Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids

In this paper, we study the large time behaviors of boundary layer solution of the inflow problem on the half space for a class of isentropic compressible non-Newtonian fluids. We establish the existence and uniqueness of the boundary layer solution to the non-Newtonian fluids. Especially, it is shown that such a boundary layer solution have a maximal interval of existence. Then we prove that if the strength of the boundary layer solution and the initial perturbation are suitably small, the unique global solution in time to the non-Newtonian fluids exists and asymptotically tends toward the boundary layer solution. The proof is given by the elementary energy method.

The global existence of sufficiently regular solutions to two-dimensional and threedimensional equations of compressible non-Newtonian fluids have been established by Mamontov [14]. Yuan and his cooperators [25,26] obtained the existence and uniqueness of local and global solutions for one dimensional initial boundary value problem. Fang and Guo [3] gave the blow-up criterion for the local strong solutions, constructed an analytical solutions to a class of compressible non-Newtonian fluids with free boundaries in [4], and considered the existence and uniqueness of global classical solution for a initial boundary problem [6]. For weak solutions to the non-Newtonian fluids, Zhikov and Pastukhova [27] obtained the existence of weak solutions of initial boundary value problem for multidimensional cases. Guo and Zhu [9] investigated the partial regularity of the suitable weak solutions. Feireisl, Liao and Málek [8] studied mathematical properties of unsteady for three dimensional compressible non-Newtonian fluids in bounded domains and shown the long-time and large-data existence result of weak solutions with strictly positive density. The existence of weak solutions to a one-dimensional full compressible non-Newtonian fluids has been investigated by Fang, Kong and Liu [7]. Recently, Fang and Guo [5], Shi, Wang and Zhang [21] discussed the stability of rarefaction waves for the isentropic and nonisentropic compressible non-Newtonian fluids, respectively.
In this paper, we study the large time behaviors of boundary layer solutions of the inflow problem to the system (1.1) with the initial data (ρ, u)(0, x) = (ρ 0 , u 0 )(x) → (ρ + , u + ), as x → +∞, (1.5) and the condition on the boundary x = 0: u(t, 0) = u − > 0, ρ(t, 0) = ρ − > 0, t > 0, (1.6) where ρ ± and u ± are prescribed constants. The inflow problem was proposed by Matsumura [16] for a one-dimensional isentropic compressible Navier-Stokes equations and the author classified all possible large time behaviors of the solutions in terms of the boundary values. Then Matsumura and Nishihara [17] gave the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave. Huang, Matsumura and Shi [11] established the asymptotic stability on both the viscous shock wave and a superposition of the viscous shock wave and the boundary layer solution under some smallness conditions for the inflow problem to the isentropic Navier-Stokes equations. For the full Navier-Stokes equations, Huang, Li and Shi [10] proved the existence of the subsonic boundary layer solution and the stability of this boundary layer solution and its superposition with the 3-rarefaction wave under some smallness assumptions. Qin and Wang [19] showed the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem under some smallness conditions. Then they [20] also proved the asymptotic stability of the superposition of the transonic boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave, and the 3-rarefaction wave to the inflow problem under some smallness conditions. We also refer to [18] for the asymptotic stability of stationary solution to the inflow problem.
The nonlinear waves can be identified in the non-Newtonian fluids, such as shock phenomena in the blood fluids [1]. However, since the nonlinear constitutive relation between viscous stress tensor and rate of strain in non-Newtonian fluids, the nonlinear waves may exhibit some properties which are different from those in the Newtonian fluids. Our main goal is to investigate the boundary layer solution of the non-Newtonian fluids (1.1) and to show that it is asymptotically stable under small initial perturbations.
In the present paper, we establish the existence and uniqueness of the boundary layer solution to the inflow problem of the non-Newtonian fluids (1.1). In contrast with the case of the Newtonian fluids [17], we identify a peculiar feature of boundary layer solution, i.e., it is shown that the boundary layer solution to the non-Newtonian fluids (1.1) has a maximal interval of existence (see Section 2 below for the details). This characteristic of boundary layer solution do not develop in the Newtonian fluids. The effect of the nonlinear constitutive relation between viscous stress tensor and rate of strain for the non-Newtonian fluids (1.1) gives rise to such phenomenon of boundary layer solution.
Then we prove that if the initial data are close to the boundary layer solution and the strength of boundary layer solution is suitably small, a unique solution to the inflow problem of the non-Newtonian fluids (1.1) exists globally in time and tends toward the boundary layer solution as the time goes to infinity. The main point in our analysis is based on the maximal interval of existence of the boundary layer solution to the non-Newtonian fluids (1.1), which allows us to use the elemental energy method to study the asymptotic stability of the boundary layer solution for any q > 2.
The rest of this paper is organized as follows. In Section 2, we study the existence of the boundary layer solution for the non-Newtonian fluids (1.1) and state our main theorems. Sections 3 reformulates the original problem to obtain a initial-boundaryvalue problem, and establishes the a priori estimates for proving the main theorems. In Section 4, we give the proof of the a priori estimates and complete the proof of the main theorems.

