Processing math: 98%
Research article Special Issues

Persistent and carcinogenic polycyclic aromatic hydrocarbons in the north-western coastal marine environment of India

  • Received: 16 March 2021 Accepted: 28 April 2021 Published: 10 May 2021
  • The carcinogenic and endocrine-disrupting PAHs were investigated in surface water of the north-western creeks of India. The concentrations of ΣPAHs were found to vary in the range of 114.32-347.04 μg L-1, (mean 224.78 ± 8.85 μg L-1), out of which 49.12% contribution is due to ΣC-PAHs. The assessment of toxicity and biological risk arising due to individual C-PAHs was made by calculating their toxic equivalent quantity. The level of individual C-PAHs was reported exceeding the final chronic values, Canadian water quality guideline values and Netherlands maximum permissible concentration values set for the protection of aquatic life. The mean BaP concentration (10.32 ± 2.75 μg L-1) was above the European Directive 2008/105/EC Environmental Quality Standards (EQS) value; while the sum of BkF + BbF (26.76 μg L-1) and BghiP + InP (19.59 μg L-1) were significantly higher than that set by the EQS. The results of the present study will help in understanding the global distribution and fate of PAHs which is required for implementing the necessary steps towards mitigation of the ecotoxicological risk arising due to the existence of such contaminants in the aquatic environment across the world.

    Citation: P. U. Singare, J.P. Shirodkar. Persistent and carcinogenic polycyclic aromatic hydrocarbons in the north-western coastal marine environment of India[J]. AIMS Environmental Science, 2021, 8(2): 169-189. doi: 10.3934/environsci.2021012

    Related Papers:

    [1] Hirotada Honda . Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12(1): 25-57. doi: 10.3934/nhm.2017002
    [2] Seung-Yeal Ha, Shi Jin, Jinwook Jung . A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks and Heterogeneous Media, 2019, 14(2): 317-340. doi: 10.3934/nhm.2019013
    [3] Hirotada Honda . On Kuramoto-Sakaguchi-type Fokker-Planck equation with delay. Networks and Heterogeneous Media, 2024, 19(1): 1-23. doi: 10.3934/nhm.2024001
    [4] Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li . Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks and Heterogeneous Media, 2017, 12(1): 1-24. doi: 10.3934/nhm.2017001
    [5] Seung-Yeal Ha, Yongduck Kim, Zhuchun Li . Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks and Heterogeneous Media, 2014, 9(1): 33-64. doi: 10.3934/nhm.2014.9.33
    [6] Seung-Yeal Ha, Hansol Park, Yinglong Zhang . Nonlinear stability of stationary solutions to the Kuramoto-Sakaguchi equation with frustration. Networks and Heterogeneous Media, 2020, 15(3): 427-461. doi: 10.3934/nhm.2020026
    [7] Tingting Zhu . Synchronization of the generalized Kuramoto model with time delay and frustration. Networks and Heterogeneous Media, 2023, 18(4): 1772-1798. doi: 10.3934/nhm.2023077
    [8] Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue . Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks and Heterogeneous Media, 2018, 13(2): 323-337. doi: 10.3934/nhm.2018014
    [9] Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun . Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks and Heterogeneous Media, 2013, 8(4): 943-968. doi: 10.3934/nhm.2013.8.943
    [10] Vladimir Jaćimović, Aladin Crnkić . The General Non-Abelian Kuramoto Model on the 3-sphere. Networks and Heterogeneous Media, 2020, 15(1): 111-124. doi: 10.3934/nhm.2020005
  • The carcinogenic and endocrine-disrupting PAHs were investigated in surface water of the north-western creeks of India. The concentrations of ΣPAHs were found to vary in the range of 114.32-347.04 μg L-1, (mean 224.78 ± 8.85 μg L-1), out of which 49.12% contribution is due to ΣC-PAHs. The assessment of toxicity and biological risk arising due to individual C-PAHs was made by calculating their toxic equivalent quantity. The level of individual C-PAHs was reported exceeding the final chronic values, Canadian water quality guideline values and Netherlands maximum permissible concentration values set for the protection of aquatic life. The mean BaP concentration (10.32 ± 2.75 μg L-1) was above the European Directive 2008/105/EC Environmental Quality Standards (EQS) value; while the sum of BkF + BbF (26.76 μg L-1) and BghiP + InP (19.59 μg L-1) were significantly higher than that set by the EQS. The results of the present study will help in understanding the global distribution and fate of PAHs which is required for implementing the necessary steps towards mitigation of the ecotoxicological risk arising due to the existence of such contaminants in the aquatic environment across the world.



    The theoretical study of weakly coupled limit cycle oscillators is being actively developed in several research fields. For example, in statistical physics, various models are being developed, whereas in network science, synchronization on complex networks is attracting attention. As we state later, mathematical analysis of this field is being promoted, especially by those who are concerned with functional equations. Furthermore, this phenomenon is applied to various areas of engineering, including neural networks, bio-sciences, and network engineering[9].

    Limit cycle oscillators, which are also called nonlinear oscillators, are different from harmonic oscillators, whose limit cycle is vulnerable to the perturbation forces from outside. The synchronization phenomenon is another feature of coupled limit cycle oscillators under specific conditions.

    As is well known, such a phenomenon was first discovered by Huygens in the 17th century, who devised the pendulum clock for navigation officers. The physical formulation of this phenomenon was rigorously discussed later in the 1960's. In 1967, Winfree[37] proposed an attractive formulation, in which he successfully and rigorously defined the phase of the oscillator. He also revealed the region of synchronization in the sense of the phase space.

    Based on Winfree's contribution, Kuramoto[19] proposed a simplified but more sophisticated model, which is called the Kuramoto model. The features of his approach are phase reduction and mean field approximation. His discussion begins with the usual dynamical system:

    dXdt=F(X),

    where X is the n-dimensional vector, and F is the n-dimensional vector valued function. Based on Winfree's theory, Kuramoto defined the phase of this system as that satisfying

    dϕdt=ω,

    where ω=2π/T0 is the frequency of the oscillator with period T0. From this definition, he derived the temporal behavior of the phase as

    dϕdt=ϕF(X),

    which is the start point of phase reduction. In an N-oscillator system, he described the temporal evolution of the phase (or disturbance of phase) of each oscillator:

    dϕjdt=ωj+Nk=1Kjksin(ϕjϕk)(j=1,2,N), (1.1)

    where ϕj(j=1,2,,N) is the phase or phase disturbance of the j-th oscillator, ωj, the natural frequency (for the definition of the natural frequency, see p.67 in[19], for instance), and Kjk, the coupling strength between the jth and kth oscillators.

    Kuramoto[17][19] further sophisticated (1.1) by applying the mean field approximation and the assumption Kjk=K/N, which means that all oscillators couple with uniform strength (Kuramoto called this model as global coupling[20]):

    {dϕjdt=ωj+Krsin(ηϕj)(j=1,2,N),rexp(iη)=1NNj=1exp(iϕj), (1.2)

    where r=r(t) is the order parameter, and η is the average phase. Note that i=1 hereafter. As is well known, r[0,1] measures the coherence strength of the oscillators.

    Since this model is sufficiently simple for rigorously analyzing and simulating on computers, numerous investigations have been conducted on it. For instance, Daido[8] derived a model that replaces the term sinθ by a more general function, and Bonilla[4] revealed stability with the aid of asymptotic expansion. Recently, Li[26] considered the Kuramoto-type model with intrinsic frustrations.

    As for the stability analysis, the work by Strogatz and Mirollo[34] is the first one, which concerns the spectrum of the incoherent state as the number of oscillators tends to infinity. Later, Crawford[6][7] applied the center manifold reduction to verify the stability of incoherence in detail.

    Due to the limitation of space, we refer the reader to the survey by Acebrón[3] of this research area.

    From the perspective of network science, it is interesting to generalize the network topology and distribution of coupling strength. Ichinomiya[14] proposed a model on random networks, and Nakao[29] numerically analyzed a model on a complex network.

    Although (1.2) is a system of ordinary differential equations, adding white noise to it makes it possible to apply partial differential equation-based analysis. We consider

    {dϕjdt=ωj+Krsin(ηϕj)+ξj(t)      V1(ϕj,t,ωj)+ξj(t)(j=1,2,N),rexp(iη)=1NNj=1exp(iϕj), (1.3)

    where {ξj(t)}Nj=1 are the independent Wiener processes satisfying

    <ξj(t)>=0,  <ξj(t)ξk(τ)>=2Dδ(tτ)δjk,

    <> stands for the mean value, D>0, the diffusion coefficient, δ(), the Dirac's delta function, and δjk, the Kronecker's delta. In virtue of the theory of stochastic processes (see Remark 1 below), the Fokker-Planck equation, which describes the temporal behavior of the probability density of particles subject to (1.3)[10][30][33], is written as

    {ϱt+θ[(ω+Krsin(ηθ))ϱ]D2ϱθ2=0,rexp(iη)=R2π0exp(iθ)ϱ(θ,t;ω)g(ω)dϕdω, (1.4)

    where ϱ(θ,t;ω) is the probability distribution function of the oscillators' phase θ at time t with natural frequency ω, and g(ω) is the probability distribution function of ω. The independent variables θ and t represent the phase of oscillators and time, respectively, and we regard ω as a parameter.

    Remark 1. Let X(t) be a Markovian process defined on t>0, and {Kj(x)}j=1 be a series of its intensity defined by

    Kj(x)=limτ0(X(τ)X(0))j/τ.

    Then, the probability distribution function w(x,t) of X(t) at time t satisfies[33]

    w(x,t)t=j=11j!(x)j{Kj(x)w(x,t)}. (1.5)

    A Markovian process satisfying

    Kj(x)=0  (j=3,4,)

    is called to be continuous. One example of this type of stochastic process is the one driven by the Brownian motion, like (1.3). In this case, (1.5) becomes

    w(x,t)t=x[K1(x)w(x,t)]+122x2[K2(x)w(x,t)],

    which is the Fokker-Planck equation. Applying this argument to (1.3), where K1=V1 and K2=2D, the corresponding Fokker-Planck equation is written as

    ϱt+θ(V1ϱ)D2ϱθ2=0,

    which corresponds to (1.4)1 (hereafter, we represent the ith equation of (a.b) as (a.b)i).

