Processing math: 50%
Research article Special Issues

Interpolating estimates with applications to some quantitative symmetry results

  • Received: 09 September 2021 Revised: 06 December 2021 Accepted: 07 December 2021 Published: 10 January 2022
  • We prove interpolating estimates providing a bound for the oscillation of a function in terms of two Lp norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.

    Citation: Rolando Magnanini, Giorgio Poggesi. Interpolating estimates with applications to some quantitative symmetry results[J]. Mathematics in Engineering, 2023, 5(1): 1-21. doi: 10.3934/mine.2023002

    Related Papers:

    [1] Yasir Rehman, Cindy Zhang, Haolin Ye, Lionel Fernandes, Mathieu Marek, Andrada Cretu, William Parkinson . The extent of the neurocognitive impairment in elderly survivors of war suffering from PTSD: meta-analysis and literature review. AIMS Neuroscience, 2021, 8(1): 47-73. doi: 10.3934/Neuroscience.2021003
    [2] Ian M Rowbotham, Franco F Orsucci, Mohamed F Mansour, Samuel R Chamberlain, Haroon Y Raja . Relevance of Brain-derived Neurotrophic Factor Levels in Schizophrenia: A Systematic Review and Meta-Analysis. AIMS Neuroscience, 2015, 2(4): 280-293. doi: 10.3934/Neuroscience.2015.4.280
    [3] Scott Mendoza . The role of tetrahydrocannabivarin (THCV) in metabolic disorders: A promising cannabinoid for diabetes and weight management. AIMS Neuroscience, 2025, 12(1): 32-43. doi: 10.3934/Neuroscience.2025003
    [4] Antonio G. Lentoor, Tiro B. Motsamai, Thandokuhle Nxiweni, Bongumusa Mdletshe, Siyasanga Mdingi . Protocol for a systematic review of the effects of gardening physical activity on neuroplasticity and cognitive function. AIMS Neuroscience, 2023, 10(2): 118-129. doi: 10.3934/Neuroscience.2023009
    [5] Graziella Orrù, Marina Baroni, Ciro Conversano, Angelo Gemignani . Exploring the therapeutic potential of tDCS, TMS and DBS in overcoming tobacco use disorder: an umbrella review. AIMS Neuroscience, 2024, 11(4): 449-467. doi: 10.3934/Neuroscience.2024027
    [6] Nicholas J. D. Wright . A review of the direct targets of the cannabinoids cannabidiol, Δ9-tetrahydrocannabinol, N-arachidonoylethanolamine and 2-arachidonoylglycerol. AIMS Neuroscience, 2024, 11(2): 144-165. doi: 10.3934/Neuroscience.2024009
    [7] Fiammetta Iannuzzo, Fabrizio Turiaco, Vincenzo Messina, Alessandro Magazzù Minutoli, Maria Catena Silvestri, Maria Rosaria Anna Muscatello, Antonio Bruno . Non-invasive brain stimulation for suicidal ideation: a systematic review and metanalysis of the current literature. AIMS Neuroscience, 2025, 12(3): 332-350. doi: 10.3934/Neuroscience.2025018
    [8] Mahmood Moosazadeh, Fatemeh Nabinezhad-Male, Mahdi Afshari, Mohammad Mehdi Nasehi, Mohammad Shabani, Motahareh Kheradmand, Iraj Aghaei . Vitamin D status and disability among patients with multiple sclerosis: a systematic review and meta-analysis. AIMS Neuroscience, 2021, 8(2): 239-253. doi: 10.3934/Neuroscience.2021013
    [9] Nisha Shantakumari, Musaab Ahmed . Whole body vibration therapy and cognitive functions: a systematic review. AIMS Neuroscience, 2023, 10(2): 130-143. doi: 10.3934/Neuroscience.2023010
    [10] Sanaz Mehrabani, Mohsen Rastkar, Narges Ebrahimi, Mahsa Ghajarzadeh . Microbiomes and Pediatric onset multiple sclerosis (MS): A systematic review. AIMS Neuroscience, 2023, 10(4): 423-432. doi: 10.3934/Neuroscience.2023031
  • We prove interpolating estimates providing a bound for the oscillation of a function in terms of two Lp norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.



