ZERO DISSIPATION LIMIT TO RAREFACTION WAVE WITH VACUUM FOR A ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUID

. In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian ﬂuid for general initial data, whose far ﬁelds are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian ﬂuid which converge to the above rarefaction wave with vacuum as the viscosity coeﬃcient (cid:15) tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.


(Communicated by Volker Elling)
Abstract. In this paper, we study the zero dissipation limit toward rarefaction waves for solutions to a one-dimensional compressible non-Newtonian fluid for general initial data, whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we construct a sequence of solutions to the one-dimensional compressible non-Newtonian fluid which converge to the above rarefaction wave with vacuum as the viscosity coefficient tends to zero. Moreover, the uniform convergence rate is obtained, based on one fact that the viscosity constant can control the degeneracies caused by the vacuum in rarefaction waves and another fact that the energy estimates are obtained under some a priori assumption.
We define i−Riemann (i = 1, 2) invariants as follows There are several literatures about mathematical studies on compressible non-Newtonian fluid with various initial and boundary conditions. For example, Zhikov and Pastukhova [30] obtained the existence of weak solutions of initial boundary value problem for multidimensional non-Newtonian fluids under some restrictions. Mamontov [18] established the global existence of sufficiently regular solutions to two-dimensional and three-dimensional equations of compressible non-Newtonian fluids. Yuan and his cooperators obtained the existence result on the local strong solutions of initial-boundary-value problem in one dimensional non-Newtonian fluids see [28,29] and the references therein. Later Fang and Guo [5] gave the blowup criterion for the local strong solutions obtained in [28], constructed an analytical solutions to a class of compressible non-Newtonian fluids in [4], and considered the initial boundary problem for a compressible non-Newtonian fluid with densitydependent viscosity [6]. For more related results, we refer the reader to [3,27] and the references therein.
As pointed out by Liu and Smoller [12], among the two non-linear waves, i.e., shock and rarefaction waves, to the one-dimensional compressible isentropic Euler equation, only rarefaction waves can be connected to vacuum when vacuum appears. On the stability of the rarefaction waves to the one-dimensional compressible Navier-Stokes equations, there has been great interest and intensive studies completed in the development of the theory of viscous conservation laws from 1985; this started with studies on the nonlinear stability of viscous shock profiles by Matsumura and Nishihara [14]. Along with new phenomena of compressible fluids have been discovered, new techniques, such as weighted characteristic energy methods and uniform approximate Green's functions, have been developed based on the intrinsic properties of the underlying nonlinear waves see [13,14,16,20] and the references therein. In [26], Xin first studied the stability toward contact waves for solutions of systems of viscous conservation laws. Later, Liu and Xin [13] obtained pointwise asymptotic behavior towards viscous contact waves, which leads to the nonlinear stability of the viscous contact wave in L p −norms for all p ≥ 1.
However, the qualitative behaviors of the solutions to compressible non-Newtonian fluids were addressed only recently. In order to study the large time behavior of compressible non-Newtonian fluid, Shi, Wang and Zhang [22] obtained the asymptotical stability of the rarefaction wave solution for a compressible non-Newtonian fluid, provided the initial disturbance is suitably small. But, the rarefaction wave itself is away from the vacuum in [22]. To our knowledge, there are no results on the zero dissipation limit of the system (1) in the case when the Euler system (2) contains the rarefaction wave connected to the vacuum. In this paper, our concern is this fundamental problem and we intend to obtain the decay rate with respect to the given viscosity constant . Now, we give a description of the rarefaction wave connected to the vacuum to the compressible Euler equations (2) see also [23] and [24]. If the compressible Euler system (2) is investigated with the Riemann initial data with the left side being the vacuum state and ρ + > 0, u + being prescribed constants on the right state, then the Riemann problem (2)-(3) admits a 2-rarefaction wave connected to the vacuum on the left side. By the fact that along the 2-rarefaction wave curve, 2-Riemann invariant Σ 2 (ρ, u) is constant in (x, t). So, the velocity u = Σ 2 (ρ + , u + ) coming into the vacuum ρ = 0 to (ρ + , u + ) is the self-similar solution (ρ r2 , u r2 )(ξ) of (2) defined by and where ξ = x t . Thus, the momentum of 2-rarefaction wave is defined by In the present paper, we intend to construct a sequence of solutions (ρ , m )(x, t) to the system (1) which converge to the 2-rarefaction wave (ρ r 2 , m r 2 )( x t ) defined over as tends to zero. The effects of initial layers will be ignored by choosing the wellprepared initial data depending on the viscosity for the compressible non-Newtonian fluids.
