
Citation: Morteza Fotouhi, Andreas Minne, Henrik Shahgholian, Georg S. Weiss. Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type[J]. Mathematics in Engineering, 2020, 2(4): 698-708. doi: 10.3934/mine.2020032
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In this paper, we consider the non-isentropic compressible Euler equations with a source term in the following Euler coordinate system:
{ρt+(ρu)x=0,(ρu)t+(ρu2+p(ρ,S))x=βρ|u|αu,St+uSx=0, | (1.1) |
whereρ,u,S andp(ρ,S) are the density, velocity, entropy and pressure of the considered gas, respectively. x∈[0,L] is the spatial variable, and L>0 is a constant denoting the duct's length. p(ρ,S)=aeSργ, with constants a>0 and γ>1. And, the term βρ|u|αu represents the source term with α,β∈R. Especially, the source term denotes friction when β<0.
System (1.1) is equipped with initial data:
(ρ,u,S)⊤|t=0=(ρ0(x),u0(x),S0(x))⊤, | (1.2) |
and boundary conditions:
(ρ,u,S)⊤|x=0=(ρl(t),ul(t),Sl(t))⊤. | (1.3) |
If S=Const., the system (1.1) is the isentropic Euler equations with a source term. In the past few decades, the problems related to the isentropic compressible Euler equations with different kinds of source terms have been studied intensively. We refer the reader to [1,2,3,4,5,6,7,8,9,10] to find the existence and decay rates of small smooth (or large weak) solutions to Euler equations with damping. The global stability of steady supersonic solutions of 1-D compressible Euler equations with friction βρ|u|u was studied in [11]. For the singularity formation of smooth solutions, we can see [12,13,14,15] and the references therein. Moreover, the authors in [16] established the finite-time blow-up results for compressible Euler system with space-dependent damping in 1-D. Recently, time-periodic solutions have attracted much attention. However, most of these temporal periodic solutions are driven by the time-periodic external force; see [17,18] for examples. The first result on the existence and stability of time-periodic supersonic solutions triggered by boundary conditions was considered in [19]. Then, the authors of [20] studied the global existence and stability of the time-periodic solution of the isentropic compressible Euler equations with source term βρ|u|αu.
If S≠Const., much less is known. In [21,22,23,24,25,26], the authors used characteristics analysis and energy estimate methods to study 1-D non-isentropic p-systems with damping in Lagrangian coordinates. Specifically, the global existence of smooth solutions for the Cauchy problem with small initial data has been investigated in [21,22]. The influence of the damping mechanism on the large time behavior of solutions was considered in [23,24]. For the results of the initial-boundary value problem, see [25,26]. The stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data was obtained in [27]. As for the problems about non-isentropic compressible Euler equations with a vacuum boundary, we refer the reader to [28,29]. In [30,31,32], the relaxation limit problems for non-isentropic compressible Euler equations with source terms in multiple space dimensions were discussed.
In this paper, we are interested in the dynamics of non-isentropic Euler equations with friction. Exactly speaking, we want to prove the global existence and stability of temporal periodic solutions around the supersonic steady state to non-isentropic compressible Euler equations with the general friction term βρ|u|αu for any α,β∈R. It is worth pointing out that the temporal periodic non-isentropic supersonic solution considered in this paper is driven by periodic boundary conditions.
We choose the steady solution ˜W(x)=(˜ρ(x),˜u(x),˜S(x))⊤ (with ˜u(x)>0) as a background solution, which satisfies
{(˜ρ˜u)x=0,(˜ρ˜u2+p(˜ρ,˜S))x=β˜ρ˜uα+1,˜u˜Sx=0,(˜ρ,˜u,˜S)⊤|x=0=(ρ−,u−,S0)⊤. | (1.4) |
The equation (1.4)3 indicates that the static entropy in the duct must be a constant. That is, ˜S(x)=S0. Moreover, when (α,β) lies in different regions of R2, the source term β˜ρ˜uα+1 affects the movement of flow dramatically. We analyze the influence meticulously and gain the allowable maximal duct length for subsonic or supersonic inflow.
Based on the steady solution, we are interested in two problems. The first one is, if ρl(t)−ρ−, ul(t)−u−, Sl(t)−S0 and ρ0(x)−˜ρ(x), u0(x)−˜u(x), S0(x)−S0 are small in some norm sense, can we obtain a classical solution of the problem described by (1.1)–(1.3) for [0,∞)×[0,L] while this classical solution remains close to the background solution? If the first question holds, our second one is whether the small classical solution is temporal-periodic as long as the inflow is time-periodic at the entrance of ducts?
We use ˉW(t,x)=(ˉρ(t,x),ˉu(t,x),ˉS(t,x))⊤=(ρ(t,x)−˜ρ(x),u(t,x)−˜u(x),S(t,x)−S0)⊤ to denote the perturbation around the background solution, and, correspondingly,
ˉW0(x)=(ˉρ0(x),ˉu0(x),ˉS0(x))=(ρ0(x)−˜ρ(x),u0(x)−˜u(x),S0(x)−S0), |
ˉWl(t)=(ˉρl(t),ˉul(t),ˉSl(t))=(ρl(t)−ρ−,ul(t)−u−,Sl(t)−S0), |
that is,
t=0:{ρ0(x)=ˉρ0(x)+˜ρ(x),u0(x)=ˉu0(x)+˜u(x),0≤x≤L,S0(x)=ˉS0(x)+S0, | (1.5) |
and
x=0:{ρl(t)=ˉρl(t)+ρ−,ul(t)=ˉul(t)+u−,t≥0.Sl(t)=ˉSl(t)+S0. | (1.6) |
The main conclusions of this article are as follows:
Theorem 1.1. For any fixed non-sonic upstream state (ρ−,u−,S0) with ρ−≠ρ∗=[(ρ−u−)2aγeS0]1γ+1>0 and u−>0, the following holds:
1) There exists a maximal duct length Lm, which only depends on α,β,γ and (ρ−,u−,S0), such that the steady solution ˜W(x)=(˜ρ(x),˜u(x),S0)⊤ of the problem (1.1) exists in [0,L] for any L<Lm;
2) The steady solution (˜ρ(x),˜u(x),S0)⊤ keeps the upstream supersonic/subsonic state and ˜ρ˜u=ρ−u−>0;
3) ‖(˜ρ(x),˜u(x),S0)‖C2([0,L])<M0, where M0 is a constant only depending on α, β, γ, ρ−, u−, S0 and L;
4) If β>0, α≤1 and the upstream is supersonic, the maximal duct length Lm can be infinite and a vacuum cannot appear in any finite place of ducts;
5) When β>0, α≥−γ and the upstream is subsonic, the maximal duct length Lm can also be infinite, and the flow cannot stop in any place of ducts.
Theorem 1.2. Assume that the length of duct L<Lm and the steady flow is supersonic at the entrance of a duct, i.e., ρ−<ρ∗=[(ρ−u−)2aγeS0]1γ+1. Then, there are constants ε0 and K0 such that, if
‖ˉW0(x)‖C1([0,L])=‖(ρ0(x)−˜ρ(x),u0(x)−˜u(x),S0(x)−S0)‖C1([0,L])≤ε<ε0, | (1.7) |
‖ˉWl(t)‖C1([0,+∞))=‖ρl(t)−ρ−,ul(t)−u−,Sl(t)−S0‖C1([0,+∞))≤ε<ε0, | (1.8) |
and the C0,C1 compatibility conditions are satisfied at point (0,0), there is a unique C1 solution W(t,x)=(ρ(t,x),u(t,x),S(t,x))⊤ for the mixed initial-boundary value problems (1.1)–(1.3) in the domain G={(t,x)|t≥0,x∈[0,L]}, satisfying
‖ˉW(t,x)‖C1(G)=‖ρ(t,x)−˜ρ(x),u(t,x)−˜u(x),S(t,x)−S0‖C1(G)≤K0ε. | (1.9) |
Remark 1.1. Since the flows at {x=L} are entirely determined by the initial data on x∈[0,L] and the boundary conditions on {x=0} under the supersonic conditions, we only need to present the boundary conditions on {x=0} in Theorem 1.2.
If we further assume that the boundaries ρl(t),ul(t),Sl(t) are periodic, then the C1 solution obtained in Theorem 1.2 is a temporal periodic solution:
Theorem 1.3. Suppose that the assumptions of Theorem 1.2 are fulfilled and the flow at the entrance x=0 is temporal-periodic, i.e., Wl(t+P)=Wl(t); then, the C1 solution W(t,x)=(ρ(t,x),u(t,x),S(t,x))⊤ of the problem described by (1.1)–(1.3) is also temporal-periodic, namely,
W(t+P,x)=W(t,x) | (1.10) |
for any t>T1 and x∈[0,L], where T1 is a constant defined in (4.3).
The organization of this article is as follows. In the next section, we study the steady-state supersonic and subsonic flow. The wave decomposition for non-isentropic Euler equations is introduced in Section 3. In Section 4, based on wave decomposition, we prove the global existence and stability of smooth solutions under small perturbations around the steady-state supersonic flow. And, in Section 5, with the help of Gronwall's inequality, we obtain that the smooth supersonic solution is a temporal periodic solution, after a certain start-up time, with the same period as the boundary conditions.
In this section, the steady-state flow is considered for some positive constants upstream (ρ−,u−,S0) on the left side. In [11], the authors considered the differential equation in which the Mach number varies with the length of the duct. In [20], the authors investigated the steady-state equation with sound speed and flow velocity. Different from the methods used in [11] and [20], and motivated by [33], we rewrite (1.4) as the equations related to momentum and density in this paper, namely,
{˜mx=0,(˜m2˜ρ+p(˜ρ,S0))x=β˜mα+1˜ρα,(˜ρ,˜m)⊤|x=0=(ρ−,ρ−u−), | (2.1) |
where ˜m=˜ρ˜u represents momentum. The advantage of this method is that the vacuum and stagnant states can be considered. Now, we analyze this problem in three cases:
Case 1: α≠1 and α≠−γ.
