Low Mach number limit for the compressible Navier–Stokes equations with density-dependent viscosity and vorticity-slip boundary condition

In this paper, we consider the three-dimensional compressible Navier–Stokes equations with density-dependent viscosity and vorticity-slip boundary condition in a bounded smooth domain. The main idea is to derive the uniform estimates for both time and the Mach number. The difficulty is dealing with density-dependent viscosity terms carefully. With the uniform estimates, we can verify the low Mach limit of the global strong solutions of compressible Navier–Stokes equations and the global existence and uniqueness of the strong solution of incompressible Navier–Stokes equations around a steady state.


Introduction
In this paper, we study the low Mach number limit for the initial-boundary value problem of the following three-dimensional compressible Navier-Stokes equations in a bounded domain ⊂ R 3 with a smooth boundary: pressure p(ρ) satisfies the barotropic law, namely, where a > 0 and γ > 1 are constants. Formally, as the Mach number vanishes, the solution to (1)- (2) will converge to the one of the following incompressible Navier-Stokes equations: div u = 0.
It is known as the low Mach number limit. Since the large parameter -2 appears in (2), this limit process is singular. The fact that both the uniform estimates in Mach number and the convergence to the incompressible model are usually difficult to obtain creates a serious difficulty for the rigorous justification of this limit. The low Mach number limit of local smooth solutions to the Navier-Stokes equations (or the Euler equations) in R n or T n with "well-prepared" initial data was proved by Klainerman and Majda in [17,18]. They established the general framework for studying the low Mach number limit for local strong or smooth solutions. For bounded domain, Lions and Masmoudi [19] investigated the low Mach number limit for the weak solutions to the Navier-Stokes equations with the "vorticity-slip" boundary condition, that is, on the boundary ∂ ⊂ R n , u · n = 0, curl u = 0 for n = 2, or (6) u · n = 0, n × curl u = 0 for n = 3, where curl u = (∂ 2 u 1 , -∂ 1 u 2 ) t for n = 2 and curl u = (∂ 2 u 3 -∂ 3 u 2 , ∂ 3 u 1 -∂ 1 u 3 , ∂ 1 u 2 -∂ 2 u 1 ) t for n = 3. There are abundant results about the low Mach number limit for local solutions to the isentropic Navier-Stokes equations, the reader may refer to [6-10, 23, 24] and the references therein, for instance. The low Mach number limit for global solutions to the isentropic Navier-Stokes equations have been considered by many authors; see [3,14,21,22]. Compared with the study of the low Mach number limit for local solutions, one must get the uniform estimates with respect to both the Mach number and t ∈ [0, +∞). Thus this is challenging. D. Hoff [14] verified the low Mach number limit for the global solutions in R 3 × [0, +∞) with general large initial data. For bounded domain, H. Bessaih [3] investigated the low Mach number limit of regular solutions to the compressible Navier-Stokes equations with no-slip boundary conditions and slightly compressible initial data. In [21], Ou obtained the low Mach number limit of regular solutions to the compressible Navier-Stokes equations (1)-(2) with slightly compressible initial data in a 2-D bounded domain with the "vorticity-slip" boundary condition (6). [22] investigated the low Mach number limit of strong solutions to 3-D Navier-Stokes equations with Navier's slip boundary condition for all time.
Concerning with the low mach number limit of the compressible non-isentropic Navier-Stokes equations, many results was presented in [2,5,12,13,15,16,20], and the references therein. After learning this progress on the low mach number limit carefully, we find the fact that most of it was concerned with the constant viscosity coefficients.
The purpose of this paper is to verify rigorously the corresponding low Mach number limit for all time of the 3-D isentropic Navier-Stokes equations with density-dependent viscosity and the "vorticity-slip" boundary condition. We establish the uniform estimates of strong solutions with respect to the Mach number and justify rigorously the low Mach number limit for all time when the non-constant viscosity coefficients are present, in contrast with [21]. Because the viscosity depends on the density, the uniform estimates of strong solutions are much more difficult to obtain.
To verify the low Mach number limit, we shall consider the density varies slightly around a constant state, namely, We reformulate the problem (1)-(2) as The initial data for the system (1)-(2) are defined as We impose the "vorticity-slip" boundary condition for the velocity, that is, on the boundary ∂ of ⊂ R 3 , where curl u = (∂ 2 u 3 -∂ 3 u 2 , ∂ 3 u 1 -∂ 1 u 3 , ∂ 1 u 2 -∂ 2 u 1 ) t and n is the unit outer normal vector to the boundary. We state the main results of this paper as follows.
Assume that the assumptions in Theorem 1.1 are satisfied. Then one can get the local existence of the initial-boundary problem (8)-(11) by the method of characteristics, the Galerkin method and the Schauder fixed point theorem, that is, there exists a T > 0, such that for T ≤ T the problem (8)-(11) admits a solution satisfying The boundary conditions (11) are "complementing" boundary conditions in the sense of Agmon-Douglis-Nirenberg [1]. The local existence result can be proved by the frame in [22], so we omit the details of the proof here.
Theorem 1.2 (Incompressible limit) Let the assumptions in Theorem 1.1 be satisfied, and u be the global strong solution established in Theorem (1.1). Suppose that the initial data u 0 → v 0 as → 0 in H s for any 0 ≤ s < 2. Then we have u → v in C(R + loc , H s ) as → 0, for any 0 ≤ s < 2. Moreover, there exists a function P(x, t), such that (v, P) is the unique global strong solution to the following initial-boundary value problem of incompressible Navier-Stokes equations: Before ending this section, we introduce the notations throughout this paper. We use the constant C to denote various positive constants independent of and t, use the constant C η to emphasize the dependence on η. Moreover, we denote by H m and · H m the Sobolev space H m ( ) ≡ W m,2 ( ) and its norm, by L p and · L p the Lebesgue space L p ( ) and its norm.