    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,βR. 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 SConst., 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,βR. 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),0xL,S0(x)=ˉS0(x)+S0, (1.5)

    and

    x=0:{ρl(t)=ˉρl(t)+ρ,ul(t)=ˉul(t)+u,t0.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 ρρ=[(ρu)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=ρu>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, α1 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, αγ 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=[(ρu)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)S0C1([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)|t0,x[0,L]}, satisfying

    ˉW(t,x)C1(G)=ρ(t,x)˜ρ(x),u(t,x)˜u(x),S(t,x)S0C1(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,ρu), (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: α1 and αγ.

    In this case, (2.1) becomes

    {˜m=const.=ρu,F1(˜ρ,˜m)x=β˜mα+1, (2.2)

    where

    F1(˜ρ,˜m)=˜m2α1˜ρα1+aγeS0γ+α˜ργ+α. (2.3)

    Then, we get

    F1(˜ρ,˜m)˜ρ=˜ρα(˜m2˜ρ2+aγeS0˜ργ1)=˜ρα2(˜ρ2p˜ρ˜m2). (2.4)

    Let G(˜ρ,˜m)=˜ρ2p˜ρ˜m2. For any fixed ˜m>0, we have that lim˜ρ0G(˜ρ,˜m)=˜m2<0. From the definition of p(˜ρ,S0), we obtain

    ˜ρ2p˜ρ is a strictly increasing function for˜ρ>0.

    Thus, when ˜ρ+, G(˜ρ,˜m)+. Then, there exists ρ=[(ρu)2aγeS0]1γ+1>0 such that G(ρ,˜m)=0 (i.e., (ρ)2p˜ρ(ρ)=˜m2). That is, when ˜ρ=ρ, the fluid velocity is equal to the sound speed (i.e., ˜u=˜c=p˜ρ=aγeS02˜ργ12). Therefore, we have

    F1(˜ρ,˜m)˜ρ=˜ρα2(p˜ρ˜ρ2˜m2)<0p˜ρ˜ρ2<˜m2 (2.5)

    and

    F1(˜ρ,˜m)˜ρ=˜ρα2(p˜ρ˜ρ2˜m2)>0p˜ρ˜ρ2>˜m2. (2.6)

    We conclude that F1(˜ρ,˜m)˜ρ<0 for ˜ρ<ρ and F1(˜ρ,˜m)˜ρ>0 for ˜ρ>ρ. Furthermore, we have

    lim˜ρ0F1(˜ρ,˜m)=0,lim˜ρ+F1(˜ρ,˜m)=+,F1(ρ,˜m)<0,forα>1; (2.7)
    lim˜ρ0F1(˜ρ,˜m)=+,lim˜ρ+F1(˜ρ,˜m)=+,F1(ρ,˜m)>0,forγ<α<1; (2.8)

    and

    lim˜ρ0F1(˜ρ,˜m)=+,lim˜ρ+F1(˜ρ,˜m)=0,F1(ρ,˜m)<0,forα<γ. (2.9)

    Then, for any fixed ˜m=ρu>0, according to different regions of αR, we draw the graphs of F1(˜ρ,˜m). See Figure 1 below.

    Figure 1.  Plot of ˜ρF1(˜ρ,m).

    Integrating (2.2)2 over (0,x), we obtain

    F1(˜ρ(x),˜m)F1(ρ,˜m)=β˜mα+1x. (2.10)

    If β<0, by (2.10), F1(˜ρ,˜m) will decrease as the length of ducts increases, until it arrives at the minimum F1(ρ,˜m), no matter whether the upstream is supersonic (i.e., ρ<ρ) or subsonic (i.e., ρ>ρ). Therefore, we get the maximal length of ducts

    Lm=1β[u1α1α+aγeS0γ+αργ1uα1+(aγeS0)1αγ+1(ρu)(1α)(γ1)γ+1(1α11γ+α)] (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 β>0, α>1 and the upstream is supersonic (i.e., ρ<ρ), by (2.7), (2.10) and Figure 1 (i), we know that ˜ρ is decreasing as duct length x increases. Then, we get the maximal length of ducts