In this case, (2.1) becomes
{˜m=const.=ρ−u−,F1(˜ρ,˜m)x=β˜mα+1, | (2.2) |
where
F1(˜ρ,˜m)=−˜m2α−1˜ρα−1+aγeS0γ+α˜ργ+α. | (2.3) |
Then, we get
∂F1(˜ρ,˜m)∂˜ρ=˜ρα(−˜m2˜ρ2+aγeS0˜ργ−1)=˜ρα−2(˜ρ2p˜ρ−˜m2). | (2.4) |
Let G(˜ρ,˜m)=˜ρ2p˜ρ−˜m2. For any fixed ˜m>0, we have that lim. From the definition of p(\tilde{\rho}, S_{0}) , we obtain
\tilde{\rho}^{2} p_{\tilde{\rho}} \;{\text{ is a strictly increasing function for}}\; \tilde{\rho} > 0. |
Thus, when \tilde{\rho}\rightarrow +\infty , G(\tilde{\rho}, \tilde{m})\rightarrow +\infty . Then, there exists \rho^{*} = \left[\frac{(\rho_{-} u_{-})^2}{a \gamma e^{S_{0}} }\right]^ {\frac{1}{\gamma +1}} > 0 such that G\left(\rho^{*}, \tilde{m}\right) = 0 (i.e., \left(\rho^*\right)^2 p_{\tilde{\rho}} \left(\rho^*\right) = \tilde{m}^{2} ). That is, when \tilde{\rho} = \rho^{*} , the fluid velocity is equal to the sound speed (i.e., \tilde{u} = \tilde{c} = \sqrt{\frac {\partial p}{\partial \tilde{\rho}}} = \sqrt{a\gamma} e^{\frac{S_{0}}{2}} \tilde{\rho}^{\frac{\gamma -1}{2}} ). Therefore, we have
\begin{equation} \frac{\partial F_1(\tilde{\rho},\tilde{m})}{\partial \tilde{\rho}} = \tilde{\rho}^{\alpha-2} \left(p_{\tilde{\rho}} \tilde{\rho}^2-\tilde{m}^{2}\right) < 0 \Leftrightarrow p_{\tilde{\rho}} \tilde{\rho}^2 < \tilde{m}^{2} \end{equation} | (2.5) |
and
\begin{equation} \frac{\partial F_1(\tilde{\rho},\tilde{m})}{\partial \tilde{\rho}} = \tilde{\rho}^{\alpha-2} \left(p_{\tilde{\rho}} \tilde{\rho}^2-\tilde{m}^{2}\right) > 0 \Leftrightarrow p_{\tilde{\rho}} \tilde{\rho}^2 > \tilde{m}^{2}. \end{equation} | (2.6) |
We conclude that \frac{\partial F_1(\tilde{\rho}, \tilde{m})}{\partial \tilde{\rho}} < 0 for \tilde{\rho} < \rho^{*} and \frac{\partial F_1(\tilde{\rho}, \tilde{m})}{\partial \tilde{\rho}} > 0 for \tilde{\rho} > \rho^{*} . Furthermore, we have
\begin{equation} \lim\limits _{\tilde{\rho }\rightarrow 0} F_1(\tilde{\rho},\tilde{m}) = 0,\; \lim\limits _{\tilde{\rho }\rightarrow +\infty} F_1(\tilde{\rho},\tilde{m}) = +\infty,\; F_{1}(\rho^{*},\tilde{m}) < 0,\quad {\rm for}\; \alpha > 1; \end{equation} | (2.7) |
\begin{equation} \lim\limits _{\tilde{\rho} \rightarrow 0} F_1(\tilde{\rho},\tilde{m}) = +\infty,\; \lim\limits _{\tilde{\rho }\rightarrow +\infty} F_1(\tilde{\rho},\tilde{m}) = +\infty,\; F_{1}(\rho^{*},\tilde{m}) > 0,\quad {\rm for}\; -\gamma < \alpha < 1; \end{equation} | (2.8) |
and
\begin{equation} \lim\limits _{\tilde{\rho} \rightarrow 0} F_1(\tilde{\rho},\tilde{m}) = +\infty,\; \lim\limits _{\tilde{\rho }\rightarrow +\infty} F_1(\tilde{\rho},\tilde{m}) = 0, \; F_{1}(\rho^{*},\tilde{m}) < 0,\quad{\rm for}\; \alpha < -\gamma. \end{equation} | (2.9) |
Then, for any fixed \tilde{m} = \rho_{-}u_{-} > 0 , according to different regions of \alpha \in R , we draw the graphs of F_1(\tilde{\rho}, \tilde{m}) . See Figure 1 below.
Integrating (2.2)_{2} over (0, x) , we obtain
\begin{equation} F_1\left(\tilde{\rho}(x),\tilde{m}\right)-F_1(\rho_{-},\tilde{m}) = \beta \tilde{m}^{\alpha+1}x. \end{equation} | (2.10) |
If \beta < 0 , by (2.10) , F_{1}(\tilde{\rho}, \tilde{m}) will decrease as the length of ducts increases, until it arrives at the minimum F_{1}(\rho^{*}, \tilde{m}) , no matter whether the upstream is supersonic (i.e., \rho_{-} < \rho^{*} ) or subsonic (i.e., \rho_{-} > \rho^{*} ). Therefore, we get the maximal length of ducts
\begin{equation} L_m = -\frac{1}{\beta}\left[\frac{u_{-}^{1-\alpha}}{1-\alpha} +\frac{a \gamma e^{S_{0}}}{\gamma+\alpha} \rho_{-}^{\gamma-1} u_{-}^{-\alpha-1}+\left(a \gamma e^{S_{0}}\right)^{\frac{1-\alpha}{\gamma+1}}\left(\rho_{-} u_{-}\right)^{\frac{(1-\alpha)(\gamma-1)} {\gamma+1}}\left(\frac{1}{\alpha-1} -\frac{1}{\gamma+\alpha}\right)\right] \end{equation} | (2.11) |
for a supersonic or subsonic flow before it gets choked, which is the state where the flow speed is equal to the sonic speed.
However, if \beta > 0 , \alpha > 1 and the upstream is supersonic (i.e., \rho_{-} < \rho^{*} ), by (2.7), (2.10) and Figure 1 (i), we know that \tilde{\rho} is decreasing as duct length x increases. Then, we get the maximal length of ducts
\begin{equation} L_m = \frac{1}{\beta}\left(\frac{u_{-}^{1-\alpha}}{\alpha-1}-\frac{a \gamma e^{S_{0}}}{\gamma+\alpha} \rho_{-}^{\gamma-1} u_{-}^{-\alpha-1}\right) \end{equation} | (2.12) |
for a supersonic flow before it reaches the vacuum state. If -\gamma < \alpha < 1 or \alpha < -\gamma , by (2.8)–(2.10) and Figure 1(ii) and (iii), \tilde{\rho} is decreasing as the duct length x increases for supersonic upstream, too. But, the vacuum will never occur for any duct length L .
Moreover, if \beta > 0 , \alpha < -\gamma and the upstream is subsonic (i.e., \rho_{-} > \rho^{*} ), combining (2.9), (2.10) with Figure 1(iii), \tilde{\rho} is increasing as the duct length x increases. At the same time, F_{1}(\tilde{\rho}, \tilde{m}) is increasing and approaching its supremum 0. Then, we get the maximal length of the duct L_{m} , which is still as shown in (2.12). When L > L_{m} , the fluid velocity is zero, that is, the fluid stagnates in a finite place. While, if -\gamma < \alpha < 1 or \alpha > 1 , again, by (2.7), (2.8), (2.10) and Figure 1(i) and (ii), \tilde{\rho} is also increasing as the duct length x increases, but F_{1}(\tilde{\rho}, \tilde{m}) goes to infinity as \tilde{\rho} grows. In this case, although the fluid is slowing down, it does not stagnate at any finite place.
Case 2: \alpha = 1 .
Now, (2.2) turns into
\begin{equation} \left\{\begin{array}{l} \tilde{m} = \rho_{-}u_{-}, \\ F_{2} \left(\tilde{\rho},\tilde{m}\right)_{x} = \beta \tilde{m}^{2}, \end{array}\right. \end{equation} | (2.13) |
where
F_2\left(\tilde{\rho},\tilde{m}\right) = -\tilde{m}^{2} \ln \tilde{\rho}+\frac{a \gamma e^{S_{0}}}{\gamma+1} \tilde{\rho}^{\gamma+1}. |
And, we get
\begin{equation} \lim\limits _{\tilde{\rho} \rightarrow 0} F_2(\tilde{\rho},\tilde{m}) = +\infty, \lim\limits _{\tilde{\rho} \rightarrow +\infty} F_2(\tilde{\rho},\tilde{m}) = +\infty, \end{equation} | (2.14) |
\begin{equation} \frac{\partial F_2(\tilde{\rho},\tilde{m})}{\partial \tilde{\rho}} = \tilde{\rho}\left(-\frac{\tilde{m}^{2}}{\tilde{\rho}^2}+a \gamma e^{S_{0}} \tilde{\rho} ^{\gamma-1}\right), \end{equation} | (2.15) |
and
\begin{equation} F_{2}\left(\tilde{\rho}(x),\tilde{m}\right)-F_{2}(\rho_{-},\tilde{m}) = \beta \tilde{m}^{2}x. \end{equation} | (2.16) |
Similarly, the function F_{2}\left(\tilde{\rho}(x), \tilde{m}\right) gets its minimum at point \tilde{\rho} = \rho^{*} . If \beta < 0 , combining (2.14) with (2.16), we get the maximal length of ducts
\begin{equation} L_m = -\frac{1}{\beta(\gamma+1)} \left(\ln \frac{\rho_{-}^{1-\gamma} u_{-}^{2}}{a \gamma e^{S_{0}}}+a \gamma e^{S_{0}} \rho_{-}^{\gamma-1} u_{-}^{-2}-1\right) \end{equation} | (2.17) |
for a supersonic or subsonic flow before it gets choked. While, if \beta > 0 , the flow remains in its entrance state for any duct length L > 0 , no matter whether it is supersonic or subsonic.
Case 3: \alpha = -\gamma .
