Incompressible inviscid limit of the viscous two-fluid model with general initial data

Abstract

In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space ${\mathbb {R}}^3$ with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.

1 Introduction

In this paper, we consider a compressible model of two-phase fluids in the following form [3, 26, 33]:

$$\begin{aligned}&\partial _t n + \mathrm{div}(n \mathbf{u} ) = 0, \end{aligned}$$

(1.1)

$$\begin{aligned}&\partial _t \varrho + \mathrm{div}(\varrho \mathbf{u} ) = 0, \end{aligned}$$

(1.2)

$$\begin{aligned}&\partial _t [(n+\varrho ) \mathbf{u} ] + \mathrm{div}[ (n+\varrho ) \mathbf{u} \otimes \mathbf{u} ] +\nabla P(n,\varrho )\nonumber \\&\quad =\mu \Delta \mathbf{u} +(\mu +\nu )\nabla \mathrm{div} \mathbf{u} , \end{aligned}$$

(1.3)

where the unknowns n and $\varrho$ denote the densities of the fluids, and $\mathbf{u} \in {\mathbb {R}}^3$ denotes the common velocity of the fluids. Here we assume that the two fluids obey the same velocity for simplicity. The parameters $\mu$ and $\lambda$ denote the viscosity coefficients satisfying $2\mu +\nu \ge 0$, and the pressure P takes the form $P(n,\varrho )=\frac{1}{\alpha }n^\alpha +\frac{1}{\gamma }\varrho ^\gamma$ with $\gamma >1$ and $\alpha > 1$.

The system (1.1)–(1.3) can be derived from the general two-fluid model [17, 28] or from a coupled system of the compressible Navier–Stokes equation and a Vlasov–Fokker–Planck equation by taking an asymptotic limit [3, 26]. The system (1.1)–(1.3) is also related to the compressible Oldroyd-B type model with stress diffusion [2].

Compared with the classical isentropic Navier–Stokes equations, the main difference is that the pressure law $P(n,\varrho )$ depends on two variables n and $\varrho$.

In a series of papers [7,8,9,10] by S. Evje and his collaborators, they obtained the global existence of weak solutions to the system (1.1)–(1.3) in one-dimensional case. Yao, et al. [35] studied the existence of asymptotic behaviour of global weak solutions to a 2D viscous liquid–gas two-phase flow model with small initial data. Recently, Vasseur, et al. [33] obtained the global weak solution to the viscous two-fluid model (1.1)–(1.3) with finite energy in the framework of DiPerna–Lions’ theory. Later, their results were extended to more general case by Novotný and Pokorný [28].

Yao et al. [36] studied the incompressible limit of the system (1.1)–(1.3) in the torus $\mathbb {T}^3$ with well-prepared initial data under the framework of local classical solutions. They obtained the convergence of the local strong solutions to the two-fluid model to that of the incompressible Navier–Stokes equations with an convergence rate.

In this paper, we study the incompressible limit of the compressible models of two-phase fluids (1.1)–(1.3) in the whole space ${\mathbb {R}}^3$ with general initial data in the framework of weak solutions established in [33]. We shall derive rigorously the incompressible Euler or Navier–Stokes equations based on the refined relative entropy method and the detailed analysis on the oscillations of the densities and the velocity. To begin with, we introduce the scaling

$$\begin{aligned} x\mapsto x,~t\mapsto \epsilon t,~ \mathbf{u} \mapsto \epsilon \mathbf{u} _\epsilon ,~\mu \mapsto \epsilon \mu _\epsilon ,~\nu \mapsto \epsilon \nu _\epsilon . \end{aligned}$$

(1.4)

Then the system (1.1)–(1.3) can be rewritten as

$$\begin{aligned}&\partial _t n_{\epsilon } + \mathrm{div}(n_{\epsilon } \mathbf{u} _\epsilon ) = 0, \end{aligned}$$

(1.5)

$$\begin{aligned}&\partial _t \varrho _{\epsilon } + \mathrm{div}(\varrho _{\epsilon } \mathbf{u} _\epsilon ) = 0, \end{aligned}$$

(1.6)

$$\begin{aligned}&\partial _t [(n_{\epsilon }+\varrho _{\epsilon }) \mathbf{u} _\epsilon ] + \mathrm{div}[ (n_{\epsilon }+\varrho _{\epsilon }) \mathbf{u} _\epsilon \otimes \mathbf{u} _\epsilon ] +\frac{1}{\epsilon ^2}\nabla \left( \frac{1}{\alpha }n_{\epsilon }^\alpha +\frac{1}{\gamma }\varrho _{\epsilon }^\gamma \right) \nonumber \\&\quad =\mu _\epsilon \Delta \mathbf{u} _\epsilon +(\mu _\epsilon +\nu _\epsilon )\nabla \mathrm{div} \mathbf{u} _\epsilon ,\end{aligned}$$

(1.7)

where we assume the initial data at the infinity:

$$\begin{aligned} n_\epsilon \rightarrow 1,~~\varrho _\epsilon \rightarrow 1,~~ \mathbf{u} _\epsilon \rightarrow 0, \end{aligned}$$

when $|x|\rightarrow \infty .$

Formally, by taking $\epsilon \rightarrow 0$ in (1.7), we get $n_{\epsilon } \rightarrow n(t)$ and $\varrho _{\epsilon } \rightarrow \varrho (t)$. If we further assume that the initial densities $n_{0,\epsilon }$ and $\varrho _{0,\epsilon }$ are small perturbation of some positive constant, say 1, we can also expect that $n_{\epsilon } \rightarrow 1$, $\varrho _{\epsilon } \rightarrow 1$ as $\epsilon \rightarrow 0$. Moreover, if we assume that the shear and bulk viscosity coefficients satisfy

$$\begin{aligned} \mu _\epsilon \rightarrow 0,\quad \nu _\epsilon \rightarrow 0\quad \text {as} \quad \epsilon \rightarrow 0, \end{aligned}$$

(1.8)

then the system (1.5)–(1.7) is reduced to the classical incompressible Euler equations (suppose that the limits $n_\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u}$ and $\varrho _\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u}$ exist):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \mathrm {div} \mathbf{u} =0,}\\ \displaystyle { \partial _t \mathbf{u} +( \mathbf{u} \cdot \nabla ) \mathbf{u} +\nabla \Pi =0. } \end{array} \right. \end{aligned}$$

(1.9)

In addition, if we suppose that

$$\begin{aligned} \mu _\epsilon \rightarrow {\bar{\mu }}>0,\quad \nu _\epsilon \rightarrow {\bar{\nu }} \quad \text {as} \quad \epsilon \rightarrow 0, \end{aligned}$$

(1.10)

then the system (1.5)–(1.7) is reduced to the classical incompressible Navier–Stokes equations (suppose that the limits $n_\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u}$ and $\varrho _\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u}$ exist):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \mathrm {div} \mathbf{u} =0,}\\ \displaystyle { \partial _t \mathbf{u} +( \mathbf{u} \cdot \nabla ) \mathbf{u} -{\bar{\mu }}\Delta \mathbf{u} +\nabla \Pi =0. } \end{array} \right. \end{aligned}$$

(1.11)

.

Our goal of this paper is to investigate the above limits rigorously in some suitable sense.

The outline of this article is as follows: In Sect. 2, we present the result of global weak solutions for the compressible models of two-phase fluids (1.5)–(1.7) and state our main results. In Sect. 3, we give the proofs of our results.

Before ending Introduction, we point out that the incompressible limits of the compressible Navier–Stokes equations and related models are very interesting topics and there are a lot of works on it. Among others, we mention [5, 15, 23, 25, 27] on the isentropic Navier–Stokes equations, [1, 12,13,14] on the full Navier–Stokes–Fourier system, [24, 34] on quantum isentropic Navier–Stokes equations, [16, 18, 19] on isentropic compressible magnetohydrodynamic equations, and [20, 21] on the full compressible magnetohydrodynamic equations.

2 Main results

In this section, we introduce our main results of incompressible limit for the compressible model of two-phase fluids (1.5)–(1.7) in the whole space ${\mathbb {R}}^3$.

For any vector field $\mathbf {v}$, we use $\mathbf{P}$ and $\mathbf{Q}$ to denote the divergence-free part of $\mathbf {v}$ and the gradient part of $\mathbf {v}$, respectively, i.e. $\mathbf{P} (\mathbf {v})=\mathbf {v}- \mathbf{Q} (\mathbf {v})$ and $\mathbf{Q} (\mathbf {v})=\nabla \Delta ^{-1}\mathrm{div}\mathbf {v}.$ Below the letter C denotes a generic positive constant, independent of $\epsilon$, and may change from line to line. And the letter $C_T$ denotes a generic positive constant, dependent on T.