Preliminaries
In this paper, we will use the following lemmas frequently.

Lemma 2.1 (See [4])
Let be a bounded domain in R N with smooth boundary ∂ and outward normal n. Then there exists a constant C > 0 independent of u, such that for any u ∈ H s ( ) N .

Lemma 2.2 (See [26])
Let be a bounded domain in R N with smooth boundary ∂U and outward normal n. Then there exists a constant C > 0 independent of u, such that for all u ∈ H s ( ) N .
Moreover, we assume that is simply connected and non-axisymmetric. Then, for any u ∈ H 1 ( ) satisfying u · n| ∂ = 0, one has and where C is a constant independent of u.
Then the problem Lemma 2.5 (See [4]) Let k ≥ 2 be an integer, and let 1 ≤ p ≤ q ≤ +∞ be such that p < +∞ and k > N p + 1. Let f ∈ W k,p ( ), then the mapping g − → g • f is continuous from D k,p ( ) into W k,p ( ).

Energy estimates
In order to extend the local solution of the initial-boundary value problem (8)-(11) globally in time, we shall establish a differential inequality which provides us the uniform estimates of solutions for both time and the Mach number. Suppose that (σ , u) is the local solution to the initial-boundary value problem (8)- (11) in × (0, T), for 0 < T < ∞. Moreover, we assume that 1/c ≤ ρ = 1 + σ ≤ c for some constant c > 1. Then the viscosity coefficients can be estimated as follows:

L 2 estimate
where γ 1 is a positive constant independent of .
Proof We integrate the product of (8) and p (ρ)σ to get Due to the boundary conditions (11) and Lemma 2.3, we have Integrating the product of (9) and u, we get Using (11) again and integration by parts, we have Summing up the above equalities and choosing η small enough, we get the lemma.

Estimates of first order derivatives
where η is to be determined later.
Proof First, by differentiating (9) with respect to t, we have Multiplying (20) by u in L 2 , integrating by parts and using the boundary conditions (11), we deduce that We multiply (8) by p (ρ)σ t , integrate by parts and use the boundary conditions (11) again to infer that Summing up the above estimates, we obtain the above lemma. (8)- (11), we have

Lemma 3.3 For the solution to
where η is to be determined later.
Proof Applying ∇ to (8), multiplying the resulting equation by ∇σ , integrating in L 2 , we obtain Now, we apply (9), p (ρ) -1 ∇ div u to derive that where with the aid of curl ∇ = 0 and curl u × n| ∂ = 0, Putting the above estimates together, we get this lemma.