In this case, (2.1) changes into
\begin{equation} \left\{\begin{array}{l} \tilde{m} = \rho_{-}u_{-}, \\ F_{3} \left(\tilde{\rho},\tilde{m}\right)_{x} = \beta \tilde{m}^{1-\gamma}, \end{array}\right. \end{equation} | (2.18) |
where
F_3(\tilde{\rho},\tilde{m}) = \frac{\tilde{m}^2}{1+\gamma} \tilde{\rho}^{-\gamma-1}+a \gamma e^{S_{0}} \ln \tilde{\rho}. |
Then, we have
\begin{equation} \lim\limits _{\tilde{\rho} \rightarrow 0} F_3(\tilde{\rho},\tilde{m}) = +\infty, \lim\limits _{\tilde{\rho} \rightarrow +\infty} F_3(\tilde{\rho},\tilde{m}) = +\infty, \end{equation} | (2.19) |
\begin{equation} \frac{\partial F_3\left(\tilde{\rho},\tilde{m}\right)}{\partial \tilde{\rho}} = \tilde{\rho}^{-\gamma} \left(-\frac{\tilde{m}^{2}}{\tilde{\rho}^2}+a \gamma e^{S_{0}} \tilde{\rho}^{\gamma-1}\right), \end{equation} | (2.20) |
and
\begin{equation} F_{3}\left(\tilde{\rho}(x),\tilde{m}\right)-F_{3}(\rho_{-},\tilde{m}) = \beta \tilde{m}^{1-\gamma}x. \end{equation} | (2.21) |
Similar to the other two cases, the function F_{3}\left(\tilde{\rho}(x), \tilde{m}\right) gets its minimum at point \tilde{\rho} = \rho^{*} . If \beta < 0 , by (2.19) and (2.21), we obtain the maximal length of ducts
\begin{equation} L_m = -\frac{1}{\beta(1+\gamma)} \left[u_{-}^{\gamma+1}+ a \gamma e^{S_{0}}\left(\rho_{-} u_{-}\right)^{\gamma-1} \ln \left(a \gamma e^{S_{0}-1} \rho_{-}^{\gamma-1} u_{-}^{-2} \right) \right] \end{equation} | (2.22) |
for a supersonic or subsonic flow before it gets choked. While, if \beta > 0 , again, by (2.19) and (2.21), the flow also keeps the upstream supersonic or subsonic state for any duct length L > 0 .
To sum up, we draw the following conclusion from the above analysis:
Lemma 2.1. If \rho_{-} \neq \rho^{*} > 0, \; u_{-} > 0, \; c^{*} = (a \gamma e^{S_{0}})^{\frac{1}{\gamma+1}}(\rho_{-} u_{-})^{\frac{\gamma-1}{\gamma+1}} > 0 and the duct length L < L_{m} , where L_{m} is the maximal allowable duct length given in (2.11), (2.12), (2.17) and (2.22), then the Cauchy problem (1.4) admits a unique smooth positive solution (\tilde{\rho}(x), \tilde{u}(x), S_{0}) ^{\top} which satisfies the following properties:
1) 0 < \rho_{-} < \tilde{\rho}(x) < \rho^{*} and c^{*} < \tilde{u}(x) < u_{-} , \quad if \beta < 0 and \rho_{-} < \rho^{*} ;
2) 0 < \rho^{*} < \tilde{\rho}(x) < \rho_{-} and u_{-} < \tilde{u}(x) < c^{*} , \quad if \beta < 0 and \rho_{-} > \rho^{*} ;
3) 0 < \tilde{\rho}(x) < \rho_{-} and c^{*} < u_{-} < \tilde{u}(x) < +\infty , \quad if \beta > 0 and \rho_{-} < \rho^{*} ;
4) 0 < \rho_{-} < \tilde{\rho}(x) < +\infty and 0 < \tilde{u}(x) < u_{-} < c^{*} , \quad if \beta > 0 and \rho_{-} > \rho^{*} ;
5) \tilde{\rho} \tilde{u} = \rho_{-}u_{-} ;
6) \|(\tilde{\rho}(x), \tilde{u}(x), S_{0})\|_{C^{2}([0, L])} < M_{0}, where M_{0} is a constant only depending on \alpha , \beta , \gamma , \rho_{-} , u_{-} , S_{0} and L .
Remark 2.1. The following is worth pointing out:
1) When \beta > 0 and the upstream is supersonic, a vacuum can occur at the finite place for \alpha > 1 , while a vacuum will never happen in any finite ducts for \alpha\leq 1 ;
2) When \beta > 0 and the upstream is subsonic, fluid velocity can be zero at the finite place for \alpha < -\gamma , while the movement of fluid will never stop in the duct for \alpha \geq -\gamma ;
3) For the case of \beta = 0 , we refer the reader to [19] for details.
Thus, from Lemma 2.1 and Remark 2.1, we can directly get Theorem 1.1.
In order to answer the two problems proposed in the introduction, we introduce a wave decomposition for system (1.1) in this section. Here, we choose the steady supersonic solution \tilde{W}(x) = (\tilde{\rho}(x), \tilde{u}(x), \tilde{S}(x))^{\top} (with \tilde{u}(x) > 0 ) as the background solution, which satisfies (1.4). For system (1.1), the corresponding simplification system has the form
\begin{equation} \left\{\begin{array}{l} \rho_{t}+\rho_{x} u+\rho u_{x} = 0, \\ u_{t}+u u_{x}+a \gamma e^{S} \rho ^{\gamma -2} \rho_{x}+a e^{S} \rho ^{\gamma -1} S_{x} = \beta u ^{\alpha+1},\\ S_{t}+u S_{x} = 0. \end{array}\right. \end{equation} | (3.1) |
Let us denote W(t, x) = \bar{W}(t, x)+\tilde{W}(x) , where \bar{W} = (\bar{\rho}, \bar{u}, \bar{S})^\top is the perturbation around the background solution. Substituting
\begin{equation} \rho(t,x) = \bar{\rho}(t,x)+\tilde{\rho}(x),\; \; u(t,x) = \bar{u}(t,x)+\tilde{u}(x),\; \; S(t,x) = \bar{S}(t,x)+S_{0} \end{equation} | (3.2) |
into (3.1) yields
\begin{equation} \left\{\begin{array}{l} \bar{\rho}_{t}+ u \bar{\rho}_{x}+\rho \bar{u}_{x}+ \tilde{\rho}_{x}\bar{u}+ \tilde{u}_{x}\bar{\rho}+\tilde{u} \tilde{\rho}_{x}+\tilde{\rho} \tilde{u}_{x} = 0, \\ \bar{u}_{t}+u \bar{u}_{x}+\bar{u} \tilde{u}_{x}+ \tilde{u}_{x}\tilde{u}+a \gamma e^{S} \rho^{\gamma-2} (\bar{\rho}_{x}+ \tilde{\rho}_{x})+a e^{S} \rho ^{\gamma -1} \bar{S}_{x} = \beta(\bar{u}+\tilde{u})^{\alpha+1},\\ \bar{S_{t}}+u\bar{S}_{x} = 0. \end{array}\right. \end{equation} | (3.3) |
Combining this with (1.4), system (3.3) can be simplified as
\begin{equation} \left\{\begin{array}{l} \bar{\rho}_{t}+u \bar{\rho}_{x} +\rho \bar{u}_{x} = - \tilde{u}_{x}\bar{\rho}-\tilde{\rho}_{x}\bar{u}, \\ \begin{aligned} \bar{u}_{t}+u \bar{u}_{x}+a \gamma e^{S} \rho^{\gamma-2} \bar{\rho}_{x}+a e^{S} \rho ^{\gamma -1} \bar{S}_{x} = & -\Theta (\rho, \tilde{\rho}, S, S_{0}) e^{\bar{S}} \bar{\rho} \tilde{\rho}_{x} - \tilde{u}_{x} \bar{u} -g(u,\tilde{u}) \bar{u},\\ \end{aligned}\\ \bar{S_{t}}+u\bar{S}_{x} = 0, \end{array}\right. \end{equation} | (3.4) |
where \Theta(\rho, \tilde{\rho}, S, S_{0}) e^{\bar{S}} \bar{\rho} = a \gamma (e^{S} \rho ^{\gamma-2}-e^{S_{0}} \tilde{\rho}^{\gamma-2}) and g(u, \tilde{u}) \bar{u} = -\beta [(\bar{u}+\tilde{u})^{\alpha+1}-\tilde{u}^{\alpha+1}] . g(u, \tilde{u}) can be represented as follows:
g(u,\tilde{u}) = -\beta(\alpha+1) \int_{0}^{1} (\theta \bar{u}+\tilde{u})^{\alpha} {\mathrm{d}} \theta. |
Obviously, system (3.4) can be expressed as the following quasi-linear equations:
\begin{equation} \bar{W}_{t}+A(W) \bar{W}_{x}+H(\tilde{W}) \bar{W} = 0, \end{equation} | (3.5) |
where
\begin{equation} A(W) = \left(\begin{array}{ccc} u & \rho & 0 \\ a \gamma e^{S} \rho^{\gamma-2} & u & a e^{S} \rho ^{\gamma -1}\\ 0 & 0 & u \end{array}\right), \end{equation} | (3.6) |
\begin{equation} H(\tilde{W}) = \left(\begin{array}{ccc} \tilde{u}_{x} & \tilde{\rho}_{x} & 0\\ \Theta(\rho, \tilde{\rho}, S, \tilde{S}) e^{\bar{S}} \tilde{\rho}_{x} & \tilde{u}_{x}+g(u,\tilde{u}) & 0 \\ 0 & 0 & 0 \end{array}\right). \end{equation} | (3.7) |
Through simple calculations, the three eigenvalues of system (3.5) are
\begin{equation} \lambda_{1}(W) = u-c, \quad \lambda_{2}(W) = u, \quad \lambda_{3}(W) = u+c, \end{equation} | (3.8) |
where c = \sqrt{a\gamma} e^{\frac{S}{2}} \rho^{\frac{\gamma -1}{2}} . The three right eigenvectors r_{i}(W)\; (i = 1, 2, 3) corresponding to \lambda_{i}\; (i = 1, 2, 3) are
\begin{equation} \left\{\begin{array}{l} r_{1}(W) = \frac{1}{\sqrt{\rho^{2}+c^{2}}}(\rho,-c,0)^{\top}, \\ r_{2}(W) = \frac{1}{\sqrt{\rho^{2}+\gamma^{2}}}(\rho, 0, -\gamma)^{\top}, \\ r_{3}(W) = \frac{1}{\sqrt{\rho^{2}+c^{2}}}(\rho, c, 0)^{\top}. \end{array}\right. \end{equation} | (3.9) |
The left eigenvectors l_{i}(W)\; (i = 1, 2, 3) satisfy
\begin{equation} l_{i}(W) r_{j}(W) \equiv \delta_{i j}, \quad r_{i}(W)^{\top} r_{i}(W) \equiv 1, \quad(i, j = 1,2,3), \end{equation} | (3.10) |
where \delta_{i j} represents the Kroneckeros symbol. It is easy to get the expression for l_{i}(W) as follows:
\begin{equation} \left\{\begin{array}{l} l_{1}(W) = \frac{\sqrt{\rho^{2}+c^{2}}}{2}(\rho^{-1},-c^{-1},0), \\ l_{2}(W) = \frac{\sqrt{\rho^{2}+\gamma^{2}}}{2}(\rho^{-1}, 0, -\gamma^{-1}), \\ l_{3}(W) = \frac{\sqrt{\rho^{2}+c^{2}}}{2}(\rho^{-1}, c^{-1}, 0). \end{array}\right. \end{equation} | (3.11) |
Besides, l_{i}(W) and r_{i}(W) have the same regularity.