We first recall the global weak solutions of the system (1.5)–(1.7). We assume that the initial data $(n_\epsilon ,\varrho _{\epsilon }, \mathbf{u} _\epsilon )|_{t=0}=(n_{0,\epsilon },\varrho _{0,\epsilon }, \mathbf{u} _{0,\epsilon })$ satisfy

$$\begin{aligned}&\int _{{\mathbb {R}}^3}\Big [\frac{(n_{0,\epsilon }+\varrho _{0,\epsilon })| \mathbf{u} _{0,\epsilon }|^2}{2}+G_\alpha (n_{0,\epsilon })+H_\gamma (\varrho _{0,\epsilon }) \Big ]\mathrm{d}x<\infty , \end{aligned}$$

(2.1)

$$\begin{aligned}&\inf _{x\in {{\mathbb {R}}^3}}n_{0,\epsilon }\ge 0,~\inf _{x\in {{\mathbb {R}}^3}}\varrho _{0,\epsilon }\ge 0,~G_\alpha (n_{0,\epsilon }),~H_\gamma (\varrho _{0,\epsilon })\in L^1({{\mathbb {R}}^3}), \end{aligned}$$

(2.2)

where $n_{0,\epsilon }\rightarrow 1,~\varrho _{0,\epsilon }\rightarrow 1,~ \mathbf{u} _{0,\epsilon }\rightarrow 0$ when $|x|\rightarrow \infty$ and $n_{0,\epsilon }-1\in L^\alpha ({{\mathbb {R}}^3}),~\varrho _{0,\epsilon }-1\in L^\gamma ({{\mathbb {R}}^3}).$ Furthermore, we assume

$$\begin{aligned} \frac{M_{0,\epsilon }}{\sqrt{n_{0,\epsilon }+\varrho _{0,\epsilon }}}\in L^2({{\mathbb {R}}^3})~\mathrm {if~}\frac{M_{0,\epsilon }}{\sqrt{n_{0,\epsilon }+\varrho _{0,\epsilon }}}=0\mathrm {~on~}\{x\in {{\mathbb {R}}^3} | n_{0,\epsilon }(x)+\varrho _{0,\epsilon }(x)=0\}, \end{aligned}$$

(2.3)

where

$$\begin{aligned} G_\alpha (n_{\epsilon }) = \frac{1}{\alpha (\alpha -1)\epsilon ^2}(n_\epsilon ^\alpha -1-\alpha (n_\epsilon -1)), \\ H_\gamma (\varrho _{\epsilon }) = \frac{1}{\gamma (\gamma -1)\epsilon ^2}(\varrho _\epsilon ^\gamma -1-\gamma (\varrho _\epsilon -1)), \end{aligned}$$

and $M_{0,\epsilon }=n_{0,\epsilon }+\varrho _{0,\epsilon }.$

Proposition 2.1

[33] Let $\epsilon >0$ be a fixed number. Suppose that the initial data $(n_{0,\epsilon },\varrho _{0,\epsilon }, \mathbf{u} _{0,\epsilon })$ satisfy (2.1)–(2.3). If

$$\begin{aligned} \alpha \ge 1,\quad \gamma >\frac{9}{5} \end{aligned}$$

and the initial data $n_{0,\epsilon }$ and $\varrho _{0,\epsilon }$ verify

$$\begin{aligned} \frac{1}{C_0}\varrho _{0,\epsilon }\le n_{0,\epsilon }\le C_0 \varrho _{0,\epsilon }\ \ \mathrm {~on~}\ \ {{\mathbb {R}}^3} \end{aligned}$$

(2.4)

for some positive constant $C_0$, then there exists a weak solution of the system (1.5)–(1.7) in the sense of distribution verifying the following energy inequality:

$$\begin{aligned} {E}_\epsilon (\tau )+\int _0^T\int _{{\mathbb {R}}^3}\Big (\mu _\epsilon |\nabla \mathbf{u} _\epsilon |^2+(\mu _\epsilon +\nu _\epsilon )|\mathrm{div} \mathbf{u} _\epsilon |^2 \Big )\mathrm{d}x\mathrm{d}t\le E_{0,\epsilon }, \end{aligned}$$

(2.5)

where

$$\begin{aligned}&{E}_\epsilon (\tau ):=\int _{{\mathbb {R}}^3}\Big (\frac{1}{2}(n_\epsilon +\varrho _\epsilon )| \mathbf{u} _\epsilon |^2+G_\alpha (n_\epsilon )+H_\gamma (\varrho _\epsilon )\Big )\mathrm{d}x,\\&{E}_{0,\epsilon }:=\int _{{\mathbb {R}}^3}\Big (\frac{1}{2}(n_{0,\epsilon }+\varrho _{0,\epsilon })| \mathbf{u} _{0,\epsilon }|^2+G_\alpha (n_{0,\epsilon })+H_\gamma (\varrho _{0,\epsilon }) \Big )\mathrm{d}x. \end{aligned}$$

Remark 2.1

Although the proof of global existence of the system (1.5)–(1.7) is given in [33] on bounded domains, we can prove a similar global weak solution of it in the whole space ${\mathbb {R}}^3$ based on the same spirit with the standard expanding domain techniques [29]. For the value of density at the infinity, we get: for fixed number $M\gg 1$, we define $\varrho _M:[0,\infty )\rightarrow [0,\infty )$ by

$$\begin{aligned} \varrho _M\in C^\infty ,~\varphi _M(x)=x ~\mathrm {if}~|x|\le M,~\varphi _M(x)=1~\mathrm {if}~|x|\ge 2M. \end{aligned}$$

Let $\overline{\varrho }_{0}(x)=\varphi _M (\varrho _0(x)).$ Then we construct a weak solution with the initial data $\overline{\varrho }_{0}(x)$ on $B_M(0)=\{ x| |x|\le M\}.$ Then, when $M\rightarrow \infty$, obviously, we get $\rho _0(x)\rightarrow 1$ when $|x|\rightarrow \infty .$ Similarly, we can also prove that $n_0(x)\rightarrow 1$ when $|x|\rightarrow \infty .$

Remark 2.2

In fact, [33] is a particular case of [28]. In particular, in [28], they have constructed solutions for $\alpha > 0$ and $\gamma \ge 9/5$. However, we here use the result of [33] for the convenient presentation such that it is more convenient to deal with the pressure terms of the relative entropy. Actually, the arguments in the proof of Theorem 2.1 still hold for all $\alpha >1$ and $\gamma >1$ provided that the system (1.5)–(1.7) has a global weak solution as stated in Proposition 2.1.

Remark 2.3

In [33], the authors pointed that the results in Proposition 2.1 still hold without the condition (2.4) on the initial densities if we have further restrictions on the powers of the pressure:

$$\begin{aligned} \alpha ,\gamma >\frac{9}{5}\mathrm {~and~}\max \left\{ \frac{3\gamma }{4},\gamma -1,\frac{3(\gamma +1)}{5}\right\}<\alpha <\max \left\{ \frac{3\gamma }{4},\gamma +1,\frac{5\gamma }{3}-1\right\} . \end{aligned}$$

Since our main concerns here are on the incompressible limit to the system (1.5)–(1.7), it is easy to check that our assumptions in Theorem 2.1 satisfy the condition (2.4).

Next, we recall the classical results on the incompressible Euler equations (1.9).

Proposition 2.2

[22, 32] Assume that the initial datum $\mathbf {u}(x,t)|_{t=0}=\mathbf {u}_0(x)$ satisfies

$$\begin{aligned} \mathbf {u}_0(x)\in H^{s}({\mathbb {R}}^3),\quad \text{ div }\, \mathbf {u}_0=0,\quad s> 5/2. \end{aligned}$$

(2.6)

Then there exist a $T^*\in (0,\infty )$ and a unique solution $\mathbf {u}$ to the incompressible Euler equations

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \mathrm {div} \, \mathbf{u} =0,}\\ \displaystyle { \partial _t \mathbf{u} +( \mathbf{u} \cdot \nabla ) \mathbf{u} +\nabla \Pi =0 } \end{array} \right. \end{aligned}$$

(2.7)

satisfying the following estimates

$$\begin{aligned} \sup _{0<t\le T}\big (\Vert \mathbf {u}\Vert _{H^s({\mathbb {R}}^3)}+\Vert \partial _t\mathbf {u}\Vert _{H^{s-1}({\mathbb {R}}^3)}+\Vert \nabla \Pi \Vert _{H^s({\mathbb {R}}^3)}+\Vert \partial _t\nabla \Pi \Vert _{H^{s-1}({\mathbb {R}}^3)}\big )\le C(T) \end{aligned}$$

(2.8)

with $C(T)>0$, a constant for any $0<T<T^*$.

Finally, we denote $n_\epsilon =1+\epsilon \varphi _\epsilon ,\varrho _\epsilon =1+\epsilon \psi _\epsilon$ and

$$\begin{aligned} \Phi _\epsilon =\sqrt{2G_\alpha (n_\epsilon )},~ \Phi _{0,\epsilon }=\sqrt{2G_\alpha (n_{0,\epsilon })},~\Psi _\epsilon =\sqrt{2H_\gamma (\varrho _\epsilon )},~ \Psi _{0,\epsilon }=\sqrt{2H_\gamma (\varrho _{0,\epsilon })}. \end{aligned}$$

Now we are in a position to state our results of this paper.

Theorem 2.1

Let $\alpha >1$ and $\gamma >9/5$ and

$$\begin{aligned} \mu _\epsilon = \nu _\epsilon =\epsilon ^a, \quad a>0. \end{aligned}$$

(2.9)

Assume that the initial data $(n_{0,\epsilon },\varrho _{0,\epsilon }, \mathbf{u} _{0,\epsilon })$ satisfy the conditions in Proposition 2.1 and $n_{0,\epsilon }=1+\epsilon \varphi _{0,\epsilon },~\varrho _{0,\epsilon }=1+\epsilon \psi _{0,\epsilon }$,

$$\begin{aligned}&\Vert \Phi _{0,\epsilon }-\varphi _{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})}+\Vert \Psi _{0,\epsilon }-\psi _{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\qquad +\,\Vert \sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon }-{\hat{ \mathbf{u} }}_0\Vert ^2_{L^2({{\mathbb {R}}^3})}+\Vert \sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon }-{\hat{ \mathbf{u} }}_0\Vert ^2_{L^2({{\mathbb {R}}^3})}\le C\epsilon ^b, \end{aligned}$$

(2.10)

$$\begin{aligned}&\Vert \varphi _{0,\epsilon }-\psi _{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})}\le C\epsilon ^c \end{aligned}$$

(2.11)

for some constants $b>0$ and $c>0$, where

$$\begin{aligned} \Vert \varphi _{0,\epsilon }\Vert _{L^\infty \cap L^2 (\mathbb {R}^3)} +\Vert \psi _{0,\epsilon }\Vert _{ L^\infty \cap L^2 (\mathbb {R}^3)}\le c_0,~~ \mathbf{P} ({\hat{ \mathbf{u} }}_0)\in H^{k}({{\mathbb {R}}^3};\mathbb {R}^3), \end{aligned}$$