Lemma 3.4 For the solution to
where 0 < η < 1 is to be determined later, and γ 2 is a positive constant independent of .
Proof Applying ∂ t to (8), multiplying the resulting equation by p (1)σ t , integrating in L 2 , we get Applying ∂ t to (9), we have Taking (22), u t and using the boundary conditions (11), we find that Hence, by choosing η appropriately small and using Korn's inequality, we obtain the estimate (21).
Next, we estimate the vorticity of the velocity, which is denoted by ω = curl u. By virtue of (8) and (9), it is easy to see that ω satisfies the following systems: where Then we have the following.
where 0 < η < 1 is a positive constant which is to be determined.
Proof Multiplying (23) 1 by ω, with the aid of the boundary condition (23) 2 we infer that where With the aid of Lemma 2.2, it is easy to verify that Using Holder's inequality and Young's inequality, we have Inserting the above two inequalities into (25) and choosing η appropriately small, we get the above lemma.
Definition 3.1 Now, defining two functions: we conclude from Lemmas 3.1-3.5 that, for small , there is a positive constant C 1 , such that d dt

Boundedness of second order derivatives
First, we show the following lemma.

Lemma 3.6
For the solution to (8)-(11), we have where 0 < η < 1 is a small positive constant which is to be determined.
Proof Differentiating (9) with respect to t, we have Multiplying (29) by ∇ div u and integrating in L 2 , we get where we have used the following estimate: Similarly, we take ∇(8), p (1)∇σ t to infer that Summing up the above inequalities together and choosing η small, we get the above estimate.

Lemma 3.7 We have
where 0 < η < 1 is a small positive constant which is to be determined.
We take (29), ρ -1 ∇ div u t and ∂ t ∇(8), ρ -1 p (1)∇σ t , summing up the resulting equations to obtain the above lemma. Here we use the following estimate: Next, we estimate the derivatives of curl u.
where γ 3 > 0 is a positive constant and 0 < η < 1 is a small positive constant which is to be determined.
where 0 < η < 1 is a small positive constant which is to be determined.
Proof We take (23) 1 , ω tδ ω (in which δ is a positive constant to be determined later) to get where we use the following estimate: Thus, we choose δ and η suitably small to conclude the lemma. In order to close the estimates, we have to estimate σ H 2 . To this end, we obtain from the continuity equation (8) and the boundary condition u · n| ∂ = 0 From Eqs. (9) and Poincare's inequality, we have In addition, in order to control the terms u t H 2 and u H 3 , we use the following fact which is obtained from Lemmas 2.1-2.2 and the boundary condition (11): curl u H 2 ≤ C curl curl u H 1 + curl u H 1 , curl curl u H 1 ≤ C curl u L 2 + curl curl u × n H 1/2 (∂ ) + curl u L 2 , u t H 2 ≤ C div u t H 1 + curl u t H 1 + u t H 1 , where the estimate of the term curl curl u × n H 1/2 (∂ ) is crucial for the proof. We can estimate it by the strategy in [22] as follows: In order to derive an estimate near the boundary, Next, we denote the vorticity near the boundary as w := ( w 1 , w 2 , w 3 ) t := w(t, (y)). So we get curl w · n = (a k2 D k w 3a k3 D k w 2 , a k3 D k w 1a k1 D k w 3 , a k1 D k w 2a k2 D k w 1 ) · (a 31 , a 32 , a 33 ) = (a 32 a 13a 33 a 12 )D 1 + (a 32 a 23a 33 a 22 )D 2 w 1 + (a 33 a 11a 31 a 13 )D 1 + (a 33 a 21a 31 a 23 )D 2 w 2 + (a 31 a 12a 32 a 11 )D 1 + (a 31 a 22a 32 a 21 )D 2 w 3 Thus, with the boundary condition (11) we get the estimate Definition 3. 2 We define where 1 (t) and 1 (t) are defined by (26), and Combining Lemmas 3.1-3.10 with the estimates (35)-(37) and choosing a suitable constant C, and small enough constants and η, we finally conclude that d dt (t) + (t) ≤ c 0 (t) (t) + 2 (t) , where c 0 ≥ 1 is a constant independent of .