    Lm=1β(u1αα1aγeS0γ+αργ1uα1) (2.12)

    for a supersonic flow before it reaches the vacuum state. If γ<α<1 or α<γ, by (2.8)–(2.10) and Figure 1(ii) and (iii), ˜ρ 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 β>0, α<γ and the upstream is subsonic (i.e., ρ>ρ), combining (2.9), (2.10) with Figure 1(iii), ˜ρ is increasing as the duct length x increases. At the same time, F1(˜ρ,˜m) is increasing and approaching its supremum 0. Then, we get the maximal length of the duct Lm, which is still as shown in (2.12). When L>Lm, the fluid velocity is zero, that is, the fluid stagnates in a finite place. While, if γ<α<1 or α>1, again, by (2.7), (2.8), (2.10) and Figure 1(i) and (ii), ˜ρ is also increasing as the duct length x increases, but F1(˜ρ,˜m) goes to infinity as ˜ρ grows. In this case, although the fluid is slowing down, it does not stagnate at any finite place.

    Case 2: α=1.

    Now, (2.2) turns into

    {˜m=ρu,F2(˜ρ,˜m)x=β˜m2, (2.13)

    where

    F2(˜ρ,˜m)=˜m2ln˜ρ+aγeS0γ+1˜ργ+1.

    And, we get

    lim˜ρ0F2(˜ρ,˜m)=+,lim˜ρ+F2(˜ρ,˜m)=+, (2.14)
    F2(˜ρ,˜m)˜ρ=˜ρ(˜m2˜ρ2+aγeS0˜ργ1), (2.15)

    and

    F2(˜ρ(x),˜m)F2(ρ,˜m)=β˜m2x. (2.16)

    Similarly, the function F2(˜ρ(x),˜m) gets its minimum at point ˜ρ=ρ. If β<0, combining (2.14) with (2.16), we get the maximal length of ducts

    Lm=1β(γ+1)(lnρ1γu2aγeS0+aγeS0ργ1u21) (2.17)

    for a supersonic or subsonic flow before it gets choked. While, if β>0, the flow remains in its entrance state for any duct length L>0, no matter whether it is supersonic or subsonic.

    Case 3: α=γ.

    In this case, (2.1) changes into

    {˜m=ρu,F3(˜ρ,˜m)x=β˜m1γ, (2.18)

    where

    F3(˜ρ,˜m)=˜m21+γ˜ργ1+aγeS0ln˜ρ.

    Then, we have

    lim˜ρ0F3(˜ρ,˜m)=+,lim˜ρ+F3(˜ρ,˜m)=+, (2.19)
    F3(˜ρ,˜m)˜ρ=˜ργ(˜m2˜ρ2+aγeS0˜ργ1), (2.20)

    and

    F3(˜ρ(x),˜m)F3(ρ,˜m)=β˜m1γx. (2.21)

    Similar to the other two cases, the function F3(˜ρ(x),˜m) gets its minimum at point ˜ρ=ρ. If β<0, by (2.19) and (2.21), we obtain the maximal length of ducts

    Lm=1β(1+γ)[uγ+1+aγeS0(ρu)γ1ln(aγeS01ργ1u2)] (2.22)

    for a supersonic or subsonic flow before it gets choked. While, if β>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 ρρ>0,u>0,c=(aγeS0)1γ+1(ρu)γ1γ+1>0 and the duct length L<Lm, where Lm 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 (˜ρ(x),˜u(x),S0) which satisfies the following properties:

    1) 0<ρ<˜ρ(x)<ρ and c<˜u(x)<u, \quad if β<0 and ρ<ρ;

    2) 0<ρ<˜ρ(x)<ρ and u<˜u(x)<c, \quad if β<0 and ρ>ρ;

    3) 0<˜ρ(x)<ρ and c<u<˜u(x)<+, \quad if β>0 and ρ<ρ;

    4) 0<ρ<˜ρ(x)<+ and 0<˜u(x)<u<c, \quad if β>0 and ρ>ρ;

    5) ˜ρ˜u=ρu;

    6) (˜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.