ZHENHUA GUO, WENCHAO DONG AND JINJING LIU
C denotes the generic positive constants which are independent of time t unless otherwise stated.
2. Boundary layer solution and main results. In this section, we study the boundary layer solution of (1.1) and state our main results. We transform the inflow problem (1.1), (1.5) and (1.6) to the moving boundary problem in the Lagrangian coordinate where v = 1 ρ represents the specific volume, the boundary moves with the constant The corresponding hyperbolic system without viscosity is which has two eigenvalues We define the sound speed c(v) by Let Ξ = {(v, u) ∈ R + × R + } be the phase plane of (v, u) and abbreviate (v, u) to w. For any fixed left state (v − , u − ) := w − ∈ Ξ, solving the Riemann problem of (2.3) on Ξ, we can get the 1-rarefaction wave (2.6) and the 2-rarefaction wave On the other hand, from the Rankine-Hugoniot condition where [q] = q r − q l with q l = q(t, x(t) − 0) and q r = q(t, x(t) + 0), s is the velocity of shock waves, and the Lax entropy condition s < λ 1 (w l ), λ 1 (w r ) < s < λ 2 (w r ) for 1-shock; λ 1 (w l ) < s < λ 2 (w l ), λ 2 (w r ) < s for 2-shock, we can obtain 1-shock curve and 2-shock curve (2.10) 2.1. Existence and uniqueness of boundary layer solution. We divide the domain Ξ into three regions ( Figure 1): Call them the supersonic, transonic and subsonic regions, respectively. When (v − , u − ) ∈ Ξ sub , λ 1 (v − ) < s 0 < 0, i.e., the first wave speed λ 1 (v − ) is less than the boundary speed s 0 , hence the existence of a traveling wave solution where = d/dξ and ξ = x − s 0 t > 0. We call this solution W (ξ) = (U, V )(ξ) the boundary layer solution (BL-solution).
Seek the condition for the existence of the BL-solution and investigate its properties. Integrating (2.11) over (0, +∞) and (ξ, +∞) yields and respectively. From (2.12) 1 and (2.13) 1 , we get that (2.14) Thus we define BL-line through w − ∈ Ξ sub by The line BL(w − ) always intersects the transonic line Γ trans at the point w * = (v * , u * ). See Figure 2. By (2.13), we obtain the ordinary differential equation of V  (i) If w − ∈ Ξ sub ∪ Γ trans and u + > u * , then there is no BL-solution.

21)
and (iii) If w − ∈ Ξ sub and u − < u + < u * , then there exists a unique BL-solution of (2.11). Furthermore, there exists a positive constant ξ + such that

25)
and which contradicts with the condition v * < v + . Therefore, in this case, there is no BL-solution of (2.11). For part (ii), in this case, we have and from which, we can get a initial value problem of the ordinary differential equation This equation can be solved using a small ruse [22].
Since G(V ) is strictly monotone, it can be inverted and we obtain a unique solution in a neighborhood of ξ = 0: Now let us investigate the maximal interval of existence of this solution. Define Returning to (2.35), we have where C 1 , C 2 are some positive constants. Next, according to the quantities of q, we discuss the BL-solution for the following three cases.
For part (iv), in a similar analysis as above, we can conclude that (iv) holds.
Remark 2.1. From the analysis above, it is clear to see that if q = 2, i.e., the case of Newtonian fluids, we always have ξ + = +∞. However, in the non-Newtonian fluids (1.1), the nonlinear constitutive relation between viscous stress tensor and rate of strain leads to the maximal interval of existence of the BL-solutions.

Stability of boundary layer solution.
For any w − ∈ Ξ sub and w + ∈ BL(w − )(u − < u + < u * ), we assume that the initial data in (2.1) satisfies and the compatibility condition Then, we can prove the other main result of this paper.
Theorem 2.2 (Stability of BL-solution). For any w − ∈ Ξ sub and w + ∈ BL(w − ) ( u − < u + < u * ), suppose that the assumptions (2.50) and (2.51) hold. Then there exists a small positive constant σ 0 such that if |w + − w − | is suitably small and , and the asymptotic behavior lim Remark 2.2. Our result and method can also be generalized to discuss the stability of BL-solutions to (2.1) for the cases (ii) and (iv) shown in Theorem 2.1 and similar results to Theorem 2.2 are also expected for the two cases.
(3.4) The global smooth solution in X(0, T ) is constructed by the combination of the local existence and the a priori estimate. The proof of the existence of local solution is standard, and we are mainly concerned about the following a priori estimate.
Proposition 3.1 (A priori estimate). For any q > 2, let (φ, ψ) ∈ X(0, T ) be the solution of (3.2) for a positive T . Then there exists a constant σ > 0 such that, if δ 1 is suitably small and N (T ) ≤ σ, then it holds, for t ∈ [0, T ], that Once the Proposition 3.1 is obtained, we can show the following global existence theorem, which implies Theorem 2.2 by the definition (3.1).
Theorem 3.2. For any q > 2, there exists a small positive constant σ 0 such that if |w + − w − | is suitably small and In what follows, the analysis is always carried out under the a priori assumption for sufficiently small σ: Throughout this paper, we denote Ω 1 = (0, ξ + ) and Ω 2 = (ξ + , +∞).
Proof. Since Then according to the definition of limit, we complete the proof of this lemma.
Using a same argument as in the proof of Lemma 4.1, we can get the following two lemmas.
Remark 4.1. It can be seen from the analysis above that, thanks to the properties of the maximal interval of existence for the BL-solution, we can obtain the desired basic energy estimate (4.3) for any q > 2 by elemental energy method.
where 1 is a small and fixed constant, C is a positive constant depend on .
Lemma 4.6. Under the assumptions of Proposition 3.1, it holds that where 1 is a small and fixed constant, C is a positive constant depend on .