Inviscid Limit of 3D Nonhomogeneous Navier–Stokes Equations with Slip Boundary Conditions

Abstract

In this paper, we consider the inviscid limit of a nonhomogeneous incompressible Navier–Stokes system with a slip-without-friction boundary condition. We study the convergence in strong norms for a solution and obtain the convergence rate in space $W^{2,p}(\Omega )$ when the boundary is flat. We need to establish the uniform bound of the solution in space $W^{3,p}(\Omega )$, and the key of proofs is to obtain a priori estimation of ${\partial }_{t}u$ in space $W^{1,p}(\Omega )$.

1. Introduction

The 3D nonhomogeneous incompressible Navier–Stokes equations can be written as

$$\begin{array}{ccc}\hfill & {\partial }_t\rho +u·\nabla \rho =0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(1)

$$\begin{array}{ccc}\hfill & \rho {\partial }_{t}u-\nu \Delta u+\rho u·\nabla u+\nabla \pi =0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(2)

$$\begin{array}{ccc}\hfill & \mathrm{div}u=0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(3)

with initial data and slip boundary conditions

$$\begin{array}{ccc}\hfill & \rho (x,0)={\rho }_0,u(x,0)=u_0\hfill & \hfill \mathrm{in}\Omega ,\end{array}$$

(4)

$$\begin{array}{ccc}\hfill & u·n=0,\mathrm{curl}u\times n=0\hfill & \hfill \mathrm{on}\partial \Omega \times (0,T),\end{array}$$

(5)

where $\rho ,u=(u_1,u_2,u_3)$ and $\pi$ denote the mass density, velocity and pressure of the nonhomogeneous fluids, respectively. Let $\Omega \subset {\mathbb{R}}^3$ be a class of bounded smooth domains with flat boundary. The positive constant $\nu$ is the viscosity coefficient, the vector n is the unit outward normal to boundary.

Significantly, there is a considerable number of papers on the study of nonhomogeneous incompressible Navier–Stokes equations, for example, see [1,2,3,4,5,6]. The global existence of weak solutions to systems (1)–(3) was studied by Simon [7,8] and Lions [9]. Some more advances concerning the existence and uniqueness of strong solutions have been made in the framework of the so-called critical spaces, for example, see [1,2,10,11,12]. Concerning the nonhomogeneous incompressible Navier–Stokes system, the local existence of strong solutions was first established in [13], with ${\rho }_0$ satisfying

$$\begin{array}{c}\hfill 0<\underline{\rho }\le {\rho }_0\le \overline{\rho },\end{array}$$

(6)

and the uniqueness was later obtained in [14,15]. The authors addressed the vanishing viscosity limit without vacuum and proved the convergence of local strong solutions in Hilbert and Sobolev spaces in [16,17,18,19].

The vanishing viscosity limit for the incompressible Navier–Stokes system was studied in [20,21,22,23,24]. The issue of the vanishing viscosity limit resulting in the whole space can be found, for instance, in [22,25,26,27,28,29,30]. In [23], the vanishing viscosity limit for solutions has been studied by Xiao and Xin in smooth domains; the strong convergence was established and the rate convergence was obtained when the boundary was flat. Berselli and Spirito in [31] showed strong convergence for a solution starting with special initial data in a general domain with a nonflat boundary by a perturbation argument. We mainly focus on 3D Navier–Stokes equations in ‘smooth solution’ situations, for some main results concerning inviscid limits in ‘non-smooth situations’, we refer the reader to [20,32]. In the homogeneous case, the authors in [33] established the convergence in $W^{k,p}$ space for arbitrarily large k and p. However, in the inhomogeneous case, we can only obtain a similar result in $W^{3,p}$.

Obviously, the vanishing viscosity limit for the nonhomogeneous incompressible Navier–Stokes equation seems more complex because of the close coupling between density and velocity. One of the difficulties is to show that the time of solution is independent of the viscosity. Consequently, we assume

$$\begin{array}{c}\hfill \nabla {\rho }_0·n=0\mathrm{on}\partial \Omega .\end{array}$$

(7)

Hence, we obtain $\nabla \rho ·n=0\mathrm{on}\partial \Omega$ by the transport equation; see [34]. In addition, we establish a priori estimates (26), and this is the key of this paper.

The corresponding Euler system is the following:

$$\begin{array}{ccc}\hfill & {\partial }_t{\rho }^0+u^0·\nabla {\rho }^0=0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(8)

$$\begin{array}{ccc}\hfill & {\rho }^0{\partial }_t{u}^0+{\rho }^0{u}^0·\nabla u^0+\nabla p^0=0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(9)

$$\begin{array}{ccc}\hfill & \mathrm{div}{u}^0=0\hfill & \hfill \mathrm{in}\Omega \times (0,T),\end{array}$$

(10)

with initial data and slip boundary condition

$$\begin{array}{ccc}\hfill & {\rho }^0(x,0)={\rho }_0,u^0(x,0)=u_0\hfill & \hfill \mathrm{in}\Omega ,\end{array}$$

(11)

$$\begin{array}{ccc}\hfill & u^0·n=0\hfill & \hfill \mathrm{on}\partial \Omega \times (0,T).\end{array}$$

(12)

The local existence and uniqueness of the strong solution of the above Euler system were obtained in [35], also referred to in [10,36,37].

Now, let us state our main result of our paper as follows:

Theorem 1.