    Remark 2.1. The following is worth pointing out:

    1) When β>0 and the upstream is supersonic, a vacuum can occur at the finite place for α>1, while a vacuum will never happen in any finite ducts for α1;

    2) When β>0 and the upstream is subsonic, fluid velocity can be zero at the finite place for α<γ, while the movement of fluid will never stop in the duct for αγ;

    3) For the case of β=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 ˜W(x)=(˜ρ(x),˜u(x),˜S(x)) (with ˜u(x)>0) as the background solution, which satisfies (1.4). For system (1.1), the corresponding simplification system has the form

    {ρt+ρxu+ρux=0,ut+uux+aγeSργ2ρx+aeSργ1Sx=βuα+1,St+uSx=0. (3.1)

    Let us denote W(t,x)=ˉW(t,x)+˜W(x), where ˉW=(ˉρ,ˉu,ˉS) is the perturbation around the background solution. Substituting

    ρ(t,x)=ˉρ(t,x)+˜ρ(x),u(t,x)=ˉu(t,x)+˜u(x),S(t,x)=ˉS(t,x)+S0 (3.2)

    into (3.1) yields

    {ˉρt+uˉρx+ρˉux+˜ρxˉu+˜uxˉρ+˜u˜ρx+˜ρ˜ux=0,ˉut+uˉux+ˉu˜ux+˜ux˜u+aγeSργ2(ˉρx+˜ρx)+aeSργ1ˉSx=β(ˉu+˜u)α+1,¯St+uˉSx=0. (3.3)

    Combining this with (1.4), system (3.3) can be simplified as

    {ˉρt+uˉρx+ρˉux=˜uxˉρ˜ρxˉu,ˉut+uˉux+aγeSργ2ˉρx+aeSργ1ˉSx=Θ(ρ,˜ρ,S,S0)eˉSˉρ˜ρx˜uxˉug(u,˜u)ˉu,¯St+uˉSx=0, (3.4)

    where Θ(ρ,˜ρ,S,S0)eˉSˉρ=aγ(eSργ2eS0˜ργ2) and g(u,˜u)ˉu=β[(ˉu+˜u)α+1˜uα+1]. g(u,˜u) can be represented as follows:

    g(u,˜u)=β(α+1)10(θˉu+˜u)αdθ.

    Obviously, system (3.4) can be expressed as the following quasi-linear equations:

    ˉWt+A(W)ˉWx+H(˜W)ˉW=0, (3.5)

    where

    A(W)=(uρ0aγeSργ2uaeSργ100u), (3.6)
    H(˜W)=(˜ux˜ρx0Θ(ρ,˜ρ,S,˜S)eˉS˜ρx˜ux+g(u,˜u)0000). (3.7)

    Through simple calculations, the three eigenvalues of system (3.5) are

    λ1(W)=uc,λ2(W)=u,λ3(W)=u+c, (3.8)

    where c=aγeS2ργ12. 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.