(2.12)

for some $k>{7}/{2}$. Let

$$\begin{aligned} \sigma =\left\{ \begin{array}{ll} \min \{\frac{1}{4},\frac{b}{2},a,c\},&{}\quad 1<\alpha \le 4, ~~ 9/5<\gamma \le 4,\\ \min \{\frac{1}{4},\frac{b}{2},\frac{\alpha -4}{2\alpha },a,c\},&{}\quad \alpha>4,~~9/5<\gamma \le 4, \\ \min \{\frac{1}{4},\frac{b}{2},\frac{\gamma -4}{2\gamma },a,c\},&{}\quad 1 <\alpha \le 4,~~\gamma>4, \\ \min \{\frac{b}{2},\frac{\alpha -4}{2\alpha },\frac{\gamma -4}{2\gamma },a,c\},\qquad &{}\quad \alpha>4,~~\gamma > 4, \end{array}\right. \end{aligned}$$

(2.13)

and $( \mathbf{u} ,\Pi )$ be the local strong solution, on the time interval $0,T^*)$, to the incompressible Euler equations (2.7) with the initial datum $\mathbf{u} (x,0)= \mathbf{u} _0=\mathbf { P}({\hat{ \mathbf{u} }}_0)$. Then, for any $T<T_*$, the weak solution $(n_\epsilon ,\varrho _\epsilon , \mathbf{u} _\epsilon )$ of the system (1.5)–(1.7) established in Proposition 2.1 satisfies:

$$\begin{aligned}&\Vert \sqrt{n_\epsilon }-1\Vert _{L^\infty (0,T;L^2({\mathbb {R}}^3))}+ \Vert \sqrt{\varrho ^\epsilon }-1\Vert _{L^\infty (0,T;L^2({\mathbb {R}}^3))}\le C_T\epsilon , \end{aligned}$$

(2.14)

$$\begin{aligned}&\Vert \mathbf{P} (\sqrt{n_\epsilon } \mathbf{u} _\epsilon )- \mathbf{u} \Vert _{L^\infty (0,T; L^2({\mathbb {R}}^3))} + \Vert \mathbf{P} (\sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon )- \mathbf{u} \Vert _{L^\infty (0,T; L^2({\mathbb {R}}^3))}\le C_T \epsilon ^\sigma , \end{aligned}$$

(2.15)

$$\begin{aligned}&\Vert \sqrt{n_\epsilon } \mathbf{u} _{\epsilon }- \mathbf{u} \Vert _{L^r(0,T;L^2_{\mathrm {loc}}({{\mathbb {R}}^3}))}+\Vert \sqrt{\varrho _\epsilon } \mathbf{u} _{\epsilon }- \mathbf{u} \Vert _{L^r(0,T;L^2_{\mathrm {loc}}({{\mathbb {R}}^3}))} \le C_T \epsilon ^d \end{aligned}$$

(2.16)

for all $2<r<\infty$. Here $d=\min \big \{\frac{\sigma }{2},\frac{1}{r}\big \}$.

Remark 2.4

If we replace the assumption (2.9) by (1.10), we can obtain a similar result to Theorem 2.1. Here the target equations are the incompressible Navier–Stokes equations (1.11). Since the proof is similar to that of Theorem 2.1, we omit the details here for brevity.

Remark 2.5

When the initial data are well prepared, it is much easier to show that the weak solutions of the compressible two-fluid model converge to the local strong solution of the incompressible Euler or Navier–Stokes equations in the time interval provided that the latter exists. Moreover, we can obtain the convergence rates. In fact, in this case, because there are no oscillations, we do not need to use the Strichartz’s estimates of linear wave equations. Thus, the relative entropy to the system (1.5)–(1.7) becomes

$$\begin{aligned} \mathcal {E}_\epsilon (\tau )=\frac{1}{2}\int _{{\mathbb {R}}^3}\Big ((n_\epsilon +\varrho _\epsilon )| \mathbf{u} _\epsilon - \mathbf{u} |^2+|\Phi _\epsilon |^2+|\Psi _\epsilon |^2\Big )\mathrm{d}x, \end{aligned}$$

(2.17)

where $\mathbf{u}$ is the local strong solution of the incompressible Euler or Navier–Stokes equations. Since the proof is much simpler than that in Theorem 2.1, we will omit the details here. The readers can refer [15] on the discussion of isentropic Navier–Stokes equations.

Remark 2.6

For well-prepared initial data, we can also study the incompressible limit to the system (1.5)–(1.7) in the torus $\mathbb {T}^3$ or bounded domain and obtain a similar convergence result stated in the above remark. In fact, no Poincaré inequality is needed in our arguments. Thus, our results can be regarded as an extension and improvement in that in [36], where the incompressible limit is studied only for local strong solutions with well-prepared initial data in the $\mathbb {T}^3$. However, if we consider the general initial data case in the torus, the oscillations will survive forever and satisfy a parabolic equations; the readers can refer [27, 30] on the discussion of isentropic Navier–Stokes equations in the torus $\mathbb {T}^3$. See also [5] on the isentropic Navier–Stokes equations in the bounded domain case.

We give some comments on the proof of Theorem 2.1. We will make full use of the energy inequality, compact arguments, the refined relative entropy method (see [11, 13, 27]), and the Strichartz’s estimates of linear wave equations (see [6]). Thanks to the dispersive effects of the linear wave equations in the whole space ${\mathbb {R}}^3$, we can further obtain the convergence rate of the incompressible limit. Compared with the previous results on the isentropic Navier–Stokes equations (see [4, 15, 25, 27]), the main ingredient of this paper is that our wave equation includes the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolutions of the solutions. In fact, we have used the oscillation of $\frac{\rho _\epsilon -1+n_\epsilon -1}{2}$ for the density to figure out the pressure term in the relative entropy.

3 Proof of Theorem 2.1

In this section, we are going to give a rigorous proof of Theorem 2.1. We will make full use of the energy inequality, the refined relative entropy inequality, and the Strichartz’s estimates of linear wave equations to obtain the convergence rate of the solutions. From now on, we work on any time $T<T_*$, where $T_*$ is the maximal existing time of solutions to the incompressible Euler equations (2.7).

3.1 Uniform bounds

In this subsection, we are going to derive some estimates on the sequence $\{(n_\epsilon ,\varrho _\epsilon , \mathbf{u} _\epsilon )\}_{\epsilon >0}$.

From the energy inequality (2.5), we obtain that

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Vert \sqrt{n_\epsilon } \mathbf{u} _\epsilon (t)\Vert _{L^2({{\mathbb {R}}^3})}\le C_T, \end{aligned}$$

(3.1)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Vert \sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon (t)\Vert _{L^2({{\mathbb {R}}^3})}\le C_T, \end{aligned}$$

(3.2)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Vert n_\epsilon ^\alpha -1-\alpha (n_\epsilon -1)\Vert _{L^1({{\mathbb {R}}^3})}\le C_T\epsilon ^2, \end{aligned}$$

(3.3)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Vert \varrho _\epsilon ^\gamma -1-\gamma (\varrho _\epsilon -1)\Vert _{L^1({{\mathbb {R}}^3})}\le C_T\epsilon ^2. \end{aligned}$$

(3.4)

We consider the properties of convex function

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {s^\gamma -1-\gamma (s-1)\ge C|s-1|^2\quad \mathrm {if} \quad \gamma \ge 2,}\\ \displaystyle {s^\gamma -1-\gamma (s-1)\ge C|s-1|^2\quad \mathrm {if}\quad 1<\gamma< 2 \quad \mathrm { and } \quad 0<s\le R,}\\ \displaystyle {s^\gamma -1-\gamma (s-1)\ge C|s-1|^\gamma \quad \mathrm {if}\quad 1<\gamma < 2 \quad \mathrm { and }\quad s\ge R} \end{array}\right. \end{aligned}$$

(3.5)

for some constant $R>0$.

Let us introduce the set of the essential and residual values

$$\begin{aligned} g=[g]_{\mathrm {ess}}+[g]_{\mathrm {res}}, \end{aligned}$$

where $[g]_{\mathrm {ess}}=\chi (\varrho _\epsilon )g,~[g]_{\mathrm {res}}=(1-\chi (\varrho _\epsilon ))g$, and $\chi$ is defined as

$$\begin{aligned} \chi (r)= \left\{ \begin{array}{ll} 1, &{}\quad r\in [1/2,2],\\ 0, &{}\quad \mathrm {otherwise}. \end{array} \right. \end{aligned}$$

Following the estimates (3.3), (3.4), and the convexity (3.5), we get

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Big \Vert \Big [\frac{n_\epsilon -1}{\epsilon }\Big ]_{\mathrm {ess}}\Big \Vert _{L^2({{\mathbb {R}}^3})}\le C_T, \end{aligned}$$

(3.6)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\Big \Vert \Big [\frac{\varrho _\epsilon -1}{\epsilon }\Big ]_{\mathrm {ess}}\Big \Vert _{L^2({{\mathbb {R}}^3})}\le C_T,\end{aligned}$$

(3.7)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\int _{{\mathbb {R}}^3}[1+n_\epsilon ^\alpha ]_{\mathrm {res}}\mathrm{d}x\le C_T\epsilon ^2\end{aligned}$$

(3.8)

$$\begin{aligned}&\mathrm {ess}\sup _{t\in (0,T)}\int _{{\mathbb {R}}^3}[1+\varrho _\epsilon ^\gamma ]_{\mathrm {res}}\mathrm{d}x\le C_T\epsilon ^2. \end{aligned}$$

(3.9)

Noting that the following two elementary inequalities

$$\begin{aligned}&|\sqrt{x}-1|^2\le M|x-1|^\gamma ,\;\;\;|x-1|\ge \delta , \;\; \gamma \ge 1,\end{aligned}$$

(3.10)

$$\begin{aligned}&|\sqrt{x}-1|^2\le M|x-1|^2,\;\;\;x\ge 0, \end{aligned}$$

(3.11)

for some positive constant M and $0<\delta <1$, it is easy to obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^3}|\sqrt{n_\epsilon }-1|^2 =&\int _{{\mathbb {R}}^3}|\sqrt{n_\epsilon }-1|^2 1_{\{|n_\epsilon -1|\le 1/2\}}+\int _{{\mathbb {R}}^3}|\sqrt{n_\epsilon }-1|^2 1_{\{|n_\epsilon -1|> 1/2\}}\nonumber \\ \le&\,M\int _{{\mathbb {R}}^3}|\sqrt{n_\epsilon }-1|^2 1_{\{|n_\epsilon -1|\le 1/2\}}+M\int _{{\mathbb {R}}^3}|\sqrt{n_\epsilon }-1|^\gamma 1_{\{|n_\epsilon -1|> 1/2\}}\nonumber \\ \le&\,CM\epsilon ^2. \end{aligned}$$

(3.12)

It then follows that

$$\begin{aligned} \Vert \sqrt{n_\epsilon }-1\Vert _{L^\infty (0,T;L^2({\mathbb {R}}^3))}\le C_T\epsilon . \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert \sqrt{\varrho ^\epsilon }-1\Vert _{L^\infty (0,T;L^2({\mathbb {R}}^3))}\le C_T\epsilon . \end{aligned}$$

Hence, (2.14) holds.

In accordance with (3.6) and (3.8), we obtain that

$$\begin{aligned} n_\epsilon \rightarrow 1\ \ \mathrm {in}\ \ L^\infty (0,T;(L^2+L^p)({{\mathbb {R}}^3})),\quad p=\min \{\alpha ,2\}. \end{aligned}$$

(3.13)

Similarly,

$$\begin{aligned} \varrho _\epsilon \rightarrow 1\ \ \mathrm {in}\ \ L^\infty (0,T;(L^2+L^p)({{\mathbb {R}}^3})),\quad p=\min \{\gamma ,2\}. \end{aligned}$$

(3.14)

(3.13) and (3.14), together with (3.1) and (3.2), imply that, up to extraction an subsequence, still denoted by $n_\epsilon \mathbf{u} _\epsilon$ and $\varrho _\epsilon \mathbf{u} _\epsilon$,

$$\begin{aligned}&n_\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u} \ \ ~~\mathrm {weakly-\!*\ \ in}\ \ L^\infty (0,T;(L^2+L^{2\alpha /(\alpha +1})({{\mathbb {R}}^3};\mathbb {R}^3)),\end{aligned}$$

(3.15)

$$\begin{aligned}&\varrho _\epsilon \mathbf{u} _\epsilon \rightarrow \mathbf{u} \ \ ~~\mathrm {weakly-\!*\ \ in}\ \ L^\infty (0,T;(L^2+L^{2\gamma /(\gamma +1})({{\mathbb {R}}^3};\mathbb {R}^3)). \end{aligned}$$

(3.16)

Combining this with continuity Eqs. (1.5) or (1.6), we deduce that, up to extraction an subsequence,

$$\begin{aligned} \mathrm{div} \mathbf{u} _\epsilon \ \ \mathrm {converges~ weakly~to } 0 \mathrm { in } H^{-1}((0,T)\times {{\mathbb {R}}^3}). \end{aligned}$$

(3.17)

3.2 Stricharz’s estimates

In this subsection, we introduce the Strichartz’s estimate of linear wave equations to handle the oscillations of the densities and the velocity. To do this, we consider the following acoustic system:

$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon \partial _t s_\epsilon +\Delta q_\epsilon =0, &{} s_\epsilon (0,\cdot )=s_{0,\epsilon }=\frac{1}{2}(\Phi _{0,\epsilon }+\Psi _{0,\epsilon }),\\ \epsilon \partial _t \nabla q_\epsilon +\nabla s_\epsilon =0,&{} \nabla q_\epsilon (0,\cdot )=\nabla q_{0,\epsilon }={\frac{1}{2}( \mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })+ \mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })).} \end{array} \right. \end{aligned}$$

(3.18)

Here we have used $\mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })$ and $\mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })$ as an approximation of $\mathbf{Q} ({\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })$ and $\mathbf{Q} ({n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })$, respectively, since

$$\begin{aligned} \Vert \mathbf{Q} ({\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })- \mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })\Vert _{L^1({\mathbb {R}}^3)} \le C\epsilon ,\end{aligned}$$

(3.19)

$$\begin{aligned} \Vert \mathbf{Q} ({n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })- \mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })\Vert _{L^1({\mathbb {R}}^3)} \le C\epsilon . \end{aligned}$$

(3.20)

We shall regularize the initial data $(\frac{1}{2}(\Psi _{0,\epsilon }+\Psi _{0,\epsilon }),{ \frac{1}{2}( \mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon }) + \mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })) )}$ to remove the interruption of computation of convergence. Let us choose the following initial data:

$$\begin{aligned} \left\{ \begin{array}{l} s_{0,\epsilon }=s_{0,\epsilon ,\delta }=\chi _\delta *[\frac{1}{2}(\Phi _{0,\epsilon }+\Psi _{0,\epsilon })],\\ { \nabla q_{0,\epsilon } =\nabla q_{0,\epsilon ,\delta }=\chi _\delta *\big [\frac{1}{2}( \mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })+ \mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })) \big ].} \end{array}\right. \end{aligned}$$

(3.21)

Here $\chi ^\delta (x)=(1/\delta ^3)\chi (x/\delta )$ and $\chi \in C^\infty _0({\mathbb {R}}^3)$ is the Friedrich’s mollifier, i.e. $\int _{{\mathbb {R}}^3} \chi \mathrm{d}x =1$. From now on, we remove $\delta$ to proceed the convenient presentation. Then we have the Strichartz’s estimates:

Proposition 3.1

[6] Let $(s_\epsilon ,\nabla q_\epsilon )$ be the solution of system (3.18) with initial data $(s^j_{\epsilon ,0}, q^j_{\epsilon ,0})$ given in (3.21). Then, one has

$$\begin{aligned}&\Vert s_\epsilon (\cdot ,t)\Vert ^2_{H^k(\mathbb {R}^3)}+\Vert \nabla q_\epsilon ( \cdot ,t)\Vert ^2_{H^k(\mathbb {R}^3;\mathbb {R}^3)}\nonumber \\&\quad \le \Vert s_{0,\epsilon }\Vert ^2_{H^k(\mathbb {R}^3)}+\Vert \nabla q_{0,\epsilon } \Vert ^2_{H^k(\mathbb {R}^3;\mathbb {R}^3)}, \end{aligned}$$

(3.22)

$$\begin{aligned}&\Vert s_\epsilon \Vert _{L^l(\mathbb {R};W^{k,p}(\mathbb {R}^3;\mathbb {R}^3))}+\Vert \nabla q_\epsilon \Vert _{L^l(\mathbb {R};W^{k,p}(\mathbb {R}^3;\mathbb {R}^3))}\nonumber \\&\quad \le C\epsilon ^{1/l}\big ( \Vert s_{0,\epsilon }\Vert _{H^{k+2}(\mathbb {R}^3;\mathbb {R}^3)}+\Vert \nabla q_{0,\epsilon }\Vert _{H^{k+2}(\mathbb {R}^3;\mathbb {R}^3)} \big ) \end{aligned}$$

(3.23)

with

$$\begin{aligned} 2< p,\,l\le \infty ,~~\frac{1}{p}+\frac{1}{l}=\frac{1}{2},~~k=0,1,2,\dots . \end{aligned}$$

3.3 Relative entropy inequality

Recall that we have assumed that

$$\begin{aligned} \mu _\epsilon =\nu _\epsilon =\epsilon ^a, \quad a>0. \end{aligned}$$

In order to prove the convergence of Theorem 2.1, we will introduce the relative entropy to the two-fluid system (1.5)–(1.7):

$$\begin{aligned} \mathcal {E}_\epsilon (\tau )=\frac{1}{2}\int _{{\mathbb {R}}^3}\Big ((n_\epsilon +\varrho _\epsilon )| \mathbf{u} _\epsilon - \mathbf{U} |^2+|\Phi _\epsilon -s_\epsilon |^2+|\Psi _\epsilon -s_\epsilon |^2\Big )\mathrm{d}x, \end{aligned}$$

where $\mathbf{U} = \mathbf{u} +\nabla q_\epsilon .$

Let first recall the energy inequality to the system (1.5)–(1.7):

$$\begin{aligned} {E}_\epsilon (\tau )+\int _0^T\int _{{\mathbb {R}}^3}\Big (\mu _\epsilon |\nabla \mathbf{u} _\epsilon |^2+(\mu _\epsilon +\nu _\epsilon )|\mathrm{div} \mathbf{u} _\epsilon |^2 \Big )\mathrm{d}x\mathrm{d}t\le E_{0,\epsilon }, \end{aligned}$$

(3.24)

where

$$\begin{aligned}&{E}_\epsilon (\tau ):=\int _{{\mathbb {R}}^3}\Big (\frac{1}{2}(n_\epsilon +\varrho _\epsilon )| \mathbf{u} _\epsilon |^2+G_\alpha (n_\epsilon )+H_\gamma (\varrho _\epsilon )\Big )\mathrm{d}x,\\&{E}_{0,\epsilon }:=\int _{{\mathbb {R}}^3}\Big (\frac{1}{2}(n_{0,\epsilon }+\varrho _{\epsilon ,0})| \mathbf{u} _{0,\epsilon }|^2+G_\alpha (n_{0,\epsilon })+H_\gamma (\varrho _{0,\epsilon }) \Big )\mathrm{d}x. \end{aligned}$$

The conservation of energy for the incompressible Euler equations (2.7) reads

$$\begin{aligned} \frac{1}{2}\int _{{\mathbb {R}}^3}|u(t)|^2 \mathrm{d} x= \frac{1}{2}\int _{{\mathbb {R}}^3}|u_0|^2 \mathrm{d} x. \end{aligned}$$

(3.25)

From the system (3.18), we get

$$\begin{aligned}&\int _{{\mathbb {R}}^3}(|s_\epsilon |^2(\tau )+|\nabla q_\epsilon |^2(\tau ))\mathrm{d}x= \int _{{\mathbb {R}}^3}(|s_{0,\epsilon }|^2+|\nabla q_{0,\epsilon }|^2)\mathrm{d}x. \end{aligned}$$

(3.26)

We adapt $\mathbf{U}$ as test functions to the moment equation (1.7):

$$\begin{aligned}&\displaystyle {-\int _ {{\mathbb {R}}^3}((n_\epsilon +\varrho _\epsilon ) \mathbf{u} _\epsilon \cdot \mathbf{U} )(\tau )\mathrm{d}x=-\int _ {{\mathbb {R}}^3}((n_{0,\epsilon }+\varrho _{0,\epsilon }) \mathbf{U} _{0,\epsilon })\cdot \mathbf{u} _{0,\epsilon }\mathrm{d}x}\nonumber \\&\displaystyle {\qquad -\int _0^\tau \!\int _ {{\mathbb {R}}^3}\big [((n_\epsilon +\varrho _\epsilon ) \mathbf{u} _\epsilon \otimes \mathbf{u} _\epsilon :\nabla \mathbf{U} -\mu _\epsilon \nabla \mathbf{u} _\epsilon :\nabla \mathbf{U} \big ]\mathrm{d}x\mathrm{d}t}\nonumber \\&\displaystyle {\qquad +\int _0^\tau \!\int _ {{\mathbb {R}}^3}(\mu _\epsilon +\nu _\epsilon )\mathrm{div} \mathbf{u} _\epsilon ~\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t-\int _0^\tau \!\int _ {{\mathbb {R}}^3}(n_\epsilon +\varrho _\epsilon ) \mathbf{u} _\epsilon \cdot \partial _t \mathbf{U} \mathrm{d}x\mathrm{d}t}\nonumber \\&\qquad -\frac{1}{\epsilon }\int _0^\tau \!\int _{{\mathbb {R}}^3} (\varphi _\epsilon + \psi _\epsilon )\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t-\frac{\alpha -1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3}|\Phi _\epsilon |^2 \mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t,\nonumber \\&\qquad -\frac{\gamma -1}{2}\int _0^\tau \!\int _\Omega |\Psi _\epsilon |^2\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.27)

We also use $\frac{1}{2}| \mathbf{U} |^2$ as a test function to continuity Eqs. (1.5) and (1.6), respectively, to deduce that

$$\begin{aligned}&\frac{1}{2}\int _ {{\mathbb {R}}^3} (n_\epsilon +\varrho _\epsilon ) | \mathbf{U} |^2\mathrm{d}x=\frac{1}{2}\int _ {{\mathbb {R}}^3}( n_{0,\epsilon }+\varrho _{0,\epsilon }) | \mathbf{U} _0|^2\mathrm{d}x\nonumber \\&+\int _0^\tau \!\int _ {{\mathbb {R}}^3} (n_\epsilon +\varrho _\epsilon )\partial _t \mathbf{U} \cdot \mathbf{U} \mathrm{d}x\mathrm{d}t+\int _0^\tau \!\int _ {{\mathbb {R}}^3} (n_\epsilon +\varrho _\epsilon ) \mathbf{u} _\epsilon \cdot \nabla \mathbf{U} \cdot \mathbf{U} \mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.28)

Using $s_\epsilon$ as a test function to continuity Eqs. (1.5) and (1.6), respectively, we get

$$\begin{aligned} -\int _ {{\mathbb {R}}^3} \varphi _\epsilon s_\epsilon \mathrm{d}x=-\int _ {{\mathbb {R}}^3} \varphi _{0,\epsilon } \varphi _0\mathrm{d}x-\int ^\tau _0\!\int _ {{\mathbb {R}}^3} \varphi _\epsilon \partial _t s_\epsilon \mathrm{d}x\mathrm{d}t-\frac{1}{\epsilon }\int _0^\tau \!\int _ {{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \nabla s_\epsilon \mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.29)

$$\begin{aligned} -\int _ {{\mathbb {R}}^3} \psi _\epsilon s_\epsilon \mathrm{d}x=-\int _ {{\mathbb {R}}^3} \psi _{0,\epsilon } \varphi _0\mathrm{d}x-\int ^\tau _0\!\int _ {{\mathbb {R}}^3} \psi _\epsilon \partial _t s_\epsilon \mathrm{d}x\mathrm{d}t-\frac{1}{\epsilon }\int _0^\tau \!\int _\Omega \varrho _\epsilon \mathbf{u} _\epsilon \cdot \nabla s_\epsilon \mathrm{d}x\mathrm{d}t. \end{aligned}$$

(3.30)

Thus, we deduce, after adding up (3.24)–(3.30), the following inequality:

$$\begin{aligned} \begin{array}{l} \displaystyle {\mathcal {E}_\epsilon (\tau )+\int _0^\tau \!\int _{{\mathbb {R}}^3}\Big (\mu _\epsilon |\nabla \mathbf{u} _\epsilon |^2+(\mu _\epsilon +\nu _\epsilon )|\mathrm{div} \mathbf{u} _\epsilon |^2\Big )\mathrm{d}x\mathrm{d}t\le \Sigma _{j=1}^7 A_j^\epsilon }, \end{array} \end{aligned}$$

(3.31)

where

$$\begin{aligned}&\displaystyle {A_1^\epsilon =\mathcal {E}_\epsilon (0) -\Big [\int _{{\mathbb {R}}^3}|\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau },\\&\displaystyle {A_2^\epsilon = -\int _{{\mathbb {R}}^3}(\Phi _\epsilon (\tau )-\varphi _\epsilon (\tau ))s_\epsilon \mathrm{d}x+\int _{{\mathbb {R}}^3}(\Phi _\epsilon (0)-\varphi _\epsilon (0))s_{0,\epsilon }\mathrm{d}x},\\&\displaystyle {A_3^\epsilon = -\int _{{\mathbb {R}}^3}(\Psi _\epsilon (\tau )-\psi _\epsilon (\tau ))s_\epsilon \mathrm{d}x+\int _{{\mathbb {R}}^3}(\Psi _\epsilon (0)-\psi _\epsilon (0))s_{0,\epsilon }\mathrm{d}x},\\&\displaystyle {A_4^\epsilon =\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon +\varrho _\epsilon )\Big (\partial _t \mathbf{U} + \mathbf{u} _\epsilon \cdot \nabla \mathbf{U} \Big )\cdot ( \mathbf{U} - \mathbf{u} _\epsilon )\mathrm{d}x\mathrm{d}t},\\&\displaystyle {A_5^\epsilon =\int _0^\tau \!\int _{{\mathbb {R}}^3}\mu _\epsilon \nabla \mathbf{u} _\epsilon :\nabla \mathbf{U} \mathrm{d}x\mathrm{d}t+\int _0^\tau \!\int _{{\mathbb {R}}^3}(\mu _\epsilon +\nu _\epsilon )\mathrm{div} \mathbf{u} _\epsilon ~\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t},\\&\displaystyle {A_6^\epsilon =-\frac{1}{\epsilon }\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \nabla s_\epsilon \mathrm{d}x\mathrm{d}t-\frac{\alpha -1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3}|\Phi _\epsilon |^2\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t},\\&\displaystyle {A_7^\epsilon =-\frac{1}{\epsilon }\int _0^\tau \!\int _{{\mathbb {R}}^3} \varrho _\epsilon \mathbf{u} _\epsilon \cdot \nabla s_\epsilon \mathrm{d}x\mathrm{d}t-\frac{\gamma -1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3}|\Psi _\epsilon |^2\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t.} \end{aligned}$$

Note that thanks to $\mathrm{div} \mathbf{u} =0$ and the wave equation (3.18), the terms $-\int ^\tau _0\int _{{\mathbb {R}}^3} \varphi _\epsilon \partial _t s_\epsilon \mathrm{d}x\mathrm{d}t$ in (3.29), $-\int ^\tau _0\int _ {{\mathbb {R}}^3} \psi _\epsilon \partial _t s_\epsilon \mathrm{d}x\mathrm{d}t$ in (3.30), and the term $-\frac{1}{\epsilon }\int _0^\tau \!\int _{{\mathbb {R}}^3} (\varphi _\epsilon + \psi _\epsilon )\mathrm{div} \mathbf{U} \mathrm{d}x\mathrm{d}t$ in (3.27) are cancelled.

3.4 Computation of relative entropy

In this subsection, we are going to compute the estimates of relative entropy. Let us now carry out on the estimates of $\{A_j^\epsilon \}_{j=1}^7.$ Thanks to (2.10), (3.19), (3.20), the regularity of $\mathbf{u} _0$ and $\hat{ \mathbf{u} }_0$, and Hausdorff–Young’s inequality, we first deal with the initial data part of $A_1^\epsilon$. We have

$$\begin{aligned}&\Vert \sqrt{n_{0,\epsilon }}( \mathbf{u} _{0,\epsilon }- \mathbf{u} _0-\nabla q_{0,\epsilon })\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&= \Vert \sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon }-\hat{ \mathbf{u} }_0+\hat{ \mathbf{u} }_0 -\sqrt{n_{0,\epsilon }}\hat{ \mathbf{u} }_0+\sqrt{n_{0,\epsilon }}(\hat{ \mathbf{u} }_0 - \mathbf{u} _0-\nabla q_{0,\epsilon })\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\le C \Vert \sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon }-\hat{ \mathbf{u} }_0\Vert ^2_{L^2({{\mathbb {R}}^3})}+C\Vert (1 -\sqrt{n_{0,\epsilon }})\hat{ \mathbf{u} }_0\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\quad +C \Vert \sqrt{n_{0,\epsilon }}(\hat{ \mathbf{u} }_0 - \mathbf{u} _0-\nabla q_{0,\epsilon })\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\le C\epsilon ^b +C\epsilon ^2+ C \Vert \sqrt{n_{0,\epsilon }}(\hat{ \mathbf{u} }_0 - \mathbf{u} _0-\nabla q_{0,\epsilon })\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\le C\epsilon ^b +C\epsilon ^2+C \Vert \sqrt{n_{0,\epsilon }}(\hat{ \mathbf{u} }_0- \mathbf{u} _0- \mathbf{Q} (\hat{ \mathbf{u} }_0)*\chi ^\delta )\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\quad +C\Vert \sqrt{n_{0,\epsilon }}( \mathbf{Q} (\hat{ \mathbf{u} }_0)*\chi ^\delta - \mathbf{Q} (\sqrt{n_{0,\epsilon }} \mathbf{u} _{0,\epsilon })*\chi ^\delta )\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\quad +C\Vert \sqrt{n_{0,\epsilon }}( \mathbf{Q} (\hat{ \mathbf{u} }_0)*\chi ^\delta - \mathbf{Q} (\sqrt{\varrho _{0,\epsilon }} \mathbf{u} _{0,\epsilon })*\chi ^\delta )\Vert ^2_{L^2({{\mathbb {R}}^3})}\nonumber \\&\le C\epsilon ^b+C\epsilon ^2+\chi (\delta ). \end{aligned}$$