Assume the initial data ${\rho }_0,u_0$ belongs to $W^{3,p}(\Omega )$, where $p>3$, and ${\rho }_0$ satisfies (6) and (7), $u_0$ is divergence-free in Ω and satisfies the boundary conditions (5). For each $\nu >0$, $(\rho ,u)$ denotes the strong solution to the initial boundary value problem (1)–(5) in Ω. Then, there exist $T_0$ only depending on $\parallel ({\rho }_0,u_0){\parallel }_{3,p}$; we have the following convergence result, as $\nu \to 0$.

$$\begin{array}{cc}\hfill & (\rho ,u)\stackrel{*}{\rightharpoonup }({\rho }^0,u^0)\mathrm{in}{L}^{\infty }(0,T_0;W^{3,p}),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\partial }_t\rho \stackrel{*}{\rightharpoonup }{\partial }_t{\rho }^0\mathrm{in}{L}^{\infty }(0,T_0;W^{2,p}),{\partial }_{t}u\stackrel{*}{\rightharpoonup }{\partial }_t{u}^0\mathrm{in}{L}^{\infty }(0,T_0;W^{1,p}),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & (\rho ,u)\to ({\rho }^0,u^0)\mathrm{in}C(0,T_0;W^{s,p}),foreachs<3,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \parallel \rho -{\rho }^0{\parallel }_{2,p}+{\parallel u-u^0\parallel }_{2,p}\le C{\nu }^{\frac{1}{2}},t\in [0,T_0],\hfill \end{array}$$

(16)

where $({\rho }^0,u^0)$ is the unique solution of Euler systems (8)–(12).

2. Preliminary

In this section, we introduce some notations and state several lemmas which are used in the rest of this paper. We denote by $(·,·)$ the usual inner product in $L^2(\Omega )$. $L^p(\Omega ),W^{k,p}(\Omega )$ stand for classical Lebesgue spaces and Sobolev spaces with norms ${\parallel ·\parallel }_{L^q(\Omega )}$ and ${\parallel ·\parallel }_{W^{k,p}(\Omega )}$, separately. For simplicity, ${\parallel ·\parallel }_{L^q(\Omega )},{\parallel ·\parallel }_{W^{k,p}(\Omega )}$ are replaced by $={\parallel ·\parallel }_q,{\parallel ·\parallel }_{W^{k,p}(\Omega )}$, respectively. Furthermore, we write $(u·\nabla )w$ as $u·\nabla w$, and denote by $\mathrm{div},\mathrm{curl}$ the div and curl operators, respectively. For notational convenience, $\Omega$ may be omitted when we write the spaces and integration over the $\Omega$ without confusion. In the sequel, we set

$$\begin{array}{c}\hfill \mathrm{curl}u=w,\mathrm{curl}w=\zeta ,\mathrm{curl}\zeta =\chi .\end{array}$$

(17)

We recall the following basic vector identities:

$$\begin{array}{cc}\hfill {\mathrm{curl}}^{2}a& =\nabla \left(\mathrm{div}a\right)-\Delta a,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \mathrm{curl}\left(fa\right)& =f\mathrm{curl}a+\nabla f\times a,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \mathrm{div}(a\times b)& =b·\left(\mathrm{curl}a\right)-a·\left(\mathrm{curl}b\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \mathrm{curl}(a\times b)& =b·\nabla a-a·\nabla b+a\left(\mathrm{div}b\right)-b\left(\mathrm{div}a\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \nabla (a·b)& =a·\nabla b+b·\nabla a+a\times \left(\mathrm{curl}b\right)+b\times \left(\mathrm{curl}a\right),\hfill \end{array}$$

(22)

where f is a scalar function, and a and b are vector functions. After that, we introduce an auxiliary result by integration by parts.

Lemma 1

(cf. [38]). Let Ω be a regular, open bounded set in ${\mathbb{R}}^3$. Then, for each $p>1$, and a smooth enough vector field v, the following identity holds true:

$$\begin{array}{cc}\hfill -\int \Delta v·v\mid v{\mid }^{p-2}dx=& \frac{1}{2}\int \mid \nabla v{\mid }^2\mid v{\mid }^{p-2}dx+\frac{4(p-2)}{p^2}\int \mid \nabla \mid v{\mid }^{\frac{p}{2}}{\mid }^{2}dx\hfill \\ \hfill & -{\int }_{\partial \Omega }\mid v{\mid }^{p-2}{\partial }_{n}v·vdS.\hfill \end{array}$$

(23)

As remarked in [39], one has

$$\begin{array}{c}\hfill \mid \nabla \mid v{\mid }^{\frac{p}{2}}{\mid }^2\le {\left(\frac{p}{2}\right)}^2\mid \nabla v{\mid }^2\mid v{\mid }^{p-2}.\end{array}$$

(24)

Therefore, for $p\ge 2$, the second term on the right-hand side of Equation (23) is controlled by the first term, the third term requires the analysis of boundary conditions. Taking the curl of Equation (1), by (19), we obtain equality (25); the following lemmas are inspired by some results in [38].

Lemma 2.

Assume, $u·n=0,w\times n=0,\nabla \rho ·n=0$ on $\partial \Omega$, and that w satisfies the equation

$$\begin{array}{c}\hfill \rho ({\partial }_{t}w+u·\nabla w-w·\nabla u)+\nabla \rho \times ({\partial }_{t}u+u·\nabla u)=\nu \Delta w,\end{array}$$

(25)

where $\nu >0$. Then, $\chi \times n=0$ on $\partial \Omega$.

Proof. 