    [1] R. A. Adams, Sobolev spaces, New York: Academic Press, 1975.
    [2] A. Aftalion, J. Busca, W. Reichel, Approximate radial symmetry for overdetermined boundary value problems, Adv. Differential Equations, 4 (1999), 907–932.
    [3] L. Cavallina, G. Poggesi, T. Yachimura, Quantitative stability estimates for a two-phase Serrin-type overdetermined problem, 2021, arXiv: 2107.05889.
    [4] S. Dipierro, G. Poggesi, E. Valdinoci, A Serrin-type problem with partial knowledge of the domain, Nonlinear Anal., 208 (2021), 112330. http://dx.doi.org/10.1016/j.na.2021.112330 doi: 10.1016/j.na.2021.112330
    [5] A. Gilsbach, M. Onodera, Linear stability estimates for Serrin's problem via a modified implicit function theorem, Calc. Var., 60 (2021), 241. http://dx.doi.org/10.1007/s00526-021-02107-1 doi: 10.1007/s00526-021-02107-1
    [6] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Berlin, Heidelberg: Springer, 2001. http://dx.doi.org/10.1007/978-3-642-61798-0
    [7] R. Magnanini, Alexandrov, Serrin, Weinberger, Reilly: symmetry and stability by integral identities, Bruno Pini Mathematical Analysis Seminar, 8 (2017), 121–141. http://dx.doi.org/10.6092/issn.2240-2829/7800 doi: 10.6092/issn.2240-2829/7800
    [8] R. Magnanini, G. Poggesi, On the stability for Alexandrov's Soap Bubble theorem, JAMA, 139 (2019), 179–205. http://dx.doi.org/10.1007/s11854-019-0058-y doi: 10.1007/s11854-019-0058-y
    [9] R. Magnanini, G. Poggesi, Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities, Indiana Univ. Math. J., 69 (2020), 1181–1205. http://dx.doi.org/10.1512/iumj.2020.69.7925 doi: 10.1512/iumj.2020.69.7925
    [10] R. Magnanini, G. Poggesi, Nearly optimal stability for Serrin's problem and the Soap Bubble theorem, Calc. Var., 59 (2020), 35. http://dx.doi.org/10.1007/s00526-019-1689-7 doi: 10.1007/s00526-019-1689-7
    [11] R. Magnanini, G. Poggesi, An interpolating inequality for solutions of uniformly elliptic equations, In: Geometric properties for parabolic and elliptic PDE's, Cham: Springer. http://dx.doi.org/10.1007/978-3-030-73363-6_11
    [12] R. Magnanini, G. Poggesi, The location of hot spots and other extremal points, Math. Ann., 2021, in press. http://dx.doi.org/10.1007/s00208-021-02290-8
    [13] Y. Okamoto, M. Onodera, Stability analysis of an overdetermined fourth order boundary value problem via an integral identity, J. Differ. Equations, 301 (2021), 97–111. http://dx.doi.org/10.1016/j.jde.2021.08.017 doi: 10.1016/j.jde.2021.08.017
    [14] G. Poggesi, Radial symmetry for p-harmonic functions in exterior and punctured domains, Appl. Anal., 98 (2019), 1785–1798. http://dx.doi.org/10.1080/00036811.2018.1460819 doi: 10.1080/00036811.2018.1460819
    [15] G. Poggesi, The Soap Bubble Theorem and Serrin's problem: quantitative symmetry, PhD Thesis, Università di Firenze, 2019, arXiv: 1902.08584.
    [16] J. Scheuer, Stability from rigidity via umbilicity, 2021, arXiv: 2103.07178.
    [17] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304–318. http://dx.doi.org/10.1007/BF00250468 doi: 10.1007/BF00250468
    [18] H. F. Weinberger, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal., 43 (1971), 319–320. http://dx.doi.org/10.1007/BF00250469 doi: 10.1007/BF00250469
  • This article has been cited by:

