Research
Open Access
Local existence of strong solutions to the k-ε model equations for turbulent flows
- Baoquan Yuan1Email author and
- Guoquan Qin1
Received: 12 November 2015
Accepted: 13 January 2016
Published: 29 January 2016
Abstract
In this paper, we are concerned with the local existence of strong solutions to the k-ε model equations for turbulent flows in a bounded domain \(\Omega\subset\mathbb{R}^{3}\). We prove the existence of unique local strong solutions under the assumption that the turbulent kinetic energy and the initial density both have lower bounds away from zero.
Keywords
k-ε model equations strong solution local well-posednessMSC
35Q35 76F60 76N101 Introduction
Turbulence is a natural phenomenon, which occurs inevitably when the Reynolds number of flows becomes high enough (106 or more). In this paper, we consider the k-ε model equations [1, 2] for turbulent flows in a bounded domain Ω⊂ \(\mathbb{R}^{3}\) with smooth boundary,
with
where \(\delta_{ij}=0\) if \(i\neq j\), \(\delta_{ij}=1\) if \(i=j\), and μ, \(\mu_{t}\), \(\mu_{e}\), \(C_{1}\), and \(C_{2}\) are five positive constants satisfying \(\mu+\mu_{t}=\mu_{e}\), and n⃗ is the unit outward normal to ∂Ω.
$$\begin{aligned}& \rho_{t}+\nabla\cdot(\rho u)=0, \end{aligned}$$
(1.1)
$$\begin{aligned}& (\rho u)_{t}+\nabla\cdot(\rho u\otimes u)-\Delta u-\nabla(\nabla \cdot u)+\nabla p=-\frac{2}{3}\nabla(\rho k), \end{aligned}$$
(1.2)
$$\begin{aligned}& (\rho h)_{t}+\nabla\cdot(\rho uh)-\Delta h=p_{t}+u\cdot \nabla p+S_{k}, \end{aligned}$$
(1.3)
$$\begin{aligned}& (\rho k)_{t}+\nabla\cdot(\rho uk)-\Delta k=G-\rho\varepsilon , \end{aligned}$$
(1.4)
$$\begin{aligned}& (\rho\varepsilon)_{t}+\nabla\cdot(\rho u\varepsilon)-\Delta \varepsilon= \frac{C_{1}G\varepsilon}{k}-\frac{C_{2}\rho\varepsilon ^{2}}{k}, \end{aligned}$$
(1.5)
$$\begin{aligned}& (\rho,u,h,k,\varepsilon) (x,0)=\bigl(\rho _{0}(x),u_{0}(x),h_{0}(x),k_{0}(x), \varepsilon_{0}(x)\bigr), \end{aligned}$$
(1.6)
$$\begin{aligned}& \biggl(u\cdot\vec{n},h, \frac{\partial k}{\partial\vec{n}},\frac{\partial \varepsilon}{\partial\vec{n}} \biggr)\bigg|_{\partial\Omega }=(0,0, 0, 0), \end{aligned}$$
(1.7)
$$\begin{aligned}& S_{k}= \biggl[\mu \biggl(\frac{\partial u^{i}}{\partial x_{j}}+\frac {\partial u^{j}}{ \partial x_{i}} \biggr)-\frac{2}{3}\delta_{ij}\frac{\partial u^{k}}{ \partial x_{k}} \biggr] \frac{\partial u^{i}}{\partial x_{j}} +\frac{\mu_{t}}{\rho^{2}}\frac{\partial p}{ \partial x_{j}}\frac{\partial\rho}{\partial x_{j}}, \end{aligned}$$
(1.8)
$$\begin{aligned}& G=\frac{\partial u^{i}}{\partial x_{j}} \biggl[\mu_{e} \biggl(\frac {\partial u^{i}}{ \partial x_{j}}+ \frac{\partial u^{j}}{\partial x_{i}} \biggr)-\frac {2}{3}\delta_{ij} \biggl(\rho k+ \mu_{e}\frac{\partial u^{k}}{\partial x_{k}} \biggr) \biggr], \end{aligned}$$
(1.9)
$$\begin{aligned}& p=\rho^{\gamma}, \end{aligned}$$
(1.10)
Equations (1.1)-(1.10) are derived from combining the effect of turbulence on the time-averaged Navier-Stokes equations with the k-ε model equations. The unknown functions ρ, u, h, k, and ε denote the density, velocity, total enthalpy, turbulent kinetic energy, and the rate of viscous dissipation of turbulent flows, respectively. The expression of the pressure p has been simplified here, which indeed has no bad effect on our study.
In partial differential equations, the k-ε equations belong to the compressible ones. In this regard, we will refer to the classical compressible Navier-Stokes equations and compressible MHD equations, which are also research mainstreams, to carry out our study.
For compressible isentropic Navier-Stokes equations, the first question provoking our interest is the existence of the weak solutions. Lions [3, 4] proved the global existence of weak solutions under the condition that \(\gamma >\frac{3n}{n+2}\), where γ is the same as in (1.10) and n is the dimension of space. Later, Feireisl [5, 6] improved his result to \(\gamma>\frac{n}{2}\). The condition satisfied by γ is to prove the existence of renormalized solutions, which were introduced by DiPerna and Lions [7]. When the initial data are general small perturbations of non-vacuum resting state, Hoff [8] proved the global existence of weak solutions provided \(\gamma>1\). The existence of strong solutions is another problem provoking our interest in the research of Navier-Stokes equations. It has been proved that the density will be away from vacuum at least in a small time interval provided the initial density is positive. If the initial data have better regularity, the compressible isentropic Navier-Stokes equations will admit a unique local strong solution under various boundary conditions [9–12]. However, when the initial vacuum is allowed, it was shown recently in [9] that the isentropic one will have a local strong solution in the case that some compatibility conditions are satisfied initially. Choe and Kim [13] obtained the unique local strong solutions for full compressible polytropic Navier-Stokes equations under a similar condition in [9]. In [13], the technique the authors used is mainly the standard iteration argument and the key point of their success is the estimate for the \(L^{2}\) norm of the gradient of the pressure. In the process of studying the condition of a local solution becoming a global one, Xin [14] proved that the smooth solutions will blow up in finite time when an initial vacuum is allowed.
As for compressible MHD equations, the research directions, which mainly contain first the existence of weak and strong solutions and second the condition of weak solutions becoming a strong or even classical one and the local becoming a global one, are similar to that of Navier-Stokes equations. For example, Hu and Wang [15–17] obtained the local existence of weak solutions to the compressible isentropic MHD equations. Rozanova [18] proved the local existence of classical solutions to the compressible barotropic MHD equations provided both the mass and energy are finite. Fan and Yu in [19] proved the existence and uniqueness of strong solutions to the full compressible MHD equations. The method Fan and Yu [19] used is similar to that in [13], for example, they are both dependent on the standard iteration argument and the estimate for the \(L^{2}\) norm of the gradient of the pressure.
In this paper, we consider the existence of strong solutions to the k-ε model equations (1.1)-(1.10) in a bounded domain \(\Omega\subset\mathbb{R}^{3}\). Our method is similar to that in [19] and [13]. However, in the process of applying the method to the k-ε model equations, we find that the regularity of the solutions should be higher, which is induced by the higher nonlinearity in the compressible Navier-Stokes equations and compressible MHD equations than that in [19] and [13]. In fact, when we make the difference of the nth and the \((n+1)\)th cases of equation (2.4) and integrating the result, we inevitably arrive at the term \(\int\partial_{j}\overline{\rho}^{n+1}\partial_{j}\rho^{n+1}\cdot \overline{h}^{n+1}\). Therefore, we have to use integration by parts, which leads to two terms as \(\int\overline{\rho}^{n+1}\partial _{j}\partial_{j}{\rho}^{n+1}\cdot\overline{h}^{n+1}\) and \(\int\overline{\rho}^{n+1}\partial_{j}{\rho}^{n+1}\cdot \partial_{j}\overline{h}^{n+1}\). Then, by the Hölder and Young inequalities, it turns out that \(\Vert \nabla^{2}\rho^{n+1}\Vert _{L^{3}}\) and \(\Vert \nabla\rho^{n+1}\Vert _{L^{\infty}}\) should be bounded. Thus, we need \(\Vert \rho \Vert _{H^{3}}\) to be bounded for an a priori estimate. Therefore, from the mass equation enough regularity of the velocity field should be imposed. Moreover, due to the strong-coupling property of the k-ε equations, we need a corresponding high regularity of the unknown functions k and ε.
Stated simply, the high nonlinearity of the k-ε equations leads to the necessity of high regularity of some unknown functions and thus leads to much difficulties for the a priori estimates. Besides, physically, when the turbulent kinetic energy k vanish, the turbulence will disappear and the k-ε model equations will degenerate into the Navier-Stokes equation. Therefore, without loss of generality, we assume throughout this paper that the turbulent kinetic energy k has a positive lower bound away from zero, namely, \(0< m< k\) with m a constant.
To conclude this introduction, we give the outline of the rest of this paper: In Section 2, we consider a linearized problem of the k-ε equations and derive some local-in-time estimates for the solutions of the linearized problem. In Section 3, we prove the existence theorem of the local strong solution of the original nonlinear problem.
2 A priori estimates for a linearized problem
Using the density equation (1.1), we could change (1.1)-(1.10) into the following equivalent form:
$$ \textstyle\begin{cases} \rho_{t}+\nabla\cdot(\rho u)=0,\\ \rho u_{t}+\rho u\cdot\nabla u-\Delta u-\nabla\operatorname{div} u+\nabla p=-\frac{2}{3}\nabla(\rho k),\\ \rho h_{t}+\rho u\cdot\nabla h-\Delta h=p_{t}+u\cdot\nabla p+S_{k},\\ \rho k_{t}+\rho u\cdot\nabla k-\Delta k=G-\rho\varepsilon,\\ \rho\varepsilon_{t}+\rho u\cdot\nabla\varepsilon-\Delta \varepsilon= \frac{C_{1}G\varepsilon}{k}-\frac{C_{2}\rho\varepsilon^{2}}{k},\\ (\rho,u,h,k,\varepsilon)(x,0)=(\rho _{0}(x),u_{0}(x),h_{0}(x),k_{0}(x),\varepsilon_{0}(x)),\\ (u\cdot\vec{n},h, \frac{\partial k}{\partial\vec{n}},\frac{\partial \varepsilon}{\partial\vec{n}})|_{\partial\Omega}=(0, 0, 0, 0). \end{cases} $$
(2.1)
Then we consider the following linearized problem of (2.1):
with where v, π, and θ are known quantities on \((0,T_{1})\times\Omega\) with \(T_{1}>0\).
$$\begin{aligned}& \rho_{t}+\nabla\cdot(\rho v)=0, \end{aligned}$$
(2.2)
$$\begin{aligned}& \rho u_{t}+\rho v\cdot\nabla u-\Delta u-\nabla\operatorname{div} u+\nabla p=-\frac{2}{3}\nabla(\rho\pi), \end{aligned}$$
(2.3)
$$\begin{aligned}& \rho h_{t}+\rho v\cdot\nabla h-\Delta h=p_{t}+u\cdot \nabla p+S_{k}^{\prime}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \rho k_{t}+\rho v\cdot\nabla k-\Delta k=G^{\prime}-\rho \theta, \end{aligned}$$
(2.5)
$$\begin{aligned}& \rho\varepsilon_{t}+\rho v\cdot\nabla\varepsilon-\Delta \varepsilon= \frac{C_{1}G^{\prime}\theta}{\pi}-\frac{C_{2}\rho\theta^{2}}{\pi }, \end{aligned}$$
(2.6)
$$\begin{aligned}& (\rho,v,h,\pi,\theta) (x,0)=\bigl(\rho _{0}(x),u_{0}(x),h_{0}(x),k_{0}(x), \varepsilon_{0}(x)\bigr), \end{aligned}$$
(2.7)
$$\begin{aligned}& \biggl(v\cdot\vec{n},h, \frac{\partial\pi}{\partial\vec{n}},\frac{\partial \theta}{\partial\vec{n}} \biggr)\bigg|_{\partial\Omega}=(0, 0, 0, 0), \end{aligned}$$
(2.8)
$$\begin{aligned}& S_{k}^{\prime}= \biggl[\mu \biggl(\frac{\partial v^{i}}{\partial x_{j}}+ \frac{\partial v^{j}}{\partial x_{i}} \biggr)-\frac{2}{3}\delta_{ij} \frac{\partial v^{k}}{\partial x_{k}} \biggr]\frac{\partial v^{i}}{\partial x_{j}}\ +\frac{\mu_{t}}{\rho^{2}} \frac{\partial p}{\partial x_{j}}\frac {\partial\rho}{\partial x_{j}}, \\& G^{\prime}=\frac{\partial v^{i}}{\partial x_{j}} \biggl[\mu_{e} \biggl( \frac{\partial v^{i}}{\partial x_{j}}+\frac{\partial v^{j}}{ \partial x_{i}} \biggr)-\frac{2}{3} \delta_{ij} \biggl(\rho\pi+ \mu_{e}\frac{\partial v^{k}}{\partial x_{k}} \biggr) \biggr], \end{aligned}$$
Here we also impose the following regularity conditions on the initial data:
$$ \textstyle\begin{cases} 0< m< \rho_{0},\quad \rho_{0}\in H^{3}(\Omega),\\ u_{0}\in H^{3}(\Omega),\\ (h_{0}, k_{0}, \varepsilon_{0})\in H^{2}(\Omega),\\ (u_{0}\cdot\vec{n},h_{0}, \frac{\partial k_{0}}{\partial\vec{n}},\frac{\partial \varepsilon_{0}}{\partial\vec{n}} ) |_{\partial \Omega}=(0, 0, 0, 0),\\ 0< m< k_{0}. \end{cases} $$
(2.9)
For the known quantities v, π, θ, we assume that \(v(0)=u_{0}\), \(\pi(0)=k_{0}\), \(\theta(0)=\varepsilon_{0}\), and for some fixed constants \(c_{i}\) satisfying \(1< c_{0}< c_{i}\) (\(i=1,2,\ldots,6\)) and some time \(T_{2}>0\). Here For simplicity, we set another small time T as \(T =\min\{ c_{0}^{-6\gamma-16}c_{1}^{-10}c_{2}^{-8}c_{3}^{-8}c_{4}^{-2} c_{5}^{-2}c_{6}^{-4} , T_{1} , T_{2}\}\) and all of the T in Section 2 are defined as this.
$$ \textstyle\begin{cases} \sup _{0\leq t\leq T_{2}}(\Vert v\Vert _{H^{1}} +\Vert \pi \Vert _{H^{1}}+\Vert \theta \Vert _{H^{1}})\\ \quad{}+\int_{0}^{T_{2}}(\Vert \pi \Vert _{H^{3}}^{2} +\Vert v_{t}\Vert _{H^{1}}^{2}+\Vert \pi_{t}\Vert _{H^{1}}^{2}+\Vert \theta_{t}\Vert _{H^{1}}^{2})\,\mathrm{d}t\leq c_{1},\\ \sup _{0\leq t\leq T_{2}}\Vert v\Vert _{H^{2}}\leq c_{2},\\ \sup _{0\leq t\leq T_{2}}\Vert v\Vert _{H^{3}}\leq c_{3},\\ \int_{0}^{T_{2}}\Vert v\Vert _{H^{4}}^{2}\,\mathrm{d}t\leq c_{4},\\ \sup _{0\leq t\leq T_{2}}\Vert \pi \Vert _{H^{2}}\leq c_{5},\\ \sup _{0\leq t\leq T_{2}}\Vert \theta \Vert _{H^{2}}\leq c_{6}, \end{cases} $$
(2.10)
$$c_{0}=2+\bigl\Vert (\rho_{0}, u_{0})\bigr\Vert _{H^{3}}+\bigl\Vert (h_{0},k_{0}, \varepsilon_{0})\bigr\Vert _{H^{2}}. $$
Remark 2.1
Here it should be emphasized that throughout this paper, C denotes a generic positive constant which is only dependent on m, γ, and \(\vert \Omega \vert \), but independent of \(c_{i}\) (\(i=0,1,2,\ldots,6\)).