(3.32)

Here and below we use $\chi (\delta )$ to denote a generic function of $\delta$ satisfying

$$\begin{aligned} \lim _{\delta \rightarrow 0}\chi (\delta )=0. \end{aligned}$$

(3.33)

Similarly,

$$\begin{aligned} \Vert \sqrt{\varrho _{0,\epsilon }}( \mathbf{u} _{0,\epsilon }- \mathbf{u} _0-\nabla q_{0,\epsilon })\Vert ^2_{L^2({{\mathbb {R}}^3})} \le&C\epsilon ^b+C\epsilon ^2+\chi (\delta ). \end{aligned}$$

(3.34)

For the term $\Vert \Phi _{0,\epsilon }-s_{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})}$, it can be treated as

$$\begin{aligned} \Vert \Phi _{0,\epsilon }-s_{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})}&=\Vert \Phi _{0,\epsilon }-\varphi _{0,\epsilon }+\varphi _{0,\epsilon }-\varphi _{0,\epsilon }*\chi ^\delta +\varphi _{0,\epsilon }*\chi ^\delta -s_{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\le C\Vert \Phi _{0,\epsilon }-\varphi _{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)} +C\Vert \varphi _{0,\epsilon }-\varphi _{0,\epsilon }*\chi ^\delta \Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\quad +C \Vert \varphi _{0,\epsilon }*\chi ^\delta -\Phi _{0,\epsilon }*\chi ^\delta \Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\quad +C \Vert \psi _{0,\epsilon }*\chi ^\delta -\Psi _{0,\epsilon }*\chi ^\delta \Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\quad +C \Vert \varphi _{0,\epsilon }*\chi ^\delta -\psi _{0,\epsilon }*\chi ^\delta \Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\le C\Vert \Phi _{0,\epsilon }-\varphi _{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)} +C \Vert \varphi _{0,\epsilon } -\Phi _{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)}\nonumber \\&\quad +C \Vert \psi _{0,\epsilon } -\Psi _{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)} +C \Vert \varphi _{0,\epsilon } -\psi _{0,\epsilon }\Vert ^2_{L^2({\mathbb {R}}^3)}+ \chi (\delta )\nonumber \\&\le C\epsilon ^{{b}}+C\epsilon ^{{c}} +\chi (\delta ). \end{aligned}$$

(3.35)

Similarly, we have

$$\begin{aligned} \Vert \Psi _{0,\epsilon }-s_{0,\epsilon }\Vert ^2_{L^2({{\mathbb {R}}^3})} \le&C\epsilon ^{{b}}+C\epsilon ^{{c}} +\chi (\delta ). \end{aligned}$$

(3.36)

Thus, we have

$$\begin{aligned} A_1^\epsilon \le -\left[ \int _{{\mathbb {R}}^3}|\nabla q_\epsilon |^2\mathrm{d}x\right] _0^\tau +C\epsilon ^{{b}}+C\epsilon ^{{c}} +\chi (\delta )+C\epsilon ^2+\chi (\delta ). \end{aligned}$$

(3.37)

We remark that the first term on the right-hand side of (3.37) will be cancelled later.

For the terms $A_2^\epsilon$ and $A_3^\epsilon$, by the arguments of ( [31], pp. 13–14), we have

$$\begin{aligned} A^{\epsilon }_{2}+A^{\epsilon }_{3} \le \left\{ \begin{array}{ll} C_T\epsilon ^{\min \{1,\frac{b}{2}\}},&{}\quad 1<\alpha \le 4, ~~ 9/5<\gamma \le 4,\\ C_T\epsilon ^{\min \{1,\frac{b}{2},\frac{\alpha -4}{2\alpha }\}},&{}\quad \alpha>4,~~9/5<\gamma \le 4, \\ C_T\epsilon ^{\min \{1,\frac{b}{2},\frac{\gamma -4}{2\gamma }\}},&{}\quad 1 <\alpha \le 4,~~\gamma>4, \\ C_T\epsilon ^{\min \{\frac{b}{2},\frac{\alpha -4}{2\alpha },\frac{\gamma -4}{2\gamma }\}},&{}\quad \alpha>4,~~\gamma > 4. \end{array}\right. \end{aligned}$$

(3.38)

We next control $A_4^\epsilon$ and denote it by $A_4^\epsilon =B_1^\epsilon +B_2^\epsilon$ where

$$\begin{aligned} B^\epsilon _1&=\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon (\partial _t \mathbf{U} + \mathbf{u} _\epsilon \cdot \nabla \mathbf{U} )( \mathbf{U} - \mathbf{u} _\epsilon )\mathrm{d}x\mathrm{d}t,\\ B^\epsilon _2&=\int _0^\tau \!\int _{{\mathbb {R}}^3} \varrho _\epsilon (\partial _t \mathbf{U} + \mathbf{u} _\epsilon \cdot \nabla \mathbf{U} )( \mathbf{U} - \mathbf{u} _\epsilon )\mathrm{d}x\mathrm{d}t. \end{aligned}$$

For the term $B^\epsilon _1$, we have

$$\begin{aligned} B^\epsilon _1&=\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon (\partial _t \mathbf{U} + \mathbf{u} _\epsilon \cdot \nabla \mathbf{U} )( \mathbf{U} - \mathbf{u} _\epsilon )\mathrm{d}x\mathrm{d}t\\&=\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\otimes ( \mathbf{U} - \mathbf{u} _\epsilon ):\nabla \mathbf{U} ~\mathrm{d}x\mathrm{d}t\\&\quad +\int _0^\tau \!\int _{{\mathbb {R}}^3}\Big (n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot \partial _t \mathbf{U} +\varrho _\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon ):\nabla \mathbf{U} \Big )\mathrm{d}x\mathrm{d}t\\&\le C_T\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+\sum _{k=1}^5 J_k^\epsilon , \end{aligned}$$

where

$$\begin{aligned}&\displaystyle {J_1^\epsilon =\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot (\partial _t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} )\mathrm{d}x\mathrm{d}t},\\&\displaystyle {J_2^\epsilon =\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot \partial _t\nabla q_{\epsilon }\mathrm{d}x\mathrm{d}t},\\&\displaystyle {J_3^\epsilon =\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\otimes \nabla q_{\epsilon }:\nabla \mathbf{v} \mathrm{d}x\mathrm{d}t},\\&\displaystyle {J_4^\epsilon = \int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\otimes \mathbf{v} :\nabla ^2 q_{\epsilon }\mathrm{d}x\mathrm{d}t},\\&\displaystyle {J_5^\epsilon = \frac{1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot \nabla |\nabla q_{\epsilon }|^2\mathrm{d}x\mathrm{d}t}. \end{aligned}$$

For $J_1^\epsilon$, recalling that $\mathbf{U} = \mathbf{u} +\nabla q_\epsilon$ and using the incompressible Euler equations (1.9) and $\mathrm{div} \mathbf{u} =0$, we have

$$\begin{aligned} |J_1^\epsilon |&\le \Big |\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot (\partial _t \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} )\mathrm{d}x\mathrm{d}t\Big |\nonumber \\&\le \Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon -1) \mathbf{U} \cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |+\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}\nabla q_{\epsilon }\cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |\nonumber \\&\quad +\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |. \end{aligned}$$

(3.39)

For the first term on right-hand side of (3.39), we use the estimates (3.6) and (3.8) to obtain that

$$\begin{aligned}&\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon -1) \mathbf{U} \cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |\\&\le C\Vert (n_\epsilon -1)\mathrm {1}_{\{\{|n_\epsilon -1|\le 1/2\}}\Vert _{L^2(0,T;L^2({{\mathbb {R}}^3}))}\Vert \mathbf{U} \Vert _{L^\infty (0,T;L^\infty ({{\mathbb {R}}^3}))}\Vert \nabla \Pi \Vert _{L^2(0,T;L^2({{\mathbb {R}}^3}))}\\&\quad +\Vert (n_\epsilon -1)\mathrm {1}_{\{|n_\epsilon -1|\ge 1/2\}}\Vert _{L^\alpha (0,T; L^\alpha ({{\mathbb {R}}^3}))}\\&\quad \times \Vert \mathbf{U} \Vert _{L^\infty (0,T;L^\infty ({{\mathbb {R}}^3}))} \Vert \nabla \Pi \Vert _{L^{\frac{\alpha }{\alpha -1}}(0,T;L^{\frac{\alpha }{\alpha -1}}({{\mathbb {R}}^3}))}\\&\le C_T(\epsilon +\epsilon ^\kappa )\le C_T\epsilon , \end{aligned}$$

where $\kappa :=\min \{2,\alpha \}$ and we have used the condition $k>7/2$ and $\alpha >1$.

For the second term on the right-hand side of (3.39), together with using the equation (3.18) and the dispersive regularity (3.1), it can be estimated as follows:

$$\begin{aligned} \Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}\nabla q_{\epsilon }\cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |&\le \epsilon \Big [\int _{{\mathbb {R}}^3} |s_{\epsilon }||\Pi |\mathrm{d}x\Big ]^\tau _0 +\epsilon \int _0^\tau \!\int _{{\mathbb {R}}^3}|s_\epsilon ||\partial _t\Pi |\mathrm{d}x\mathrm{d}t\\&\le \epsilon \big (\Vert s_\epsilon \Vert _{L^\infty (0,T;L^{2}({{\mathbb {R}}^3}))}\Vert \Pi \Vert _{L^\infty (0,T;L^{2}({{\mathbb {R}}^3}))}\\&\quad +\Vert s_{0,\epsilon }\Vert _{L^\infty (0,T;L^{2}({{\mathbb {R}}^3}))}\Vert \Pi _{0}\Vert _{L^\infty (0,T;L^{2}({{\mathbb {R}}^3}))}\\&\quad +\Vert s_\epsilon \Vert _{L^2(0,T;L^{2}({{\mathbb {R}}^3}))}\Vert \partial _t\Pi \Vert _{L^2(0,T;L^{2}({{\mathbb {R}}^3}))}\big )\\&\le C_T\epsilon . \end{aligned}$$

For the third term on the right-hand side of (3.39), together with using continuity Eqs. (1.5), (3.6), and (3.8), it can be bounded as follows:

$$\begin{aligned}&\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \nabla \Pi \mathrm{d}x\mathrm{d}t\Big |\\&\le \Big | \int _{{\mathbb {R}}^3}\big \{((n_\epsilon -1)\Pi )(t)-((n_\epsilon -1) \Pi )(0)\big \}\mathrm{d}x+\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon -1)\cdot \partial _t\Pi \mathrm{d}x\mathrm{d}t\Big |\\&\le \Vert (n_\epsilon -1)\mathrm {1}_{\{|n_\epsilon -1|\le 1/2\}}\Vert _{L^\infty (0,T;L^2({{\mathbb {R}}^3}))}+\Vert (n_\epsilon -1)\mathrm {1}_{\{|n_\epsilon -1|\ge 1/2\}}\Vert _{L^\infty (0,T; L^\alpha ({{\mathbb {R}}^3}))} \\&\quad +\Vert (n_\epsilon -1)\mathrm {1}_{\{|n_\epsilon -1|\le 1/2\}}\Vert _{L^2(0,T;L^2({{\mathbb {R}}^3}))}\Vert \partial _t\Pi \Vert _{L^2(0,T;L^2({{\mathbb {R}}^3}))}\\&\quad +\Vert (n_\epsilon -1)\mathrm {1}_{\{|n_\epsilon -1|\ge 1/2\}}\Vert _{L^\alpha (0,T; L^\alpha ({{\mathbb {R}}^3}))}\Vert \mathbf{U} \Vert _{L^\infty (0,T;L^\infty ({{\mathbb {R}}^3}))} \Vert \partial _t\Pi \Vert _{L^{\frac{\alpha }{\alpha -1}}(0,T;L^{\frac{\alpha }{\alpha -1}}({{\mathbb {R}}^3}))}\\&\quad +C_T\epsilon \\&\le C_T(\epsilon +\epsilon ^\kappa )\le C_T\epsilon , \end{aligned}$$

where we have here used (2.12). Thus, we get

$$\begin{aligned} |J_1^\epsilon |\le C_T\epsilon . \end{aligned}$$

Next, we have

$$\begin{aligned} J^\epsilon _2&=-\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \partial _t\nabla q_\epsilon \mathrm{d}x\mathrm{d}t+\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} \cdot \partial _t\nabla q_\epsilon \mathrm{d}x\mathrm{d}t\nonumber \\&\quad \displaystyle {+\frac{1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \partial _t|\nabla q_\epsilon |^2\mathrm{d}x\mathrm{d}t}. \end{aligned}$$

(3.40)

In virtue of $\mathrm{div} \mathbf{u} =0$, (3.1), (3.6), and (3.8), we get

$$\begin{aligned}&\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} \cdot \partial _t\nabla q_\epsilon \mathrm{d}x\mathrm{d}t\Big |\\&=\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon -1) \mathbf{u} \cdot \partial _t\nabla q_\epsilon \mathrm{d}x\mathrm{d}t\Big |\\&\le \Big \Vert \frac{n_\epsilon -1}{\epsilon }\mathrm {1}_{\{|n_\epsilon -1|\le 1/2\}}\Big \Vert _{L^\infty (0,T;L^2({{\mathbb {R}}^3}))}\Vert \mathbf{u} \Vert _{L^{4/3}(0,T;L^4({{\mathbb {R}}^3}))}\Vert \nabla s_\epsilon \Vert _{L^4(0,T;L^4({{\mathbb {R}}^3}))}\\&\quad +\frac{C}{\epsilon }\Vert (n_\epsilon ^\alpha +1)\mathrm {1}_{\{|n_\epsilon -1|>1/2\}}\Vert _{L^\infty (0,T;L^1({{\mathbb {R}}^3}))}\\&\le C_T(\epsilon ^{1/4}+\epsilon ) , \end{aligned}$$

where we have used the facts that

$$\begin{aligned} \Vert \mathbf{u} \Vert _{L^\infty ((0,T)\times {{\mathbb {R}}^3})}\le C_T,\quad \Vert \nabla s_\epsilon \Vert _{L^\infty ((0,T)\times {{\mathbb {R}}^3})}\le C_T. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \begin{array}{l} \displaystyle { \frac{1}{2}\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \partial _t|\nabla q_\epsilon |^2\mathrm{d}x\mathrm{d}t\le \frac{1}{2}\Big [\int _{{\mathbb {R}}^3}|\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau + C_T(\epsilon ^{1/4}+\epsilon ).} \end{array} \end{aligned}$$

Thus, the term $J_2^\epsilon$ is bounded by:

$$\begin{aligned} J_2^\epsilon&=\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon ( \mathbf{U} - \mathbf{u} _\epsilon )\cdot \partial _t\nabla q_\epsilon \mathrm{d}x\mathrm{d}t\\&\le \frac{1}{2}\Big [\int _{{\mathbb {R}}^3} |\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau -\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \partial _t\nabla q_{\epsilon }\mathrm{d}x\mathrm{d}t+C_T(\epsilon ^{1/4}+\epsilon ). \end{aligned}$$

Using the regularity (2.8), the dispersive property (3.23), (3.1), and (3.12), the term $J^\epsilon _3$ can be estimated as:

$$\begin{aligned} J_3^\epsilon&=\int _0^\tau \!\int _{{\mathbb {R}}^3}(n_\epsilon -1) \mathbf{U} \otimes \nabla q_\epsilon :\nabla \mathbf{u} \mathrm{d}x\mathrm{d}t+\int _0^\tau \!\int _{{\mathbb {R}}^3} \mathbf{U} \otimes \nabla q_\epsilon :\nabla \mathbf{u} \mathrm{d}x\mathrm{d}t\\&\quad +\int _0^\tau \!\int _{{\mathbb {R}}^3}(\sqrt{n_\epsilon }-1)\sqrt{n_\epsilon } \mathbf{u} _\epsilon \otimes \nabla q_\epsilon :\nabla \mathbf{u} \mathrm{d}x\mathrm{d}t+\int _0^\tau \!\int _{{\mathbb {R}}^3}\sqrt{n_\epsilon } \mathbf{u} _\epsilon \otimes \nabla q_\epsilon :\nabla \mathbf{u} \mathrm{d}x\mathrm{d}t\\&\le C_T(\epsilon ^{1/4}+\epsilon ). \end{aligned}$$

Similarly, we get

$$\begin{aligned} J_4^\epsilon +J_5^\epsilon \le C(\epsilon ^{1/4}+\epsilon ). \end{aligned}$$

So, the term $B^\epsilon _1$ can be estimated as follows:

$$\begin{aligned} B^\epsilon _1&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+\frac{1}{2}\Big [\int _{{\mathbb {R}}^3} |\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau -\int _0^\tau \!\int _{{\mathbb {R}}^3} n_\epsilon \mathbf{u} _\epsilon \cdot \partial _t\nabla q_{\epsilon }\mathrm{d}x\mathrm{d}t\\&\quad +C(\epsilon ^{1/4}+\epsilon ), \end{aligned}$$

and with the same arguments, $B^\epsilon _2$ can be handled as follows:

$$\begin{aligned} B^\epsilon _2&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+\frac{1}{2}\Big [\int _{{\mathbb {R}}^3} |\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau -\int _0^\tau \!\int _{{\mathbb {R}}^3} \varrho _\epsilon \mathbf{u} _\epsilon \cdot \partial _t\nabla q_{\epsilon }\mathrm{d}x\mathrm{d}t\\&\quad +C_T(\epsilon ^{1/4}+\epsilon ). \end{aligned}$$

The above inequalities imply that

$$\begin{aligned} A^\epsilon _4&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+\Big [\int _{{\mathbb {R}}^3} |\nabla q_\epsilon |^2\mathrm{d}x\Big ]_0^\tau -\int _0^\tau \!\int _{{\mathbb {R}}^3} (\varrho _\epsilon +n_\epsilon ) \mathbf{u} _\epsilon \cdot \partial _t\nabla q_{\epsilon }\mathrm{d}x\mathrm{d}t\\&\quad +C_T(\epsilon ^{1/4}+\epsilon ). \end{aligned}$$