Let
\begin{equation} \mu_{i} = l_{i}(W) \bar{W}, \quad \varpi_{i} = l_{i}(W) \bar{W}_{x}, \quad \mu = \left(\mu_{1}, \mu_{2}, \mu_{3}\right)^{\top}, \quad \varpi = \left(\varpi_{1}, \varpi_{2}, \varpi_{3}\right)^{\top}; \end{equation} | (3.12) |
then,
\begin{equation} \bar{W} = \sum\limits_{k = 1}^{3} \mu_{k} r_{k}(W), \quad \frac{\partial \bar{W}}{\partial x} = \sum\limits_{k = 1}^{3} \varpi_{k} r_{k}(W). \end{equation} | (3.13) |
Noticing (3.5) and (3.12), we have
\begin{equation} \begin{aligned} \frac{d \mu_{i}}{d_{i} t} & = \frac{d \left(l_{i} (W) \bar{W}\right)}{d_{i} t}\\ & = \frac{d \left( \bar{W} \right)}{d_{i} t} \nabla l_{i} (W) \bar{W}+\lambda_{i}(W)\tilde{W}^{\prime}\nabla l_{i} (W) \bar{W}-l_{i} (W) H(\tilde{W})\bar{W}, \end{aligned} \end{equation} | (3.14) |
where
\begin{equation} \nabla l_{i} (W) = \begin{pmatrix} \frac{\partial}{\partial W_{1}}(l_{i} (W)) \\ \frac{\partial}{\partial W_{2}}(l_{i} (W)) \\ \frac{\partial}{\partial W_{3}}(l_{i} (W)) \end{pmatrix}. \end{equation} | (3.15) |
By using (3.5) and (3.13), we get
\begin{equation} \begin{aligned} \frac{d (\bar{W})}{d_{i} t} & = \frac{\partial \bar{W}}{\partial t} + \lambda_{i}(W)\frac{\partial(\bar{W})}{\partial x} \\ & = \sum\limits_{k = 1}^{3}(\lambda_{i}(W)-\lambda_{k}(W))\varpi_{k}r_{k}(W)-H(\tilde{W})\bar{W}. \end{aligned} \end{equation} | (3.16) |
Thus, noting \nabla(l_{i}(W) r_{j}(W)) = 0 and \nabla l_{i}(W) r_{j}(W) = -l_{i}(W) \nabla r_{j}(W) , we get
\begin{equation} \begin{aligned} \frac{d \mu_{i}}{d_{i} t} = & \frac{\partial \mu_{i}}{\partial t}+\lambda_{i}(W) \frac{\partial \mu_{i}}{\partial x}\\ = &\sum\limits_{j, k = 1}^{3} \Phi_{i j k}(W) \varpi_{j} \mu_{k}+\sum\limits_{j, k = 1}^{3} \tilde{\Phi}_{i j k}(W) \mu_{j} \mu_{k}-\sum\limits_{k = 1}^{3} \tilde{\tilde{\Phi}}_{i k}(W) \mu_{k}, \end{aligned} \end{equation} | (3.17) |
where
\begin{equation} \begin{array}{l} \Phi_{i j k}(W) = \left(\lambda_{j}(W)-\lambda_{i}(W)\right) l_{i}(W) \nabla_{W} r_{j}(W) r_{k}(W), \\ \tilde{\Phi}_{i j k}(W) = l_{i}(W) H(\tilde{W}) \nabla_{W} r_{j}(W) r_{k}(W), \\ \tilde{\tilde{\Phi}}_{i k}(W) = \lambda_{i}(W) l_{i}(W) \tilde{W}^{\prime} \nabla_{W} r_{k}(W)+l_{i}(W)H(\tilde{W}) r_{k}(W), \end{array} \end{equation} | (3.18) |
and
\begin{equation} \Phi_{i i k}(W)\equiv 0 ,\quad\forall k = 1,2,3. \end{equation} | (3.19) |
Similarly, we have from (3.10) and (3.13) that
\begin{equation} \begin{aligned} \frac{d \varpi_{i}}{d_{i} t} & = \frac{d (l_{i} (W) \bar{W}_{x})}{d_{i} t}\\ & = \sum\limits_{k = 1}^{3} \varpi_{k} \frac{d (l_{i} (W))}{d_{i} t} r_{k} (W)+l_{i} (W) \frac{d( \bar{W}_{x} )}{d_{i} t},\\ \end{aligned} \end{equation} | (3.20) |
and
\begin{equation} \begin{aligned} \frac{d \left(l_{i} (W) \right)}{d_{i} t} r_{k} (W) & = -l_{i} (W) \frac{d \left( r_{k} (W) \right)}{d_{i} t} = -\sum\limits_{s = 1}^{3} l_{i} (W) \frac{\partial \left( r_{k} (W) \right)}{\partial W_{s}} \frac{d \left( W_{s} \right)}{d_{i} t}\\ & = -\sum\limits_{s = 1}^{3} C_{ksi} (W) \left( \frac{d \bar{W}_{s}}{d_{i} t}+\frac{d \tilde{W}_{s}}{d_{i} t}\right),\\ \end{aligned} \end{equation} | (3.21) |
where C_{ksi} (W) = l_{i} (W) \frac{\partial \left(r_{k} (W) \right)}{\partial W_{s}} . It is concluded from (3.16) that
\begin{equation} \frac{d (\bar{W}_{s})}{d_{i} t} = \sum\limits_{j = 1}^{3}(\lambda_{i}(W)-\lambda_{j}(W))\varpi_{j}r_{js}(W)-H(\tilde{W})\bar{W}. \end{equation} | (3.22) |
Therefore,
\begin{equation} \begin{aligned} \sum\limits_{k = 1}^{3} \varpi_{k} \frac{d \left(l_{i} (W) \right)}{d_{i} t} r_{k} (W) = &\sum\limits_{j,k,s = 1}^{3} \varpi_{k} C_{ksi} (\lambda_{j}(W)-\lambda_{i}(W))\varpi_{j}r_{js}(W)\\ &-\sum\limits_{k,s = 1}^{3} C_{ksi} \lambda_{i} \frac{\partial \tilde{W}_{s}}{\partial x} \varpi_{k}+\sum\limits_{k,s = 1}^{3} C_{ksi} \varpi_{k}H(\tilde{W})\bar{W}.\\ \end{aligned} \end{equation} | (3.23) |
Then,
\begin{equation} \begin{aligned} l_{i} (W) \frac{d \bar{W}_{x}}{d_{i} t} & = l_{i} (W)\left( \frac{\partial \bar{W}_{x}}{\partial t} +A(W) \frac{\partial \bar{W}_{x}}{\partial x} \right)\\ & = -\sum\limits_{k,s = 1}^{3} l_{i} (W) \frac{\partial \left( A(W) \right)}{\partial W_{s}} \left( \bar{W}_{s}+\tilde{W}_{s}\right)_{x} \varpi_{k} r_{k}-l_{i} (W) \left(H(\tilde{W})\bar{W}\right)_{x}, \end{aligned} \end{equation} | (3.24) |
where we used (3.5). By differentiating
A(W) r_{k}(W) = \lambda_{k} (W) r_{k}(W) |
with respect to W_{s} and multiplying the result by l_{i}(W) , we get
\begin{equation} \begin{aligned} l_{i} (W) \frac{\partial \left( A(W) \right)}{\partial W_{s}} r_{k}& = l_{i} (W) \frac{\partial \left(\lambda_{k} \right)}{\partial W_{s}} r_{k}+ l_{i} (W) \lambda_{k} \frac{\partial \left(r_{k} \right)}{\partial W_{s}}-l_{i} (W) A(W)\frac{\partial \left(r_{k} \right)}{\partial W_{s}}\\ & = \frac{\partial \left(\lambda_{k} \right)}{\partial W_{s}} \delta_{ik}+(\lambda_{k}-\lambda_{i})C_{ksi} (W). \end{aligned} \end{equation} | (3.25) |
Thus,
\begin{equation} \begin{aligned} \frac{d \varpi_{i}}{d_{i} t} = & \sum\limits_{k = 1}^{3} \varpi_{k} \frac{d \left(l_{i} (W) \right)}{d_{i} t} r_{k} (W)+l_{i} (W) \frac{d ( \bar{W}_{x} )}{d_{i} t} \\ = &\sum\limits_{j, k = 1}^{3} \Upsilon_{i j k}(W) \varpi_{j} \varpi_{k}+\sum\limits_{j, k = 1}^{3} \tilde{\Upsilon}_{i j k}(W) \varpi_{k}-l_{i}(W) H(\tilde{W})_{x} \bar{W}, \end{aligned} \end{equation} | (3.26) |
where
\begin{aligned} \Upsilon_{i j k}(W) = & \left(\lambda_{j}(W)-\lambda_{k}(W)\right) l_{i}(W) \nabla_{W} r_{k}(W) r_{j}(W)-\nabla_{W} \lambda_{k}(W) r_{j}(W) \delta_{i k}, \\ \tilde{\Upsilon}_{i j k}(W) = & -\lambda_{k}(W) l_{i}(W) \nabla_{W} r_{k}(W) \tilde{W}^{\prime}+l_{i}(W) \nabla_{W} r_{k} H(\tilde{W}) r_{j} \mu_{j}(W) \\ & -\nabla_{W} \lambda_{k}(W) \delta_{ik} \tilde{W}^{\prime}-l_{i}(W) H(\tilde{W}) r_{k}(W). \end{aligned} |
In view of Lemma 2.1, it is clear that the term H(\tilde{W})_{x} in (3.26) is meaningful.
For the convenience of the later proof, we can rewrite system (3.5) as
\begin{equation} \bar{W}_{x}+A^{-1}(W) \bar{W}_{t}+A^{-1}(W) H(\tilde{W}) \bar{W} = 0 \end{equation} | (3.27) |
by swapping the variables t and x . Here, we represent the eigenvalues, left eigenvectors and right eigenvectors of the matrix A^{-1}(W) as \hat{\lambda}_{i} , \hat{l}_{i}(W) and \hat{r}_{i}(W), i = 1, 2, 3 , respectively.