    By substituting (1.4)2 into (1.4)1, we have a single integro-differential equation with suitable boundary and initial conditions:

    {ϱt+ωϱθ+Kθ[ϱ(θ,t;ω)R2π0sin(ϕθ)ϱ(ϕ,t;ω)g(ω)dϕdω]       D2ϱθ2=0θ(0,2π),t>0,ωR,iϱθ|θ=0=iϱθ|θ=2π(i=1,2),t>0,ωR,ϱ|t=0=ϱ0θ(0,2π),ωR, (1.6)

    which is the so-called Kuramoto-Sakaguchi equation[24]. This approach enables macroscopic analysis when a system consists of numerous oscillators. As we state later, (1.6) with D=0 has also been discussed in past arguments.

    Kuramoto also presented a direction to take the spatial validity of coupling strength into account in his original model, which he called the non-local coupling model. In it, the strength of the connection between oscillators depends on the distance between them[21]. Due to the varying strength of connections, it was shown that characteristic patterns, such as a chimera pattern, emerge[21][22]. Numerical studies of the chimera state have also been conducted, mainly by Abrams and Strogatz[1][2].

    In spite of numerous contributions concerning numerical simulations, there have been few studies regarding the mathematical analysis of this model. In this paper, we discuss the existence and uniqueness of the global-in-time solution to this model. We also discuss the nonlinear stability of the trivial stationary solution and existence of the vanishing diffusion limit. We note that the vanishing diffusion limit is discussed in the function spaces of higher order derivative than that by Ha and Xiao's[11] by applying a different approach.

    The remainder of this paper is organized as follows. In the next section, we formulate the problem of the phase reduction of non-local coupling oscillators, including both with and without diffusion. In Section 3, we introduce related work. In Sections 4 and 5, we are concerned with the local and global-in-time solvability of the problem and the stability of the stationary incoherent solution, respectively. We discuss the existence of the solution in the vanishing diffusion limit in Section 6, and provide a conclusion and discuss remaining issues in Section 7.

    In this section, we formulate the problem to be considered.We begin the discussion with the temporal evolution of the phase with additive noise under non-local coupling [15][20][31]:

    dϕdt(x,t)=ω+RG(xy)Γ(ϕ(x,t)ϕ(y,t))dy+ξ(x,t), (2.1)

    where x and y stand for the location of each oscillator; Γ(), the phase coupling function which is periodic with respect to its argument and depend only on the difference of each oscillator's phase, and G(), its strength; ω, the natural frequency of each oscillator, and ξ(x,t), the Wiener process satisfying:

    <ξ(x,t)ξ(y,τ)>=2Dδ(xy)δ(tτ)

    with D being the diffusion coefficient. Kuramoto mentioned that the non-local coupling model encompasses the original Kuramoto model as a specific case of (2.1) when G(x) is a constant (p.132 in[20], [21]).

    It is also to be noted that sin(ϕjϕk) in (1.1) is the first Fourier mode approximation of the periodic function Γ(ϕjϕk)[19]. It is possible to adopt sin() for the non-local coupling case as in[16], but in many cases, they apply Γ() for the non-local coupling model. By regarding ω and ϕ as random variables, the average of the second term in the right-hand side is written as[15][16][18]

    RG(xy)dyRg(ω)dω2π0Γ(ϕϕ)ϱ(ϕ,t;y,ω)dϕ,

    where g() is the probability distribution function of the natural frequency, and ϱ=ϱ(,t;x,ω) is the probability distribution function of the phase at time t, with location x and natural frequency ω of each oscillator. We replace the second term in the right-hand side of (2.1) with the representation above to obtain:

    dϕdt=ω+RG(xy)dyRg(ω)dω2π0Γ(ϕϕ)ϱ(ϕ,t;y,ω)dϕ+ξ(x,t)    V2(ϕ,t,x,ω)+ξ(x,t). (2.2)

    If we consider the case that G(x) is a constant K in (2.1), the average of the second term in the right-hand side is

    KRg(ω)dω2π0Γ(ϕϕ)ϱ(ϕ,t;ω)dϕ,

    and (2.2) becomes

    dϕdt=ω+KRg(ω)dω2π0Γ(ϕϕ)ϱ(ϕ,t;ω)dϕ+ξ(x,t),

    which corresponds to (1.3) as a infinite population limit of it.

    Note that (2.2) is the mean field approximation to (2.1), in the sense that we replace the second term of the right-hand side of (2.1) with its average. However, while (1.4)2 is derived by taking the infinite limit of oscillators' population, this representation of the average is theoretically exact.

    By tracing the same argument as we derived (1.4), the Fokker-Planck equation corresponding to (2.2) is written as (hereafter we denote the phase by θ)

    ϱt+θ(V2ϱ)D2ϱθ2=0.