Thanks to $\mathrm{div} \mathbf{u} =0$, the estimate (3.22) and the viscosity term $A_5^\epsilon (\tau )$ can be bounded by the following:

$$\begin{aligned} A_5^\epsilon (\tau )&=\int _0^\tau \int _{{\mathbb {R}}^3}\Big (\sqrt{\mu _\epsilon }\nabla \mathbf{u} _\epsilon :\sqrt{\mu _\epsilon }\nabla \mathbf{U} +\sqrt{\mu _\epsilon +\nu _\epsilon }\mathrm{div} \mathbf{u} _\epsilon ~\sqrt{\mu _\epsilon +\nu _\epsilon }\mathrm{div} \mathbf{U} \Big )\mathrm{d}x\mathrm{d}t\nonumber \\&\le \frac{\mu _\epsilon }{2} \Vert \nabla \mathbf{u} _\epsilon \Vert ^2_{L^2((0,T)\times {{\mathbb {R}}^3})}+\frac{\mu _\epsilon +\nu _\epsilon }{2} \Vert \mathrm{div} \mathbf{u} _\epsilon \Vert ^2_{L^2((0,T)\times {{\mathbb {R}}^3})}+C_T(\mu _\epsilon +\nu _\epsilon ). \end{aligned}$$

(3.41)

Noticing that after adding the second term on the right-hand side of $J^\epsilon _2$ in (3.40) to $A^\epsilon _6$, we get that the first term of $A^\epsilon _6$ vanishes. The second term of $A^\epsilon _6$ can be estimated as follows:

$$\begin{aligned}&\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}|\Phi _\epsilon |^2\Delta q_\epsilon \mathrm{d}x\mathrm{d}t\Big |\nonumber \\&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+C\Big |\int _0^\tau \!\int _{{\mathbb {R}}^3} |s_\epsilon |^2\Delta q_\epsilon \mathrm{d}x\mathrm{d}t\Big |\nonumber \\&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+C\Vert s_\epsilon \Vert _{L^2(0,T;L^2({{\mathbb {R}}^3}))}\Vert s_\epsilon \Vert _{L^4(0,T;L^4({{\mathbb {R}}^3}))}\Vert \nabla q_\epsilon \Vert _{L^4(0,T;W^{1,4}({{\mathbb {R}}^3}))}\nonumber \\&\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+C_T\epsilon ^{1/4}. \end{aligned}$$

(3.42)

Here we have used the regularity of $s_\epsilon$ and $q_\epsilon$ in (3.22), the dispersive regularity (3.23), and the computation in [12] together with the Strichartz’s estimate (3.23).

Similarly, the second term of $A^\epsilon _7$ can be estimated as follows:

$$\begin{aligned}&\displaystyle { \Big |\int _0^\tau \!\int _{{\mathbb {R}}^3}|\Psi _\epsilon |^2\Delta q_\epsilon \mathrm{d}x\mathrm{d}t\Big |\le C\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+C_T\epsilon ^{1/4}.} \end{aligned}$$

(3.43)

Consequently, putting all the above estimates related to $\{A_j^\epsilon \}_{j=1}^7$ into the inequality (3.31), we obtain that

$$\begin{aligned}&\displaystyle {\mathcal {E}_\epsilon (\tau )+\frac{\mu _\epsilon }{2}\int _0^\tau \!\int _{{\mathbb {R}}^3} |\nabla \mathbf{u} _\epsilon |^2\mathrm{d}x\mathrm{d}t+\frac{\mu _\epsilon +\nu _\epsilon }{2}\int _0^\tau \!\int _{{\mathbb {R}}^3} |\mathrm{div} \mathbf{u} _\epsilon |^2\mathrm{d}x\mathrm{d}t}\nonumber \\&\ \qquad \quad \le C_T\int _0^\tau \mathcal {E}_\epsilon (t)\mathrm{d}t+C_T\epsilon ^{\sigma } +\chi (\delta ) , \end{aligned}$$

(3.44)

where the number $\omega$ is defined as follows:

$$\begin{aligned} \sigma =\left\{ \begin{array}{ll} \min \{\frac{1}{4},\frac{b}{2},a,c\},&{}\quad 1<\alpha \le 4, ~~ 9/5<\gamma \le 4,\\ \min \{\frac{1}{4},\frac{b}{2},\frac{\alpha -4}{2\alpha },a,c\},&{}\quad \alpha>4,~~9/5<\gamma \le 4, \\ \min \{\frac{1}{4},\frac{b}{2},\frac{\gamma -4}{2\gamma },a,c\},&{}\quad 1 <\alpha \le 4,~~\gamma>4, \\ \min \{\frac{b}{2},\frac{\alpha -4}{2\alpha },\frac{\gamma -4}{2\gamma },a,c\}, &{}\quad \alpha>4,~~\gamma > 4, \end{array}\right. \end{aligned}$$

(3.45)

and we have used the condition (2.9) and the facts that:

$$\begin{aligned} \epsilon ^{1/4}\ge \epsilon ^{1/3}\ge \epsilon \end{aligned}$$

for sufficiently small $0<\epsilon <1.$

Applying the Gronwall’s inequality to (3.44) gives

$$\begin{aligned} \begin{array}{l} \mathcal {E}_\epsilon (\tau )\le C_T\epsilon ^{\sigma } +\chi (\delta ) \end{array} \end{aligned}$$

(3.46)

for any $\tau \in (0,T].$

We are now able to prove the convergence of $\sqrt{n_\epsilon } \mathbf{u} _\epsilon$ and $\sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon$. Note that the projection $\mathbf{P}$ is a bounded linear mapping from $L^2({\mathbb {R}}^3)$ to $L^2({\mathbb {R}}^3)$. Hence, we have

$$\begin{aligned} \sup _{0\le t\le T}\Vert \mathbf{P} (\sqrt{n_\epsilon } \mathbf{u} _\epsilon )- \mathbf{u} \Vert _{L^2({\mathbb {R}}^3)}&=\sup _{0\le t\le T}\Big \Vert \mathbf{P} \Big (\sqrt{n_\epsilon } \mathbf{u} _\epsilon - \mathbf{u} -\nabla q_\epsilon )\Big )\Big \Vert _{L^2({\mathbb {R}}^3)}\nonumber \\&\le C\sup _{0\le t\le T}\Vert \sqrt{n_\epsilon } \mathbf{u} _\epsilon - \mathbf{u} -\nabla q_\epsilon \Vert _{L^2({\mathbb {R}}^3)}. \end{aligned}$$

(3.47)

Thus, combining (2.10) and (3.46) and letting $\delta \rightarrow 0$, we further derive that

$$\begin{aligned} \Vert \mathbf{P} (\sqrt{n_\epsilon } \mathbf{u} _\epsilon )- \mathbf{u} \Vert _{L^\infty (0,T; L^2({\mathbb {R}}^3))} \le C_T(\epsilon ^\frac{b}{2}+\epsilon ^\frac{\sigma }{2}) \le C_T\epsilon ^\frac{\sigma }{2}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert \mathbf{P} (\sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon )- \mathbf{u} \Vert _{L^\infty (0,T; L^2({\mathbb {R}}^3))} \le C_T\epsilon ^\frac{\sigma }{2}. \end{aligned}$$

It is not difficult to show the local strong convergence of $\sqrt{n_\epsilon } \mathbf{u} ^\epsilon$ and $\sqrt{\varrho _\epsilon } \mathbf{u} ^\epsilon$ to $\mathbf{u}$ in $L^{r}(0,T;L^2(\Omega ))$ for all $2<r<+\infty$ on any bounded domain $K \subset {\mathbb {R}}^3$. In fact, for any $t\in (0,T]$ and any compact subset $K\subset {{\mathbb {R}}^3}$, we have

$$\begin{aligned}&\displaystyle {\int _K|\sqrt{n_\epsilon } \mathbf{u} _\epsilon - \mathbf{u} |^2\mathrm{d}x}\nonumber \\&\le \displaystyle { \int _K|\sqrt{n_\epsilon }( \mathbf{u} _\epsilon - \mathbf{u} -\nabla q_\epsilon )-(1-\sqrt{n_\epsilon })( \mathbf{v} +\nabla q_\epsilon )+\nabla q_\epsilon |^2\mathrm{d}x}\nonumber \\&\le \mathcal {E}_\epsilon (\tau )+C(K)\left( \int _K |\nabla q_\epsilon |^{\frac{2r}{r-2}}\mathrm{d}x\right) ^{\frac{r-2}{2r}}+C_T\epsilon ^{\sigma }+\chi (\delta ) \end{aligned}$$

(3.48)

for any $r>2$, where we have here used (3.12) and (3.23). Similarly, we have

$$\begin{aligned}&\displaystyle {\int _K|\sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon - \mathbf{u} |^2\mathrm{d}x\le \mathcal {E}_\epsilon (\tau )+C(K)\left( \int _K |\nabla q_\epsilon |^{\frac{2r}{r-2}}\mathrm{d}x\right) ^{\frac{r-2}{2r}}+C\epsilon ^{\sigma }+\chi (\delta )}. \end{aligned}$$

(3.49)

Consequently, using (3.46), (3.48), and (3.49) together with (3.23) and passing to the limit for $\delta \rightarrow 0$, we get

$$\begin{aligned} \Vert \sqrt{n_\epsilon } \mathbf{u} _\epsilon - \mathbf{v} \Vert _{L^r(0,T;L^2_{\mathrm {loc}}({{\mathbb {R}}^3}))}+\Vert \sqrt{\varrho _\epsilon } \mathbf{u} _\epsilon - \mathbf{u} \Vert _{L^r(0,T;L^2_{\mathrm {loc}}({{\mathbb {R}}^3}))}\le C(\epsilon ^{\frac{\sigma }{2}}+\epsilon ^{\frac{1}{r}})\le C_T \epsilon ^d \end{aligned}$$

with $d=\min \big \{\frac{\sigma }{2}, \frac{1}{r}\big \}$. Thus, we prove (2.15) and (2.16) where the terms of constant depend on

$$\begin{aligned} \Vert \nabla q_{0,\epsilon }\Vert _{H^{k+2}({{\mathbb {R}}^3};\mathbb {R}^3)}+\Vert s_{0,\epsilon }\Vert _{H^{k+2}({{\mathbb {R}}^3};\mathbb {R}^3)}, \end{aligned}$$

and it is uniformly bounded by a constant number when $\delta \rightarrow 0.$ This completes the proof of Theorem 2.1.