any fixed T > 0, then we do not need to impose the smallness condition on the contact wave strength δ CD because one can apply the Gronwall inequality to get an estimate depending on time T rather than the uniform estimate in time.

Boltzmann equation.
We now turn to the Boltzmann equation. Similarly, we also define individual wave pattern, and then the superposition and finally state the main result in this subsection.

FEIMIN HUANG, YI WANG AND TONG YANG
Then the leading order of the energy equation (1.21) 4 is By using the mass equation (1.21) 1 and v ≈ Rθ p+ , we obtain the following nonlinear diffusion equation with the following boundary condi- with some positive constant c depending only on θ ± . Now the contact wave (V CD , U CD , Θ CD )(t, x) can be defined by (2.32) Note that the contact wave (V CD , U CD , Θ CD )(t, x) satisfies the following system   .32) is different from the one used in [16]. Here, this ansatz is chosen such that the momentum equation is satisfied with a higher order error term. This is also different from the compressible Navier-Stokes equations where the ansatz satisfies the momentum equation exactly. But similar to the compressible Navier-Stokes cases, the approximate energy equation that the viscous contact wave satisfies is not in the conservative form.
(2.36) Therefore, given in Lemma 2.3 will also be used later.

42) and
Here, Q 11 and Q 21 represent the interaction of waves in different families, Q 12 and Q 22 represent the error terms coming from the approximate rarefaction wave profiles, and Q CD i (i = 1, 2) are the error terms defined in (2.34) and (2.35) due to the viscous contact wave.
Similar to the compressible Navier-Stokes equations case, for the interaction terms, we have for some positive constants C.
We now reformulate the system by introducing a scaling for the independent variables. Set y = x ε , τ = t ε as in the previous section for the compressible Navier-Stokes equations. We also use the notations (v, u, θ)(τ, y), G(τ, y, ξ), Π 1 (τ, y, ξ) and (V, U, Θ)(τ, y) in the scaled independent variables. Set the perturbation around the composite wave (V, U, Θ)(τ, y) by Under this scaling, the hydrodynamic limit problem is reduced to a time asymptotic stability problem of the composite wave to the Boltzmann equation. Notice that the hydrodynamic limit proved here is global in time compared to the case on shock profile studied in [32] which is locally in time. From (1.21) and (2.42), we have the following system for the perturbation (φ, ψ, ζ) (2.45) where the error terms Q i (i = 1, 2) are given in (2.42) and (2.43) respectively.

FEIMIN HUANG, YI WANG AND TONG YANG
Then G 1 (τ, y, ξ) satisfies (2.51) Notice that in (2.50) and (2.51), G 0 is subtracted from G because (Θ y , U y ) 2 ∼ (1 + ε 1 2 τ ) −1/2 is not integrable globally in τ . Finally, from (1.15) and the scaling transformation (2.23), we have The estimation on the fluid and non-fluid components governed by the above systems will be given in the last section.