Let
\begin{equation} \hat{\mu}_{i} = \hat{l}_{i}(W) \bar{W}, \quad \hat{\varpi}_{i} = \hat{l}_{i}(W) \bar{W}_{t}, \quad \hat{\mu} = \left(\hat{\mu}_{1}, \hat{\mu}_{2}, \hat{\mu}_{3} \right)^{\top}, \quad \hat{\varpi} = \left(\hat{\varpi}_{1}, \hat{\varpi}_{2}, \hat{\varpi}_{3}\right)^{\top} . \end{equation} | (3.28) |
Similar to the above arguments, we can get similar results by combining (3.27) and (3.28):
\begin{equation} \begin{aligned} \frac{{\mathrm{d}} \hat{\mu}_{i}}{{\mathrm{\; d}}_{i} t} = & \frac{\partial \hat{\mu}_{i}}{\partial x}+\hat{\lambda}_{i}(W) \frac{\partial \hat{\mu}_{i}}{\partial t} \\ = & \sum\limits_{j, k = 1}^{3} \hat{\Phi}_{i j k}(W) \hat{\varpi}_{j} \hat{\mu}_{k}+\sum\limits_{j, k = 1}^{3} \hat{\tilde{\Phi}}_{i j k}(W) \hat{\mu}_{j} \hat{\mu}_{k}-\sum\limits_{k = 1}^{3} \hat{\tilde{\tilde{\Phi}}}_{i k}(W) \hat{\mu}_{k}, \end{aligned} \end{equation} | (3.29) |
with
\begin{array}{l} \hat{\Phi}_{i j k}(W) = \left(\hat{\lambda}_{j}(W)-\hat{\lambda}_{i}(W)\right) \hat{l}_{i}(W) \nabla_{W} \hat{r}_{j}(W) \hat{r}_{k}(W), \\ \hat{\tilde{\Phi}}_{i j k}(W) = \hat{\lambda}_{j}(W) \hat{l}_{i}(W) H(\tilde{W}) \nabla_{W} \hat{r}_{j}(W) \hat{r}_{k}(W), \\ \hat{\tilde{\tilde{\Phi}}}_{i k}(W) = \hat{l}_{i}(W) \tilde{W}^{\prime} \nabla_{W} \hat{r}_{k}(W)+\hat{\lambda}_{i}(W) \hat{l}_{i}(W) H(\tilde{W}) \hat{r}_{k}(W), \end{array} |
and
\begin{equation} \begin{aligned} \frac{{\mathrm{d}} \hat{\varpi}_{i}}{{\mathrm{\; d}}_{i} t} = &\frac{\partial \hat{\varpi}_{i}}{\partial x}+\hat{\lambda}_{i}(W) \frac{\partial \hat{\varpi}_{i}}{\partial t} \\ = &\sum\limits_{j, k = 1}^{3} \hat{\Upsilon}_{i j k}(W) \cdot \hat{\varpi}_{j} \hat{\varpi}_{k}+\sum\limits_{j, k = 1}^{3} \hat{\tilde{\Upsilon}}_{i j k}(W) \cdot \hat{\varpi}_{k}-\hat{l}_{i}(W)\left(A^{-1} H(\tilde{W})\right)_{t} \bar{W}, \end{aligned} \end{equation} | (3.30) |
where
\begin{aligned} \hat{\Upsilon}_{i j k}(W) = &\left(\hat{\lambda}_{j}(W)-\hat{\lambda}_{k}(W)\right) \hat{l}_{i}(W) \nabla_{W} \hat{r}_{k}(W) \hat{r}_{j}(W)-\nabla_{W} \hat{\lambda}_{k}(W) \hat{r}_{j}(W) \delta_{i k}, \\ \hat{\tilde{\Upsilon}}_{i j k}(W) = & -\hat{l}_{i}(W) \nabla_{W} \hat{r}_{k}(W) \tilde{W}^{\prime}+\hat{l}_{i}(W) \nabla_{W} \hat{r}_{k}(W) A^{-1} H(\tilde{W}) \hat{r}_{j} \hat{\mu}_{j}(W) \\ & -\hat{l}_{i}(W) A^{-1}(W) H(\tilde{W}) \hat{r}_{k}(W). \end{aligned} |
The wave decomposition for the initial data
\bar{W}(t,x)|_{t = 0} = \bar{W}_{0}(x) = (\bar{\rho}_{0}(x),\bar{u}_{0}(x),\bar{S}_{0}(x))^{\top} |
and boundary conditions
\bar{W}(t,x)|_{x = 0} = \bar{W}_{l}(t) = (\bar{\rho}_{l}(t), \bar{u}_{l}(t),\bar{S}_{l}(t)) ^{\top} |
have the following form:
\begin{equation} \mu_{0} = \left(\mu_{10}, \mu_{20}, \mu_{30}\right)^{\top},\; \varpi_{0} = \left(\varpi_{10}, \varpi_{20}, \varpi_{30}\right)^{\top}, \; \hat{\mu}_{l} = \left(\hat{\mu}_{1l}, \hat{\mu}_{2l}, \hat{\mu}_{3l} \right)^{\top}, \; \hat{\varpi}_{l} = \left(\hat{\varpi}_{1l}, \hat{\varpi}_{2l}, \hat{\varpi}_{3l} \right)^{\top}, \end{equation} | (3.31) |
\begin{equation} \mu_{l} = \left(\mu_{1l}, \mu_{2l}, \mu_{3l}\right)^{\top},\; \varpi_{l} = \left(\varpi_{1l}, \varpi_{2l}, \varpi_{3l}\right)^{\top}, \end{equation} | (3.32) |
with
\begin{equation} \mu_{i0} = l_{i}(W_{0}) \bar{W_{0}},\; \varpi_{i0} = l_{i}(W_{0}) \partial_{x}(\bar{W}_{0}),\; \hat{\mu}_{il} = \hat{l}_{i}(W_{l}) \bar{W}_{l}, \; \hat{\varpi}_{il} = \hat{l}_{i}(W_{l}) \partial_{t}(\bar{W}_{l}), \end{equation} | (3.33) |
\begin{equation} \mu_{il} = l_{i}(W_{l}) \bar{W_{l}},\; \varpi_{il} = l_{i}(W_{l}) \partial_{x}(\bar{W}_{l}) , \end{equation} | (3.34) |
where
W_{0} = (\rho_{0},u_{0},S_{0})^{\top},\; W_{l} = (\rho_{l},u_{l},S_{l})^{\top}. |
In this section, based on wave decomposition, we prove the global existence and stability of smooth solutions under small perturbations around the steady-state supersonic flow in region G = \{(t, x)|t\geq0, \; x\in[0, L]\} . The initial data and boundary conditions satisfy the compatibility conditions at point (0, 0) (see [11]).
In order to verify Theorem 1.2, we first establish a uniform prior estimate of the supersonic classical solution. That is, we assume that
\begin{equation} \left|\mu_{i}(t, x)\right| \leq K \varepsilon, \quad\left|\varpi_{i}(t, x)\right| \leq K \varepsilon, \quad \forall (t, x) \in G,\; i = 1,2,3, \end{equation} | (4.1) |
when
\begin{equation} \left\| (\bar{\rho}_{0}, \bar{u}_{0}, \bar{S}_{0}) \right\|_{C^{1}([0,L])} < \varepsilon, \quad \left\| (\bar{\rho}_{l}, \bar{u}_{l}, \bar{S}_{l}) \right\|_{C^{1}([0,+\infty))} < \varepsilon , \end{equation} | (4.2) |
where \varepsilon is a suitably small positive constant. Here and hereafter, K , K_{i} and K_{i}^{*} are constants that depend only on L , \varepsilon , \left\| (\tilde{\rho}, \tilde{u}, S_{0})) \right\|_{C^{2}([0, L])} and T_{1} , defined by
\begin{equation} T_{1} = \min _{\substack{t \geq 0, x \in[0, L] \\ i = 1,2,3}} \frac{L}{\lambda_{i}(W(t, x))} > 0.\\ \end{equation} | (4.3) |
Here, \lambda_{1} , \lambda_{2} and \lambda_{3} are the three eigenvalues of system (3.5). Combining (3.9) and (3.13), (4.1) means
\begin{equation} \left|\bar{W}(t, x)\right| \leq K \varepsilon, \quad\left|\frac{\partial \bar{W}}{\partial x}(t, x)\right| \leq K \varepsilon, \quad \forall (t, x) \in G. \end{equation} | (4.4) |
In what follows, we will show the validity of the hypothesis given by (4.1).
Let x = x^{*}_{j}(t)\; (j = 1, 2, 3) be the characteristic curve of \lambda_{j} that passes through (0, 0):
\begin{equation} \frac{d x_{j}^{*}(t)}{{\mathrm{d}} t} = \lambda_{j}\left(W\left(t, x_{j}^{*}(t)\right)\right), \quad x_{j}^{*}(0) = 0. \end{equation} | (4.5) |
Since \lambda_{3}(W) > \lambda_{2}(W) > \lambda_{1}(W) , we have that x = x^{*}_{3}(t) lies below x = x^{*}_{2}(t) and x = x^{*}_{2}(t) lies below x = x^{*}_{1}(t) . In what follows, we divide domain G = \{(t, x)|t\geq0, \; x\in[0, L]\} into several different regions.
Region 1: The region G_{1} = \left\{(t, x) \mid 0 \leq t \leq T_{1}, \; 0 \leq x \leq L, \; x \geq x_{3}^{*}(t)\right\} .
For any point (t, x) \in G_{1} , integrating the i -th equation in (3.17) along the i -th characteristic curve about t from 0 to t , we have
\begin{equation} \begin{aligned} \left|\mu_{i}(t, x(t))\right| = &\left|\mu_{i}\left(0, b_{i}\right)\right|+\int_{0}^{t} \sum\limits_{j, k = 1}^{3}\left|\Phi_{i j k}(W) \varpi_{j} \mu_{k}\right| {\mathrm{d}} \tau\\ &+ \int_{0}^{t} \sum\limits_{j, k = 1}^{3}\left|\tilde{\Phi}_{i j k}(W) \mu_{j} \mu_{k}\right| {\mathrm{d}} \tau +\int_{0}^{t} \sum\limits_{k = 1}^{3}\left|\tilde{\tilde{\Phi}}_{i k}(W) \mu_{k}\right| {\mathrm{d}} \tau\\ \leq & \left|\mu_{i0}\left(b_{i}\right)\right|+K_{1} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau,\quad i = 1,2,3, \end{aligned} \end{equation} | (4.6) |
where we have used (4.3) and (4.4) and assumed that the line intersects the x axis at (0, b_{i}) . Similarly, integrating the i -th equation in (3.26) along the i -th characteristic curve and assuming that the line intersects the x axis at (0, b_{i}) again, we get
\begin{equation} \begin{aligned} \left|\varpi_{i}(t, x(t))\right| = &\left|\varpi_{i}\left(0, b_{i}\right)\right|+\int_{0}^{t} \sum\limits_{j, k = 1}^{3}\left|\Upsilon_{i j k}(W) \varpi_{j} \varpi_{k}\right| {\mathrm{d}} \tau\\ &+\int_{0}^{t} \sum\limits_{j, k = 1}^{3}\left|\tilde{\Upsilon}_{i j k}(W) \varpi_{k}\right| {\mathrm{d}} \tau +\int_{0}^{t}\left|l_{i}(W) H(\tilde{W})_{x} \bar{W}\right| {\mathrm{d}} \tau \\ \leq & \left|\varpi_{i0}\left( b_{i}\right)\right| + K_{2} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+\int_{0}^{t} [\frac{1}{2}|\tilde{u}_{xx}\mp \frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}- \frac{c\tilde{\rho}_{xx}}{\rho} \pm \tilde{u}_{xx}\\ &\pm g_{x}(u,\tilde{u}) | |\mu_{1}|+ \frac{1}{2}|\tilde{u}_{xx} \mp \frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}+ \frac{c\tilde{\rho}_{xx}} {\rho} \mp \tilde{u}_{xx}\mp g_{x}(u,\tilde{u}) | |\mu_{3}| ] {\mathrm{d}} \tau \\ \leq & \left|\varpi_{i0}\left( b_{i}\right)\right|+K_{2} \int_{0}^{t} |\varpi(\tau,x(\tau)) |{\mathrm{d}} \tau+K_{2}^{*} \int_{0}^{t} |\mu(\tau,x(\tau)) |{\mathrm{d}} \tau,\quad i = 1,3, \end{aligned} \end{equation} | (4.7) |
and
\begin{equation} \begin{aligned} \left|\varpi_{2}(t, x(t))\right| = & \left|\varpi_{2}\left(0, b_{2}\right)\right|+\int_{0}^{t} \sum\limits_{j, k = 1}^{3}\left|\Upsilon_{i j k}(W) \varpi_{j} \varpi_{k}\right| {\mathrm{d}} \tau\\ &+\int_{0}^{t}\sum\limits_{j, k = 1}^{3}\left|\tilde{\Upsilon}_{i j k}(W) \varpi_{k}\right| {\mathrm{d}} \tau +\int_{0}^{t}\left|l_{2}(W) H(\tilde{W})_{x} \bar{W}\right| {\mathrm{d}} \tau \\ \leq & \left|\varpi_{20}\left(b_{2}\right)\right| + K_{3}\int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+ \int_{0}^{t} \; [\frac { \sqrt{\rho^{2}+ \gamma^{2}}}{2 \sqrt{\rho^{2}+ c^{2}}} ( |\tilde{u}_{xx}-\frac{c\tilde{\rho}_{xx}}{\rho}||\mu_{1}|\\ &+ |\tilde{u}_{xx}+\frac{c\tilde{\rho}_{xx}}{\rho}||\mu_{3}| ) ] {\mathrm{d}} \tau \\ \leq & \left|\varpi_{20}\left(b_{2}\right)\right|+K_{3} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{3}^{*}\int_{0}^{t}| \mu(\tau,x(\tau)) |{\mathrm{d}} \tau, \end{aligned} \end{equation} | (4.8) |
where \Theta = \Theta(\rho, \tilde{\rho}, S, S_{0}) . Adding (4.6)–(4.8) together, for any i = 1, 2, 3 , and using Gronwall's inequality, one gets
\begin{equation} |\mu(t, x)|+|\varpi(t, x)| \leq e^{K_{4}T_{1}} \left(\left\|\mu_{0}\right\|_{C^{0}([0,L])}+\left\|\varpi_{0}\right\|_{C^{0}([0,L])}\right). \end{equation} | (4.9) |
Due to the boundedness of T_{1} , the arbitrariness of (t, x)\in G_{1} and (4.9), it holds that
\begin{equation} \max\limits _{(t, x) \in G_{1}} \{|\mu(t, x)|+|\varpi(t, x)|\} \leq K\left(\left\|\mu_{0}\right\|_{C^{0}([0,L])}+\left\|\varpi_{0}\right\| _{C^{0} ([0,L]) }\right). \end{equation} | (4.10) |
Region 2: The region G_{2} = \left\{(t, x) \mid t \geq 0, \; 0 \leq x \leq L, \; 0 \leq x \leq x_{1}^{*}(t)\right\} .
We make the change of variables t and x . For any point (t, x) \in G_{2} , integrating (3.29) along the i -th characteristic curve about x , it follows that
\begin{equation} \begin{aligned} \left|\hat{\mu}_{i}(t(x), x)\right| \leq | \hat{\mu}_{il}(t_{i})|+K_{5} \int_{0}^{x}| \hat{\mu}(t(\varsigma),\varsigma) |{\mathrm{d}} \varsigma,\quad i = 1,2,3, \end{aligned} \end{equation} | (4.11) |
where we assumed that the line intersects the t axis at the point (t_{i}, 0) . Similarly, repeating the above procedure for (3.30), we get
\begin{equation} \begin{aligned} \left|\hat{\varpi}_{i}(t(x), x)\right| \leq | \hat{\varpi}_{il}(t_{i})|+K_{6} \int_{0}^{x}| \hat{\varpi}(t(\varsigma),\varsigma) |{\mathrm{d}} \varsigma+K_{6}^{*} \int_{0}^{x}| \hat{\mu}(t(\varsigma),\varsigma)| {\mathrm{d}} \varsigma ,\quad i = 1,2,3. \end{aligned} \end{equation} | (4.12) |
Summing up (4.11) and (4.12) for i = 1, 2, 3 and applying Gronwall's inequality, we obtain
\begin{equation} \max\limits _{(t, x) \in G_{2}} \{ |\hat{\mu}(t, x)|+|\hat{\varpi}(t, x)| \}\leq K \left(\left\|\hat{\mu}_{l}\right\|_{C^{0}([0,+\infty))}+ \left\|\hat{\varpi}_{l}\right\|_{C^{0} ([0,+\infty))}\right), \quad \forall(t, x) \in G_{2}, \end{equation} | (4.13) |
where we exploit the arbitrariness of (t, x)\in G_{2} .
Region 3: The region G_{3} = \left\{(t, x) \mid 0 \leq t \leq T_{1}, \; 0 \leq x \leq L, \; x_{2}^{*}(t) \leq x \leq x_{3}^{*}(t)\right\} .
For any point (t, x) \in G_{3} , integrating the 1st and 2nd equations in (3.17) and (3.26) along the 1st and 2nd characteristic curve, we get
\begin{equation} \left|\mu_{1}(t, x(t))\right| \leq \left|\mu_{10}\left(x^{\prime}_{1}\right)\right|+ K_{7} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.14) |
\begin{equation} \begin{aligned} \left|\varpi_{1}(t, x(t))\right| \leq & \left|\varpi_{10}\left(x^{\prime}_{1}\right)\right| + K_{8}\int_{0}^{t}| \varpi(\tau,x(\tau)) |{\mathrm{d}} \tau+\int_{0}^{t} [\frac{1}{2}|2\tilde{u}_{xx}- \frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}- \frac{c\tilde{\rho}_{xx}}{\rho} \\ &+ g_{x}(u,\tilde{u}) | |\mu_{1}|+ \frac{1}{2}| - \frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}+ \frac{c\tilde{\rho}_{xx}} {\rho} - g_{x}(u,\tilde{u}) | |\mu_{3}| ] {\mathrm{d}} \tau \\ \leq & \left|\varpi_{10}\left( x^{\prime}_{1}\right)\right|+K_{8} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{8}^{*} \int_{0}^{t}| \mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{aligned} \end{equation} | (4.15) |
\begin{equation} \left|\mu_{2}(t, x(t))\right| \leq \left|\mu_{20}\left( x^{\prime}_{2}\right)\right|+K_{9} \int_{0}^{t}| \mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.16) |
and
\begin{equation} \begin{aligned} \left|\varpi_{2}(t, x(t))\right| \leq \left|\varpi_{20}\left( x^{\prime}_{2}\right)\right|+K_{10} \int_{0}^{t}| \varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{10}^{*} \int_{0}^{t} |\mu(\tau,x(\tau)) |{\mathrm{d}} \tau, \end{aligned} \end{equation} | (4.17) |
where we assumed that the line intersects the x axis at points (0, x^{\prime}_{1}) and (0, x^{\prime}_{2}) , respectively. Similarly, integrating the 3rd equations in (3.17) and (3.26) along the 3rd characteristic curve, one has
\begin{equation} \begin{aligned} \left|\mu_{3}(t, x(t))\right| &\leq \left|\mu_{3l}\left(t^{\prime}_{3}\right)\right|+ K_{11} \int_{t^{\prime}_{3}}^{t} |\mu(\tau,x(\tau)) |{\mathrm{d}} \tau\\ &\leq \left|\mu_{3l}\left(t^{\prime}_{3}\right)\right|+ K_{11} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{aligned} \end{equation} | (4.18) |
and
\begin{equation} \begin{aligned} \left|\varpi_{3}(t, x(t))\right| \leq & \left|\varpi_{3l}\left(t^{\prime}_{3}\right)\right|+ K_{12} \int_{t^{\prime}_{3}}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+\int_{t^{\prime}_{3}}^{t} [\frac{1}{2}| \frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}- \frac{c\tilde{\rho}_{xx}}{\rho} \\ &- g_{x}(u,\tilde{u}) | |\mu_{1}|+ \frac{1}{2}|2\tilde{u}_{xx} +\frac{\rho}{c} (\Theta e^{\bar{S}} \tilde{\rho}_{x})_{x}+ \frac{c\tilde{\rho}_{xx}} {\rho} + g_{x}(u,\tilde{u}) | |\mu_{3}| ] {\mathrm{d}} \tau\\ \leq & \left|\varpi_{3l}\left(t^{\prime}_{3}\right)\right|+K_{12} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{12}^{*} \int_{0}^{t} |\mu(\tau,x(\tau)) | {\mathrm{d}} \tau, \end{aligned} \end{equation} | (4.19) |
where the point (t^{\prime}_{3}, 0) is the intersection of the line and the t axis.