Remark 2.2
From the physical viewpoint, we assume that the turbulent kinetic energy k has a positive lower bound away from zero, namely, \(0< m< k\) with m a constant. We do not know whether \(0< m< k\) holds afterwards if the initial turbulent kinetic energy \(k_{0}>m\).
In this section we aim to prove the following local existence theorem of the linearized system (2.2)-(2.6).
Theorem 2.1
There exists a unique strong solution
\((\rho,u,h,k,\varepsilon)\)
to the linearized problem (2.2)-(2.8) and (2.9) in
\([0,T]\)
satisfying the estimates (2.99) and (2.100) as well as the regularity
$$\begin{aligned}& \rho\in C\bigl(0,T;H^{3}\bigr),\quad\quad \rho_{t} \in C \bigl(0,T;H^{1}\bigr),\quad\quad u \in C\bigl(0,T; H^{3}\bigr)\cap L^{2}\bigl(0,T; H^{4}\bigr), \\& u_{t}\in L^{2}\bigl(0,T;H^{1}\bigr), \quad\quad k\in C \bigl(0,T; H^{2}\bigr)\cap L^{2}\bigl(0,T;H^{3} \bigr),\quad\quad k_{t}\in L^{2}\bigl(0,T;H^{1}\bigr), \\& \varepsilon\in C\bigl(0,T; H^{2}\bigr),\quad\quad \varepsilon_{t} \in L^{2}\bigl(0,T;H^{1}\bigr),\quad\quad h\in C\bigl(0,T; H^{2}\bigr),\quad\quad h_{t}\in L^{2}\bigl(0,T;H^{1} \bigr), \\& (\sqrt{\rho}u_{t},\sqrt{\rho}k_{t},\sqrt{\rho} \varepsilon _{t},\sqrt{\rho}h_{t})\in L^{\infty} \bigl(0,T;L^{2}\bigr). \end{aligned}$$
In the following part, we decompose the proof of Theorem 2.1 into some lemmas.
Lemma 2.1
Proof
First, applying the particle trajectory method to the equation (2.3), we easily deduce and thus Second, by simple calculation, we have applying the Gronwall and Hölder’s inequalities, one gets for \(0\leq t\leq T\).
$$\rho\geq\rho_{0}\exp \biggl(- \int_{0}^{T}\Vert \nabla v\Vert _{L^{\infty }} \,\mathrm{d}t \biggr) \geq\rho_{0}\exp(-c_{3}T)\geq \frac{\rho_{0}}{e}\geq\frac {m}{e} $$
$$\frac{1}{\rho}\leq\frac{e}{m}\leq C. $$
$$\frac{\mathrm{d}}{\mathrm{d}t}\Vert \rho \Vert _{H^{3}}\leq C\Vert v\Vert _{H^{3}}\Vert \rho \Vert _{H^{3}}+C\bigl\Vert \nabla^{4}v\bigr\Vert _{L^{2}}, $$
$$\Vert \rho \Vert _{H^{3}}\leq \biggl[\exp \biggl(C \int_{0}^{t}\Vert v\Vert _{H^{3}}\, \mathrm{d}t \biggr) \biggr] \biggl(\Vert \rho_{0}\Vert _{H^{3}} +C \int_{0}^{t}\Vert v\Vert _{H^{4}}\, \mathrm{d}t \biggr)\leq Cc_{0} $$
Next, from the equation (2.2), one obtains for \(0\leq t\leq T\).
$$\Vert \rho_{t}\Vert _{H^{1}}=\bigl\Vert \nabla\cdot(\rho v)\bigr\Vert _{H^{1}}\leq C\Vert \rho \Vert _{H^{3}}\Vert v \Vert _{H^{2}}\leq Cc_{0}c_{2} $$
Thus, we complete the proof of Lemma 2.1. □
Next, we estimate the velocity field u.
Lemma 2.2
There exists a unique strong solution u to the initial boundary value problem (2.3) and (2.9) such that
for
\(0\leq t\leq T\).
$$\begin{aligned}& \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2} + \Vert u\Vert _{H^{1}}^{2}+ \int_{0}^{t}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s \leq Cc_{0}^{5+2\gamma},\quad\quad \Vert u\Vert _{H^{2}}\leq Cc_{0}^{\frac {5}{2}+3\gamma}c_{1}^{2}, \end{aligned}$$
(2.12)
$$\begin{aligned}& \Vert u\Vert _{H^{3}}\leq Cc_{0}^{\frac{13}{2}+3\gamma}c_{1}^{4}c_{2}c_{5},\quad\quad \int_{0}^{t}\Vert u\Vert _{H^{4}}^{2} \,\mathrm{d}s \leq Cc_{0}^{9+6\gamma }c_{1}^{5}c_{2}^{2} \end{aligned}$$
(2.13)
Proof
We only need to prove the estimates. Differentiating the equation (2.3) with respect to t, then multiplying both sides of the result by \(u_{t}\) and integrating over Ω, we derive that where we have used the equation (2.2) and integration by parts. We will estimate \(I_{i}\) (\(i=1,2,\ldots,5\)) item by item.
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int\rho u_{t}^{2}\,\mathrm{d}x+\Vert \nabla u_{t}\Vert _{L^{2}}^{2}+\Vert \operatorname{div} u_{t}\Vert _{L^{2}}^{2} \\& \quad=- \int\rho_{t}v\cdot\nabla u\cdot u_{t}- \int\rho v_{t}\cdot \nabla u\cdot u_{t}-2 \int\rho v\cdot\nabla u_{t}\cdot u_{t} \\& \quad\quad{}- \int \nabla p_{t}\cdot u_{t}-\frac{2}{3} \int\bigl[\nabla(\rho\pi)\bigr]_{t}\cdot u_{t} \\& \quad=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}, \end{aligned}$$
(2.14)
First, because ρ has a lower bound away from zero, we easily deduce \(\Vert u_{t}\Vert _{L^{2}}\leq C\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}\). Therefore, using the Hölder, Sobolev, and Young inequalities and (2.10), we have
where \(\eta>0\) is a small number to be determined later.
$$\begin{aligned}& I_{1}\leq C\Vert v\Vert _{L^{\infty}} \Vert \rho_{t}\Vert _{L^{3}}\Vert \nabla u\Vert _{L^{2}} \Vert u_{t}\Vert _{L^{6}} \leq C\Vert v\Vert _{L^{\infty}} \Vert \rho_{t}\Vert _{L^{3}}\Vert \nabla u \Vert _{L^{2}}\bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}+\Vert \nabla u_{t}\Vert _{L^{2}}\bigr) \\& \hphantom{I_{1}}\leq Cc_{0}^{2}c_{2}^{4}\Vert \nabla u\Vert _{L^{2}}^{2}+C\Vert \sqrt{\rho }u_{t}\Vert _{L^{2}}^{2}+\frac{1}{8}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.15)
$$\begin{aligned}& I_{3}\leq C\Vert \rho \Vert _{L^{\infty}}^{\frac{1}{2}}\Vert v \Vert _{L^{\infty}} \Vert \nabla u_{t}\Vert _{L^{2}} \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}} \leq Cc_{0}c_{2}^{2}\Vert \sqrt{\rho}u_{t} \Vert _{L^{2}}^{2}+\frac{1}{8}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.16)
$$\begin{aligned}& I_{2} \leq C\Vert \rho \Vert _{L^{\infty}}^{\frac{1}{2}}\Vert v_{t}\Vert _{L^{6}}\Vert \nabla u\Vert _{L^{3}} \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}} \leq C \eta^{-1}c_{0}\Vert \nabla u\Vert _{L^{3}}^{2}+ \eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{ \rho}u_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.17)
Next, to evaluate \(\Vert \nabla u\Vert _{L^{3}}^{2}\) in (2.17), we can first the Sobolev interpolation inequality to get Then applying the standard elliptic regularity result to the equation (2.3) and using (2.18), we have thus the Young inequality and (2.10) yield Combining (2.17), (2.18), and (2.19), and using the Young inequality, we get
$$ \Vert \nabla u\Vert _{L^{3}}^{2}\leq C\Vert \nabla u \Vert _{L^{2}}\Vert \nabla u\Vert _{L^{6}}\leq C\Vert \nabla u\Vert _{L^{2}}\Vert \nabla u\Vert _{H^{1}}. $$
(2.18)
$$\begin{aligned} \Vert \nabla u\Vert _{H^{1}} \leq& Cc_{0}^{\gamma} \bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}+\Vert v\Vert _{L^{6}}\Vert \nabla u\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla u\Vert _{H^{1}}^{\frac{1}{2}} +\Vert \nabla\rho \Vert _{L^{2}} \\ &{}+\Vert \nabla\rho \Vert _{L^{4}}\Vert \pi \Vert _{L^{4}}+\Vert \nabla\pi \Vert _{L^{2}}\bigr), \end{aligned}$$
$$ \Vert \nabla u\Vert _{H^{1}}\leq Cc_{0}^{2\gamma} \bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}+c_{1}^{2} \Vert \nabla u\Vert _{L^{2}}+c_{0}c_{1} \bigr). $$
(2.19)
$$ I_{2} \leq C\eta^{-1}c_{0}^{2\gamma+1}\bigl( \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+c_{1}^{2} \Vert \nabla u\Vert _{L^{2}}^{2} +c_{0}^{2}c_{1}^{2} \bigr) +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}. $$
(2.20)
By integration by parts, we have
On the other hand, we easily have and Combining (2.14)-(2.16) and (2.20)-(2.24), we get setting \(\eta=\frac{1}{c_{1}}\) and using the Gronwall inequality, we derive for \(0\leq t\leq T\), where we have used the fact that \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+\Vert u\Vert _{H^{1}}^{2})\leq Cc_{0}^{5+2\gamma}\).
$$\begin{aligned}& I_{4}= \int p_{t}\operatorname{div} u_{t} \leq Cc_{0}^{\gamma-1}\Vert \rho_{t}\Vert _{L^{2}}\Vert \nabla u_{t}\Vert _{L^{2}}\leq Cc_{0}^{2\gamma}c_{2}^{2}+\frac{1}{8} \Vert \nabla u_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.21)
$$\begin{aligned}& I_{5} =\frac{2}{3} \int\rho_{t}\pi\nabla\cdot u_{t}-\frac {2}{3} \int\pi_{t}\nabla\rho\cdot u_{t} -\frac{2}{3} \int\rho\nabla\pi_{t}\cdot u_{t} \\& \hphantom{I_{5} } \leq C\Vert \rho_{t}\Vert _{L^{3}}\Vert \pi \Vert _{L^{6}}\Vert \nabla u_{t}\Vert _{L^{2}} +Cc_{0}^{\frac{1}{2}}\Vert \nabla\rho \Vert _{L^{3}}\Vert \pi_{t}\Vert _{L^{6}}\Vert \sqrt{\rho}u_{t} \Vert _{L^{2}} +Cc_{0}^{\frac{1}{2}}\Vert \nabla \pi_{t}\Vert _{L^{2}}\Vert \sqrt{\rho}u_{t} \Vert _{L^{2}} \\& \hphantom{I_{5} }\leq Cc_{0}^{2}c_{1}^{2}c_{2}^{2}+C \eta^{-1}c_{0}^{3}+C\eta \Vert \pi _{t} \Vert _{H^{1}}^{2}\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2} +\frac{1}{8}\Vert \nabla u_{t} \Vert _{L^{2}}^{2}. \end{aligned}$$
(2.22)
$$ \frac{\mathrm{d}}{\mathrm{d}t} \int \vert \nabla u\vert ^{2} =2 \int\nabla u\cdot\nabla u_{t}\leq\frac{1}{8}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}+C\Vert \nabla u \Vert _{L^{2}}^{2} $$
(2.23)
$$ \frac{\mathrm{d}}{\mathrm{d}t} \int \vert u\vert ^{2}\leq Cc_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}\Vert u\Vert _{L^{2}} \leq Cc_{0}\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+C\Vert u\Vert _{L^{2}}^{2}. $$
(2.24)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+\Vert u\Vert _{H^{1}}^{2} \bigr)+\Vert \nabla u_{t}\Vert _{L^{2}}^{2} \\& \quad\leq C\bigl(c_{0}^{2}c_{2}^{4}+ \eta^{-1}c_{0}^{2\gamma+1}c_{1}^{2}+ \eta \Vert \pi_{t}\Vert _{H^{1}}^{2} +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\bigr) \bigl(\Vert \sqrt{ \rho}u_{t}\Vert _{L^{2}}^{2}+\Vert u\Vert _{H^{1}}^{2}\bigr) \\& \quad\quad{}+C\bigl(c_{0}^{2\gamma}c_{1}^{2}c_{2}^{2}+ \eta^{-1}c_{0}^{2\gamma +3}c_{1}^{2} \bigr), \end{aligned}$$
(2.25)
$$ \Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+\Vert u\Vert _{H^{1}}^{2}+ \int_{0}^{t}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5+2\gamma} $$
(2.26)
Next, by (2.19) and (2.26), we deduce which implies (2.12) by (2.26).
$$ \Vert \nabla u\Vert _{H^{1}}\leq Cc_{0}^{\frac{5}{2}+3\gamma}c_{1}^{2}, $$
(2.27)
Next, we will estimate \(\int_{0}^{t}\Vert u\Vert _{H^{4}}^{2}\,\mathrm{d}t\). By the standard elliptic regularity result of the equation (2.3), we have By simple calculation, the first term of the right-hand side of (2.28) can be controlled as In order to estimate \(\Vert \nabla^{2}u_{t}\Vert _{L^{2}}\), differentiating the equation (2.3) with respect to t yields applying the standard elliptic regularity result to (2.30) and using (2.26), one obtains therefore, the key point is to estimate \(\Vert \rho u_{tt}\Vert _{L^{2}}\). Because we have the fact \(\Vert \rho u_{tt}\Vert _{L^{2}}\leq C\Vert \sqrt{\rho} u_{tt}\Vert _{L^{2}}\), we could first estimate \(\Vert \sqrt{\rho} u_{tt}\Vert _{L^{2}}\) as follows.
$$ \bigl\Vert \nabla^{4}u\bigr\Vert _{L^{2}}\leq \Vert \rho u_{t}\Vert _{H^{2}}+\Vert \rho v\cdot \nabla u\Vert _{H^{2}}+\Vert \nabla p\Vert _{H^{2}}+\biggl\Vert \frac{2}{3}\nabla(\rho\pi )\biggr\Vert _{H^{2}}. $$
(2.28)
$$ \Vert \rho u_{t}\Vert _{H^{2}}\leq C\bigl(\Vert \rho u_{t}\Vert _{L^{2}}+\Vert \rho \Vert _{H^{2}} \Vert u_{t}\Vert _{H^{2}}\bigr)\leq Cc_{0}\Vert u_{t}\Vert _{H^{2}}. $$
(2.29)
$$\begin{aligned} \Delta u_{t}+\nabla\operatorname{div} u_{t} =&\rho_{t}u_{t}+\rho u_{tt}+ \rho_{t}v\cdot\nabla u+\rho v_{t}\cdot \nabla u +\rho v\cdot \nabla u_{t}+\nabla p_{t} \\ &{}+\frac{2}{3}(\nabla \rho_{t}\pi +\rho_{t}\nabla\pi+\nabla\rho\pi_{t}+ \rho\nabla\pi _{t}), \end{aligned}$$
(2.30)
$$\begin{aligned} \bigl\Vert \nabla^{2}u_{t}\bigr\Vert _{L^{2}} \leq& C\bigl(\Vert \rho_{t}\Vert _{L^{4}}\Vert u_{t}\Vert _{L^{4}}+\Vert \rho u_{tt}\Vert _{L^{2}}+\Vert \rho_{t}\Vert _{L^{4}}\Vert v \Vert _{L^{\infty}} \Vert \nabla u\Vert _{L^{4}} \\ &{}+\Vert \rho \Vert _{L^{\infty}} \Vert v_{t}\Vert _{L^{4}}\Vert \nabla u\Vert _{L^{4}} +\Vert v\Vert _{L^{\infty}} \Vert u_{t}\Vert _{H^{1}} +\Vert \rho \Vert _{H^{2}}^{\gamma} \Vert \rho_{t}\Vert _{H^{1}} +\Vert \pi \Vert _{L^{\infty}} \Vert \rho_{t}\Vert _{H^{1}} \\ &{}+\Vert \rho_{t} \Vert _{L^{4}}\Vert \nabla\pi \Vert _{L^{4}} +\Vert \nabla\rho \Vert _{L^{4}}\Vert \pi_{t}\Vert _{L^{4}} +\Vert \rho \Vert _{L^{\infty}} \Vert \nabla \pi_{t}\Vert _{L^{2}}\bigr) \\ \leq& C\bigl(\Vert \rho u_{tt}\Vert _{L^{2}}+c_{0}^{\frac{7}{2}+3\gamma }c_{1}^{2}c_{2}^{2}c_{5}+c_{0}^{\frac{7}{2}+3\gamma}c_{1}^{2} \Vert v_{t}\Vert _{H^{1}} \\ &{} +c_{0}c_{2} \Vert u_{t}\Vert _{H^{1}}+c_{0}\Vert \pi_{t}\Vert _{H^{1}}\bigr), \end{aligned}$$
(2.31)
Multiplying both sides of (2.30) by \(u_{tt}\) and integrating the result over Ω yield Using the Hölder, Sobolev, and Young inequalities and (2.10) and (2.26), we get
inserting (2.33)-(2.38) to (2.32), then integrating the result over \((0,t)\), we derive where we have used the equation (2.3) to get \(\lim_{t\rightarrow 0}\Vert \nabla u_{t}(t)\Vert _{L^{2}}^{2}\leq Cc_{0}^{2\gamma+4}\).
$$\begin{aligned}& \int\rho u_{tt}^{2}\,\mathrm{d}x+\frac{1}{2} \frac{\mathrm{d}}{\mathrm {d}t}\Vert \nabla u_{t}\Vert _{L^{2}}^{2} +\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \operatorname{div} u_{t} \Vert _{L^{2}}^{2} \\& \quad=- \int\rho_{t}u_{t}\cdot u_{tt}- \int\rho_{t}v\cdot\nabla u\cdot u_{tt}- \int\rho v_{t}\cdot\nabla u\cdot u_{tt} - \int\rho v\cdot\nabla u_{t}\cdot u_{tt}- \int\nabla p_{t}\cdot u_{tt} \\& \quad\quad{}-\frac{2}{3} \int(\pi\nabla\rho_{t}+\rho_{t}\nabla\pi+\pi _{t}\nabla\rho+\rho\nabla\pi_{t})\cdot u_{tt} \\& \quad=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}. \end{aligned}$$
(2.32)
$$\begin{aligned}& J_{1} \leq Cc_{0}^{\frac{1}{2}}\Vert \rho_{t}\Vert _{L^{3}}\Vert u_{t}\Vert _{L^{6}}\Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}} \leq Cc_{0}^{\frac{1}{2}}\Vert \rho_{t}\Vert _{L^{3}}\bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}+ \Vert \nabla u_{t}\Vert _{L^{2}}\bigr)\Vert \sqrt{ \rho}u_{tt}\Vert _{L^{2}} \\& \hphantom{ J_{1}}\leq Cc_{0}^{3}c_{2}^{2}\Vert \nabla u_{t}\Vert _{L^{2}}^{2}+Cc_{0}^{8+2\gamma}c_{2}^{2}+ \frac{1}{18}\Vert \sqrt{\rho }u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.33)
$$\begin{aligned}& J_{2}\leq Cc_{0}^{\frac{1}{2}}\Vert \sqrt{ \rho}u_{tt}\Vert _{L^{2}}\Vert \rho _{t}\Vert _{L^{3}} \Vert v\Vert _{L^{\infty}} \Vert \nabla u\Vert _{L^{6}} \leq Cc_{0}^{8+6\gamma}c_{1}^{4}c_{2}^{4}+ \frac{1}{18}\Vert \sqrt{\rho }u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.34)
$$\begin{aligned}& J_{3}\leq Cc_{0}^{\frac{1}{2}}\Vert \sqrt{ \rho}u_{tt}\Vert _{L^{2}}\Vert v_{t}\Vert _{L^{3}}\Vert \nabla u\Vert _{L^{6}} \leq Cc_{0}^{6+6\gamma}c_{1}^{4} \Vert v_{t}\Vert _{H^{1}}^{2}+\frac{1}{18} \Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.35)
$$\begin{aligned}& J_{4}\leq Cc_{0}^{\frac{1}{2}}\Vert v\Vert _{L^{\infty}} \Vert \sqrt{\rho }u_{tt}\Vert _{L^{2}}\Vert \nabla u_{t}\Vert _{L^{2}} \leq Cc_{0}c_{2}^{2} \Vert \nabla u_{t}\Vert _{L^{2}}^{2}+ \frac{1}{18}\Vert \sqrt {\rho}u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.36)
$$\begin{aligned}& J_{5}\leq Cc_{0}^{\frac{1}{2}}\Vert \sqrt{ \rho}u_{tt}\Vert _{L^{2}}\Vert \nabla p_{t}\Vert _{L^{2}} \leq Cc_{0}^{2\gamma+1}c_{2}^{2}+ \frac{1}{18}\Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.37)
$$\begin{aligned}& J_{6} \leq Cc_{0}^{\frac{1}{2}}\Vert \pi \Vert _{L^{\infty}} \Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}\Vert \nabla\rho_{t}\Vert _{L^{2}}+ Cc_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}\Vert \nabla\pi \Vert _{L^{4}}\Vert \rho_{t}\Vert _{L^{4}} \\& \hphantom{ J_{6}=}{} +Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}\Vert \nabla\rho \Vert _{L^{\infty}} \Vert \pi_{t}\Vert _{L^{2}} +Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}\Vert \nabla \pi_{t}\Vert _{L^{2}} \\& \hphantom{ J_{6}} \leq Cc_{0}^{3}c_{2}^{2}c_{5}^{2}+Cc_{0}^{3} \Vert \pi_{t}\Vert _{H^{1}}^{2}+\frac{2}{9} \Vert \sqrt{\rho}u_{tt}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.38)
$$ \int_{0}^{t} \int_{\Omega}\rho u_{tt}^{2}\,\mathrm{d}x\,\mathrm{d}t+ \Vert \nabla u_{t}\Vert _{L^{2}}^{2}\leq Cc_{0}^{6+6\gamma}c_{1}^{5}c_{2}^{2}, $$
(2.39)
So, combining (2.29), (2.31), and (2.39), we obtain In the following, we shall estimate the rest terms of the inequality (2.28).
$$ \int_{0}^{t}\Vert \rho u_{t}\Vert _{H^{2}}^{2}\leq Cc_{0}^{9+6\gamma }c_{1}^{5}c_{2}^{2}. $$
(2.40)
For the second term of the inequality (2.28), direct calculation yields therefore, we have to evaluate \(\Vert u\Vert _{H^{3}}\). In fact, applying the standard elliptic regularity result to the equation (2.3), we obtain we could estimate the right-hand side of (2.42) item by item.
$$ \Vert \rho v\cdot\nabla u\Vert _{H^{2}}\leq C\Vert \rho \Vert _{H^{2}}\Vert v\Vert _{H^{2}}\Vert u\Vert _{H^{3}} \leq Cc_{0}c_{2}\Vert u\Vert _{H^{3}}, $$
(2.41)
$$ \bigl\Vert \nabla^{3}u\bigr\Vert _{L^{2}}\leq C\bigl( \Vert \rho u_{t}\Vert _{H^{1}}+\Vert \rho v\cdot \nabla u \Vert _{H^{1}}+\Vert \nabla p\Vert _{H^{1}}+\bigl\Vert \nabla(\rho\pi)\bigr\Vert _{H^{1}}\bigr), $$
(2.42)
First, from (2.26), we have \(\Vert u_{t}\Vert _{L^{2}}\leq Cc_{0}^{\frac{5}{2}+\gamma}\), thus Second, using the Sobolev interpolation inequality and the Young inequality, we get Third, due to (2.11), we easily derive Last, by simple calculation, one gets Combining (2.39) and (2.42)-(2.46), we deduce Next, by simple calculation, the third and fourth terms on the right-hand side of (2.28) can be estimated as Combining (2.26), (2.28), (2.40), (2.41), and (2.47)-(2.48), one deduces for \(0\leq t\leq T\).
$$\begin{aligned} \Vert \rho u_{t}\Vert _{H^{1}} \leq& Cc_{0} \Vert u_{t}\Vert _{L^{2}}+\Vert \nabla\rho \Vert _{L^{\infty}} \Vert u_{t}\Vert _{L^{2}}+Cc_{0} \Vert \nabla u_{t}\Vert _{L^{2}} \\ \leq& Cc_{0}^{\frac{7}{2}+\gamma}+Cc_{0} \Vert \nabla u_{t}\Vert _{L^{2}}. \end{aligned}$$
(2.43)
$$\begin{aligned}& \Vert \rho v\cdot\nabla u\Vert _{H^{1}} \\& \quad\leq C\bigl(\Vert \rho v \cdot\nabla u\Vert _{L^{2}}+\bigl\Vert \nabla(\rho v\cdot\nabla u)\bigr\Vert _{L^{2}}\bigr) \\& \quad\leq C\bigl(c_{0}\Vert v\Vert _{L^{\infty}} \Vert \nabla u \Vert _{L^{2}}+\Vert \nabla\rho \Vert _{L^{\infty}} \Vert v\Vert _{L^{\infty}} \Vert \nabla u\Vert _{L^{2}} +c_{0}\Vert \nabla v\Vert _{L^{2}}\Vert \nabla u\Vert _{L^{2}}^{\frac{1}{4}} \bigl\Vert \nabla ^{3} u\bigr\Vert _{L^{2}}^{\frac{3}{4}} \\& \quad\quad{}+c_{0}\Vert v\Vert _{L^{\infty}}\bigl\Vert \nabla^{2}u\bigr\Vert _{L^{2}}\bigr) \\& \quad \leq Cc_{0}^{\frac{13}{2}+3\gamma}c_{1}^{4}c_{2}+ \frac{3}{4}\Vert u\Vert _{H^{3}}. \end{aligned}$$
(2.44)
$$ \Vert \nabla p\Vert _{H^{1}}\leq Cc_{0}^{2}. $$
(2.45)
$$ \bigl\Vert \nabla(\rho\pi)\bigr\Vert _{H^{1}} \leq C\Vert \rho \Vert _{H^{3}}\Vert \pi \Vert _{H^{2}} \leq Cc_{0}c_{5}. $$
(2.46)
$$ \Vert u\Vert _{H^{3}} \leq Cc_{0}^{\frac{13}{2}+3\gamma}c_{1}^{4}c_{2}c_{5}. $$
(2.47)
$$ \Vert \nabla p\Vert _{H^{2}}\leq Cc_{0}^{3}, \quad\quad \bigl\Vert \nabla(\rho\pi)\bigr\Vert _{H^{2}}\leq Cc_{0} \Vert \pi \Vert _{H^{3}}. $$
(2.48)
$$ \int_{0}^{t}\Vert u\Vert _{H^{4}}^{2} \,\mathrm{d}t\leq Cc_{0}^{9+6\gamma }c_{1}^{5}c_{2}^{2}, $$
(2.49)
Thus, we complete the proof of Lemma 2.2. □
In the following part, we estimate the turbulent kinetic energy k.
Lemma 2.3
There exists a unique strong solution k to the initial boundary value problem (2.5) and (2.9) such that
for
\(0\leq t\leq T\).
$$\begin{aligned}& \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2} + \Vert k\Vert _{H^{1}}^{2} + \int_{0}^{t}\Vert \nabla k_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5}, \end{aligned}$$
(2.50)
$$\begin{aligned}& \Vert k\Vert _{H^{2}}\leq Cc_{0}^{\frac{7}{2}}c_{1}c_{2}^{2},\quad\quad \int_{0}^{t}\Vert k\Vert _{H^{3}}^{2} \,\mathrm{d}s\leq Cc_{0}^{7}, \end{aligned}$$
(2.51)
Proof
We only need to prove the estimates. Differentiating the equation (2.5) with respect to t, then multiplying both sides of the resulting equation by \(k_{t}\) and integrating over Ω, we get we could evaluate \(K_{i}\) (\(i=1,\ldots,6\)) as follows.
$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2} +\Vert \nabla k_{t}\Vert _{L^{2}}^{2} =&- \int\rho_{t}v\cdot\nabla k\cdot k_{t} - \int\rho v_{t}\cdot\nabla k\cdot k_{t} -2 \int\rho v\cdot\nabla k_{t}\cdot k_{t} \\ &{} + \int G_{t}^{\prime}\cdot k_{t} - \int\rho_{t}\theta\cdot k_{t} - \int\rho\theta_{t}\cdot k_{t} \\ =& \sum _{i=1}^{6}K_{i}, \end{aligned}$$
(2.52)
First, using a similar method to deriving (2.15), (2.20), (2.16), respectively, one has
Next, differentiating \(G^{\prime}\) with respect to t and inserting the result thus obtained into \(K_{4}\) yield Last, direct calculation leads to
On the other hand, we easily get
Combining (2.52)-(2.60), we obtain setting \(\eta=c_{1}^{-1}\) and using the Gronwall inequality, we deduce for \(0\leq t\leq T\), where we have used the fact that \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+\Vert k\Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
$$\begin{aligned}& K_{1} \leq Cc_{0}^{2}c_{2}^{4} \Vert \nabla k\Vert _{L^{2}}^{2}+C\Vert \sqrt{ \rho}k_{t}\Vert _{L^{2}}^{2}+\frac{1}{10} \Vert \nabla k_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.53)
$$\begin{aligned}& K_{2}\leq C\eta^{-1}c_{0}^{2\gamma+1}\bigl( \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+c_{1}^{2} \Vert \nabla k\Vert _{L^{2}}^{2} +c_{0}^{2}c_{1}^{2}c_{2}^{4} \bigr) +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.54)
$$\begin{aligned}& K_{3} \leq Cc_{0}c_{2}^{2}\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+ \frac{1}{10}\Vert \nabla k_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.55)
$$\begin{aligned} K_{4} \leq& C \int \vert \nabla v_{t}\vert \vert \nabla v\vert \vert k_{t}\vert +C \int \vert \rho \vert \vert \pi \vert \vert \nabla v_{t}\vert \vert k_{t}\vert +C \int \vert \rho_{t}\vert \vert \pi \vert \vert \nabla v\vert \vert k_{t}\vert \\ &{}+C \int \vert \rho \vert \vert \pi_{t}\vert \vert \nabla v\vert \vert k_{t}\vert \\ \leq& Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho}k_{t} \Vert _{L^{2}}\Vert \nabla v_{t}\Vert _{L^{2}} \Vert \nabla v\Vert _{L^{\infty}} +Cc_{0}^{\frac{1}{2}}\Vert \pi \Vert _{L^{\infty}} \Vert \nabla v_{t}\Vert _{L^{2}} \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}} \\ &{}+C\Vert \pi \Vert _{L^{\infty}} \Vert \rho_{t}\Vert _{L^{3}}\Vert \nabla v\Vert _{L^{2}}\Vert k_{t} \Vert _{L^{6}} +Cc_{0}^{\frac{1}{2}}\Vert \sqrt{ \rho}k_{t}\Vert _{L^{2}}\Vert \pi_{t}\Vert _{L^{6}}\Vert \nabla v\Vert _{L^{3}} \\ \leq& C\eta^{-1}c_{0}c_{3}^{2}c_{5}^{2}+Cc_{0}^{2}c_{1}^{2}c_{2}^{2}c_{5}^{2} +C\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2} +C\eta \bigl(\Vert v_{t}\Vert _{H^{1}}^{2} +\Vert \pi_{t}\Vert _{H^{1}}^{2}\bigr)\Vert \sqrt{ \rho}k_{t}\Vert _{L^{2}}^{2} \\ &{} +\frac{1}{10} \Vert \nabla k_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.56)
$$\begin{aligned}& K_{5}\leq \Vert \rho_{t}\Vert _{L^{3}}\Vert \theta \Vert _{L^{2}}\Vert k_{t}\Vert _{L^{6}} \leq Cc_{0}^{2}c_{1}^{2}c_{2}^{2} +C\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2} + \frac{1}{10}\Vert \nabla k_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.57)
$$\begin{aligned}& K_{6}\leq Cc_{0}^{\frac{1}{2}}\Vert \sqrt{ \rho}k_{t}\Vert _{L^{2}}\Vert \theta _{t}\Vert _{L^{2}} \leq C\eta^{-1}c_{0}+\eta \Vert \theta_{t}\Vert _{L^{2}}^{2}\Vert \sqrt{\rho }k_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.58)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla k\Vert _{L^{2}}^{2}\leq \frac {1}{10}\Vert \nabla k_{t}\Vert _{L^{2}}^{2} +C\Vert \nabla k\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.59)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\Vert k\Vert _{L^{2}}^{2} \leq Cc_{0}\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+C \Vert k\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.60)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+\Vert k\Vert _{H^{1}}^{2} \bigr)+\Vert \nabla k_{t}\Vert _{L^{2}}^{2} \\& \quad\leq C\bigl(c_{0}^{2}c_{2}^{4}+ \eta^{-1}c_{0}^{2\gamma+1}c_{1}^{2}+ \eta \Vert v_{t}\Vert _{H^{1}}^{2}+\eta \Vert \pi_{t}\Vert _{H^{1}}^{2} +\eta \Vert \theta_{t}\Vert _{L^{2}}^{2}\bigr) \bigl(\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2} +\Vert k\Vert _{H^{1}}^{2}\bigr) \\& \quad\quad{}+C\bigl(\eta ^{-1}c_{0}^{2}c_{1}^{2}c_{2}^{4} c_{3}^{2}c_{5}^{2}+c_{0}^{2}c_{1}^{2}c_{2}^{2}c_{5}^{2} \bigr), \end{aligned}$$
(2.61)
$$ \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+\Vert k\Vert _{H^{1}}^{2}+ \int_{0}^{t}\Vert \nabla k_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s \leq Cc_{0}^{5} $$
(2.62)
Then, by the standard elliptic regularity result of the equation (2.5) and using (2.62), we have and To evaluate \(\int_{0}^{t}\Vert k\Vert _{H^{3}}^{2}\,\mathrm{d}t\), we will estimate the right-hand side of (2.64) item by item.
$$\begin{aligned} \Vert \nabla k\Vert _{H^{1}} \leq& Cc_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}+Cc_{0} \Vert v\Vert _{L^{\infty}} \Vert \nabla k\Vert _{L^{2}}+ C\Vert \nabla v\Vert _{L^{4}}^{2} \\ &{}+Cc_{0}\Vert \pi \Vert _{L^{4}}\Vert \nabla v\Vert _{L^{4}}+Cc_{0}\Vert \theta \Vert _{L^{2}} \\ \leq& Cc_{0}^{\frac{7}{2}}c_{1}c_{2}^{2} \end{aligned}$$
(2.63)
$$ \bigl\Vert \nabla^{2} k\bigr\Vert _{H^{1}}\leq C\bigl( \Vert \rho k_{t}\Vert _{H^{1}}+\Vert \rho v\cdot \nabla k \Vert _{H^{1}}+\bigl\Vert G^{\prime}\bigr\Vert _{H^{1}}+ \Vert \rho\theta \Vert _{H^{1}}\bigr). $$
(2.64)
In fact, we derive by using (2.62) and (2.63) that
and Therefore, inserting (2.65)-(2.68) to (2.64) and integrating the result thus obtained over \((0,t)\), one gets for \(0\leq t\leq T\).
$$\begin{aligned}& \Vert \rho k_{t}\Vert _{H^{1}}\leq C\bigl(\Vert \rho k_{t}\Vert _{L^{2}}+\bigl\Vert \nabla(\rho k_{t})\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\Vert \rho k_{t}\Vert _{H^{1}}} \leq Cc_{0}^{\frac{7}{2}}+Cc_{0} \Vert \nabla k_{t}\Vert _{L^{2}}, \end{aligned}$$
(2.65)
$$\begin{aligned}& \Vert \rho v\cdot\nabla k\Vert _{H^{1}}\leq C\bigl(\Vert \rho v \cdot\nabla k\Vert _{L^{2}}+\bigl\Vert \nabla(\rho v\cdot\nabla k)\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\Vert \rho v\cdot\nabla k\Vert _{H^{1}}}\leq C\bigl(c_{0}\Vert v\Vert _{L^{\infty}} \Vert \nabla k \Vert _{L^{2}}+\Vert \nabla\rho \Vert _{L^{\infty}} \Vert v\Vert _{L^{\infty}} \Vert \nabla k\Vert _{L^{2}} \\& \hphantom{\Vert \rho v\cdot\nabla k\Vert _{H^{1}}\leq}{} +c_{0}\Vert \nabla v\Vert _{L^{4}} \Vert \nabla k \Vert _{L^{4}}+c_{0}\Vert v\Vert _{L^{\infty }}\bigl\Vert \nabla^{2}k\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\Vert \rho v\cdot\nabla k\Vert _{H^{1}} }\leq Cc_{0}^{\frac{9}{2}}c_{1}c_{2}^{3}, \end{aligned}$$
(2.66)
$$\begin{aligned}& \bigl\Vert G^{\prime}\bigr\Vert _{H^{1}}\leq C\bigl(\Vert \nabla v\Vert _{L^{4}}^{2}+\Vert \nabla v\cdot\rho \cdot\pi \Vert _{L^{2}}+\bigl\Vert \nabla v\cdot\nabla^{2}v\bigr\Vert _{L^{2}} +\bigl\Vert \nabla(\nabla v\cdot\rho\cdot\pi)\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{\bigl\Vert G^{\prime}\bigr\Vert _{H^{1}} }\leq C\bigl(\Vert \nabla v\Vert _{L^{4}}^{2}+c_{0} \Vert \pi \Vert _{L^{\infty}} \Vert \nabla v\Vert _{L^{2}}+\Vert \nabla v\Vert _{L^{4}}\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{4}} +c_{0}\Vert \pi \Vert _{L^{\infty}}\bigl\Vert \nabla^{2} v\bigr\Vert _{L^{2}} \\& \hphantom{\bigl\Vert G^{\prime}\bigr\Vert _{H^{1}}\leq}{} +\Vert \pi \Vert _{L^{\infty}} \Vert \nabla\rho \Vert _{L^{\infty}} \Vert \nabla v\Vert _{L^{2}}+c_{0}\Vert \nabla v\Vert _{L^{\infty}} \Vert \nabla\pi \Vert _{L^{2}}\bigr) \\& \hphantom{\bigl\Vert G^{\prime}\bigr\Vert _{H^{1}} } \leq Cc_{0}c_{1}c_{2}^{2}c_{3}c_{5}, \end{aligned}$$
(2.67)
$$ \Vert \rho\theta \Vert _{H^{1}}\leq C\Vert \rho \Vert _{H^{3}}\Vert \theta \Vert _{H^{1}}\leq Cc_{0}c_{1}. $$
(2.68)
$$ \int_{0}^{t}\Vert k\Vert _{H^{3}}^{2} \,\mathrm{d}t\leq Cc_{0}^{7} $$
(2.69)
Combining (2.62), (2.63), and (2.69), we complete the proof of Lemma 2.3. □
In the next part, we estimate the viscous dissipation rates of the turbulent flows ε.
Lemma 2.4
There exists a unique strong solution
ε
to the initial boundary value problem (2.6) and (2.9) such that
for
\(0\leq t\leq T\).
$$\begin{aligned}& \Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2} +\Vert \varepsilon \Vert _{H^{1}}^{2} + \int_{0}^{t}\Vert \nabla\varepsilon_{t} \Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5}, \end{aligned}$$
(2.70)
$$\begin{aligned}& \Vert \varepsilon \Vert _{H^{2}}\leq Cc_{0}^{\frac {9}{2}}c_{1}^{2}c_{2}^{2} \end{aligned}$$
(2.71)
Proof
We only need to prove the estimates. Differentiating the equation (2.6) with respect to t, then multiplying both sides of the result by \(\varepsilon_{t}\) and integrating over Ω, one obtains We could evaluate \(E_{4}\) and \(E_{5}\) in the first place. Because π has an upper and a lower bound away from zero, direct calculation yields and
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Vert \sqrt{\rho}\varepsilon _{t}\Vert _{L^{2}}^{2} +\Vert \nabla \varepsilon_{t}\Vert _{L^{2}}^{2} \\ & \quad= - \int\rho_{t}v\cdot\nabla\varepsilon\cdot\varepsilon_{t} - \int\rho v_{t}\cdot\nabla\varepsilon\cdot\varepsilon_{t} -2 \int\rho v\cdot\nabla\varepsilon_{t}\cdot\varepsilon _{t} \\ & \quad\quad{}+ \int \biggl(\frac{C_{1}G^{\prime}\theta}{\pi} \biggr)_{t}\cdot \varepsilon_{t} - \int \biggl(\frac{C_{2}\rho\theta^{2}}{\pi} \biggr)_{t}\cdot \varepsilon_{t} \\ & \quad= \sum _{i=1}^{5}E_{i}. \end{aligned}$$
(2.72)
$$\begin{aligned} E_{4} \leq& C \int\bigl(\bigl\vert G^{\prime}_{t}\theta\bigr\vert +\bigl\vert G^{\prime}\theta_{t}\bigr\vert +\bigl\vert G^{\prime}\theta \pi_{t}\bigr\vert \bigr)\vert \varepsilon_{t}\vert \\ \leq& C \int\bigl(\vert \nabla v_{t}\cdot\nabla v\vert +\vert \rho_{t}\pi\nabla v\vert +\vert \rho\pi_{t}\nabla v \vert +\vert \rho\pi\nabla v_{t}\vert \bigr)\vert \theta \vert \vert \varepsilon_{t}\vert \\ &{}+C \int\bigl(\vert \nabla v\vert ^{2}+\vert \rho\pi\nabla v \vert \bigr)\vert \theta_{t}\vert \vert \varepsilon_{t} \vert +C \int\bigl(\vert \nabla v\vert ^{2}+\vert \rho\pi\nabla v \vert \bigr)\vert \theta \vert \vert \pi _{t}\vert \vert \varepsilon_{t}\vert \\ \leq &Cc_{0}^{\frac{1}{2}}\Vert \theta \Vert _{L^{\infty}} \Vert \nabla v\Vert _{L^{\infty}} \Vert \nabla v_{t}\Vert _{L^{2}}\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}} +Cc_{0}^{\frac{1}{2}}\Vert \pi \Vert _{L^{\infty}} \Vert \sqrt{\rho}\varepsilon _{t}\Vert _{L^{2}}\Vert \rho_{t}\Vert _{L^{6}}\Vert \nabla v \Vert _{L^{6}}\Vert \theta \Vert _{L^{6}} \\ &{}+Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho}\varepsilon_{t} \Vert _{L^{2}}\Vert \pi _{t}\Vert _{L^{6}}\Vert \nabla v\Vert _{L^{6}}\Vert \theta \Vert _{L^{6}} +Cc_{0}\Vert \pi \Vert _{L^{\infty}} \Vert \theta \Vert _{L^{\infty}} \Vert \nabla v_{t}\Vert _{L^{2}}\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}} \\ &{}+C\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}\Vert \theta_{t}\Vert _{L^{2}}\Vert \nabla v\Vert _{L^{\infty}}^{2} +Cc_{0}\Vert \pi \Vert _{L^{\infty}} \Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}\Vert \theta_{t}\Vert _{L^{2}}\Vert \nabla v\Vert _{L^{\infty}} \\ &{}+Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho}\varepsilon_{t} \Vert _{L^{2}}\Vert \pi _{t}\Vert _{L^{6}}\Vert \nabla v\Vert _{L^{6}}^{2}\Vert \theta \Vert _{L^{\infty}} +Cc_{0}^{\frac{1}{2}}\Vert \pi \Vert _{L^{\infty}} \Vert \sqrt{ \rho}\varepsilon _{t}\Vert _{L^{2}}\Vert \pi_{t} \Vert _{L^{6}}\Vert \nabla v\Vert _{L^{6}}\Vert \theta \Vert _{L^{6}} \\ \leq& C\eta ^{-1}c_{0}c_{1}^{2}c_{2}^{4}c_{6}^{2}c_{3}^{4}c_{5}^{2}+Cc_{0}^{4}c_{1}^{2}c_{2}^{4}c_{5}^{2} +C\eta\bigl(\Vert \nabla v_{t}\Vert _{L^{2}}^{2}+ \Vert \pi_{t}\Vert _{L^{6}}^{2}+\Vert \theta _{t}\Vert _{L^{2}}^{2}\bigr) \Vert \sqrt{\rho} \varepsilon_{t}\Vert _{L^{2}}^{2} \\ &{}+C\Vert \sqrt{ \rho }\varepsilon_{t}\Vert _{L^{2}}^{2} \end{aligned}$$
(2.73)
$$\begin{aligned} E_{5} \leq& C \int\bigl\vert \rho_{t}\theta^{2} \varepsilon_{t}\bigr\vert +C \int \vert \theta\theta_{t}\rho\varepsilon_{t} \vert +C \int\bigl\vert \rho\theta^{2}\pi_{t} \varepsilon_{t}\bigr\vert \\ \leq& C\Vert \rho_{t}\Vert _{L^{3}}\Vert \theta \Vert _{L^{4}}^{2}\Vert \varepsilon_{t}\Vert _{L^{6}} +Cc_{0}^{\frac{1}{2}}\Vert \sqrt{\rho} \varepsilon_{t}\Vert _{L^{2}}\Vert \theta _{t} \Vert _{L^{2}}\Vert \theta \Vert _{L^{\infty}} +Cc_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}\Vert \pi _{t}\Vert _{L^{2}}\Vert \theta \Vert _{L^{\infty}}^{2} \\ \leq& C\eta^{-1}c_{0}c_{6}^{4}+Cc_{0}^{2}c_{1}^{4}c_{2}^{2}+C \Vert \sqrt {\rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+C \eta\bigl(\Vert \theta_{t}\Vert _{L^{2}}^{2} + \Vert \pi_{t}\Vert _{L^{2}}^{2}\bigr)\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2} \\ &{}+ \frac{1}{8}\Vert \nabla\varepsilon_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.74)
Next, using an argument similar to that used in deriving (2.53), (2.54), (2.55), (2.60), and (2.59), respectively, one gets
and finally
$$\begin{aligned}& E_{1} \leq Cc_{0}^{2}c_{2}^{4} \Vert \nabla\varepsilon \Vert _{L^{2}}^{2}+C\Vert \sqrt { \rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+ \frac{1}{10}\Vert \nabla\varepsilon _{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.75)
$$\begin{aligned}& E_{2}\leq C\eta^{-1}c_{0}^{2\gamma+1}\bigl( \Vert \sqrt{\rho}\varepsilon _{t}\Vert _{L^{2}}^{2}+c_{1}^{2} \Vert \nabla\varepsilon \Vert _{L^{2}}^{2} +c_{1}^{4}c_{2}^{4}\bigr) +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{\rho} \varepsilon_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.76)
$$\begin{aligned}& E_{3} \leq Cc_{0}c_{2}^{2}\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+ \frac {1}{10}\Vert \nabla\varepsilon_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.77)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\Vert \varepsilon \Vert _{L^{2}}^{2} \leq C\Vert \varepsilon \Vert _{L^{2}}^{2}+Cc_{0} \Vert \sqrt{\rho}\varepsilon _{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.78)
$$ \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla\varepsilon \Vert _{L^{2}}^{2} \leq\frac{1}{8}\Vert \nabla\varepsilon_{t}\Vert _{L^{2}}^{2} +C\Vert \nabla\varepsilon \Vert _{L^{2}}^{2}. $$
(2.79)
Combining (2.72)-(2.79), one obtains setting \(\eta=c_{1}^{-1}\) and using the Gronwall inequality, one obtains for \(0\leq t\leq T\), where we have used the fact that \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+\Vert \varepsilon \Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \sqrt{\rho}\varepsilon _{t}\Vert _{L^{2}}^{2}+ \Vert \varepsilon \Vert _{H^{1}}^{2}\bigr) +\Vert \nabla\varepsilon_{t} \Vert _{L^{2}}^{2} \\& \quad\leq C\bigl(c_{0}^{2}c_{2}^{4} + \eta^{-1}c_{0}^{2\gamma+1}c_{1}^{2} + \eta \Vert v_{t}\Vert _{H^{1}}^{2} +\eta \Vert \theta_{t}\Vert _{H^{1}}^{2} +\eta \Vert \pi_{t}\Vert _{H^{1}}^{2} \bigr) \bigl(\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+ \Vert \varepsilon \Vert _{H^{1}}^{2}\bigr) \\& \quad\quad{}+C\eta^{-1}c_{0}c_{1}^{4}c_{2}^{4}c_{6}^{4}c_{3}^{4}c_{5}^{2} +Cc_{0}^{4}c_{1}^{4}c_{2}^{4}c_{5}^{2}, \end{aligned}$$
(2.80)
$$ \Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2} +\Vert \varepsilon \Vert _{H^{1}}^{2} + \int_{0}^{t}\Vert \nabla\varepsilon_{t} \Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5} $$
(2.81)
Next, applying the standard elliptic regularity result to the equation (2.6) and using (2.81), we have therefore, by the Young inequality and (2.81), one deduces Thus, we complete the proof of Lemma 2.4. □
$$\begin{aligned} \Vert \nabla\varepsilon \Vert _{H^{1}} \leq& C\bigl(c_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}} +c_{0}\Vert v\Vert _{L^{6}}\Vert \nabla\varepsilon \Vert _{L^{3}} +\Vert \nabla v\Vert _{L^{6}}^{2}\Vert \theta \Vert _{L^{6}} \\ &{}+c_{0}\Vert \nabla v\Vert _{L^{6}}\Vert \theta \Vert _{L^{6}}\Vert \pi \Vert _{L^{6}} +c_{0}\Vert \theta \Vert _{L^{4}}^{2} \bigr) \\ \leq& C\bigl(c_{0}^{3}c_{2}^{2}c_{1}^{2}+c_{0}c_{1} \Vert \nabla\varepsilon \Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla \varepsilon \Vert _{L^{6}}^{\frac {1}{2}}\bigr), \end{aligned}$$
(2.82)
$$\Vert \varepsilon \Vert _{H^{2}}\leq Cc_{0}^{\frac{9}{2}}c_{1}^{2}c_{2}^{2}. $$
Finally, we estimate the total enthalpy h.
Lemma 2.5
There exists a unique strong solution h to the initial boundary value problem (2.4) and (2.9) such that
for
\(0\leq t\leq T\).
$$\begin{aligned}& \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2} + \Vert h\Vert _{H^{1}}^{2} + \int_{0}^{t}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5}, \end{aligned}$$
(2.83)
$$\begin{aligned}& \Vert h\Vert _{H^{2}}\leq Cc_{0}^{\frac{7}{2}+\gamma }c_{1}^{2}c_{2}^{2} \end{aligned}$$
(2.84)
Proof
We only need to prove the estimates. Differentiating equation (2.4) with respect to t, multiplying both sides of the result equation by \(h_{t}\) and integrating over Ω, one obtains First of all, using similar methods of deriving the estimates (2.15), (2.20), and (2.16), respectively, one has
Second, differentiating the equation (2.2) with respect to t yields Therefore, by direct calculation and using (2.89), we derive
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h\Vert _{H^{1}}^{2} \bigr) +\Vert \nabla h_{t}\Vert _{L^{2}}^{2} \\ & \quad=- \int\rho_{t}v\cdot\nabla h\cdot h_{t} - \int\rho v_{t}\cdot\nabla h\cdot h_{t} -2 \int\rho v\cdot\nabla h_{t}\cdot h_{t} + \int p_{tt}\cdot h_{t} \\ & \quad\quad{}+ \int u_{t}\cdot\nabla p\cdot h_{t} + \int u\cdot\nabla p_{t}\cdot h_{t} + \int S_{kt}^{\prime}\cdot h_{t} \\ & \quad = \sum _{i=1}^{7}H_{i}. \end{aligned}$$
(2.85)
$$\begin{aligned}& H_{1} \leq Cc_{0}^{2}c_{2}^{4} \Vert \nabla h\Vert _{L^{2}}^{2}+C\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}}^{2}+\frac{1}{20} \Vert \nabla h_{t}\Vert _{L^{2}}, \end{aligned}$$
(2.86)
$$\begin{aligned}& H_{2}\leq C\eta^{-1}c_{0}^{2\gamma+1} \bigl(c_{0}^{7}c_{2}^{4}+\Vert \sqrt {\rho}h_{t}\Vert _{L^{2}}^{2}+c_{1}^{2} \Vert \nabla h\Vert _{L^{2}}^{2}\bigr) +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}}^{2}, \end{aligned}$$
(2.87)
$$\begin{aligned}& H_{3} \leq Cc_{0}c_{2}^{2}\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+ \frac{1}{20}\Vert \nabla h_{t}\Vert _{L^{2}}. \end{aligned}$$
(2.88)
$$ \rho_{tt}=-\rho_{t}\nabla\cdot v +\rho\nabla\cdot v_{t} +v_{t}\cdot\nabla\rho +v\cdot\nabla \rho_{t}. $$
(2.89)
$$\begin{aligned} H_{4} =& \int\bigl[\gamma(\gamma-1)\rho^{\gamma-2}\rho_{t}^{2} -\gamma\rho^{\gamma-1}(\rho_{t}\nabla\cdot v +\rho\nabla\cdot v_{t} +v_{t}\cdot\nabla\rho +v\cdot\nabla \rho_{t})\bigr]\cdot h_{t} \\ \leq& Cc_{0}^{\gamma-\frac{3}{2}}\Vert \rho_{t}\Vert _{L^{4}}^{2}\Vert \sqrt {\rho}h_{t}\Vert _{L^{2}} +Cc_{0}^{\gamma-\frac{1}{2}}\Vert \rho_{t} \Vert _{L^{3}}\Vert \nabla v\Vert _{L^{6}}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}} \\ &{}+Cc_{0}^{\gamma-\frac{1}{2}} \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}\Vert \nabla v_{t}\Vert _{L^{2}} +Cc_{0}^{\gamma-\frac{1}{2}}\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}\Vert v_{t}\Vert _{L^{6}}\Vert \nabla\rho \Vert _{L^{3}} \\ &{}+Cc_{0}^{\gamma-\frac{1}{2}}\Vert \nabla \rho_{t}\Vert _{L^{2}}\Vert v\Vert _{L^{\infty}} \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}} \\ \leq& C\bigl(c_{0}^{2\gamma+1}c_{2}^{4}+ \eta^{-1}c_{0}^{2\gamma}+\Vert \sqrt { \rho}h_{t}\Vert _{L^{2}}^{2} +\eta \Vert v_{t}\Vert _{H^{1}}^{2}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}}^{2}\bigr)+ \frac {1}{20}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.90)
Third, simple calculation and (2.26) lead to
$$\begin{aligned} H_{5} \leq& Cc_{0}^{\gamma-\frac{1}{2}}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}}\Vert \nabla\rho \Vert _{L^{3}}\Vert u_{t}\Vert _{L^{6}} \leq Cc_{0}^{\gamma-\frac{1}{2}}\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}\Vert \nabla\rho \Vert _{L^{3}}\bigl(\Vert u_{t}\Vert _{L^{2}}+\Vert \nabla u_{t}\Vert _{L^{2}}\bigr) \\ \leq &Cc_{0}^{2\gamma+1}\Vert \sqrt{\rho}h_{t} \Vert _{L^{2}}^{2}+Cc_{0}^{2\gamma+5} +C\Vert \nabla u_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.91)
Next, by direct calculation, we know that \(\nabla p_{t}= \gamma(\gamma -1)\rho^{\gamma-2}\rho_{t}\nabla\rho +\gamma\rho^{\gamma-1}\nabla\rho_{t}\). Therefore,
$$\begin{aligned} H_{6} \leq& Cc_{0}^{\gamma-2} \int \vert \rho_{t}\vert \vert u\vert \vert \nabla \rho \vert \vert h_{t}\vert +Cc_{0}^{\gamma-1} \int \vert u\vert \vert \nabla\rho_{t}\vert \vert h_{t}\vert \\ \leq& Cc_{0}^{\gamma-2}\Vert \nabla\rho \Vert _{L^{\infty}} \Vert \rho_{t}\Vert _{L^{3}}\Vert u\Vert _{L^{2}} \bigl(\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}} +\Vert \nabla h_{t}\Vert _{L^{2}}\bigr) \\ &{}+Cc_{0}^{\gamma-1}\Vert u\Vert _{L^{3}}\Vert \nabla\rho_{t}\Vert _{L^{2}}\bigl(\Vert \sqrt { \rho}h_{t}\Vert _{L^{2}}+\Vert \nabla h_{t} \Vert _{L^{2}}\bigr) \\ \leq& Cc_{0}^{7+2\gamma}c_{2}^{2}+C\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+ \frac{1}{20}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}. \end{aligned}$$
(2.92)
Last, simple calculation yields \(\vert S_{kt}^{\prime} \vert \leq C\vert \nabla v\vert \vert \nabla v_{t}\vert +C\rho^{\gamma-1}\vert \rho_{t}\vert \vert \nabla\rho \vert ^{2} +C\rho^{\gamma-1}\vert \nabla\rho_{t}\vert \vert \nabla\rho \vert \), thus Furthermore, we easily have and Consequently, combining (2.85)-(2.95), one deduces setting \(\eta=c_{1}^{-1}\) and using the Gronwall inequality, we get for \(0\leq t \leq T\), where we have used the fact that \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h\Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
$$\begin{aligned} H_{7} \leq& C \int \vert \nabla v_{t}\vert \vert \nabla v\vert \vert h_{t}\vert +Cc_{0}^{\gamma-1} \int \vert \rho _{t}\vert \vert \nabla\rho \vert ^{2}\vert h_{t}\vert +Cc_{0}^{\gamma-1} \int \vert \nabla\rho_{t}\vert \vert \nabla\rho \vert \vert h_{t}\vert \\ \leq& Cc_{0}^{\frac{1}{2}}\Vert \nabla v\Vert _{L^{\infty}} \Vert \nabla v_{t}\Vert _{L^{2}}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}} +Cc_{0}^{\gamma-\frac{1}{2}} \Vert \rho_{t}\Vert _{L^{6}}\Vert \nabla\rho \Vert _{L^{6}}^{2}\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}} \\ &{}+Cc_{0}^{\gamma-\frac{1}{2}}\Vert \nabla\rho \Vert _{L^{\infty}} \Vert \nabla \rho_{t}\Vert _{L^{2}}\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}} \\ \leq& C\bigl(\eta^{-1}c_{0}c_{3}^{2}+c_{0}^{5+2\gamma}c_{2}^{2} +\eta \Vert \nabla v_{t}\Vert _{L^{2}}^{2}\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2} +\Vert \sqrt{ \rho}h_{t}\Vert _{L^{2}}^{2}\bigr). \end{aligned}$$
(2.93)
$$ \frac{\mathrm{d}}{\mathrm{d}t}\Vert h\Vert _{L^{2}}^{2}\leq Cc_{0}\Vert \sqrt{\rho }h_{t}\Vert _{L^{2}}^{2}+C \Vert h\Vert _{L^{2}}^{2} $$
(2.94)
$$ \frac{\mathrm{d}}{\mathrm{d}t}\Vert \nabla h\Vert _{L^{2}}^{2}\leq C \Vert \nabla h\Vert _{L^{2}}^{2}+\frac{1}{10}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}. $$
(2.95)
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h\Vert _{H^{1}}^{2} \bigr) +\Vert \nabla h_{t}\Vert _{L^{2}}^{2} \\& \quad\leq C\bigl(c_{0}^{2\gamma+1}c_{2}^{4}+ \eta^{-1}c_{0}^{2\gamma +1}c_{1}^{2}+ \eta \Vert v_{t}\Vert _{H^{1}}^{2}\bigr) \bigl( \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h \Vert _{H^{1}}^{2}\bigr) \\& \quad\quad{}+C\bigl(c_{0}^{7+2\gamma}c_{2}^{4}+ \eta^{-1}c_{0}^{8+2\gamma }c_{2}^{4}c_{3}^{2} \bigr), \end{aligned}$$
(2.96)
$$ \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h\Vert _{H^{1}}^{2} + \int_{0}^{t}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s \leq Cc_{0}^{5} $$
(2.97)
Next, using (2.97) and the standard elliptic regularity result of the equation (2.4), one obtains then the Young inequality and (2.97) yield Thus, we have finished the proof of Lemma 2.5. □
$$\begin{aligned} \Vert \nabla h\Vert _{H^{1}} \leq& C\bigl(c_{0}^{\frac{1}{2}} \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}} +c_{0} \Vert v\Vert _{L^{6}}\Vert \nabla h\Vert _{L^{3}} +c_{0}^{\gamma-1}\Vert \rho_{t}\Vert _{L^{2}}+c_{0}^{\gamma-1}\Vert u\Vert _{L^{6}} \Vert \nabla\rho \Vert _{L^{3}} \\ &{}+\Vert \nabla v\Vert _{L^{4}}^{2}+c_{0}^{\gamma-1} \Vert \nabla\rho \Vert _{L^{4}}^{2}\bigr) \\ \leq& Cc_{0}^{\frac{5}{2}+\gamma}c_{2}^{2}+Cc_{0}c_{1} \Vert \nabla h\Vert _{L^{2}}^{\frac{1}{2}}\Vert \nabla h\Vert _{H^{1}}^{\frac{1}{2}}, \end{aligned}$$
(2.98)
$$\Vert h\Vert _{H^{2}}\leq Cc_{0}^{\frac{7}{2}+\gamma }c_{1}^{2}c_{2}^{2}. $$
Next, let us define \(c_{i}\) (\(i=1,\ldots,6\)) as follows: then we conclude from Lemma 2.1 to Lemma 2.5 that and for \(0\leq t\leq T\).
$$\begin{aligned}& c_{1}=Cc_{0}^{7+2\gamma},\quad\quad c_{2}=Cc_{0}^{\frac{5}{2}+3\gamma}c_{1}^{2},\quad\quad c_{5}=Cc_{0}^{\frac{7}{2}}c_{1}c_{2}^{2},\\& c_{6}=Cc_{0}^{\frac{9}{2}}c_{1}^{2}c_{2}^{2},\quad\quad c_{3}=Cc_{0}^{\frac{13}{2}+3\gamma}c_{1}^{4}c_{2}c_{5},\quad\quad c_{4}=Cc_{0}^{9+6\gamma}c_{1}^{5}c_{2}^{2}, \end{aligned}$$
$$ \textstyle\begin{cases} \sup _{0\leq t\leq T}(\Vert u\Vert _{H^{1}} +\Vert k\Vert _{H^{1}}+\Vert \varepsilon \Vert _{H^{1}})\\ \quad{}+\int_{0}^{T}(\Vert k\Vert _{H^{3}}^{2} +\Vert u_{t}\Vert _{H^{1}}^{2}+\Vert k_{t}\Vert _{H^{1}}^{2}+\Vert \varepsilon_{t}\Vert _{H^{1}}^{2})\,\mathrm{d}t\leq c_{1},\\ \sup _{0\leq t\leq T}\Vert u\Vert _{H^{2}}\leq c_{2},\quad\quad \sup _{0\leq t\leq T}\Vert u\Vert _{H^{3}}\leq c_{3},\quad\quad \int_{0}^{T}\Vert u\Vert _{H^{4}}^{2}\,\mathrm{d}t\leq c_{4},\\ \sup _{0\leq t\leq T}\Vert k\Vert _{H^{2}}\leq c_{5},\quad\quad \sup _{0\leq t\leq T}\Vert \varepsilon \Vert _{H^{2}}\leq c_{6}, \end{cases} $$
(2.99)
$$ \textstyle\begin{cases} \Vert \rho \Vert _{H^{3}(\Omega)}\leq Cc_{0}, \quad\quad \Vert \rho_{t}\Vert _{H^{1}(\Omega )}\leq Cc_{0}c_{2},\\ \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2} +\Vert h\Vert _{H^{1}}^{2} +\int_{0}^{t}\Vert \nabla h_{t}\Vert _{L^{2}}^{2}\,\mathrm{d}s\leq Cc_{0}^{5},\\ \Vert h\Vert _{H^{2}}\leq Cc_{0}^{\frac{7}{2}+\gamma}c_{1}^{2}c_{2}^{2}, \end{cases} $$
(2.100)
Using a standard proof as that in [13], we complete the proof of Theorem 2.1. □
3 Existence of strong solutions to the k-ε equations
Theorem 3.1
There exist a small time
\(T^{*}>0\)
and a unique strong solution
\((\rho ,u,h,k,\varepsilon) \)
to the initial boundary value problem (1.1)-(1.10) such that
$$ \begin{aligned} & \rho\in C\bigl(0,T^{*};H^{3}\bigr),\quad\quad \rho_{t} \in C\bigl(0,T^{*};H^{1}\bigr),\quad\quad u \in C\bigl(0,T^{*}; H^{3}\bigr)\cap L^{2}\bigl(0,T^{*}; H^{4}\bigr), \\ & u_{t}\in L^{2}\bigl(0,T^{*};H^{1} \bigr),\quad\quad k\in C\bigl(0,T^{*}; H^{2}\bigr)\cap L^{2} \bigl(0,T^{*};H^{3}\bigr),\\& k_{t}\in L^{2}\bigl(0,T^{*};H^{1}\bigr), \quad\quad \varepsilon\in C\bigl(0,T^{*}; H^{2}\bigr),\quad\quad \varepsilon_{t}\in L^{2}\bigl(0,T^{*};H^{1} \bigr),\\ & h\in C\bigl(0,T^{*}; H^{2}\bigr), \quad\quad h_{t}\in L^{2}\bigl(0,T^{*};H^{1}\bigr), \\ & (\sqrt{\rho}u_{t},\sqrt{\rho}k_{t},\sqrt{\rho} \varepsilon _{t},\sqrt{\rho}h_{t})\in L^{\infty} \bigl(0,T^{*};L^{2}\bigr). \end{aligned} $$
(3.1)
Proof
Our proof will be based on the iteration argument and on the results in the last section (especially Theorem 2.1).
First, using the regularity effect of the classical heat equation, we can construct functions (\(u^{0}=u^{0}(x,t)\), \(k^{0}=k^{0}(x,t)\), \(\varepsilon ^{0}=\varepsilon^{0}(x,t)\)) satisfying \((u^{0}(x,0), k^{0}(x,0), \varepsilon^{0}(x,0)) =(u_{0}(x), k_{0}(x), \varepsilon_{0}(x))\) and Therefore it follows from Theorem 2.1 that there exists a unique strong solution \((\rho^{1},u^{1}, h^{1},k^{1},\varepsilon^{1})\) to the linearized problem (2.2)-(2.6) with v, π, θ replaced by \(u^{0}\), \(k^{0}\), \(\varepsilon^{0}\), respectively, which satisfies the regularity estimates (2.99) and (2.100). Similarly, we construct approximate solutions \((\rho^{n},u^{n},h^{n},k^{n},\varepsilon^{n})\), inductively, as follows: assuming that \(u^{n-1}\), \(k^{n-1}\), \(\varepsilon^{n-1}\) have been defined for \(n\geq1\), let \((\rho^{n},u^{n},h^{n},k^{n},\varepsilon^{n})\) be the unique solution to the linearized problem (2.2)-(2.6) with v, π, θ replaced by \(u^{n-1}\), \(k^{n-1}\), \(\varepsilon^{n-1}\), respectively. Then it follows from Theorem 2.1 that there exists a constant \(\widetilde{C}>1\) such that for all \(n\geq1\). Throughout the proof, we denote by C̃ a generic constant depending only on m, M, γ, \(|\Omega| \), and \(c_{0}\), but independent of n. Next, we will show that the full sequence \((\rho ^{n},u^{n},h^{n},k^{n},\varepsilon^{n})\) converges to a solution to the original nonlinear problem (1.1)-(1.10) in a strong sense.
$$ \textstyle\begin{cases} \sup _{0\leq t\leq T}(\Vert u^{0}\Vert _{H^{1}} +\Vert k^{0}\Vert _{H^{1}}+\Vert \varepsilon^{0}\Vert _{H^{1}})\\ \quad{}+\int_{0}^{T}(\Vert k^{0}\Vert _{H^{3}}^{2} +\Vert u^{0}_{t}\Vert _{H^{1}}^{2}+\Vert k^{0}_{t}\Vert _{H^{1}}^{2}+\Vert \varepsilon ^{0}_{t}\Vert _{H^{1}}^{2})\,\mathrm{d}t\leq c_{1}, \\ \sup _{0\leq t\leq T}\Vert u^{0}\Vert _{H^{2}}\leq c_{2},\quad\quad \sup _{0\leq t\leq T}\Vert u^{0}\Vert _{H^{3}}\leq c_{3},\quad\quad \int_{0}^{T}\Vert u^{0}\Vert _{H^{4}}^{2}\,\mathrm{d}t\leq c_{4}, \\ \sup _{0\leq t\leq T}\Vert k^{0}\Vert _{H^{2}}\leq c_{5},\quad\quad \sup _{0\leq t\leq T}\Vert \varepsilon^{0}\Vert _{H^{2}}\leq c_{6}. \end{cases} $$
$$\begin{aligned}& \sup_{0\leq t\leq T}\bigl(\bigl\Vert \rho^{n}\bigr\Vert _{H^{3}}+\bigl\Vert \rho^{n}_{t}\bigr\Vert _{H^{1}}\bigr) +\sup_{0\leq t\leq T}\bigl(\bigl\Vert u^{n}\bigr\Vert _{H^{3}}+\bigl\Vert k^{n}\bigr\Vert _{H^{2}}+\bigl\Vert \varepsilon^{n}\bigr\Vert _{H^{2}} +\bigl\Vert h^{n}\bigr\Vert _{H^{2}}\bigr) \\& \quad{}+\sup_{0\leq t\leq T}\bigl(\bigl\Vert \sqrt{\rho^{n}}u^{n}_{t} \bigr\Vert _{L^{2}}+ \bigl\Vert \sqrt{\rho^{n}}h^{n}_{t} \bigr\Vert _{L^{2}}+\bigl\Vert \sqrt{\rho^{n}}k^{n}_{t} \bigr\Vert _{L^{2}}+\bigl\Vert \sqrt{\rho^{n}} \varepsilon^{n}_{t}\bigr\Vert _{L^{2}}\bigr) \\& \quad{}+ \int_{0}^{T}\bigl(\bigl\Vert u^{n}_{t} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert h^{n}_{t} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert k^{n}_{t} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert \varepsilon^{n}_{t} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert u^{n}\bigr\Vert _{H^{4}}^{2} +\bigl\Vert k^{n}\bigr\Vert _{H^{3}}^{2}\bigr)\leq\widetilde{C} \end{aligned}$$
(3.2)
Define \(\overline{\rho}^{n+1}=\rho^{n+1}-\rho^{n}\), \(\overline {u}^{n+1}=u^{n+1}-u^{n}\), \(\overline{h}^{n+1}=h^{n+1}-h^{n}\), \(\overline {k}^{n+1}=k^{n+1}-k^{n}\), \(\overline{\varepsilon}^{n+1}=\varepsilon ^{n+1}-\varepsilon^{n}\), \(\overline{p}^{n+1}=p^{n+1}-p^{n}=(\rho^{n+1})^{\gamma}-(\rho ^{n})^{\gamma}\).
Then, by equations (2.2)-(2.6), we deduce that (\(\overline{\rho}^{n+1}\), \(\overline{u}^{n+1}\), \(\overline{h}^{n+1}\), \(\overline{k}^{n+1}\), \(\overline{\varepsilon }^{n+1}\), \(\overline{p}^{n+1}\)) satisfy the following equations:
where
$$\begin{aligned}& \overline{\rho}^{n+1}_{t}+\nabla\cdot\bigl(\overline{\rho }^{n+1}u^{n}+\rho^{n}\overline{u}^{n} \bigr)=0, \end{aligned}$$
(3.3)
$$\begin{aligned}& \rho^{n+1}\overline{u}^{n+1}_{t}+\overline{ \rho}^{n+1}u^{n}_{t} +\rho^{n+1}u^{n} \cdot\nabla\overline{u}^{n+1} +\overline{\rho}^{n+1}u^{n} \cdot\nabla u^{n} +\rho^{n}\overline{u}^{n}\cdot \nabla u^{n} \\& \quad{}-\Delta\overline{u}^{n+1} -\nabla\bigl(\nabla\cdot \overline{u}^{n+1}\bigr) +\nabla\overline{p}^{n+1} = \frac{-2}{3}\nabla\bigl(\overline{\rho}^{n+1}k^{n}+ \rho^{n}\overline {k}^{n}\bigr), \end{aligned}$$
(3.4)
$$\begin{aligned}& \rho^{n+1}\overline{h}^{n+1}_{t}+\overline{ \rho}^{n+1}h^{n}_{t} +\rho^{n+1}u^{n} \cdot\nabla\overline{h}^{n+1} +\overline{\rho}^{n+1}u^{n} \cdot\nabla h^{n} +\rho^{n}\overline{u}^{n}\cdot \nabla h^{n}-\Delta\overline{h}^{n+1} \\& \quad =\overline{p}^{n+1}_{t}+ \overline{u}^{n+1}\cdot\nabla p^{n+1}+u^{n}\cdot\nabla \overline{p}^{n+1} +S_{k,n+1}^{\prime}-S_{k,n}^{\prime}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \rho^{n+1}\overline{k}^{n+1}_{t}+\overline{ \rho}^{n+1}k^{n}_{t} +\rho^{n+1}u^{n} \cdot\nabla\overline{k}^{n+1} +\overline{\rho}^{n+1}u^{n} \cdot\nabla k^{n} +\rho^{n}\overline{u}^{n}\cdot \nabla k^{n}-\Delta\overline{k}^{n+1} \\& \quad =G_{n+1}^{\prime}-G_{n}^{\prime}- \bigl(\rho^{n+1}\varepsilon^{n}-\rho ^{n} \varepsilon^{n-1}\bigr), \end{aligned}$$
(3.6)
$$\begin{aligned}& \rho^{n+1}\overline{\varepsilon}^{n+1}_{t}+ \overline{\rho }^{n+1}\varepsilon^{n}_{t} + \rho^{n+1}u^{n}\cdot\nabla\overline{\varepsilon}^{n+1} +\overline{\rho}^{n+1}u^{n}\cdot\nabla\varepsilon^{n} +\rho^{n}\overline{u}^{n}\cdot\nabla\varepsilon^{n}-\Delta\overline{\varepsilon}^{n+1} \\& \quad =C_{1} \biggl( \frac{G_{n+1}^{\prime} \varepsilon^{n}}{k^{n}} -\frac{G_{n}^{\prime} \varepsilon^{n-1}}{k^{n-1}} \biggr) -C_{2} \biggl( \frac{\rho^{n+1}(\varepsilon^{n})^{2}}{k^{n}} - \frac{\rho^{n}(\varepsilon^{n-1})^{2}}{k^{n-1}} \biggr), \end{aligned}$$
(3.7)
$$\begin{aligned}& S_{k,n+1}^{\prime}=\biggl[\mu\bigl(\partial_{j}u_{i}^{n}+ \partial_{i}u_{j}^{n}\bigr) -\frac{2}{3} \delta_{ij}\mu\partial_{k}u_{k}^{n} \biggr]\partial_{j}u_{i}^{n} +\frac{\mu_{t}}{(\rho^{n+1})^{2}} \partial_{j}p^{n+1}\partial _{j}\rho^{n+1}, \end{aligned}$$
(3.8)
$$\begin{aligned}& G_{n+1}^{\prime}=\partial_{j}u_{i}^{n} \biggl[\mu_{e}\bigl(\partial _{j}u_{i}^{n}+ \partial_{i}u_{j}^{n}\bigr) -\frac{2}{3} \delta_{ij}\bigl(\rho^{n+1}k^{n}+\mu_{e} \partial_{l}u_{l}^{n}\bigr)\biggr]. \end{aligned}$$
(3.9)
To evaluate \(\Vert \overline{\rho}^{n+1}\Vert _{L^{2}}\), multiplying both sides of the equation (3.3) by \(\overline{\rho}^{n+1}\) and integrating the result over Ω, we get
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \overline{\rho}^{n+1} \bigr\Vert _{L^{2}}^{2} \\& \quad=- \int\nabla\cdot\bigl(\overline{\rho}^{n+1}u^{n}+ \rho^{n}\overline {u}^{n}\bigr)\cdot\overline{ \rho}^{n+1} \\& \quad=- \int\bigl(\overline{\rho}^{n+1}\bigr)^{2}\nabla\cdot u^{n} +\overline{\rho}^{n+1}u^{n}\cdot\nabla \overline{\rho}^{n+1} +\rho^{n}\overline{\rho}^{n+1} \nabla\cdot\overline{u}^{n} +\overline{\rho}^{n+1} \overline{u}^{n}\cdot\nabla\rho^{n}. \end{aligned}$$
(3.10)
Applying integration by parts to the second term of the second equality of (3.10) and using the Hölder, Sobolev, and Young inequalities yield where (3.2) has been used and \(0<\eta<1\) is a small constant to be determined later.
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} \leq& C\bigl(\bigl\Vert \nabla u^{n} \bigr\Vert _{L^{\infty}}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \nabla\overline{u}^{n} \bigr\Vert _{L^{2}}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}} +\bigl\Vert \overline{u}^{n}\bigr\Vert _{L^{6}}\bigl\Vert \nabla\rho^{n}\bigr\Vert _{L^{3}} \bigl\Vert \overline {\rho}^{n+1}\bigr\Vert _{L^{2}}\bigr) \\ \leq& \widetilde{C}\bigl(1+\eta^{-1}\bigr)\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\eta \bigl\Vert \nabla\overline{u}^{n}\bigr\Vert _{H^{1}}^{2}, \end{aligned}$$
(3.11)
Next, multiplying both sides of (3.4) by \(\overline{u}^{n+1}\) and integrating the result thus derived over Ω, one obtains
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho ^{n+1}} \overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \nabla\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \nabla\cdot\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} \\& \quad=- \int\overline{\rho}^{n+1}u^{n}_{t}\cdot \overline{u}^{n+1} - \int\overline{\rho}^{n+1}u^{n}\cdot\nabla u^{n}\cdot\overline{u}^{n+1} - \int\rho^{n}\overline{u}^{n}\cdot\nabla u^{n}\cdot\overline{u}^{n+1} - \int\nabla\overline{p}^{n+1}\cdot\overline{u}^{n+1} \\& \quad\quad{}+ \int\frac{-2}{3}\nabla\bigl(\overline{\rho}^{n+1}k^{n}+ \rho^{n}\overline{k}^{n+1}\bigr)\cdot\overline{u}^{n+1} \\& \quad= \sum _{i=1}^{5}L_{i}. \end{aligned}$$
(3.12)
Using the Hölder, Sobolev, and Young inequalities and (3.2), we estimate \(L_{1}\), \(L_{2}\), and \(L_{3}\), respectively, as follows:
Then one deduces by integration by parts that and Inserting (3.13)-(3.17) to (3.12) and using inequality \(\Vert \overline{u}^{n+1}\Vert _{L^{2}}\leq\widetilde{C}\Vert \sqrt{\rho ^{n+1}}\overline{u}^{n+1}\Vert _{L^{2}}\), one has
$$\begin{aligned}& L_{1}\leq C\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}\bigl\Vert u_{t}^{n}\bigr\Vert _{L^{3}}\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{L^{6}} \leq C\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}\bigl\Vert u_{t}^{n}\bigr\Vert _{L^{3}}\bigl(\bigl\Vert \sqrt {\rho^{n+1}}\overline{u}^{n+1} \bigr\Vert _{L^{2}}+\bigl\Vert \nabla\overline {u}^{n+1}\bigr\Vert _{L^{2}}\bigr) \\& \hphantom{L_{1}} \leq\widetilde{C}\bigl\Vert u_{t}^{n}\bigr\Vert _{L^{3}}^{2}\bigl\Vert \overline{\rho }^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} +\frac{1}{8}\bigl\Vert \nabla\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.13)
$$\begin{aligned}& L_{2}\leq C\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}\bigl\Vert u^{n}\bigr\Vert _{L^{6}}\bigl\Vert \nabla u^{n}\bigr\Vert _{L^{6}}\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{L^{6}} \\& \hphantom{L_{2}} \leq\widetilde{C}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2}+ \frac{1}{8}\bigl\Vert \nabla\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.14)
$$\begin{aligned}& L_{3} \leq C\bigl\Vert \overline{u}^{n}\bigr\Vert _{L^{6}} \bigl\Vert \nabla u^{n}\bigr\Vert _{L^{3}} \bigl\Vert \sqrt{\rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}} \leq\widetilde{C}\eta^{-1}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} +\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.15)
$$ L_{4}= \int\overline{p}^{n+1}\nabla\cdot\overline{u}^{n+1} \leq C \int\overline{\rho}^{n+1}\nabla\cdot\overline{u}^{n+1} \leq\widetilde{C}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2}+\frac {1}{8}\bigl\Vert \nabla \overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} $$
(3.16)
$$\begin{aligned} L_{5} =&\frac{2}{3} \int\overline{\rho}^{n+1}k^{n}\nabla\cdot \overline{u}^{n+1} -\overline{k}^{n}\nabla\rho^{n} \cdot\overline{u}^{n+1} -\rho^{n}\nabla\overline{k}^{n} \cdot\overline{u}^{n+1} \\ \leq &C\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}} \bigl\Vert \nabla\overline {u}^{n+1}\bigr\Vert _{L^{2}} +C\bigl\Vert \overline{k}^{n}\bigr\Vert _{L^{6}}\bigl\Vert \nabla \rho^{n}\bigr\Vert _{L^{3}}\bigl\Vert \sqrt { \rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}} \\ &{} +C\bigl\Vert \nabla\overline{k}^{n}\bigr\Vert _{L^{2}}\bigl\Vert \sqrt{\rho^{n+1}}\overline {u}^{n+1}\bigr\Vert _{L^{2}} \\ \leq&\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \sqrt{\rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2}\bigr) +\frac{1}{8}\bigl\Vert \nabla \overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} + \widetilde{C}\eta\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.17)
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho^{n+1}}\overline {u}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{H^{1}}^{2} \leq&\widetilde{C}\bigl(1+\eta^{-1}+\bigl\Vert u_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigr) \bigl(\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \sqrt{ \rho^{n+1}}\overline{u}^{n+1}\bigr\Vert _{L^{2}}^{2} \bigr) \\ &{} +\widetilde{C}\eta\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\widetilde{C}\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.18)
Then, multiplying both sides of (3.5) by \(\overline{h}^{n+1}\) and integrating the result thus got over Ω, one obtains First, using similar methods of deriving (3.13), (3.14), and (3.15), respectively, one easily obtains
Second, simple calculation leads to By the differential mean value theorem, the first integral of (3.23) can be controlled as By the equation (3.3), the second integral on the right-hand side of (3.24) can be estimated as Then the second integral on the right-hand side of (3.23) can be controlled as
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho ^{n+1}} \overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \nabla\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2} \\& \quad=- \int\overline{\rho}^{n+1}h^{n}_{t}\cdot \overline{h}^{n+1} - \int\overline{\rho}^{n+1}u^{n}\cdot\nabla h^{n}\cdot\overline{h}^{n+1} - \int\rho^{n}\overline{u}^{n}\cdot\nabla h^{n}\cdot\overline {h}^{n+1} \\& \quad\quad{}+ \int\bigl(\overline{p}^{n+1}_{t}+\overline{u}^{n+1} \cdot\nabla p^{n+1}+u^{n}\cdot\nabla\overline{p}^{n+1} \bigr)\cdot\overline{h}^{n+1} + \int\bigl(S_{k,n+1}^{\prime}-S_{k,n}^{\prime} \bigr)\cdot\overline{h}^{n+1} \\& \quad = \sum _{i=1}^{5}M_{i}. \end{aligned}$$
(3.19)
$$\begin{aligned}& M_{1} \leq\widetilde{C}\bigl\Vert h_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{\rho^{n+1}}\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}+\frac{1}{20}\bigl\Vert \nabla \overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.20)
$$\begin{aligned}& M_{2} \leq\widetilde{C}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}+ \frac{1}{20}\bigl\Vert \nabla\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.21)
$$\begin{aligned}& M_{3}\leq\widetilde{C}\eta^{-1}\bigl\Vert \sqrt{ \rho^{n+1}}\overline {h}^{n+1}\bigr\Vert _{L^{2}}^{2} +\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.22)
$$\begin{aligned} M_{4} =& \int\bigl[\gamma\bigl(\rho^{n+1}\bigr)^{\gamma-1} \rho^{n+1}_{t}-\gamma \bigl(\rho^{n} \bigr)^{\gamma-1}\rho^{n}_{t}\bigr]\cdot \overline{h}^{n+1} + \int\overline{u}^{n+1}\cdot\nabla p^{n+1}\overline {h}^{n+1} \\ &{}+ \int u^{n}\cdot\nabla\overline{p}^{n+1} \overline{h}^{n+1}. \end{aligned}$$
(3.23)
$$\begin{aligned}& \int\bigl[\gamma\bigl(\rho^{n+1}\bigr)^{\gamma-1} \rho^{n+1}_{t}-\gamma\bigl(\rho ^{n} \bigr)^{\gamma-1}\rho^{n}_{t}\bigr]\cdot \overline{h}^{n+1} \\& \quad\leq C \int\bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \rho^{n+1}_{t}\bigr\vert \bigl\vert \overline{h}^{n+1} \bigr\vert + \int\gamma\bigl(\rho^{n}\bigr)^{\gamma-1}\overline{ \rho}^{n+1}_{t}\cdot \overline{h}^{n+1}. \end{aligned}$$
(3.24)
$$\begin{aligned}& \int\gamma\bigl(\rho^{n}\bigr)^{\gamma-1}\overline{ \rho}^{n+1}_{t}\cdot \overline{h}^{n+1} \\& \quad = - \int\gamma\bigl(\rho^{n}\bigr)^{\gamma-1}\nabla\cdot\bigl( \overline{\rho }^{n+1}u^{n}+\rho^{n} \overline{u}^{n}\bigr)\cdot\overline {h}^{n+1} \\& \quad\leq C \int\bigl\vert \nabla\rho^{n}\bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert \bigl\vert \overline{\rho }^{n+1} \bigr\vert \bigl\vert u^{n}\bigr\vert +C \int\bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert u^{n}\bigr\vert \bigl\vert \nabla\overline {h}^{n+1}\bigr\vert \\& \quad\quad{}+C \int\bigl(\bigl\vert \nabla\rho^{n}\bigr\vert \bigl\vert \overline{u}^{n}\bigr\vert +\bigl\vert \rho^{n}\bigr\vert \bigl\vert \nabla \overline{u}^{n}\bigr\vert \bigr)\bigl\vert \overline{h}^{n+1}\bigr\vert . \end{aligned}$$
(3.25)
$$ \int\overline{u}^{n+1}\cdot\nabla p^{n+1} \overline{h}^{n+1} \leq C \int\bigl\vert \overline{u}^{n+1}\bigr\vert \bigl\vert \nabla\rho^{n+1}\bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert . $$
(3.26)
Next, applying integration by parts to the third integral on the right-hand side of (3.23), we easily get Consequently, combining (3.23)-(3.27) and using the Hölder, Sobolev, and Young inequalities and (3.2), one obtains
$$ \int u^{n}\cdot\nabla\overline{p}^{n+1} \overline{h}^{n+1} \leq C \int\bigl\vert \nabla u^{n}\bigr\vert \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \overline{h}^{n+1} \bigr\vert +C \int\bigl\vert u^{n}\bigr\vert \bigl\vert \overline{ \rho}^{n+1}\bigr\vert \bigl\vert \nabla\overline {h}^{n+1} \bigr\vert . $$
(3.27)
$$\begin{aligned} M_{4} \leq&\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} + \bigl\Vert \sqrt{\rho^{n+1}}\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+\frac{1}{4}\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{H^{1}}^{2} +\frac{1}{20}\bigl\Vert \nabla \overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}+\widetilde {C}\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.28)
Finally, we evaluate \(M_{5}\). Direct calculation yields
$$\begin{aligned} M_{5} \leq& C \int\bigl(\bigl\vert \nabla u^{n}\bigr\vert +\bigl\vert \nabla u^{n-1}\bigr\vert \bigr)\bigl\vert \nabla\overline {u}^{n}\bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert +C \int\bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \nabla\rho^{n+1}\bigr\vert ^{2}\bigl\vert \overline {h}^{n+1}\bigr\vert \\ &{}+ \int\frac{\mu_{t}}{(\rho^{n})^{2}}\partial_{j}\overline {p}^{n+1} \partial_{j}\rho^{n+1}\cdot\overline{h}^{n+1} + \int\frac{\mu_{t}}{(\rho^{n})^{2}}\partial_{j}p^{n}\partial _{j}\overline{\rho}^{n+1}\cdot\overline{h}^{n+1} \\ \leq& C \int\bigl(\bigl\vert \nabla u^{n}\bigr\vert +\bigl\vert \nabla u^{n-1}\bigr\vert \bigr)\bigl\vert \nabla\overline {u}^{n}\bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert +C \int\bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \nabla\rho^{n+1}\bigr\vert ^{2}\bigl\vert \overline {h}^{n+1}\bigr\vert \\ &{}+C \int\bigl\vert \nabla\rho^{n}\bigr\vert \bigl\vert \nabla\rho^{n+1}\bigr\vert \bigl\vert \overline{\rho }^{n+1} \bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert +C \int\bigl\vert \nabla^{2}\rho^{n+1}\bigr\vert \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \overline {h}^{n+1}\bigr\vert \\ &{}+C \int\bigl\vert \nabla\rho^{n+1}\bigr\vert \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \nabla\overline {h}^{n+1}\bigr\vert +C \int\bigl\vert \nabla\rho^{n}\bigr\vert ^{2} \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \overline {h}^{n+1}\bigr\vert \\ &{}+C \int\bigl\vert \nabla^{2}\rho^{n}\bigr\vert \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \overline{h}^{n+1}\bigr\vert +C \int\bigl\vert \nabla\rho^{n}\bigr\vert \bigl\vert \overline{\rho}^{n+1}\bigr\vert \bigl\vert \nabla\overline {h}^{n+1}\bigr\vert . \end{aligned}$$
(3.29)
Then, applying a similar method to deriving (3.28), one deduces
$$ M_{5}\leq\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} + \bigl\Vert \sqrt{\rho^{n+1}}\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2}\bigr) +\eta\bigl\Vert \overline{u}^{n} \bigr\Vert _{H^{1}}^{2}+\frac{1}{20}\bigl\Vert \nabla \overline {h}^{n+1}\bigr\Vert _{L^{2}}^{2}. $$
(3.30)
Consequently, inserting (3.20)-(3.22), (3.28), and (3.30) into (3.19), one gets
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho^{n+1}}\overline {h}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \overline{h}^{n+1}\bigr\Vert _{H^{1}}^{2} \leq&\widetilde{C}\bigl(1+\eta^{-1}+\bigl\Vert h_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigr) \bigl(\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \sqrt{ \rho^{n+1}}\overline{h}^{n+1}\bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+\frac{1}{4}\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{H^{1}}^{2} +\tilde{C}\eta\bigl\Vert \overline{u}^{n} \bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.31)
For the turbulent kinetic energy k, using a similar method of deriving (3.19), one easily deduces from equation (3.6) that
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho ^{n+1}} \overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \nabla\overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2} \\& \quad=- \int\overline{\rho}^{n+1}k^{n}_{t}\cdot \overline{k}^{n+1} - \int\overline{\rho}^{n+1}u^{n}\cdot\nabla k^{n}\cdot\overline {k}^{n+1} \\& \quad\quad{}- \int\rho^{n}\overline{u}^{n}\cdot\nabla k^{n}\cdot\overline{k}^{n+1} + \int\bigl(G_{n+1}^{\prime}-G_{n}^{\prime} \bigr)\cdot\overline{k}^{n+1} - \int\bigl(\rho^{n+1}\varepsilon^{n}- \rho^{n}\varepsilon^{n-1}\bigr)\cdot \overline{k}^{n+1} \\& \quad= \sum _{i=1}^{5}N_{i}. \end{aligned}$$
(3.32)
We first evaluate \(N_{4}\). Using the inserting items technique, one easily gets
$$\begin{aligned} N_{4} \leq& C \int\bigl(\bigl\vert \nabla u^{n}\bigr\vert +\bigl\vert \nabla u^{n-1}\bigr\vert \bigr)\bigl\vert \nabla\overline {u}^{n}\bigr\vert \bigl\vert \overline{k}^{n+1}\bigr\vert \\ &{}+C \int\bigl(\bigl\vert \nabla\overline{u}^{n}\bigr\vert + \bigl\vert \nabla u^{n-1}\bigr\vert \bigl\vert \overline{ \rho}^{n+1}\bigr\vert +\bigl\vert \nabla u^{n-1} \bigr\vert \bigl\vert \overline{k}^{n}\bigr\vert \bigr)\bigl\vert \overline{k}^{n+1}\bigr\vert . \end{aligned}$$
(3.33)
Using the Hölder, Sobolev, and Young inequalities and (3.2), we have
$$ N_{4}\leq\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} + \bigl\Vert \sqrt{\rho^{n+1}}\overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2}\bigr) +\widetilde{C}\eta\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} + \widetilde{C}\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. $$
(3.34)
Second, we estimate \(N_{5}\). Using a similar method to deriving (3.33) and (3.34), we have
$$\begin{aligned} N_{5} =& \int\bigl(\overline{\rho}^{n+1}\varepsilon^{n} + \rho^{n}\overline{\varepsilon}^{n}\bigr)\cdot \overline{k}^{n+1} \leq C\bigl(\bigl\Vert \overline{\rho}^{n+1} \bigr\Vert _{L^{2}}\bigl\Vert \varepsilon^{n}\bigr\Vert _{L^{\infty}} +\bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{L^{6}}\bigl\Vert \rho^{n}\bigr\Vert _{L^{3}}\bigr) \bigl\Vert \sqrt {\rho^{n+1}}\overline{k}^{n+1}\bigr\Vert _{L^{2}} \\ \leq&\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \sqrt{ \rho^{n+1}}\overline {k}^{n+1}\bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} \bigr) +\widetilde{C}\eta\bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.35)
Next, using a similar method to deriving the estimates of (3.13), (3.14), and (3.15), respectively, one easily gets
$$\begin{aligned}& N_{1} \leq\widetilde{C}\bigl\Vert k_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{\rho^{n+1}}\overline{k}^{n+1}\bigr\Vert _{L^{2}} +\frac{1}{8}\bigl\Vert \nabla\overline{k}^{n+1} \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.36)
$$\begin{aligned}& N_{2} \leq\widetilde{C}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2} +\frac{1}{8}\bigl\Vert \nabla\overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.37)
$$\begin{aligned}& N_{3}\leq\widetilde{C}\eta^{-1}\bigl\Vert \sqrt{ \rho^{n+1}}\overline {k}^{n+1}\bigr\Vert _{L^{2}}^{2} +\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.38)
Consequently, inserting (3.34)-(3.38) to (3.32), one deduces
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho^{n+1}}\overline {k}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \overline{k}^{n+1}\bigr\Vert _{H^{1}}^{2} \leq&\widetilde{C}\bigl(1+\eta^{-1}+\bigl\Vert k_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigr) \bigl(\bigl\Vert \sqrt { \rho^{n+1}}\overline{k}^{n+1}\bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \overline{\rho }^{n+1}\bigr\Vert _{L^{2}}^{2} \bigr) \\ &{}+\widetilde{C}\eta\bigl(\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{u}^{n} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{ \varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr). \end{aligned}$$
(3.39)
Next, multiplying both sides of (3.7) by \(\overline {\varepsilon}^{n+1}\) and integrating the result over Ω, one gets
$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho ^{n+1}} \overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} + \bigl\Vert \nabla\overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} \\& \quad =- \int\overline{\rho}^{n+1}\varepsilon^{n}_{t} \cdot\overline {\varepsilon}^{n+1} - \int\overline{\rho}^{n+1}u^{n}\cdot\nabla \varepsilon^{n}\cdot \overline{\varepsilon}^{n+1} \\& \quad\quad{}- \int\rho^{n}\overline{u}^{n}\cdot\nabla \varepsilon^{n}\cdot \overline{\varepsilon}^{n+1} +C_{1} \int \biggl(\frac{G_{n+1}^{\prime}\varepsilon^{n}}{k^{n}}-\frac {G_{n}^{\prime}\varepsilon^{n-1}}{k^{n-1}} \biggr)\cdot\overline { \varepsilon}^{n+1} \\& \quad\quad{}-C_{2} \int \biggl[\frac{\rho^{n+1}(\varepsilon^{n})^{2}}{k^{n}} -\frac{\rho^{n}(\varepsilon^{n-1})^{2}}{k^{n-1}} \biggr]\cdot \overline{\varepsilon}^{n+1} \\& \quad= \sum _{i=1}^{5}Q_{i}. \end{aligned}$$
(3.40)
Using an argument similar to that used in deriving (3.13), (3.14), and (3.15), respectively, we obtain
$$\begin{aligned}& Q_{1} \leq\widetilde{C}\bigl\Vert \varepsilon_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigl\Vert \overline { \rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{\rho^{n+1}}\overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} +\frac{1}{8}\bigl\Vert \nabla\overline{ \varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.41)
$$\begin{aligned}& Q_{2} \leq\widetilde{C}\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\bigl\Vert \sqrt{ \rho^{n+1}}\overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} +\frac{1}{8}\bigl\Vert \nabla\overline{ \varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}, \end{aligned}$$
(3.42)
$$\begin{aligned}& Q_{3} \leq\widetilde{C}\eta^{-1}\bigl\Vert \sqrt{ \rho^{n+1}}\overline {\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} +\widetilde{C}\eta\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2}. \end{aligned}$$
(3.43)
Next, direct calculation leads to
$$\begin{aligned} Q_{4} \leq& C \int\bigl(\bigl\vert \nabla\overline{u}^{n}\bigr\vert \bigl\vert \nabla u^{n}\bigr\vert + \bigl\vert \nabla \overline{u}^{n}\bigr\vert \bigl\vert \nabla u^{n-1}\bigr\vert \bigr)\bigl\vert \varepsilon^{n}\bigr\vert \bigl\vert \overline{\varepsilon}^{n+1}\bigr\vert \\ &{}+C \int\bigl(\bigl\vert \overline{\varepsilon}^{n}\bigr\vert +\bigl\vert \varepsilon^{n-1}\bigr\vert \bigl\vert \overline{k}^{n}\bigr\vert \bigr)\bigl\vert \nabla u^{n-1} \bigr\vert ^{2}\bigl\vert \overline{\varepsilon}^{n+1}\bigr\vert \\ &{}-\frac{2C_{1}}{3}\delta_{ij} \int\frac{(\partial_{j}u_{i}^{n}\rho ^{n+1}k^{n}\varepsilon^{n}k^{n-1} -\partial_{j}u_{i}^{n-1}\rho^{n}k^{n-1}\varepsilon ^{n-1}k^{n})}{k^{n}k^{n-1}}\cdot\overline{\varepsilon }^{n+1} \\ \leq& \int\bigl(\bigl\vert \nabla\overline{u}^{n}\bigr\vert \bigl\vert \nabla u^{n}\bigr\vert + \bigl\vert \nabla \overline{u}^{n}\bigr\vert \bigl\vert \nabla u^{n-1}\bigr\vert \bigr)\bigl\vert \varepsilon^{n}\bigr\vert \bigl\vert \overline{\varepsilon}^{n+1}\bigr\vert \\ &{}+C \int\bigl(\bigl\vert \overline{\varepsilon}^{n}\bigr\vert +\bigl\vert \varepsilon^{n-1}\bigr\vert \bigl\vert \overline{k}^{n}\bigr\vert \bigr)\bigl\vert \nabla u^{n-1} \bigr\vert ^{2}\bigl\vert \overline{\varepsilon}^{n+1}\bigr\vert \\ &{}+C \int\bigl(\bigl\vert \nabla\overline{u}^{n}\bigr\vert + \bigl\vert \nabla u^{n-1}\bigr\vert \bigl\vert \overline{ \rho}^{n+1}\bigr\vert +\bigl\vert \nabla u^{n-1}\bigr\vert \bigl\vert \overline{k}^{n}\bigr\vert \bigr)\bigl\vert \varepsilon^{n}\bigr\vert \bigl\vert \overline { \varepsilon}^{n+1}\bigr\vert \\ &{}+C \int\bigl(\bigl\vert \overline{\varepsilon}^{n}\bigr\vert +\bigl\vert \varepsilon^{n-1}\bigr\vert \bigl\vert \overline{k}^{n}\bigr\vert \bigr)\bigl\vert \nabla u^{n-1} \bigr\vert \bigl\vert \overline{\varepsilon}^{n+1}\bigr\vert \\ \leq&\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl(\bigl\Vert \sqrt{ \rho^{n+1}}\overline {\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \overline{\rho}^{n+1}\bigr\Vert _{L^{2}}^{2}\bigr) \\ &{}+\widetilde{C}\eta\bigl(\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{ \varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr) + \frac{1}{8}\bigl\Vert \nabla\overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}. \end{aligned}$$
(3.44)
Finally, using a similar method of deriving the estimate of \(Q_{4}\), one deduces
$$ Q_{5} \leq\widetilde{C}\bigl(1+\eta^{-1}\bigr) \bigl( \bigl\Vert \sqrt{\rho^{n+1}}\overline {\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}+\bigl\Vert \overline{\rho}^{n+1} \bigr\Vert _{L^{2}}^{2}\bigr) +\widetilde{C}\eta\bigl\Vert \nabla\overline{\varepsilon}^{n}\bigr\Vert _{L^{2}}^{2} +\frac{1}{8}\bigl\Vert \nabla\overline{\varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2}. $$
(3.45)
Consequently, inserting (3.41)-(3.45) to (3.40), one derives
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl\Vert \sqrt{\rho^{n+1}}\overline { \varepsilon}^{n+1}\bigr\Vert _{L^{2}}^{2} +\bigl\Vert \overline{\varepsilon}^{n+1}\bigr\Vert _{H^{1}}^{2} \\& \quad\leq\widetilde{C}\bigl(1+\eta^{-1}+\bigl\Vert \varepsilon_{t}^{n}\bigr\Vert _{L^{3}}^{2} \bigr) \bigl(\bigl\Vert \sqrt{\rho^{n+1}}\overline{\varepsilon}^{n+1} \bigr\Vert _{L^{2}}^{2}+\bigl\Vert \overline{ \rho}^{n+1}\bigr\Vert _{L^{2}}^{2}\bigr) \\& \quad\quad{} +\widetilde{C} \eta\bigl(\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{ \varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr). \end{aligned}$$
(3.46)
In the end, combining (3.11), (3.18), (3.31), (3.39), and (3.46), and setting \(\varphi^{n+1}(t)=\Vert \overline{\rho}^{n+1}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho^{n+1}}\overline{u}^{n+1}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho^{n+1}}\overline{h}^{n+1}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho^{n+1}}\overline{k}^{n+1}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho^{n+1}}\overline{\varepsilon}^{n+1}\Vert _{L^{2}}^{2}\), we get
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\varphi^{n+1}(t) +\bigl\Vert \overline{u}^{n+1} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{h}^{n+1} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n+1} \bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{ \varepsilon}^{n+1}\bigr\Vert _{H^{1}}^{2} \\& \quad\leq\widetilde{C}\bigl(1+\eta^{-1}+\bigl\Vert u_{t}^{n} \bigr\Vert _{L^{3}}^{2} +\bigl\Vert h_{t}^{n} \bigr\Vert _{L^{3}}^{2} +\bigl\Vert k_{t}^{n} \bigr\Vert _{L^{3}}^{2} +\bigl\Vert \varepsilon_{t}^{n} \bigr\Vert _{L^{3}}^{2}\bigr)\varphi^{n+1}(t) \\& \quad\quad {}+\widetilde{C}\eta\bigl(\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{ \varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr). \end{aligned}$$
(3.47)
Setting \(I_{\eta}^{n}(t)=\widetilde{C}(1+\eta^{-1}+\Vert u_{t}^{n}\Vert _{L^{3}}^{2} +\Vert h_{t}^{n}\Vert _{L^{3}}^{2} +\Vert k_{t}^{n}\Vert _{L^{3}}^{2} +\Vert \varepsilon_{t}^{n}\Vert _{L^{3}}^{2})\) and applying the Gronwall inequality to (3.47) yield where it should be noted that \(\varphi^{n+1}(0)=0\).
$$ \varphi^{n+1}(t) \leq\widetilde{C}\eta \biggl[\exp \biggl( \int_{0}^{t}I_{\eta }^{n}(s)\, \mathrm{d}s \biggr) \biggr] \biggl( \int_{0}^{t}\bigl(\bigl\Vert \overline {u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{H^{1}}^{2} \bigr)\,\mathrm{d}s \biggr), $$
(3.48)
Since setting \(\widetilde{T}\leq\eta<1\), we have for \(t\leq\widetilde{T}\).
$$ \int_{0}^{t}I_{\eta}^{n}(s)\, \mathrm{d}s \leq\widetilde{C}t +\widetilde{C}\eta^{-1}t + \widetilde{C}, $$
(3.49)
$$ \int_{0}^{t}I_{\eta}^{n}(s)\, \mathrm{d}s \leq C\widetilde{C} $$
(3.50)
By (3.48)-(3.50), integrating (3.47) from \([0,t]\), one derives for \(T^{*}:=\min\{T, \widetilde{T}\}\).
$$\begin{aligned}& \varphi^{n+1}(t) + \int_{0}^{t}\bigl(\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{h}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n+1}\bigr\Vert _{H^{1}}^{2} + \bigl\Vert \overline{\varepsilon}^{n+1}\bigr\Vert _{H^{1}}^{2} \bigr)\,\mathrm{d}s \\& \quad \leq C\widetilde{C}\eta \biggl( \int_{0}^{t}\bigl(\bigl\Vert \overline {u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{H^{1}}^{2} \bigr)\,\mathrm{d}s \biggr) \biggl[ \biggl( \int_{0}^{t}I_{\eta}^{n}(s)\, \mathrm{d}s \biggr) \exp \biggl( \int_{0}^{t}I_{\eta}^{n}(s)\, \mathrm{d}s \biggr)+1 \biggr] \\& \quad\leq C\eta\exp(\widetilde{C}) \int_{0}^{t}\bigl(\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} + \bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr)\,\mathrm{d}s \end{aligned}$$
(3.51)
Therefore, we have Thus, choosing η such that \(C\eta\exp (\widetilde{C})\leq \frac{1}{2}\), one deduces Therefore, we conclude that the full sequence \((\rho ^{n},u^{n},h^{n},k^{n},\varepsilon^{n})\) converges to a limit \((\rho,u,h, k,\varepsilon)\) in the following strong sense: \(\rho^{n}\rightarrow\rho\) in \(L^{\infty}(0,T;L^{2}(\Omega))\); \((u^{n},h^{n},k^{n},\varepsilon^{n})\rightarrow(u,h,k,\varepsilon)\) in \(L^{2}(0,T;H^{1}(\Omega))\). It is easy to prove that the limit \((\rho,u,h,k,\varepsilon)\) is a weak solution to the original nonlinear problem. Furthermore, it follows from (3.2) that \((\rho,u,h,k,\varepsilon)\) satisfies the following regularity estimates: This proves the existence of a strong solution. Then we can easily prove the time continuity of the solution \((\rho, u, h, k, \varepsilon)\) by adapting the arguments in [9, 13]. Finally, we prove the uniqueness. In fact, assume \((\rho_{1}, u_{1}, h_{1}, k_{1}, \varepsilon_{1})\) and \((\rho_{2}, u_{2}, h_{2}, k_{2}, \varepsilon_{2})\) be two strong solutions to the problem (1.1)-(1.10) with the regularity (3.1). Let \((\overline{\rho}, \overline{u}, \overline{h}, \overline{k}, \overline{\varepsilon}) =(\rho_{1}-\rho_{2}, u_{1}-u_{2}, h_{1}-h_{2}, k_{1}-k_{2}, \varepsilon_{1}-\varepsilon_{2})\). Then following the same argument as in the derivations of (3.11), (3.18), (3.31), (3.39), and (3.46), we can prove that for some \(R(t)\in L^{1}(0, T^{*})\). Thus, by the Gronwall inequality, we conclude that \((\overline{\rho}, \overline{u}, \overline{h}, \overline{k}, \overline{\varepsilon}) =(0, 0, 0, 0, 0)\) in \((0,T^{*})\times\Omega\). This completes the proof of Theorem 3.1. □
$$\begin{aligned}& \sum_{n=1}^{\infty}\sup_{0\leq t\leq T} \varphi^{n+1}(t) +\sum_{n=1}^{\infty} \int_{0}^{t}\bigl(\bigl\Vert \overline{u}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{h}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n+1}\bigr\Vert _{H^{1}}^{2} + \bigl\Vert \overline{\varepsilon}^{n+1}\bigr\Vert _{H^{1}}^{2} \bigr)\,\mathrm{d}s \\& \quad\leq C\eta\exp(\widetilde{C})\sum_{n=1}^{\infty} \int_{0}^{t}\bigl(\bigl\Vert \overline{u}^{n}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n}\bigr\Vert _{H^{1}}^{2} + \bigl\Vert \overline{\varepsilon}^{n}\bigr\Vert _{H^{1}}^{2}\bigr)\,\mathrm{d}s. \end{aligned}$$
(3.52)
$$\begin{aligned}& \sum_{n=1}^{\infty}\sup_{0\leq t\leq T} \varphi^{n+1}(t) +\sum_{n=1}^{\infty} \int_{0}^{t}\bigl\Vert \overline{h}^{n+1} \bigr\Vert _{H^{1}}^{2}\,\mathrm{d}s +\frac{1}{2}\sum_{n=1}^{\infty} \int_{0}^{t}\bigl(\bigl\Vert \overline {u}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{k}^{n+1}\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \overline{\varepsilon}^{n+1}\bigr\Vert _{H^{1}}^{2} \bigr)\,\mathrm{d}s \\& \quad\leq C\widetilde{C} < \infty. \end{aligned}$$
(3.53)
$$\begin{aligned}& \sup_{0\leq t\leq T^{*}}\bigl(\Vert \rho \Vert _{H^{3}}+\Vert \rho_{t}\Vert _{H^{1}}\bigr) +\sup_{0\leq t\leq T^{*}} \bigl(\Vert u\Vert _{H^{3}}+\Vert k\Vert _{H^{2}}+\Vert \varepsilon \Vert _{H^{2}} +\Vert h\Vert _{H^{2}}\bigr) \\& \quad{}+\sup_{0\leq t\leq T^{*}}\bigl(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}+ \Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}+ \Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}+\Vert \sqrt {\rho} \varepsilon_{t}\Vert _{L^{2}}\bigr) \\& \quad{}+ \int_{0}^{T^{*}}\bigl(\Vert u_{t}\Vert _{H^{1}}^{2} +\Vert h_{t}\Vert _{H^{1}}^{2} +\Vert k_{t}\Vert _{H^{1}}^{2} +\Vert \varepsilon_{t}\Vert _{H^{1}}^{2} +\Vert u\Vert _{H^{4}}^{2} + \Vert k\Vert _{H^{3}}^{2}\bigr)\leq\widetilde{C}< \infty. \end{aligned}$$
$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\Vert \overline{\rho} \Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho_{1}}\overline{u}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho_{1}}\overline{h}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho_{1}}\overline{k}\Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho _{1}}\overline{\varepsilon} \Vert _{L^{2}}^{2}\bigr) \\& \quad\leq R(t) \bigl(\Vert \overline{\rho} \Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho_{1}}\overline{u}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho_{1}}\overline{h}\Vert _{L^{2}}^{2} +\Vert \sqrt{\rho_{1}}\overline{k}\Vert _{L^{2}}^{2}+ \Vert \sqrt{\rho _{1}}\overline{\varepsilon} \Vert _{L^{2}}^{2}\bigr) \end{aligned}$$
Declarations
Acknowledgement
The research of BY was partially supported by the National Natural Science Foundation of China (No. 11471103).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics and Information Science, Henan Polytechnic University
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