    Together with suitable initial and boundary conditions, the problem corresponding to (2.2) is written as[15][31].

    {ϱt+θ(ωϱ)+θ(F[ϱ,ϱ])D2ϱθ2=0                                    θ(0,2π),t>0,(x,ω)R2,iϱθi|θ=0=iϱθi|θ=2π(i=0,1),t>0,(x,ω)R2,ϱ|t=0=ϱ0  θ(0,2π),(x,ω)R2, (2.3)

    where

    F[ϱ1,ϱ2]ϱ1(θ,t;x,ω)RG(xy)dyRg(ω)dω2π0Γ(θϕ)ϱ2(ϕ,t;y,ω)dϕ. (2.4)

    It is to be noted that, in[15], [21] and[31], they deal with problem (2.3) with F[ϱ1,ϱ2] replaced with

    ˜F[ϱ1,ϱ2]ϱ1(θ,t;x)RG(xy)dy2π0Γ(θϕ)ϱ2(ϕ,t;y)dϕ,

    which corresponds to the case g(ω)=δ(ωω). This means that all the oscillators have the same natural frequency, and ϱ does not depend on ω. In other words, ω is not regarded as a random variable. However, it is generally reasonable to apply (2.3)-(2.4) as Kawamura[16] did, as in the original Kuramoto model ((1.4) or (1.6)). Therefore, we employ (2.3) together with (2.4) in this paper.

    In (2.3), the unknown function is ϱ, and the independent variables are θ and t, which stand for the phase and time, respectively. We regard x and ω as parameters. We also use the notations

    F(k)[ϱ1,ϱ2]    ϱ1(θ,t;x,ω)RG(xy)dyRg(ω)dω2π0Γ(k)(θϕ)ϱ2(ϕ,t;y,ω)dϕ                                                                                         (k=1,2,).

    Note that we denote the j-th derivative of Γ(θ) by Γ(j)(j=1,2,), especially the first derivative by Γ(θ). Obviously, Γ(0)=Γ.

    As in the original Kuramoto model, the vanishing diffusion case is worth considering:

    {ϱt+θ(ωϱ)+θ(F[ϱ,ϱ])=0θ(0,2π),t>0,(x,ω)R2,iϱθi|θ=0=iϱθi|θ=2π  (i=0,1),t>0,(x,ω)R2,ϱ|t=0=ϱ0  θ(0,2π),(x,ω)R2. (2.5)

    It is obvious that ˉϱ=1/2π is the trivial stationary solution to (1.6), (2.3) and (2.5). Moreover, on the basis of appropriate assumptions on ϱ0, g and G, the following properties of ϱ will be derived (see, for instance, Lemma 2.1 in [11], Lemmas 1.1-1.2 in[24] and Lemma 4.3 of this paper):

    ϱ(θ,t;x,ω)0  θ(0,2π),t(0,T),(x,ω)R2,2π0ϱ(θ,t;x,ω)dθ=1  t(0,T),(x,ω)R2

    for arbitrary T>0, which are natural as the properties of the probability distribution function. In this paper, we discuss the solvability and some stability properties of (2.3), as well as the convergence of the solution of (2.3) to that of (2.5) as D tends to zero.

    Mathematical arguments concerning the solvability of the Kuramoto-Sakaguchi equation (1.6), which corresponds to the original Kuramoto model (1.3), was first presented by Lavrentiev et al.[24][25]. In their former work[24], they constructed the classical global-in-time solution when the support of g(ω) is compact.

    Later, they removed this restriction[25] by applying the a-priori estimates derived from the energy method. They also studied the regularity of the unknown function with respect to ω. Concenring the stability, a pioneering work was conducted by Strogatz and Mirollo[34], who were concerned with the linear stability of the trivial stationary solution ˉϱ=1/2π. Through the investigation of the spectrum of the linearized operator, they verified the existence of the critical coupling strength over which the coherent state becomes stable. Recently, Ha and Xiao[11] discussed the nonlinear stability of ˉϱ and convergence of the solution as D tends to zero. However, their estimate remained in the space L with respect to θ, as we discuss in detail in Section 6. They also verified the instability of ˉϱ when the support of g(ω) is sufficiently narrow[12].

    For the case of vanishing diffusion D=0 in (1.6), Chiba[5] argued the nonlinear stability of the trivial stationary solution under the assumption of unbounded support of g(ω).

    Concerning the non-local coupling model, however, there are no mathematical arguments as far as we know.

    In this section, we discuss the global-in-time solvability of (2.3). We first prepare the definitions of function spaces.

    In this subsection, we define the function spaces used throughout this paper. Let T>0, and G be an open set in R. Hereafter, L2(G) stands for a set of square-integrable functions defined on G, equipped with the norm

    uG|u(x)|2dx.

    The inner product is defined by

    (u1,u2)Gu1(x)¯u2(x)dx,

    where ˉz denotes the complex conjugate of zC.

    For simplicity, we denote the L2-norm of a function f(θ,t;x,ω) with respect to θ merely by f or f(,t;x,ω).

    Hereafter, let us use the notation Ω(0,2π) for simplicity.

    By C(G) and Ck(G)(kN{+}), we mean the spaces of real continuous and k-times continuously differentiable functions on Ω, respectively. The notation C0(G) denotes a set of C(G) functions with a compact support in G.

    For a Banach space E with the norm E, we denote the space of E-valued measurable functions u(t) on the interval (a,b) by Lp(a,b;E), whose norm is defined by

    uLp(a,b;E){(bau(t)pEdt)1/p  (1p<),esssupatbu(t)E  p=.

    Likewise, we denote by C(a,b;E) (and by Ck(a,b;E)) the space of continuous functions (resp. k-times continuously differentiable functions) from (a,b) into E.

    Subject to the definition by Temam[35], we say, for a fixed parameter ωR, a 2π-periodic function

    u(θ;ω)=n=an(ω)einθ,

    which is expanded in the Fourier series, belongs to the Sobolev space Hm(Ω)(m>0) when it satisfies

    u(;ω)2mn=(1+|n|2)m|an(ω)|2<.

    Due to the definition of the Fourier series, the Fourier coefficients an(n=±1,±2,) of a function u are defined by

    an(ω)=12πΩu(θ;ω)einθdθ.

    Note that in case mN, the norm above is equal to the usual Sobolev norm

    u(;ω)2Wm2(Ω)=kmkuθk(;ω)2L2(Ω).

    We also introduce the notation Cx,ω(R2) to denote a set of functions defined on R2, which are infinitely smoothly differentiable with respect to both x and ω. Let us introduce the following notations:

           ¯Hm{u(θ;x,ω)Cx,ω(R2;Hm)},  L(1)1{u(;x,ω)L1(Ω)|u0,Ωu(θ;x,ω)dθ=1(x,ω)R2 },L(1)1(T){u(,t;x,ω)L1(Ω)    |u0,Ωu(θ,t;x,ω)dθ=1t(0,T),(x,ω)R2 },

    where T>0 is an arbitrary number. In addition, we use notations |||u|||msupx,ωu(;x,ω)m and |||u||||||u|||0 for brevity. Hereafter, c's represent constants in the estimate of some quantities. When we denote c(t) with suffixes, it depends on t. For simplicity, we hereafter use notations f(j,k)(θ)j(t)kf(j,k=0,1,2,) for a function f=f(θ,t) in general.

    First, we state the existence and uniqueness of the local-in-time solution to problem (2.3).

    Theorem 4.1. Let us assume mN, and the following issues:

    (ⅰ) Γ(θ) is 2π-periodic, and satisfies |Γ(k)(θ)|Ck(k=0,1,2,,2m);

    (ⅱ) 2π0Γ(k)(θ)dθ=0(k=0,1,2,,2m);

    (ⅲ)Γ(k)=˜Ck<+(k=0,1,2,,2m);

    (ⅳ) g(ω)L(1)1;

    () G(x)L1(R)L2(R)C(R) satisfies G0, RG(x)dxM0;

    () ϱ0¯H2mL(1)1.

    Then, there exists a certain T>0 and a solution ϱ(θ,t;x,ω) to (2.3) on (0,T) such that

    ϱV2m(T),

    where

    V2m(T){ϱL(0,T;¯H2m)C1(0,T;¯H2m2)Cm(0,T;¯H0)L(1)1(T)|ϱ(1,0)L2(0,T;¯H2m)}.

    Before proceeding to the proof of Theorem 4.1, we prepare the following lemmas.

    Lemma 4.2. Let f=f(θ,t;x,ω) be a function satisfying |||f(t)|||<t>0 in general. Then, the following estimates hold:

    (ⅰ) If f(θ,t;x,ω)0 and Ωf(θ,t;x,ω)dθ=1 for each (t,x,ω), then

    supx,ω|RG(xy)dyRg(ω)dωΩΓ(k)(θϕ)f(ϕ,t;y,ω)dϕ|CkM0(k=0,1,2,,2m),

    (ⅱ)

    supx,ω|RG(xy)dyRg(ω)dωΩΓ(k)(θϕ)f(ϕ,t;y,ω)dϕ|˜CkM0|||f(t)|||(k=0,1,2,,2m),

    (ⅲ)

    supx,ωRG(xy)dyRg(ω)dωΩΓ(k)(θϕ)f(ϕ,t;y,ω)dϕ2π˜CkM0|||f(t)|||  (k=0,1,2,,2m).

    Proof. Here we only show the proof of (ⅱ). By virtue of the Schwarz inequality, we have

    supx,ω|RG(xy)dyRg(ω)dωΩΓ(k)(θϕ)f(ϕ,t;y,ω)dϕ|supx,ωf(,t;x,ω)Γ(k)(θ)|RG(xy)dy||g(ω)dω||||f(t)|||Γ(k)supx|RG(xy)dy|˜CkM0|||f(t)|||.

    The estimate (ⅲ) is obtained in a similar manner, and statement (ⅰ) is obtained easily.

    Hereafter, for arbitrary T>0, we use the notation

    V2m()(T){ϱL(0,T;¯H2m)C1(0,T;¯H2m2)Cm(0,T;¯H0)|ϱ(1,0)L2(0,T;¯H2m)}.

    Lemma 4.3. For an arbitrary T>0, if there exists a solution ϱ to (2.3) that belongs to V2m()(T), then the following issues hold:

    (ⅰ) ϱ(θ,t;x,ω)0a.e.onθΩ,t(0,T),(x,ω)R2

    (ⅱ) Ωϱ(θ,t;x,ω)dθ=1  t(0,T),(x,ω)R2.

    Remark 2. If m in Theorem 4.1 is large enough (m2), then the first statement in Theorem 4.1 holds in the pointwise sense due to the Sobolev's embedding theorem and maximum principle (see[11]).

    Proof. We verify the statement by using the Stampacchia's truncation method. Let us define

    ϱ+(|ϱ|+ϱ)/20,  ϱ(|ϱ|ϱ)/20.

    It is obvious that ϱ+ and ϱ stand for the positive and negative parts of ϱ, respectively, satisfying ϱ=ϱ+ϱ. Then, by multiplying (2.3)1 by ϱ and integrating over Ω, we obtain

    12ddtϱ(,t;x,ω)2+Dϱθ(,t;x,ω)2c41ϱ(,t;x,ω)2.

    Here we used the estimate:

    Ωϱ(θ,t;x,ω)θ(F[ϱ,ϱ])dθ  =12Ω|ϱ(θ,t;x,ω)|2          ×(RG(xy)dyRg(ω)dωΩΓ(θϕ)ϱ(ϕ,t;y,ω)dϕ)dθ  C1M02ϱ(,t;x,ω)2,

    which is derived by integration by parts and Lemma 4.2. Taking into account ϱ|t=0=0, we arrive at ϱ=0 on t(0,T) by virtue of the Gronwall's inequality. This implies the first statement. The second one is proved by direct calculation (see, [24]), and we omit the proof.

    We carry out the proof of Theorem 4.1 in three steps below.

    (ⅰ) Existence of a solution ϱ that belongs to V2m()(T) on a certain time interval (0,T);

    (ⅱ) proof of ϱL(1)1(T), and consequently ϱV2m(T);

    (ⅲ) uniqueness of the solution in V2m(T).

    We apply the semi-discrete approximation used by Sjöberg[32] and Tsutsumi[36] for the study of the KdV equation. Let us take NN, h=2π/N, θj=jh(j=1,2,,N) and denote as + and the difference operators defined by

    h+f(θj)=f(θj+1)f(θj),  hf(θj)=f(θj)f(θj1).

    Now, instead of problem (2.3), we consider the following differential-difference equation:

    {ϱNt+ωϱN+(FNϱN)D+ϱN=0,ϱN(θj,t)=ϱN(θj+N,t)  j=1,2,,N,t>0,ϱN(θj,0;x,ω)=ϱ0(θj;x,ω)  j=1,2,,N, (4.1)

    where

    FN(θ)=Nj=1RhG(xy)dyRg(ω)Γ(θϕj)ϱN(ϕj,t;y,ω)dω(j=1,2,,N).

    Note that this function is defined on the continuous interval with respect to θ.

    By virtue of the estimate (4.9) we will verify later, it is clear that problem (4.1) above has a unique solution ϱN for every NN. Then we derive some bounds for ϱN and its differences, which are uniform with respect to N. To do that, in the space of grid-functions we define the scalar product and the norm by

    (f1,f2)hNj=1f1(θj)¯f2(θj)h,  f2h(f,f)h,

    respectively. As Sjöberg[32] and Tsutsumi[36] did, we assume N=2n+1 with nN, and then

    {12πeikθ}nk=n

    forms an orthonormal basis with respect to the scalar products (,)h and (,).

    The following lemmas are due to the work by Sjöberg[32] and Tsutsumi[36]; thus, we omit the proof here.

    Lemma 4.4. If f1 is a real N-periodic grid-function, i.e., if f1(xj)=f1(xj+N)(j=1,2,,N), and if f2(x) is another real N-periodic grid-function, then the following equalities hold:

    (f1,+f2)h=(f1,f2)h,  (f1,f2)h=(+f1,f2)h, (4.2)
    (+f1,f1)h=h2+f12h. (4.3)

    Lemma 4.5.  Let τ1 and τ2 be non-negative integers with τ1+τ2=τ, and ψ, a function of the form

    ψ(θ)=12πnk=nakeikθ.

    Then,

    c42τψxτ2τ1+τ2ψ2=τ1+τ2ψ2hc43τψxτ2.

    holds with some constants c4i(i=2,3).

    For the proof of Lemma 4.5, see Lemma 2.2 in the work by Sjöberg[32]. In addition, the following lemma is useful.

    Lemma 4.6. Let f=f(θ), and ψ be the discrete Fourier series of f, that is,

    ψ(θ)=12πnk=nbkeikθ

    with bk=12π(eikθ,f)h. Then, the discrete version of the Parseval type equality

    τ1+τ2ψ2h=τ1+τ2f2h

    holds for non-negative integers τ1 and τ2.

    Proof. We first verify the statement when τ1=τ2=0. In fact, since

    |ψ(θ)|2=12π(nk=nbkeikθ)(nk=n¯bkeikθ),

    noting that

    Nr=1ei(kk)θr=Nδkk, (4.4)

    where δkk is the Kronecker's delta, we have

    ψ2h=2πNNr=1|ψ(θr)|2=1Nnk,k=nbk¯bkNr=1ei(kk)θr=nk=n|bk|2.

    On the other hand, since

    bk=2πNNr=1eikθr¯f(θr),

    we have

    |bk|2=(2πNNr=1eikθr¯f(θr))(2πNNr=1eikθrf(θr))=2πN2Nr,r=1¯f(θr)f(θr)eik(θrθr).

    Accordingly, by noting (4.4) again, we have

    nk=n|bk|2=2πN2Nr,r=1¯f(θr)f(θr)nk=neik(θrθr)=2πNNr=1|f(θr)|2=f2h.

    Finally, the statement holds in case τ1+τ2>0, since τ1+τ2ψ is the Fourier series of τ1+τ2f. We verify this for (τ1,τ2)=(1,0) for simplicity. In fact, by definition,

    +ψ(θr)=ψ(θr+1)ψ(θr)h=12πnk=n(eikθ,f)h(eikθr+1eikθrh)=12πnk=n(Nr=1eikθrf(θr)h)(eikθr+1eikθrh). (4.5)

    On the other hand,

    12πnk=n(+f,eikθ)heikθr=12πnk=n(Nr=1heikθr+f(θr))eikθr=12πnk=n(Nr=1eikθr{f(θr+1)f(θr)})eikθr.

    However, it is easy to see that this equals the rightmost-hand side of (4.5); therefore, +ψ is the discrete Fourier series of +f. For other pairs of (τ1,τ2), we are able to show the desired statement in the similar manners.

    On the basis of Lemma 4.6, we derive some estimates of ϱN and its differences.

    Lemma 4.7. The following estimates hold:

            ϱN(,t;x,ω)hc4(1)  t>0,(x,ω)R2,    ϱN(,t;x,ω)hc4(2)  t>0,(x,ω)R2,  j+jϱN(,t;x,ω)h˜c4(j)j=1,2,,m,t>0,(x,ω)R2,

    where c4(k)(k=1,2) and ˜c4(j)(j=1,2,,m) are positive constants independent on t, x and ω.

    Proof. Let us multiply (4.1)1 by ϱN. Then, by virtue of Lemma 4.2, we have

    12ddtϱN(,t;x,ω)2h+ωh2+ϱN(,t;x,ω)2h+DϱN(,t;x,ω)2h|((FNϱN),ϱN)h|. (4.6)

    Making use of

    (f1f2)=f2(f1)+(f1)f2

    for two real N-periodic grid-functions f1 and f2 in general, where (f1)(θj)=f1(θj1), we have

    ((FNϱN),ϱN)h=(ϱNFN,ϱN)h+(FNϱN,ϱN)h          ˜C1M0supxϱN(,t;x,ω)hϱN(,t;x,ω)2h                                                        +˜C0M0ϱN(,t;x,ω)h|(ϱN,ϱN)h|          ˜C1M0supxϱN(,t;x,ω)3h+εϱN(,t;x,ω)2h+CεϱN(,t;x,ω)2h,

    where ε is a certain positive constant, and Cε, a constant dependent on ε (hereafter we use these notations in the same meaning). Here we have used the estimates

                            Njh|Γ(θϕj)|2Γ2˜C21,|FN|Nj=1|RhG(xy)dyΓ(θjϕj)Rg(ω)ϱN(ϕj,t;y,ω)dω|        ˜C0M0ϱN(,t;x,ω)h,

    and by means of the mean value theorem and Schwarz's inequality,

    FN=FN(θj)FN(θj1)h        =1hNj=1RhG(xy)dy                        ×Rg(ω){Γ(θjϕj)Γ(θj1ϕj)}ϱN(ϕj,t;y,ω)dω       =Nj=1RhG(xy)dyRg(ω)Γ(θ0ϕj)ϱN(ϕj,t;y,ω)dω        supθNj=1RhG(xy)dyRg(ω)Γ(θϕj)ϱN(ϕj,t;y,ω)dω        ˜C1M0supyϱN(,t;y,ω)h, (4.7)

    where θ0(θji,θj). In addition, we have used the Young's inequality

    |(ϱN,ϱN)h|εϱN(,t;x,ω)2h+CεϱN(,t;x,ω)2h,

    where we take ε<D. Thus, taking the supremum with respect to x in (4.6), we arrive at

    12ddt(supxϱN(,t;x,ω)2h)                +ωh2supx+ϱN(,t;x,ω)2h+(Dε)supxϱN(,t;x,ω)2h                       ˜C1M0supxϱN(,t;x,ω)3h+CεsupxϱN(,t;x,ω)2h.