The main novelty and difficulty of the paper is determining how to control the strong non-linear term in (1) 2 and the degeneracies caused by the vacuum in the rarefaction wave. To overcome the difficulty coming from the vacuum, we first cut off the 2-rarefaction wave with vacuum along the rarefaction wave curve. More precisely, for any µ > 0 to be determined, the cut-off rarefaction wave will connect the state (ρ, u) = (µ, u µ ) and (ρ + , u + ), where u µ can be obtained uniquely by the definition of the 2-rarefaction wave curve. To deal with the difficulty from the strong non-linear term in (1) 2 , the classical iterative technology is taken. Then an approximate rarefaction wave to this cut-off rarefaction wave will be constructed through the Burgers equation. Finally, the desired solution sequences to the system (1) could be established around the approximate rarefaction wave. Theorem 1.1. Let α > 2 and (ρ r2 , m r2 )( x t ) be the 2-rarefaction wave defined by (4)-(6) with one-side vacuum state. Then there exists a small positive constant 0 such that for any ∈ (0, 0 ), a global smooth solution (ρ , m = ρ u )(x, t) with initial values (28) to the system (1) can be constructed which satisfies and (ρ , m )(x, t) converges to (ρ r2 , m r2 )( x t ) point wise except in the origin (0, 0) as → 0. Furthermore, for any given positive constant h, there exists a constant C h , independent of such that and holds for α > 2, where the positive constants a and b are given as follows if A few remarks follow.

Remark 1.
It is also interesting to study the zero dissipation limit of compressible non-Newtonian system (1) in the case when the Euler system (2) has two rarefaction waves with vacuum in middle. However, there is a strong non-linear term in system (1) and it is nontrivial to cut off these rarefaction wave with vacuum along the corresponding rarefaction wave cures. In fact, wave structure containing two rarefaction waves with the medium vacuum is destroyed and some new wave may occur in the cut-off process, which is quite different from the single rarefaction wave case considered in the present paper.
Remark 2. It is noted that the estimates for φ 2 from the potential energy hold with the weightρ γ−2 in Lemma 3.1 below, which is degenerate at vacuum when γ > 2. Therefore, the convergence rate obtained in Proposition 1 and thus in Theorem 1.1 not only depends on the value α but also dependents on γ.
Remark 3. All the estimates obtained in Theorem 1.1 is strongly dependent on the the α and γ, which can be regarded as the extending results related to compressible Navier-Stokes equations in [8] as the case of α = 2 in (1) for µ 0 > 0. If µ 0 = 0, i.e., for the pseudo-plastic fluid (α < 2) or dilatant fluid (α > 2) (see [4]), the strong nonlinearity and highly degeneracy can be produced. This is a big challenge which will be left for future. It is worth to point out that, in order to deal with the strong non-linear term ( (1), new technique is taken besides the classical iterative method.
The rest of the paper is organized as follows. We construct a smooth 2-rarefaction wave which approximates the cut-off rarefaction wave based on the inviscid Burgers equation in Section 2, and then the proof of Theorem 1.1 is given in Section 3.
2. Rarefaction waves. Consider the Riemann problem for the typical Burgers equation If w < w + , then the system (12) has a rarefaction wave solution of the form w r (x, t) = w r ( x t ) given by As in [25], the approximate rarefaction wave w(x, t) to the system (1) can be constructed by the solution of the Burgers equation where δ > 0 is a small parameter to be determined. In fact, we choose δ = a in (46) with a given by (9) and (10). Moreover, the solution w r δ (t, x) of the problem (14) is given by Then, w r δ (t, x) has properties stated in Lemma 2.1, which can be found in [25] and [8].