Since the boundary data are small enough, we sum up (4.14)-(4.19) and apply Gronwall's inequality to obtain the following:
\begin{equation} \begin{aligned} \max _{(t, x) \in G_{3}} \{|\mu(t, x)|+|\varpi(t, x)| \} \leq & K (\left\|\mu_{0}\right\|_{C^{0}([0,L])}+\left\|\varpi_{0}\right\|_{C^{0}([0,L])}+\\ &\left\|\mu_{l}\right\|_{C^{0}([0,+\infty))}+ \left\|\varpi_{l} \right\|_{C^{0} ([0,+\infty))}), \end{aligned} \end{equation} | (4.20) |
where we exploit the arbitrariness of (t, x)\in G_{3} .
Region 4: The region G_{4} = \left\{(t, x) \mid 0 \leq t \leq T_{1}, \; 0 \leq x \leq L, \; x_{1}^{*}(t) \leq x \leq x_{2}^{*}(t)\right\} .
For any point (t, x) \in G_{4} , integrating the 1st equations in (3.17) and (3.26) along the 1st characteristic curve, we get
\begin{equation} \left|\mu_{1}(t, x(t))\right| \leq \left|\mu_{10}\left( x^{\prime \prime}_{1}\right)\right|+ K_{13} \int_{0}^{t}| \mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.21) |
and
\begin{equation} \left|\varpi_{1}(t, x(t))\right| \leq \left|\varpi_{10}\left(x^{\prime \prime}_{1}\right)\right|+K_{14} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{14}^{*} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.22) |
where we assumed that the line intersects the x axis at (0, x^{\prime \prime}_{1}) . Similarly, integrating the 2nd and 3rd equations in (3.17) and (3.26) along the 2nd and 3rd characteristic curve, one has
\begin{equation} \left|\mu_{2}(t, x(t))\right| \leq \left|\mu_{2l}\left(t^{\prime \prime}_{2}\right)\right|+ K_{15} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.23) |
\begin{equation} \left|\varpi_{2}(t, x(t))\right| \leq \left|\varpi_{2l}\left(t^{\prime \prime}_{2}\right)\right|+K_{16} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{16}^{*} \int_{0}^{t}| \mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.24) |
\begin{equation} \left|\mu_{3}(t, x(t))\right| \leq \left|\mu_{3l}\left(t^{\prime \prime}_{3}\right)\right|+ K_{17} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.25) |
and
\begin{equation} \left|\varpi_{3}(t, x(t))\right| \leq \left|\varpi_{3l}\left(t^{\prime \prime}_{3}\right)\right|+K_{18} \int_{0}^{t} |\varpi(\tau,x(\tau))| {\mathrm{d}} \tau+K_{18}^{*} \int_{0}^{t} |\mu(\tau,x(\tau))| {\mathrm{d}} \tau, \end{equation} | (4.26) |
where the line intersects the t axis at points (t^{\prime \prime}_{2}, 0) and (t^{\prime \prime}_{3}, 0) , respectively.
Noticing that the boundary data are small enough, we sum (4.21) – (4.26) and then apply Gronwall's inequality to obtain
\begin{equation} \begin{aligned} \max _{(t, x) \in G_{4}} \{|\mu(t, x)|+|\varpi(t, x)| \} \leq & K (\left\|\mu_{0}\right\|_{C^{0}([0,L])}+\left\|\varpi_{0}\right\|_{C^{0}([0,L])}+\\ &\left\|\mu_{l}\right\|_{C^{0}([0,+\infty))}+ \left\|\varpi_{l} \right\|_{C^{0} ([0,+\infty))}), \end{aligned} \end{equation} | (4.27) |
where we exploit the arbitrariness of (t, x)\in G_{4} .
From (4.10), (4.13), (4.20) and (4.27), we have proved that the assumption of (4.1) is reasonable. Therefore, we have obtained a uniform C^{1} a priori estimate for the classical solution. Thanks to the classical theory in [34], we further obtain the global existence and uniqueness of C^{1} solutions (see [11,35,36,37,38,39]) for problems (1.1)–(1.3). This proves Theorem 1.2.
In this section, we show that the smooth supersonic solution W(t, x) = (\rho(t, x), u(t, x), S(t, x))^\top is temporal-periodic with a period P > 0 , after a certain start-up time T_{1} , under the temporal periodic boundary conditions. Here, we have assumed that W_{l}(t+P) = W_{l}(t) with P > 0 .
For system (1.1), Riemann invariants \xi , \eta and \zeta are introduced as follows:
\begin{equation} \xi = u- \frac{2}{\gamma -1} c,\quad \eta = S,\quad \zeta = u+ \frac{2}{\gamma -1} c. \end{equation} | (5.1) |
Then, system (1.1) can be transformed into the following form:
\begin{equation} \left\{\begin{array}{l} \xi_{t} +\lambda_{1} (\xi,\zeta) \xi_{x} = \beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x},\\ \eta_{t} +\lambda_{2} (\xi,\zeta) \eta_{x} = 0,\\ \zeta_{t}+\lambda_{3} (\xi,\zeta) \zeta_{x} = \beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x}, \end{array}\right. \end{equation} | (5.2) |
where
\lambda_{1} = u-c = \frac{\gamma +1}{4} \xi+\frac{3- \gamma }{4} \zeta,\quad \lambda_{2} = u = \frac{1}{2} (\xi+\zeta),\quad \lambda_{3} = u+c = \frac{3-\gamma }{4} \xi+\frac{\gamma +1}{4} \zeta |
are three eigenvalues of system (1.1). For supersonic flow (i.e., u > c ), we know that \lambda_{3} > \lambda_{2} > \lambda_{1} > 0 . Obviously, (1.2)–(1.3) can be written as
\begin{equation} \xi(0,x) = \xi_{0} (x),\quad \eta(0,x) = \eta_{0} (x),\quad \zeta(0,x) = \zeta_{0} (x),\quad 0 \leq x \leq L, \end{equation} | (5.3) |
\begin{equation} \xi(t,0) = \xi_{l} (t),\quad \eta(t,0) = \eta_{l} (t),\quad \zeta(t,0) = \zeta_{l} (t),\quad t \geq 0, \end{equation} | (5.4) |
where \xi_{l} (t+P) = \xi_{l} (t), \; \eta_{l} (t+P) = \eta_{l} (t) and \zeta_{l} (t+P) = \zeta_{l} (t) with P > 0 .
We swap t and x so that the problem described by (5.2)–(5.4) takes the following form:
\begin{equation} \left\{\begin{array}{l} \xi_{x} +\frac{1}{\lambda_{1}} \xi_{t} = \frac{1}{\lambda_{1}} [\beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x}],\\ \eta_{x} +\frac{1}{\lambda_{2}} \eta_{t} = 0,\\ \zeta_{x} +\frac{1}{\lambda_{3}} \zeta_{t} = \frac{1}{\lambda_{3}}[\beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x}],\\ \xi(t,0) = \xi_{l} (t),\\ \eta(t,0) = \eta_{l} (t),\\ \zeta(t,0) = \zeta_{l} (t), \end{array}\right. \end{equation} | (5.5) |
where t > 0 and x\in[0, L] . Next, we set
\begin{equation} V = (\xi-\tilde{\xi}, \eta-\tilde{\eta}, \zeta-\tilde{\zeta})^{\top},\quad \Lambda(t, x) = \left(\begin{array}{ccc} \frac{1}{\lambda_{1}(\xi(t, x), \zeta(t, x))} & 0 & 0\\ 0 & \frac{1}{\lambda_{2}(\xi(t, x), \zeta(t, x))}& 0\\ 0 & 0 & \frac{1}{\lambda_{3}(\xi(t, x), \zeta(t, x))} \end{array}\right); \end{equation} | (5.6) |
then, the Cauchy problem (5.5) can be simplified as follows:
\begin{equation} \begin{aligned} V_{x}+\Lambda(t, x) V_{t} = & \Lambda(t, x)\left(\begin{array}{c} \beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x}\\ 0\\ \beta(\frac{\xi}{2}+\frac{\zeta}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\zeta-\xi)^{2} \eta_{x} \end{array}\right)\\ & -\left(\begin{array}{c} \frac{1}{\tilde{\lambda}_{1}} [\beta(\frac{\tilde{\xi}}{2}+\frac{\tilde{\zeta}}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\tilde{\zeta} -\tilde{\xi})^{2} \tilde{\eta}^{\prime}] \\ 0\\ \frac{1}{\tilde{\lambda}_{3}} [\beta(\frac{\tilde{\xi}}{2}+\frac{\tilde{\zeta}}{2})^{\alpha+1}+\frac{\gamma-1}{16\gamma} (\tilde{\zeta} -\tilde{\xi})^{2} \tilde{\eta}^{\prime}] \end{array}\right), \end{aligned} \end{equation} | (5.7) |
where
\begin{array}{c} \tilde{\xi} = \tilde{u}-\frac{2}{\gamma-1} \tilde{c},\quad \tilde{\eta} = \tilde{S},\quad \tilde{\zeta} = \tilde{u}+\frac{2}{\gamma-1} \tilde{c},\\ \tilde{\lambda}_{1} = \lambda_{1}(\tilde{\xi}, \tilde{\zeta}) = \frac{\gamma+1}{4} \tilde{\xi}+\frac{3-\gamma}{2} \tilde{\zeta}, \\ \tilde{\lambda}_{2} = \lambda_{2}(\tilde{\xi}, \tilde{\zeta}) = \frac{1}{2}\tilde{\xi}+\frac{1}{2}\tilde{\zeta}, \\ \tilde{\lambda}_{3} = \lambda_{3}(\tilde{\xi}, \tilde{\zeta}) = \frac{3-\gamma}{4} \tilde{\xi}+\frac{\gamma+1}{4} \tilde{\zeta}. \end{array} |
According to
\|\rho-\tilde{\rho}\|_{C^{1}(G)}+\|u-\tilde{u}\|_{C^{1}(G)}+\|S-\tilde{S}\|_{C^{1}(G)} < K_{0} \varepsilon |
and (5.1), we can easily obtain
\begin{equation} \|\xi(t,x)-\tilde{\xi} (x)\|_{C^{1}(G)}+\|\eta(t,x)-\tilde{\eta} (x)\|_{C^{1}(G)}+\|\zeta(t,x)-\tilde{\zeta} (x)\|_{C^{1}(G)} < J_{1} \varepsilon,\\ \end{equation} | (5.8) |
where the constant J_{1}(>0) depends solely on \; \tilde{\rho}, \tilde{u}, \gamma and L .