In the flat boundary case, the boundary conditions (5) follow that

$$\begin{array}{c}\hfill u_3=w_1=w_2=0on\partial \Omega .\end{array}$$

Then, the vector fields $(u·\nabla )w$ and $(w·\nabla )u$ are normal to $\partial \Omega$; see [38]. Furthermore,

$$\begin{array}{c}\hfill (\nabla \rho \times ({\partial }_{t}u+u·\nabla u))\times n=(\nabla \rho ·n)({\partial }_{t}u+u·\nabla u)-(({\partial }_{t}u+u·\nabla u)·n)\nabla \rho ,\end{array}$$

then the vector field $\nabla \rho \times ({\partial }_{t}u+u·\nabla u)$ is normal to $\partial \Omega$. Hence, we have that $\Delta w$ is normal to $\partial \Omega$. This completes the proof of Lemma 2. □

In addition, we introduce some results in [38].

Lemma 3

(cf. [38]). Under the assumptions of Lemma 2, one has

$$\begin{array}{c}\hfill \left({\partial }_i{\zeta }_j\right)n_i{\zeta }_j=0\mathrm{and}\left({\partial }_i{\chi }_j\right)n_i{\chi }_j=0\mathrm{on}\partial \Omega .\end{array}$$

Lemma 4

(cf. [38]). Let $u,w,\zeta ,\chi$ be as above. Then, one has the following norm-equivalence results.

$$\begin{array}{c}\hfill {\parallel \chi \parallel }_p\simeq {\parallel \zeta \parallel }_{1,p}\simeq {\parallel w\parallel }_{2,p}\simeq {\parallel u\parallel }_{3,p}.\end{array}$$

When we take the operator ${\mathrm{curl}}^2$ to (25), ${\partial }_t\xi ,{\partial }_{t}w$ will be derived. So, then, we must estimate ${\partial }_{t}w$. The following inequality is applied frequently.

Lemma 5.

Assume $p>\frac{3}{2},({\rho }_0,u_0)\in W^{3,p}$ satisfies (6) and (7). Let $(\rho ,u)$ be a strong solution of problem (1)–(5). Then,

$$\begin{array}{c}\hfill \parallel {\partial }_t{w\parallel }_p\le C(S+S^{1+\gamma }),\end{array}$$

(26)

where $S:={\parallel \rho \parallel }_{3,p}+{\parallel u\parallel }_{3,p}$ and $\gamma =\frac{5p-6}{2p-3}$.

Proof. 

From (25), it follows that

$$\begin{array}{cc}\hfill \parallel {\partial }_t{w\parallel }_p\le & \parallel (u·\nabla )w-(w·\nabla )u-{\rho }^{-1}{\nu \Delta w\parallel }_p\hfill \\ \hfill & +\parallel {\rho }^{-1}\nabla \rho \times {\partial }_t{u\parallel }_p+{\parallel {\rho }^{-1}\nabla \rho \times (u·\nabla )u\parallel }_p.\hfill \end{array}$$

(27)

First, we estimate the first term on the right side of the above inequality.

$$\begin{array}{cc}\hfill \parallel u·\nabla w-w·\nabla u-{\rho }^{-1}{\nu \Delta w\parallel }_p& \le C(\parallel u{\parallel }_{\infty }\parallel \nabla w{\parallel }_p+\parallel w{\parallel }_{2p}\parallel \nabla u{\parallel }_{2p}+\nu \parallel {\rho }^{-1}\Delta w{\parallel }_p)\hfill \\ \hfill & \le C(\parallel u{\parallel }_{3,p}+\parallel u{\parallel }_{3,p}^2).\hfill \end{array}$$

Next, we estimate the second term on the right side of the above inequality. Noticing that

$$\begin{array}{cc}\hfill \parallel {\partial }_t{u\parallel }_2& \le C(\nu \parallel \Delta u{\parallel }_2+\parallel u·\nabla u{\parallel }_2)\hfill \\ \hfill & \le C(\nu \parallel \Delta u{\parallel }_2+\parallel u{\parallel }_2\parallel \nabla u{\parallel }_{\infty })\hfill \\ \hfill & \le {C\parallel u\parallel }_{3,p},\hfill \end{array}$$

accordingly, one obtains

$$\begin{array}{cc}\hfill \parallel {\rho }^{-1}\nabla \rho \times {\partial }_t{u\parallel }_p& \le {C\parallel \nabla \rho \parallel }_{2p}{\parallel {\partial }_{t}u\parallel }_{2p}\hfill \\ \hfill & \le {C\parallel \nabla \rho \parallel }_{1,p}\parallel {\partial }_t{u\parallel }_2^{1-\theta }{\parallel {\partial }_{t}u\parallel }_{1,p}^{\theta }\hfill \\ \hfill & \le {C\parallel \nabla \rho \parallel }_{1,p}^{\gamma }\parallel {\partial }_t{u\parallel }_2+\frac{1}{2}{\parallel {\partial }_{t}u\parallel }_{1,p}\hfill \\ \hfill & \le {C(\parallel \rho \parallel }_{3,p}^{1+\gamma }+{\parallel u\parallel }_{3,p}^{1+\gamma })+\frac{1}{2}{\parallel {\partial }_{t}w\parallel }_p,\hfill \end{array}$$

where $\theta =\frac{3(p-1)}{5p-6}$ and $\gamma =\frac{1}{1-\theta }$.

Finally, we estimate the third term on the right side of the above inequality.

$$\begin{array}{cc}\hfill \parallel {\rho }^{-1}{\nabla \rho \times (u·\nabla )u\parallel }_p& \le {C\parallel \nabla \rho \parallel }_{2p}{\parallel u\parallel }_{2p}{\parallel \nabla u\parallel }_{\infty }\hfill \\ \hfill & \le {C\parallel \nabla \rho \parallel }_{1,p}{\parallel u\parallel }_2^{1-\theta }{\parallel u\parallel }_{1,p}^{\theta }{\parallel u\parallel }_{3,p}\hfill \\ \hfill & \le C(\parallel u{\parallel }_{3,p}+\parallel \rho {\parallel }_{3,p}^{1+\gamma }+\parallel u{\parallel }_{3,p}^{1+\gamma }).\hfill \end{array}$$

since $\gamma >1+\theta$, where $\theta ,\gamma$ is the same as above.