Lemma 2.1. The problem (14) has a unique smooth global solution w r δ (x, t) for each δ > 0 such that the following hold: The following estimates hold for all t > 0 and p ∈ [1, +∞] : (3) There exists a constant δ 0 ∈ (0, 1) such that δ ∈ (0, δ 0 ], t > 0 Now, we turn to cut off the 2-rarefaction wave with vacuum along the wave curve in order to overcome the difficulty caused by the vacuum. More precisely, for any µ > 0 to be determined, we can get a state (ρ, u) = (µ, u µ ) belonging to the 2-rarefaction wave curve. From the fact that 2-Riemann invariant Σ 2 (ρ, u) is constant along the 2-rarefaction wave curve, u µ can be computed explicitly by u µ = Σ 2 (ρ + , u + ) + 2 γ−1 µ γ−1 2 . So, we get a new 2-rarefaction wave (ρ r2 µ , u r2 µ )(ξ) (ξ = x t ) connecting the state (µ, u µ ) to the state (ρ + , u + ) which can be expressed explicitly by and Correspondingly, it is reasonable to define the momentum function m r2 µ = ρ r2 µ u r2 µ . And it is easy to prove that the cut-off 2-rarefaction wave (ρ r2 µ , m r2 µ )( x t ) converges to the original 2-rarefaction wave with vacuum (ρ r2 , m r2 )( x t ) in sup-norm with the convergence rate µ as µ tends to zero, which is stated as follows and can be found in [8].
3. Proof of Theorem 1.1. To prove Theorem 1.1, the solution (ρ , u ) is constructed as the perturbation around the approximate rarefaction wave (ρ,ū). The Cauchy problem for (1) is considered with initial values Then we introduce the perturbation where y, τ are the scaled variables as and (ρ , u ) is assumed to be the solution to the problem (1). For simplicity of notation, the superscription of (ρ , u ) will be omitted as (ρ, u) from now on if there is no confusion of notation. Substituting (29) and (30) into (1) and using the definition of (ρ,ū), we obtain that We seek a global-in-time solution (φ, ψ) to the reformulated problem (31)-(33). To this end, the solution space for (31)-(33) is defined by holds for α ≥ 6, where a is given by (9) and (10). Consequently, and when γ > 2 it is held that By similar way, if γ > 1 then Here we omit the proof for the local existence of the solution to (31)-(33) for brevity, since it is standard. Note that the local existence time interval, denoted by τ 0 , may depend on , in order to get the convergence rate of the local solution with respect . The next step is to extend the local solution to the global solution in [0, +∞) for small but fixed constant . To do so, it is sufficient to obtain the following a priori estimates for fixed .
holds for α ≥ 6, where a is given by (9) and (10). Consequently, and when γ > 2 it is held that Before giving the proof of Lemma 3.1, we take Then, it is obvious that µ ≥ 2 a if 1. Proof of Lemma 3.1. In order to prove Lemma 3.1, we assume that the solution to the problem (31)-(33) satisfies Under the a priori assumption (46), we can get that where C 1 , C 2 are positive constants independent of . Now, we divide the proof of Lemma 3.1 into the following four steps.
Step 2. We make use of the idea in [22] with modifications to derive the estimation of φ y . Differentiating (31) with respect to y and then multiplying the resulting equation by φy ρ 3 yields that Then, multiplying (32) by φy ρ 2 , one has that Now, one can estimate the terms on the right-hand side of (74) in the following. By Lemma 2.3 and Cauchy's inequality, one obtains that Recalling (18) and one can obtain that From Lemma 2.3(i), one has thatρ y =ρ 3−γ 2ū y and then Similar to Π 2 , one can obtain that Similar to Π 3 , using Hölder inequality and Lemma 2.3 one can get that and Finally, one can estimate Π 11 as follows: where holds for 2 < α < 4 and holds for 4 ≤ α < 6 and holds for α ≥ 6, and Now, we estimate the last term in the right-hand side of (85).
Based on these a priori estimates, we can claim τ ( ) = ∞. If τ ( ) < ∞, then by again using the local existence at time τ = τ 1 ( ), we can find another time τ 2 ( ) > τ 1 ( ) so that the solution satisfies the assumption (47) in the time interval [0, τ 2 ( )] which contradicts the assumption that τ 1 ( ) is the maximum time. Therefore we extend the local solution to the global solution in [0, ∞) for small but fixed .
Proof of Theorem 1.1 It remains to prove (7) and (8) with a, b given in (9) and (11), respectively. From Lemma 2.2, Lemma 2.3(iii), and Proposition 1 and recalling that µ = a | ln |, δ = a , one can get that for any given positive constant h, there exists a constant C h > 0 which is independent of such that if 1 < γ < 2 Moreover, if 1 < γ < 2 Therefore, the proof of Theorem 1.1 is completed.