In order to prove that W(t+P, x) = W(t, x) , for any t > T_{1} and x \in [0, L] , we first prove that the following conclusions hold:
\begin{equation} \xi(t+P,x) = \xi(t,x),\quad \eta(t+P,x) = \eta(t,x),\quad \zeta(t+P,x) = \zeta(t,x),\quad\forall t > T_{1},\; x\in[0,L], \end{equation} | (5.9) |
where T_{1} is the start-up time, which is defined in (4.3).
Let
\; N(t,x) = V(t+P,x)-V(t,x); |
then, according to (5.7), we obtain
\begin{equation} \left\{\begin{array}{c} N_{x}+\Lambda(t,x)N_{t} = R(t,x),\\ N(t,0) = 0,\quad t > 0, \end{array}\right. \end{equation} | (5.10) |
where
\begin{equation} \begin{aligned} R(t,x) = & \Lambda(t+P, x) \left(\begin{array}{c} \beta(\frac{\xi(t+P,x)}{2}+\frac{\zeta(t+P,x)}{2})^{\alpha+1}+\frac{(\gamma-1) (\zeta(t+P,x)-\xi(t+P,x))^{2} \eta_{x}(t+P,x)}{16\gamma} \\ 0\\ \beta(\frac{\xi(t+P,x)}{2}+\frac{\zeta(t+P,x)}{2})^{\alpha+1}+\frac{(\gamma-1) (\zeta(t+P,x)-\xi(t+P,x))^{2} \eta_{x}(t+P,x)}{16\gamma} \end{array}\right)\\ &-\Lambda(t, x) \left(\begin{array}{c} \beta(\frac{\xi(t,x)}{2}+\frac{\zeta(t,x)}{2})^{\alpha+1}+\frac{(\gamma-1) (\zeta(t,x)-\xi(t,x))^{2} \eta_{x}(t,x)}{16\gamma} \\ 0\\ \beta(\frac{\xi(t,x)}{2}+\frac{\zeta(t,x)}{2})^{\alpha+1}+\frac{(\gamma-1) (\zeta(t,x)-\xi(t,x))^{2} \eta_{x}(t,x)}{16\gamma} \end{array}\right)\\ &-[\Lambda(t+P, x)-\Lambda(t, x)] V_{t}(t+P, x) . \end{aligned} \end{equation} | (5.11) |
Using the continuity of \lambda_{i}\; (i = 1, 2, 3) and (5.8), after some calculations, we obtain the following estimates:
\begin{equation} \begin{array}{l} \left|V_{t}(t+P, x)\right| \leq J_{2} \varepsilon, \\ \end{array} \end{equation} | (5.12) |
\begin{equation} \begin{array}{l} |\xi(t+P, x)+\zeta(t+P, x)| \leq J_{3}, \\ \end{array} \end{equation} | (5.13) |
\begin{equation} \begin{array}{l} |\Lambda(t, x)| \leq J_{4}, \end{array} \end{equation} | (5.14) |
\begin{equation} \begin{array}{l} |\Lambda(t+P, x)-\Lambda(t, x)| \leq J_{5}|N(t, x)|, \\ \end{array} \end{equation} | (5.15) |
\begin{equation} \begin{array}{l} \left|\Lambda_{t}(\xi(t, x), \eta(t, x))\right| \leq J_{6} \varepsilon, \end{array} \end{equation} | (5.16) |
and
\begin{equation} \begin{aligned} |R(t, x)| \leq & |\Lambda(t, x)| \cdot \left(\begin{array}{c} J_{7}|\beta| |N(t, x)|+\frac{\gamma-1}{16\gamma}J _{8} \cdot J_{9} |N(t, x)|\\ 0\\ J_{7}|\beta| |N(t, x)|+\frac{\gamma-1}{16\gamma}J _{8} \cdot J_{9} |N(t, x)| \end{array}\right)\\ &+|\Lambda(t+P, x)-\Lambda(t, x)|\cdot \left(\begin{array}{c} (\frac{J_{3}}{2}) ^{\alpha+1}|\beta| +\frac{\gamma-1}{16} J_{3}^{2} \cdot J _{8}\\ 0\\ (\frac{J_{3}}{2}) ^{\alpha+1} |\beta|+\frac{\gamma-1}{16} J_{3}^{2} \cdot J _{8} \end{array}\right)\\ & +|\Lambda(t+P, x)-\Lambda(t, x)|\cdot\left|V_{t}(t+P, x)\right| \\ \leq & J_{10}|N(t, x)|, \end{aligned} \end{equation} | (5.17) |
where the constants J_{i}\; (i = 2, \cdots, 10) depend only on \tilde{\rho}, \tilde{u}, \gamma and L .
In the above calculation, we have used
\begin{aligned} &|(\frac{\xi(t+P,x)}{2}+\frac{\zeta(t+P,x)}{2})^{\alpha+1}-(\frac{\xi(t,x)}{2}+\frac{\zeta(t,x)}{2})^{\alpha+1}|\\ = &|u^{\alpha+1}(t+P,x)-u^{\alpha+1}(t,x)|\\ = &|u(t+P,x)-u(t,x)| |(\alpha+1)| |\int_{0}^{1} [u(t,x)+\theta(u(t+P,x)-u(t,x))]^{\alpha} {\mathrm{d}} \theta|\\ \leq & J_{7} |N(t, x)|,\; \quad {\text{for}}\; \alpha \neq -1; \end{aligned} |
|(\frac{\xi(t+P,x)}{2}+\frac{\zeta(t+P,x)}{2})^{\alpha+1}-(\frac{\xi(t,x)}{2}+\frac{\zeta(t,x)}{2})^{\alpha+1}| = 0 \leq J_{7} |N(t, x)|,\; \quad {\text{for}}\; \alpha = -1. |
Now, fix a point (t^{*}, x^{*}) with t^{*} > T_{1} and 0 < x^{*} < L . Let \Gamma_{1}: t = \check{t}_{1}(x) and \Gamma_{3}: t = \check{t}_{3}(x) be two characteristic curves passing through point (t^{*}, x^{*}) , that is,
\begin{equation} \frac{d \check{t}_{1}}{d x} = \frac{1}{\lambda_{1}\left(\xi\left(\check{t}_{1}, x\right), \zeta\left(\check{t}_{1}, x\right)\right)},\quad \check{t}_{1} \left(x^{*}\right) = t^{*}, \end{equation} | (5.18) |
and
\begin{equation} \frac{d \check{t}_{3}}{d x} = \frac{1}{\lambda_{3}\left(\xi\left(\check{t}_{3}, x\right), \zeta\left(\check{t}_{3}, x\right)\right)},\quad \check{t}_{3} \left(x^{*}\right) = t^{*}, \end{equation} | (5.19) |
where x \in [0, x^{*}] . Since \lambda_{3}(W) > \lambda_{1}(W) , \Gamma_{1} lies below \Gamma_{3} . Set
\begin{equation} \Psi(x) = \frac{1}{2} \int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)}|N(t, x)|^{2} d t, \end{equation} | (5.20) |
where 0 \leq x < x^{*} . According to the definition of T_{1} , and combining t^{*} > T_{1} and 0 \leq x^{*} \leq L , we obtain that (\check{t}_{1}(0), \check{t}_{3}(0))\subset (0, +\infty) . Then, it follows from (5.10) that N(t, 0) \equiv 0 . Thus, \Psi(0) = 0 .
Taking the derivative of \Psi(x) with regard to x gives
\begin{equation} \begin{aligned} \Psi^{\prime}(x) = & \int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} N_{x}(t, x) d t+\frac{1}{2}\left|N\left(\check{t}_{3}(x), x\right)\right|^{2} \frac{1}{\lambda_{3}\left(\xi\left(\check{t}_{3}(x), x\right), \zeta\left(\check{t}_{3}, x\right)\right)} \\ &-\frac{1}{2}\left|N\left(\check{t}_{1}(x), x\right)\right|^{2} \frac{1}{\lambda_{1}\left(\xi\left(\check{t}_{1}(x), x\right), \zeta\left(\check{t}_{1}(x), x\right)\right)} \\ \leq & -\int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} \Lambda(t, x) N_{t}(t, x) d t+\int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} R(t, x) d t \\ &+\left.\frac{1}{2} N(t, x)^{\top} \Lambda(t, x) N(t, x)\right|_{t = \check{t}_{1}(x)} ^{t = \check{t}_{3}(x)}\\ = &-\frac{1}{2} \int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} \left[ \left(N(t, x)^{\top} \Lambda(t, x) N(t, x)\right)_{t}-N(t, x)^{\top} \Lambda_{t}(t, x) N(t, x)\right] d t \\ &+\int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} R(t, x) d t+\left.\frac{1}{2} N(t, x)^{\top} \Lambda(t, x) N(t,x)\right|_{t = \check{t}_{1}(x)} ^{t = \check{t}_{3}(x)} \\ = &\frac{1}{2} \int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} \Lambda_{t}(t, x) N(t, x) d t+\int_{\check{t}_{1}(x)}^{\check{t}_{3}(x)} N(t, x)^{\top} R(t, x) d t \\ \leq & \left(J_{6} \varepsilon+2 J_{10}\right) \Psi(x), \end{aligned} \end{equation} | (5.21) |
where we used (5.16) and (5.17).
Therefore, using Gronwall's inequality, we obtain that \Psi(x) \equiv 0 . In addition, according to the continuity of \Psi(x) , we obtain that \Psi(x^{*}) = 0 ; then, N(t^{*}, x^{*}) = 0 . Using the arbitrariness of (t^{*}, x^{*}) , we get
N(t, x) \equiv 0,\quad \forall t > T_{1},\; x \in [0, L]. |
Thus, (5.9) holds. Then, from (5.1) and c = \sqrt{a\gamma} e^{\frac{S}{2}} \rho^{\frac{\gamma -1}{2}} , it follows that
W(t+P,x) = W(t,x) |
for any t > T_{1} and x \in [0, L] , where T_{1} is the start-up time defined in (4.3). This proves Theorem 1.3.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported in part by the Natural Science Foundation of China Grant No. 12101372, No. 12271310, and the Natural Science Foundation of Shandong Province Grant No. ZR2022MA088.
The authors declare there is no conflict of interest.
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