    1. Michele Bedard-Gilligan, Elizabeth Lehinger, Sarah Cornell-Maier, Ash Holloway, Lori Zoellner, Effects of Cannabis on PTSD Recovery: Review of the Literature and Clinical Insights, 2022, 9, 2196-2952, 203, 10.1007/s40429-022-00414-x
    2. Anya Ragnhildstveit, Miriam Kaiyo, Matthew Brian Snyder, Laura Kate Jackson, Alex Lopez, Chasity Mayo, Alyssa Claire Miranda, River Jude August, Paul Seli, Reid Robison, Lynnette Astrid Averill, Cannabis-assisted psychotherapy for complex dissociative posttraumatic stress disorder: A case report, 2023, 14, 1664-0640, 10.3389/fpsyt.2023.1051542
    3. Michael A. Incze, A. Taylor Kelley, Phillip M. Singer, Heterogeneous State Cannabis Policies, 2021, 326, 0098-7484, 2363, 10.1001/jama.2021.21182
    4. Mona Khoury, Idan Cohen, Gil Bar-Sela, “The Two Sides of the Same Coin”—Medical Cannabis, Cannabinoids and Immunity: Pros and Cons Explained, 2022, 14, 1999-4923, 389, 10.3390/pharmaceutics14020389
    5. Gregory Malik Burnett, David A. Gorelick, Kevin P. Hill, Policy Ahead of the Science, 2022, 45, 0193953X, 347, 10.1016/j.psc.2022.05.002
    6. Grace Clouse, Samantha Penman, Michael Hadjiargyrou, David E. Komatsu, Panayotis K. Thanos, Examining the role of cannabinoids on osteoporosis: a review, 2022, 17, 1862-3522, 10.1007/s11657-022-01190-x
    7. Nitsa Nacasch, Chen Avni, Paz Toren, Medical cannabis for treatment-resistant combat PTSD, 2023, 13, 1664-0640, 10.3389/fpsyt.2022.1014630
    8. Amy L. Haycraft, Cannabis update: Anxiety disorders and post-traumatic stress disorder, 2023, Publish Ahead of Print, 2327-6924, 10.1097/JXX.0000000000000864
    9. Bridgid M. Conn, Whitney A. Brammer, Susie Choi, Ekaterina V. Fedorova, Janna Ataiants, Stephen E. Lankenau, Carolyn F. Wong, Mental and Physical Health-Related Cannabis Motives Mediate the Relationship between Childhood Trauma and Problematic Cannabis Use over Time among Emerging Adult Cannabis Users, 2024, 59, 1082-6084, 193, 10.1080/10826084.2023.2267111
    10. Lisa Burback, Suzette Brémault-Phillips, Mirjam J. Nijdam, Alexander McFarlane, Eric Vermetten, Treatment of Posttraumatic Stress Disorder: A State-of-the-art Review, 2024, 22, 1570159X, 557, 10.2174/1570159X21666230428091433
    11. 2025, 9780443235658, 471, 10.1016/B978-0-443-23565-8.09996-8
    12. Ben Senator, Mafalda Pardal, Liesbeth Vandam, Evidence synthesis of medical cannabis research: current challenges and opportunities, 2024, 0940-1334, 10.1007/s00406-024-01893-x
    13. Aaron S. Wolfgang, Charles W. Hoge, Cannabis and Cannabinoids for Pain and Posttraumatic Stress Disorder in Military Personnel and Veterans, 2023, 80, 2168-622X, 869, 10.1001/jamapsychiatry.2023.1685
    14. Hymie Anisman, Shawn Hayley, Alexander Kusnecov, 2025, 9780443235658, 293, 10.1016/B978-0-443-23565-8.00010-7
    15. Michael T Lynskey, Anne K Schlag, Alkyoni Athanasiou-Fragkouli, David Badcock, David J Nutt, Characteristics of and 3-month health outcomes for people seeking treatment with prescribed cannabis: Real-world evidence from Project Twenty21, 2023, 9, 2050-3245, 10.1177/20503245231167373
    16. Luke J Ney, Wole Akosile, Chris Davey, Louise Pitcher, Kim L Felmingham, Leah M Mayo, Matthew N Hill, Esben Strodl, Challenges and considerations for treating PTSD with medicinal cannabis: the Australian clinician’s perspective, 2023, 16, 1751-2433, 1093, 10.1080/17512433.2023.2276309
    17. Elizabeth A. Cairns, Melissa J. Benson, Miguel A. Bedoya-Pérez, Sara L. Macphail, Adith Mohan, Rhys Cohen, Perminder S. Sachdev, Iain S. McGregor, Medicinal cannabis for psychiatry-related conditions: an overview of current Australian prescribing, 2023, 14, 1663-9812, 10.3389/fphar.2023.1142680