Combine the above estimates, from (27), we finally obtain (26). This completes the proof of Lemma 5. □

For convenience, we define $M:={\partial }_{t}u+u·\nabla u$. Obviously, the following conclusions hold:

$$\begin{array}{c}\hfill {\parallel \mathrm{curl}M\parallel }_p\le C(S+S^{1+\gamma }),p>\frac{3}{2};\end{array}$$

(28)

$$\begin{array}{c}\hfill {\parallel \nabla M\parallel }_p\le C(S+S^{1+\gamma }),p>\frac{3}{2};\end{array}$$

(29)

$$\begin{array}{c}\hfill {\parallel M\parallel }_{\infty }\le C(S+S^{1+\gamma }),p>3.\end{array}$$

(30)

3. Uniform Estimate

We show the following uniform bound, which is the key of the proof of Theorem 1.

Proposition 1.

Under the assumption of Theorem 1, For each $\nu >0$, $(\rho ,u)$ is the strong solution to the initial boundary value problem (1)–(7) in Ω, then

$$\begin{array}{c}\hfill {\parallel \rho \parallel }_{3,p}+{\parallel \chi \parallel }_p\le C,\end{array}$$

for $t\in [0,T_0]$, where C is a constant independent of ν.

Proof. 

We take the curl to Equation (25), it follows that

$$\begin{array}{c}\hfill {\mathrm{curl}}^{2}M={\rho }^{-1}(\nu \Delta \zeta -\nabla \rho \times \mathrm{curl}M-\mathrm{curl}(\nabla \rho \times M)).\end{array}$$

(31)

Taking the operator ${\mathrm{curl}}^2$ to (25), by (19), we have

$$\begin{array}{cc}\hfill \rho {\partial }_t\chi -\nu \Delta \chi =& -\rho {\mathrm{curl}}^3(u·\nabla u)-\nabla \rho \times {\mathrm{curl}}^{2}M\hfill \\ \hfill & -\mathrm{curl}(\nabla \rho \times \mathrm{curl}M)-{\mathrm{curl}}^2(\nabla \rho \times M).\hfill \end{array}$$

(32)

Then, taking the inner product to (32), with $\chi \mid \chi {\mid }^{p-2}$, applying Lemma 1 and Lemma 3, it follows that

$$\begin{array}{c}\hfill \frac{1}{p}\frac{d}{dt}{\parallel \sqrt[p]{\rho }\chi \parallel }_p^p+\frac{\nu }{2}\int \mid \chi {\mid }^{p-2}\mid \nabla \chi {\mid }^{2}dx+4\nu \frac{p-2}{p^2}\int \mid \nabla \mid \chi {\mid }^{\frac{p}{2}}{\mid }^{2}dx=\sum _{i=1}^4{J}_i.\end{array}$$

(33)

For $J_1$, It is easy to obtain

$$\begin{array}{c}{J}_1: =(-\rho {\mathrm{curl}}^3(u·\nabla u)+\rho u·\nabla \chi ,\chi \mid \chi {\mid }^{p-2})\hfill \\ \begin{array}{cc}\hfill & \le C\int (\mid Du\mid \mid D^{2}w\mid +\mid D^{2}u\mid \mid Dw\mid )\mid \chi {\mid }^{p-1}dx\hfill \\ \hfill & \le {C(\parallel Du\parallel }_{\infty }\parallel D^2{w\parallel }_p+\parallel D^2{u\parallel }_{2p}^2{)\parallel \chi \parallel }_p^{p-1}\hfill \end{array}\hfill \\ \le C{S}^{p+1},\hfill \end{array}$$

(34)

where we use (1) and the estimates ${\parallel Du\parallel }_{\infty }\le C{\parallel Du\parallel }_{2,p}$, and $\parallel D^2{u\parallel }_{2p}\le C{\parallel D^{3}u\parallel }_p$, since $W^{1,p}\subset L^{2p}$ for $p>\frac{3}{2}$.

Next, we estimate $J_2$ by (31).

$$\begin{array}{cc}\hfill J_2:& =(-\nabla \rho \times {\mathrm{curl}}^{2}M,\chi \mid \chi {\mid }^{p-2})\hfill \\ \hfill & =\int \nabla \rho \times {\rho }^{-1}(-\nu \Delta \zeta +\nabla \rho \times \mathrm{curl}M+\mathrm{curl}(\nabla \rho \times M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =\sum _{i=1}^3{J}_{2i}.\hfill \end{array}$$

Noticing that $-\Delta \zeta =\mathrm{curl}\chi$, using Hölder’s inequality and Young’s inequality, we have

$$\begin{array}{cc}\hfill J_{21}:& =-\int \nabla \rho \times ({\rho }^{-1}\nu \Delta \zeta )·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le \int {\rho }^{-1}\nu \mid \nabla \rho \mid \mid \nabla \chi \mid \mid \chi {\mid }^{\frac{p-2}{2}}\mid \chi {\mid }^{\frac{p}{2}}dx\hfill \\ \hfill & \le {C\nu \parallel \nabla \rho \parallel }_{\infty }^2{\parallel \chi \parallel }_p^p+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le C\nu S^{p+2}+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\hfill \end{array}$$

By (28), the estimate to $J_{22}$ is easy, and we write it directly

$$\begin{array}{cc}\hfill J_{22}:& =\int \nabla \rho \times ({\rho }^{-1}\nabla \rho \times \mathrm{curl}M)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le {C\parallel \nabla \rho \parallel }_{\infty }^2{\parallel \mathrm{curl}M\parallel }_p{\parallel \chi \parallel }_p^{p-1}\hfill \\ \hfill & \le C(S^{p+2}+S^{p+2+\gamma }).\hfill \end{array}$$

Using Hölder’s inequality and Young’s inequality, and the identity (21), by (28) and (29), we have

$$\begin{array}{cc}\hfill J_{23}:& =\int \nabla \rho \times \left({\rho }^{-1}\mathrm{curl}(\nabla \rho \times M)\right)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le C\int \mid \nabla \rho \mid (\mid \nabla \rho \mid \mid \nabla M\mid +\mid {\nabla }^2\rho \mid \mid M\mid )\mid \chi {\mid }^{p-1}dx\hfill \\ \hfill & \le {C(\parallel \nabla \rho \parallel }_{\infty }^2{\parallel \nabla M\parallel }_p+{\parallel \nabla \rho \parallel }_{\infty }{\parallel M\parallel }_{2p}\parallel {\nabla }^2{\rho \parallel }_{2p}{)\parallel \chi \parallel }_p^{p-1}\hfill \\ \hfill & \le C(S^{p+2}+S^{p+2+\gamma }).\hfill \end{array}$$

Owing to the estimates $J_{21},J_{22},J_{23}$, we obtain

$$\begin{array}{c}\hfill J_2\le C(S^{p+1}+S^{p+2+\gamma })+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\end{array}$$

(35)

Likewise, we estimate $J_3$ by (25).

$$\begin{array}{cc}\hfill J_3:& =(-\mathrm{curl}(\nabla \rho \times \mathrm{curl}M),\chi \mid \chi {\mid }^{p-2})\hfill \\ \hfill & =\int \mathrm{curl}(\nabla \rho \times {\rho }^{-1}(\nabla \rho \times M-\nu \Delta w))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =J_{31}+J_{32}.\hfill \end{array}$$

We have

$$\begin{array}{cc}\hfill J_{31}:& =\int \mathrm{curl}(\nabla \rho \times {\rho }^{-1}(\nabla \rho \times M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le C\int (\mid D^2\rho \mid \mid \nabla \rho \mid \mid M\mid +\mid \nabla \rho {\mid }^2\mid DM\mid +\mid \nabla \rho {\mid }^3\left|M\right|)\mid \chi {\mid }^{p-1}dx\hfill \\ \hfill & \le C(\parallel D^2{\rho \parallel }_{2p}{\parallel \nabla \rho \parallel }_{\infty }{\parallel M\parallel }_{2p}+{\parallel \nabla \rho \parallel }_{\infty }^2{\parallel DM\parallel }_p+{\parallel \nabla \rho \parallel }_{\infty }^3{\parallel M\parallel }_p{)\parallel \chi \parallel }_p^{p-1}\hfill \\ \hfill & \le C(S^{p+2+\gamma }+S^{p+3+\gamma }).\hfill \end{array}$$

To estimate $J_{32}$, via integration by parts, and noticing that $n\times \chi =0$ on the boundary, we have

$$\begin{array}{cc}\hfill J_{32}:& =-\int \mathrm{curl}(\nabla \rho \times {\rho }^{-1}\nu \Delta w)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =-\int (\nabla \rho \times {\rho }^{-1}\nu \Delta w)·\left(\mathrm{curl}(\chi \mid \chi {\mid }^{p-2})\right)dx\hfill \\ \hfill & \le C\nu \int \mid \nabla \rho \mid \mid \nabla \chi \mid \mid \chi {\mid }^{p-1}dx\hfill \\ \hfill & \le C\nu \int \mid \nabla \rho {\mid }^2\mid \chi {\mid }^{p}dx+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le C\nu S^{p+2}+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\hfill \end{array}$$

Owing to the estimates $J_{31},J_{32}$, which yield

$$\begin{array}{c}\hfill J_3\le C(S^{p+1}+S^{p+3+\gamma })+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\end{array}$$

(36)

Finally, we estimate $J_4$. Applying the identity (18), we have

$$\begin{array}{cc}\hfill J_4:& =(-{\mathrm{curl}}^2(\nabla \rho \times M),\chi \mid \chi {\mid }^{p-2})\hfill \\ \hfill & =\int (\Delta (\nabla \rho \times M)-\nabla \left(\mathrm{div}(\nabla \rho \times M)\right))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =J_{41}+J_{42}.\hfill \end{array}$$

Now, we estimate $J_{41}$, applying the identity (18), one has

$$\begin{array}{cc}\hfill \int (\nabla \rho \times \Delta M)·\chi \mid \chi {\mid }^{p-2}dx=& \int (\nabla \rho \times (\nabla \mathrm{div}(u·\nabla u)-{\mathrm{curl}}^{2}M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill \le & {C\parallel \nabla \rho \parallel }_{\infty }{\parallel \nabla u\parallel }_{\infty }\parallel {\nabla }^2{u\parallel }_p{\parallel \chi \parallel }_p^{p-1}-J_2.\hfill \end{array}$$

By (30), we have

$$\begin{array}{cc}\hfill J_{41}:& =\int \Delta (\nabla \rho \times M)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le {C(\parallel \nabla \Delta \rho \parallel }_p{\parallel M\parallel }_{\infty }+\parallel D^2{\rho \parallel }_{\infty }{\parallel DM\parallel }_p{)\parallel \chi \parallel }_p^{p-1}\hfill \\ \hfill & +\int (\nabla \rho \times \Delta M)·\chi \mid \chi {\mid }^{p-2}dx)\hfill \\ \hfill & \le {C(\parallel S\parallel }^{p+1}+{\parallel S\parallel }^{p+1+\gamma })-J_2.\hfill \end{array}$$

Thanks to

$$\begin{array}{cc}\hfill & \int \nabla \rho ·\nabla \left(\mathrm{curl}M\right)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill =& \int \nabla \rho ·\nabla \left({\rho }^{-1}(\nu \Delta w-\nabla \rho \times M)\right)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill =& \int \nabla \rho ·\nabla ({\rho }^{-1}\nu \Delta w)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & -\int \nabla \rho ·\nabla ({\rho }^{-1}\nabla \rho \times M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill =& -\int {\rho }^{-1}\nu \Delta \rho \mid \chi {\mid }^{p}dx-\int \nabla \rho ·\nabla (\chi \mid \chi {\mid }^{p-2})·\left({\rho }^{-1}\nu \chi \right)dx\hfill \\ \hfill & -\int \nabla \rho ·\nabla ({\rho }^{-1}\nabla \rho \times M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill \le & C\nu (\int \mid \Delta \rho \mid \mid \chi {\mid }^{p}dx+\int \mid \nabla \rho \mid \mid \nabla \chi \mid \mid \chi {\mid }^{p-1}dx)\hfill \\ \hfill & +C\int (\mid D^2\rho \mid \mid \nabla \rho \mid \mid M\mid +\mid \nabla \rho {\mid }^2\mid DM\mid +\mid \nabla \rho {\mid }^3\mid M\mid )\mid \chi {\mid }^{p-1}dx\hfill \\ \hfill \le & {C\nu (\parallel \Delta \rho \parallel }_{\infty }{\parallel \chi \parallel }_p^p+{\parallel \nabla \rho \parallel }_{\infty }^2{\parallel \chi \parallel }_p^p)+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & +C(\parallel D^2{\rho \parallel }_{2p}{\parallel \nabla \rho \parallel }_{\infty }{\parallel M\parallel }_{2p}+{\parallel \nabla \rho \parallel }_{\infty }^2{\parallel DM\parallel }_p+{\parallel \nabla \rho \parallel }_{\infty }^3{\parallel M\parallel }_p{)\parallel \chi \parallel }_p^{p-1}\hfill \\ \hfill \le & {C(\parallel S\parallel }^{p+1}+{\parallel S\parallel }^{p+3+\gamma })+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\hfill \end{array}$$

Thus, by (20) and (22), one has

$$\begin{array}{cc}\hfill J_{42}:& =-\int \nabla \left(\mathrm{div}(\nabla \rho \times M)\right)·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =\int \nabla (\nabla \rho ·\left(\mathrm{curl}M\right))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & =\int (\nabla \rho ·\nabla \left(\mathrm{curl}M\right)+\left(\mathrm{curl}M\right)·{\nabla }^2\rho +\nabla \rho \times {\mathrm{curl}}^{2}M))·\chi \mid \chi {\mid }^{p-2}dx\hfill \\ \hfill & \le \int \nabla \rho ·\nabla \left(\mathrm{curl}M\right)·\chi \mid \chi {\mid }^{p-2}dx+{\parallel \mathrm{curl}M\parallel }_p\parallel {\nabla }^2{\rho \parallel }_{\infty }{\parallel \chi \parallel }_p^{p-1}+J_2\hfill \\ \hfill & \le {C(\parallel S\parallel }^{p+1}+{\parallel S\parallel }^{p+3+\gamma })+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx+J_2.\hfill \end{array}$$

Combined with the above inequalities, we have

$$\begin{array}{c}\hfill J_4\le {C(\parallel S\parallel }^{p+1}+{\parallel S\parallel }^{p+3+\gamma })+\frac{\nu }{8}\int \mid \nabla \chi {\mid }^2\mid \chi {\mid }^{p-2}dx.\end{array}$$

(37)

Inserting (34)–(37) into (33), by (24), we obtain

$$\begin{array}{cc}\hfill & \frac{1}{p}\frac{d}{dt}{\parallel \sqrt[p]{\rho }\chi \parallel }_p^p+c\nu \int \mid \nabla \mid \chi {\mid }^{\frac{p}{2}}{\mid }^{2}dx\le C(S^{p+1}+S^{p+3+\gamma }).\hfill \end{array}$$

(38)

Applying the operator $D^{\ell }$, and $\mid \ell \mid =3$, to both sides of Equation (1), multiplying both sides of the above equation by $D^{\ell }\rho \mid D^{\ell }\rho {\mid }^{p-2}$ and integrating in $\Omega$, one obtains

$$\begin{array}{cc}\hfill \frac{1}{p}\frac{d}{dt}{\parallel D^{\ell }\rho \parallel }_p^p& =-(u·\nabla D^{\ell }\rho ,D^{\ell }\rho \mid D^{\ell }\rho {\mid }^{p-2})\hfill \\ \hfill & -\sum _{r+s=\ell ,\mid s\mid \ge 1}(D^r\nabla \rho ·D^{s}u,D^{\ell }\rho \mid D^{\ell }\rho {\mid }^{p-2})\hfill \\ \hfill & =-\sum _{r+s=\ell ,\mid s\mid \ge 1}(D^r\nabla \rho ·D^{s}u,D^{\ell }\rho \mid D^{\ell }\rho {\mid }^{p-2})\hfill \\ \hfill & \le C(\parallel {\nabla }^3{\rho \parallel }_p{\parallel \nabla u\parallel }_{\infty }+\parallel {\nabla }^2{\rho \parallel }_p\parallel {\nabla }^2{u\parallel }_{\infty }+{\parallel \nabla \rho \parallel }_{\infty }\parallel {\nabla }^3{u\parallel }_p{)\parallel \rho \parallel }_{3,p}^{p-1}.\hfill \end{array}$$

By Young’s inequality, one obtains

$$\begin{array}{c}\hfill \frac{1}{p}\frac{d}{dt}{\parallel \rho \parallel }_{3,p}^p\le C{S}^{p+1}.\end{array}$$

(39)

Hence, it follows from (38) and (39) that

$$\begin{array}{cc}\hfill \frac{1}{p}& \frac{d}{dt}{(\parallel \rho \parallel }_{3,p}^p+\parallel \sqrt[p]{\rho }\chi {\parallel }_p^p)+c\nu \int \mid \nabla \mid \chi {\mid }^{\frac{p}{2}}{\mid }^{2}dx\le C_{\nu }(S^{p+1}+S^{p+3+\gamma }).\hfill \end{array}$$

(40)

In order to avoid the useless dependence on $\nu$ for the existence time of the solution, we assume that $\nu \le {\nu }_1$, then, $C_{\nu }$ can be denoted by general constant C.

From comparison theorems for ordinary differential equations applied to (40), it follows that ${\parallel \rho \parallel }_{3,p}+{\parallel \sqrt[p]{\rho }\chi \parallel }_p\le y\left(t\right)$, where $y\left(t\right)$ satisfies

$$\begin{array}{ccc}\hfill & y^{\prime }=C(y^2+y^{4+\gamma }),\hfill & \hfill y\left(0\right)=y_0=\parallel \sqrt[p]{\rho }{\chi \left(0\right)\parallel }_p+{\parallel \rho \left(0\right)\parallel }_{3,p}.\end{array}$$

We find that there is a time $T^{*}$, such that, for arbitrary $T_0\in (0,T^{*})$

$$\begin{array}{c}\hfill {\parallel \rho \parallel }_{3,p}+{\parallel \chi \parallel }_p\le C\end{array}$$

(41)

hold for all $t\in [0,T_0]$.

This completes the proof of Proposition 1. □

4. Proof of Theorem 1

Thanks to Proposition 1, from (26), we obtain directly

$$\begin{array}{c}\hfill \parallel {\partial }_t{w\parallel }_{L^{\infty }(0,T_0;L^p)}\le C.\end{array}$$

Taking the operator $D^2$ to both sides of equation (1), obviously,

$$\begin{array}{c}\hfill D^2(u·\nabla \rho )\in L^{\infty }(0,T_0;L^p),\end{array}$$

which is to say that

$$\begin{array}{c}\hfill \parallel {\partial }_t{\rho \parallel }_{L^{\infty }(0,T_0;W^{2,p})}\le C.\end{array}$$

This is enough to prove the week convergence results (13) and (14). Moreover, since the embedding $W^{s,p}\hookrightarrow W^{s^{\prime },p}$ for all $s^{\prime }<s$ is compact, by using the standard interpolation theory, we obtain the strong convergence (15).

We now establish the rate of convergence in the sense of $W^{2,p}$ space for a strong solution of the nonhomogeneous Navier–Stokes system with slip boundary conditions to the strong solution of the nonhomogeneous Euler system when the viscosity coefficient goes to zero.

The differences $\tilde{\rho }=\rho -{\rho }^0$, $\tilde{u}=u-u^0$, and $\tilde{p}=p-p^0$ satisfy

$$\begin{array}{cc}\hfill & {\partial }_t\tilde{\rho }=E,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\rho }^0({\partial }_t\tilde{u}+u^0·\nabla \tilde{u})+\nabla \tilde{p}=\nu \Delta u+F,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & \nabla ·\tilde{u}=0,\hfill \end{array}$$

(44)

where $E,F$ are defined as

$$\begin{array}{cc}\hfill & E=-(\tilde{u}·\nabla {\rho }^0+u·\nabla \tilde{\rho }),\hfill \\ \hfill & F=-\tilde{\rho }({\partial }_{t}u+u·\nabla u)-{\rho }^0\tilde{u}·\nabla u,\hfill \end{array}$$

with the initial boundary conditions

$$\begin{array}{cc}\hfill & (\tilde{\rho },\tilde{u}){\mid }_{t=0}=0,\mathrm{in}\Omega ,\hfill \\ \hfill & \tilde{u}·n=0,\mathrm{on}\partial \Omega \hfill \end{array}$$

and $w\times n=0,\left(\mathrm{curl}\zeta \right)\times n=0$, which follows from Lemma (2).

We first estimate the norm $\parallel \tilde{\rho }{\parallel }_{2,p}^p$ by applying the operator $D^2$ to (42) and taking the inner product with $D^2\tilde{\rho }\mid D^2\tilde{\rho }{\mid }^{p-2}$, it follows that

$$\begin{array}{cc}\hfill \frac{1}{p}\frac{d}{dt}{\parallel \tilde{\rho }\parallel }_{2,p}^p& =(D^{2}E+u·\nabla D^2\tilde{\rho },D^2\tilde{\rho }\mid D^2\tilde{\rho }{\mid }^{p-2})\hfill \\ \hfill & \le C(\parallel \tilde{\rho }{\parallel }_{2,p}^p+\parallel \tilde{\zeta }{\parallel }_p^p).\hfill \end{array}$$

(45)

Next, we estimate the norm $\parallel \tilde{\zeta }{\parallel }_p^p$ by applying the operator ${\mathrm{curl}}^2$ to (43) and taking the inner product with $\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2}$; it follows that

$$\begin{array}{c}\hfill \frac{1}{p}\frac{d}{dt}{\parallel \sqrt[p]{{\rho }^0}\tilde{\zeta }\parallel }_p^p=(\nu \Delta \zeta +{\mathrm{curl}}^{2}F+F_1,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2}),\end{array}$$

(46)

where $F_1={\mathrm{curl}}^2(-{\rho }^0({\partial }_t\tilde{u}+u·\nabla \tilde{u}))+{\rho }^0({\partial }_t\tilde{\zeta }+u·\nabla \tilde{\zeta })$.

Now, we consider the items on the right-hand side of the above equation. Using integrating by parts and Lemma 3, one can obtain

$$\begin{array}{cc}\hfill (\nu \Delta \zeta ,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})& =\nu (-\mathrm{curl}\zeta ,\mathrm{curl}(\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2}))\hfill \\ \hfill & \le \nu \int (\mid \chi \mid \mid \tilde{\chi }\mid \mid \tilde{\zeta }{\mid }^{p-2}+\mid \chi \mid \mid \tilde{\zeta }\mid \mid \nabla \mid \tilde{\zeta }{\mid }^{p-2}\mid )dx\hfill \\ \hfill & \le C\nu \int (\mid \chi \mid \mid \tilde{\chi }\mid \mid \tilde{\zeta }{\mid }^{p-2}+\mid \chi \mid \mid \nabla \tilde{\zeta }\mid \mid \tilde{\zeta }{\mid }^{p-2})dx\hfill \\ \hfill & \le {C\nu \parallel \chi \parallel }_p\parallel \tilde{\chi }{\parallel }_p{\parallel \tilde{\zeta }\parallel }_p^{p-2}\hfill \\ \hfill & \le C({\nu }^{\frac{p}{2}}+\parallel \tilde{\zeta }{\parallel }_p^p).\hfill \end{array}$$

Using Hölder inequality and uniform bound (41), we have

$$\begin{array}{cc}\hfill ({\mathrm{curl}}^{2}F,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})& =({\mathrm{curl}}^{2}F+\tilde{\rho }{\mathrm{curl}}^2{\partial }_{t}u,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})-(\tilde{\rho }{\mathrm{curl}}^2{\partial }_{t}u,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})\hfill \\ \hfill & =({\mathrm{curl}}^{2}F+\tilde{\rho }{\mathrm{curl}}^2{\partial }_{t}u,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})-(\mathrm{curl}{\partial }_{t}u,\mathrm{curl}(\tilde{\rho }\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2}))\hfill \\ \hfill & \le C(\parallel \tilde{\rho }{\parallel }_{2,p}^p+\parallel \tilde{\zeta }{\parallel }_p^p).\hfill \end{array}$$

In order to estimate the rest of the item, we restrict

$$\begin{array}{cc}\hfill \parallel {\partial }_t\tilde{w}{\parallel }_p& \le C\parallel \nu \Delta w+\mathrm{curl}(F-{\rho }^0{u}^0·\nabla \tilde{u})-\nabla {\rho }^0\times {\partial }_t\tilde{u}{\parallel }_p.\hfill \end{array}$$

By the same method in (26), we easily obtain

$$\begin{array}{cc}\hfill \parallel {\partial }_t\tilde{w}{\parallel }_p& \le C(\nu +\parallel \tilde{\rho }{\parallel }_{2,p}+\parallel \tilde{\zeta }{\parallel }_p),\hfill \end{array}$$

which follows that

$$\begin{array}{cc}\hfill \mid (F_1,\tilde{\zeta }\mid \tilde{\zeta }{\mid }^{p-2})\mid & \le \parallel F_1{\parallel }_p{\parallel \tilde{\zeta }\parallel }_p^{p-1}\hfill \\ \hfill & \le C(\parallel {\partial }_t\tilde{w}{\parallel }_p+\parallel \tilde{\zeta }{\parallel }_p)\parallel \tilde{\zeta }{\parallel }_p^{p-1}\hfill \\ \hfill & \le C(\nu +\parallel \tilde{\rho }{\parallel }_{2,p}+\parallel \tilde{\zeta }{\parallel }_p)\parallel \tilde{\zeta }{\parallel }_p^{p-1}\hfill \\ \hfill & \le C({\nu }^p+\parallel \tilde{\rho }{\parallel }_{2,p}^p+\parallel \tilde{\zeta }{\parallel }_p^p).\hfill \end{array}$$

Collecting all estimates, substituting (46) along with (45), we obtain the following differential inequality, for all $t\in [0,T_0]$,

$$\begin{array}{c}\hfill \frac{d}{dt}(\parallel \tilde{\rho }{\parallel }_{2,p}^p+\parallel \sqrt[p]{{\rho }^0}\tilde{\zeta }{\parallel }_p^p)\le C({\nu }^{\frac{p}{2}}+{\nu }^p+\parallel \tilde{\rho }{\parallel }_{2,p}^p+\parallel \tilde{\zeta }{\parallel }_p^p),\end{array}$$

then, by Grönwall’s inequality, we have

$$\begin{array}{c}\hfill \parallel \tilde{\rho }{\parallel }_{2,p}^p+{\parallel \tilde{\zeta }\parallel }_p^p\le C{\nu }^{\frac{p}{2}},\end{array}$$

which follows (16). This completes the proof of Theorem 1.