Main result to Boltzmann equation.
With the above preparation, we are now ready to state the main result on the Boltzmann equation as follows.
M⋆ L 2 ξ (R 3 ) and the positive constant C h depends only on h but is independent of ε.
Remark 5. Theorem 2.5 shows that, away from the initial time t = 0 and the contact discontinuity located at x = 0 with the expansion rate x 2 ε(1+t) , for Knudsen number ε < ε 0 , there exists a unique global solution f ε (t, x, ξ) of the Boltzmann equation (1.5) which tends to the Maxwellian M [V ,Ū,Θ] (t, x, ξ) with (V ,Ū ,Θ)(t, x) being the Riemann solution to the Euler equation with the combination of two rarefaction waves and a contact discontinuity when ε → 0. Moreover, a uniform convergence rate ε 1 5 in the norm L 2 ξ ( 1 √ M⋆ ) holds on the set Σ h for any fixed h > 0. Remark 6. Theorem 2.5 holds uniformly on the (t, x) ∈ Σ h for any h > 0 if the contact wave strength δ CD and Knudsen number ε are suitably small. But if we restrict the problem to the set Σ h ∩ {t ≤ T } for any fixed T > 0, then we don't need the smallness condition on the contact wave strength δ CD by using Gronwall inequality to get a time dependent estimate rather than the uniform estimation in time.
Notations: Throughout this paper, the positive generic constants which are independent of T, ε are denoted by c, C or C 0 . For function spaces, H l (R) denotes the l-th order Sobolev space with its norm ∂ j y f 2 ) 1 2 , and · := · L 2 (dy) , where L 2 (dz) means the L 2 integral over R with respect to the Lebesgue measure dz, and z = x or y.
3. Proof of Theorem 2.4: Zero dissipation limit of Navier-Stokes equations. We will prove Theorem 2.4 about the fluid dynamic limit for the compressible Navier-Stokes equations to the Riemann solution of the Euler equations in this section. The proof is based on the energy estimates on the perturbation in the scaled independent variables. In fact, to prove Theorem 2.4, it is sufficient to prove the following theorem.
and the Knudsen number ε satisfies ε ≤ ε 1 , then the problem (2.24) admits a unique global solution (v ε , u ε , θ ε )(τ, y) satisfying We will focus on the reformulated system (2.24). Since the local existence of the solution to (2.24) is standard, to prove the global existence, we only need to close the following a priori estimate by the continuity argument where χ is a small positive constant depending only on the initial values and the strength of the contact wave. And the proof of the above a priori estimate is given by the following energy estimations. Firstly, multiplying (2.24) 2 by ψ yields It is easy to check that Φ(1) = Φ ′ (1) = 0 and Φ(z) is strictly convex around z = 1. Moreover, On the other hand, note that (3.9) Substituting (3.7)-(3.9) into (3.5) gives where Direct calculation shows that (3.12) Thus, substituting (3.12) into (3.10) gives where (3.14) Here, (· · · ) y represents the conservative terms which vanishes after integrating in y over R.
By the strict convexity of Φ(z) around z = 1, under the a priori assumption (3.3) with sufficiently small χ > 0, there exist positive constants c 1 and c 2 such that, Thus, we have Notice that the last term ε|Q 2 ||(φ, ζ)| 2 on the right hand side of (3.16) can be estimated similarly as for the terms εQ 1 ψ and εQ 2 ζ θ under the a priori assumption (3.3). Now we estimate the terms εQ 1 ψ and εQ 2 ζ θ on the right hand side of (3.13). First, From the estimation on the interaction given in (2.21), we get and τ 0 R where τ 0 = t0 ε = ε − 4 5 , and β > 0 is a small constant to be determined later and C β is a positive constant depending on β.

FEIMIN HUANG, YI WANG AND TONG YANG
The term εQ 2 ζ θ can be estimated similarly because the only difference is about the error term Q CD coming from the viscous contact wave in Q 2 . For this, we have (3.19) By substituting (3.15)-(3.19) into (3.13) and choosing β suitably small, we can get (3.20) Now we need to estimate φ y Rewrite the equation (2.24) 2 as By multiplying (3.21) byṽ ỹ v and noticing that Integrating the above equality over [0, τ ] × R in τ and y, we obtain (3.23) The by using the equalityṽ By the estimation on Q 11 in (2.21) and Lemma 2.3, we have (3.26)

FEIMIN HUANG, YI WANG AND TONG YANG
Notice that where in the third inequality we have used the fact that V y L ∞ ≤ Cε Similarly, we have The remaining terms can be estimated directly by using (3.25) and the fact that Hence, if we take β suitably small, then we obtain (ψ y , ζ y )(τ, ·) 2 + τ 0 (ψ yy , ζ yy ) 2 dτ (3.32) The combination of (3.20), (3.27) and (3.32) yields that (3.33) In order to close the estimate, we only need to control the last term in (3.33), which comes from the viscous contact wave. For this, we will apply the following technique by using the heat kernel motivated by [13].

FLUID DYNAMIC LIMIT TO EULER EQUATIONS 711
where g a (τ, y) = ε and a > 0 is the constant to be determined later.
The proof of Lemma 3.2 is similar to the one given in [13]. The only difference here is that we need to be careful about the parameter ε in the estimation. Therefore, we omit its proof for brevity. Based on Lemma 3.2, we can obtain Lemma 3.3. There exists a constant C > 0 such that if δ CD and ε 0 are small enough, then we have where b is a positive constant to be determined later. Multiplying the equation (3.37) Note that (3.39) By using the equality we have (3.43) In order to get the desired estimate stated in Lemma 3.3, set We only need to compute the last term on the right hand side of (3.34) for this given function h. From the energy equation (2.24) 3 , we have By noticing that we can estimate τ 0 R hg 2 a H i dydτ (i = 2, · · · , 6) directly. The estimation on τ 0 R hg 2 a H 1 dydτ is more subtle. Firstly, by using the mass equation (2.24) 1 , we have Now the terms J i (i = 1, · · · , 4) can be estimated directly, cf. [13]. Here we only calculate the term J 5 . From (3.44), we have Now J 1 5 can be estimated as follows: Note that the other terms J i 5 (i = 2, · · · , 5) can be estimated directly, we omit the details for brevity. Therefore, by taking the constant a = C0 2 , we obtain With this, the Gronwall inequality gives And then we complete the proof of Theorem 3.1 by Sobolev imbedding.