    18. Mariyam Humayun, Jose I. Suarez, Vishank A. Shah, Neurological Complications of Cannabinoids, 2024, 44, 0271-8235, 430, 10.1055/s-0044-1787570
    19. Brian Yagi, Stan Veuger, Brian J Miller, Paul Larkin, Lessons for cannabinoid regulation from electronic nicotine delivery system product regulation, 2024, 2, 2976-5390, 10.1093/haschl/qxae101
    20. Scott D. Lane, Lokesh R. Shahani, Rodrigo Machado-Vieira, Jair C. Soares, Retrospective examination of cannabis vs. other substance misuse: Associations with 30-day readmission and global assessment of functioning in hospitalized patients with serious mental illness, 2024, 4, 27725987, 100167, 10.1016/j.psycom.2024.100167
    21. Luke J Ney, Wole Akosile, Chris Davey, Louise Pitcher, Kim Felmingham, Leah Mayo, Matthew Hill, Esben Strodl, Australian clinical feasibility considerations for treatment of PTSD with cannabinoid-augmented exposure therapy, 2023, 0004-8674, 10.1177/00048674231216587
    22. L. Riley Gournay, Morgan L. Ferretti, Anna‐Marie Nguyen, Sarah Bilsky, Grant S. Shields, Eric Mann, Parker Williams, Sydney Woychesin, Marcel Bonn‐Miller, Ellen W. Leen‐Feldner, The effects of acute versus repeated cannabidiol administration on trauma‐relevant emotional reactivity: A double‐blind, randomized, placebo‐controlled trial, 2024, 0894-9867, 10.1002/jts.23072
    23. Marion McNabb, Katherine A. Durante, Sarah Trocchio, David J. Ritter, Randal MacCaffrie, Ann Brum, Stephen Mandile, Steven White, Self-reported Medicinal Cannabis Use as an Alternative to Prescription and Over-the-counter Medication Use Among US Military Veterans, 2023, 45, 01492918, 562, 10.1016/j.clinthera.2023.04.003
    24. Guido Cammà, Monika P. Verdouw, Pim B. van der Meer, Lucianne Groenink, Albert Batalla, Therapeutic potential of minor cannabinoids in psychiatric disorders: A systematic review, 2025, 91, 0924977X, 9, 10.1016/j.euroneuro.2024.10.006
    25. Petter Grahl Johnstad, Unhealthy behaviors associated with mental health disorders: a systematic comparative review of diet quality, sedentary behavior, and cannabis and tobacco use, 2024, 11, 2296-2565, 10.3389/fpubh.2023.1268339
    26. Aleksandra Wrona, Amy C. Justice, Janet P. Tate, Christopher T. Rentsch, Kirsha S. Gordon, Farah Kidwai-Khan, Michael J. Silverberg, Derek D. Satre, Vincent C. Marconi, Suzanne M. Ingle, Jonathan A. C. Sterne, Matthias Cavassini, Kendall Bryant, Kathleen A. McGinnis, Cannabis Use and Self-Reported Bothersome Symptoms in People with HIV, 2025, 25780026, 10.26828/cannabis/2025/000269
    27. Mitchell L. Doucette, Dipak Hemraj, D. Luke Macfarlan, Junella Chin, Emily Fisher, The Cost Effectiveness of Adjunctive Medical Cannabis Therapy in the Treatment of Moderate Post-Traumatic Stress Disorder, 2025, 1173-2563, 10.1007/s40261-025-01424-z
    28. Laurent Perrault‐Sequeira, Michael Pugliese, Rachael MacDonald‐Spracklin, Jennifer Xiao, Stephen McCarthy, Daniel T. Myran, Cannabis involvement in posttraumatic stress disorder emergency department visits after cannabis legalization, 2025, 1055-0496, 10.1111/ajad.70014
    29. Russ J. Scott, Ian A. Scott, Medicinal cannabis: is current use clinically justified?, 2025, 1444-0903, 10.1111/imj.70094
    30. Bertha K. Madras, Paul J. Larkin, Rescheduling Cannabis—Medicine or Politics?, 2025, 2168-622X, 10.1001/jamapsychiatry.2025.1116
    31. Jordan P. Davis, Shaddy K. Saba, Daniel Leightley, Eric R. Pedersen, John Prindle, Carl A. Castro, Bistra Dilkina, Emily Dworkin, Jonathan Cantor, Angeles Sedano, Daily associations between posttraumatic stress disorder, cannabis use, and negative affect among veterans, 2025, 01651781, 116626, 10.1016/j.psychres.2025.116626
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2523) PDF downloads(224) Cited by(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog