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Local existence of strong solutions to the k-ε model equations for turbulent flows
Boundary Value Problems volume 2016, Article number: 27 (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.
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.
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.
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:
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 \(T_{1}>0\).
Here we also impose the following regularity conditions on the initial data:
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.
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
In the following part, we decompose the proof of Theorem 2.1 into some lemmas.
Lemma 2.1
There exists a unique strong solution ρ to the linear transport problem (2.2) and (2.9) such that
for \(0\leq t\leq T\).
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 \(0\leq t\leq T\).
Next, from the equation (2.2), one obtains
for \(0\leq t\leq T\).
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 \(0\leq t\leq T\).
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.
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.
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
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 \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}u_{t}\Vert _{L^{2}}^{2}+\Vert u\Vert _{H^{1}}^{2})\leq Cc_{0}^{5+2\gamma}\).
Next, by (2.19) and (2.26), we deduce
which implies (2.12) by (2.26).
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.
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 \(\lim_{t\rightarrow 0}\Vert \nabla u_{t}(t)\Vert _{L^{2}}^{2}\leq Cc_{0}^{2\gamma+4}\).
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).
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.
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 \(0\leq t\leq T\).
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 \(0\leq t\leq T\).
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.
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 \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}k_{t}\Vert _{L^{2}}^{2}+\Vert k\Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
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.
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 \(0\leq t\leq T\).
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 \(0\leq t\leq T\).
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
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
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 \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}\varepsilon_{t}\Vert _{L^{2}}^{2}+\Vert \varepsilon \Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
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. □
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 \(0\leq t\leq T\).
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
Third, simple calculation and (2.26) lead to
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,
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 \(\lim _{t\rightarrow0}(\Vert \sqrt{\rho}h_{t}\Vert _{L^{2}}^{2}+\Vert h\Vert _{H^{1}}^{2})\leq Cc_{0}^{5}\).
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. □
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 \(0\leq t\leq T\).
Using a standard proof as that in [13], we complete the proof of Theorem 2.1. □
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
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.
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
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
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.
Next, multiplying both sides of (3.4) by \(\overline{u}^{n+1}\) and integrating the result thus derived over Ω, one obtains
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
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
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
Finally, we evaluate \(M_{5}\). Direct calculation yields
Then, applying a similar method to deriving (3.28), one deduces
Consequently, inserting (3.20)-(3.22), (3.28), and (3.30) into (3.19), one gets
For the turbulent kinetic energy k, using a similar method of deriving (3.19), one easily deduces from equation (3.6) that
We first evaluate \(N_{4}\). Using the inserting items technique, one easily gets
Using the Hölder, Sobolev, and Young inequalities and (3.2), we have
Second, we estimate \(N_{5}\). Using a similar method to deriving (3.33) and (3.34), we have
Next, using a similar method to deriving the estimates of (3.13), (3.14), and (3.15), respectively, one easily gets
Consequently, inserting (3.34)-(3.38) to (3.32), one deduces
Next, multiplying both sides of (3.7) by \(\overline {\varepsilon}^{n+1}\) and integrating the result over Ω, one gets
Using an argument similar to that used in deriving (3.13), (3.14), and (3.15), respectively, we obtain
Next, direct calculation leads to
Finally, using a similar method of deriving the estimate of \(Q_{4}\), one deduces
Consequently, inserting (3.41)-(3.45) to (3.40), one derives
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
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 \(\varphi^{n+1}(0)=0\).
Since
setting \(\widetilde{T}\leq\eta<1\), we have
for \(t\leq\widetilde{T}\).
By (3.48)-(3.50), integrating (3.47) from \([0,t]\), one derives
for \(T^{*}:=\min\{T, \widetilde{T}\}\).
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. □
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Acknowledgement
The research of BY was partially supported by the National Natural Science Foundation of China (No. 11471103).
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Yuan, B., Qin, G. Local existence of strong solutions to the k-ε model equations for turbulent flows. Bound Value Probl 2016, 27 (2016). https://doi.org/10.1186/s13661-016-0532-8
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DOI: https://doi.org/10.1186/s13661-016-0532-8
MSC
- 35Q35
- 76F60
- 76N10
Keywords
- k-ε model equations
- strong solution
- local well-posedness