    By virtue of the comparison theorem, supxϱN(,t;x,ω)h is estimated from above by the solution of the ordinary differential equation

    {12y=˜C1M0y32+Cεy,y|t=0=y0supxϱ0(;x,ω)2h,

    which has a solution on t(0,T) with some T>0, and satisfies the estimate of the form

    supxϱN(,t;x,ω)hc44  t(0,T) (4.8)

    with a constant c44 independent on ω. Next, we multiply (4.1)1 by +ϱN. With the aid of (4.2)-(4.3), we show some of the elementary calculations below:

               (dϱNdt,+ϱN)h=12ddtϱN2h,      (ωϱN,+ϱN)h=ωh2+ϱN2h,    (D+ϱN,+ϱN)h=D+ϱN2h,((FNϱN),+ϱN)h=(FNϱN,+ϱN)h+(ϱNFN,+ϱN)h                                      ˜C0M0h2supx+ϱN(,t;x,ω)2h                                                  +˜C1M0supxϱN(,t;x,ω)2h.

    Combining these, we have

    12ddt(supxϱN(,t;x,ω)2h)+supxωh2+ϱN(,t;x,ω)2h                 +Dsupx+ϱN(,t;x,ω)2h                       ˜C0M0h2supx+ϱN(,t;x,ω)2h+˜C1M0supxϱN(,t;x,ω)2h.

    By taking h sufficiently small, and due to estimate (4.8) we have already obtained, we derive

    12ddt(supxϱN(,t;x,ω)2h)c45supxϱN(,t;x,ω)2h,

    which yields

    supxϱN(,t;x,ω)2hc46

    for t(0,T) due to the preceding discussion.

    Similarly, multiplying (4.1)1 by 2+2ϱN leads to

    ddt+ϱN(,t;x,ω)2hωh2+2ϱN(,t;x,ω)2h+2+ϱN(,t;x,ω)2h+((FNϱN),2+2ϱN)h=0.

    We expand the last term in the left-hand side as

    ((FNϱN),2+2ϱN)h=((ϱN)(FN),+2ϱN)h(ϱN(2FN),+2ϱN)h(FNϱN,2+2ϱN)h3j=1Ij.

    Each term is estimated as follows.

                    |I1|˜C1M0ϱN(,t;x,ω)h|(ϱN,+2ϱN)h|                    ˜C1M0ϱN(,t;x,ω)h+ϱN(,t;x,ω)2h,                |I2|˜C1M0ϱN(,t;x,ω)h|(ϱN,+2ϱN)h|                    ˜C1M0h2ϱN(,t;x,ω)h+ϱN(,t;x,ω)2h,|I3|˜C0M0ϱN(,t;x,ω)h|(ϱN,2+2ϱN)h|    ˜C0M0ϱN(,t;x,ω)h(ε+ϱN(,t;x,ω)2h+Cε2+ϱN(,t;x,ω)2h).

    Here we applied a similar estimate as (4.7). Thus, by using (4.8) we have the estimate of the form

    12ddt(supx+ϱN(,t;x,ω)2h)ωh2+ϱN(,t;x,ω)2h                +(Dε)2+ϱN(,t;x,ω)2h            c47+ϱN(,t;x,ω)2h,

    which yields the boundedness of +ϱN(,t;x,ω)2h on (0,T). In a similar manner, we obtain the estimates of higher difference terms with respect to θ.

    When m2, by virtue of Lemma 4.5, we easily obtain

    supxdϱNdt(,t;x,ω)hc48,supxm+mdϱNdt(,t;x,ω)hc49 (4.9)

    under the assumptions of Theorem 4.1. Then, as Sjöberg[32] and Tsutsumi[36] did, we consider the discrete Fourier series of ϱN, which is denoted as {ϕN}:

    ϕN(θ,t;x,ω)=12πnk=nak(t;x,ω)eikθ,  ak(t;x,ω)=12π(eikθ,ϱN)h.

    Estimate (4.9) and Lemmas 4.5-4.7 yield that the sequence of functions {ϕN} is uniformly bounded and equicontinuous on 0θ2π, 0tT. With the aid of the Arzera-Ascoli theorem, we see that {ϕN} contains a subsequence which converges to a certain function ϱ as N+. In addition, it is clear that

    (θ)mϕN(θ)mϱ(N+)

    in L2(Ω) for each (t,x,ω). Therefore, this ϱ is the desired solution to (4.1).

    Finally, we discuss the regularity of ϱ with respect to x. Let us define

    vϱ(θ,t;x+x,ω)ϱ(θ,t;x,ω)x,GG(x+x)G(x)x,

    which clearly satisfy

    vt=θ[ωv+ϱ(RG(xy)dyRg(ω)dωΩΓ(θϕ)ϱ(ϕ,t;y,ω)dϕ)]+F[v,ϱ]D2ϱθ2=0.

    Thus, under the assumptions of Theorem 4.1, we can show that

    ϱ(θ,t;,ω)C(R)

    with respect to x for each (θ,t) and ω in the same line with the arguments by Lavrentiev[24]. The same argument holds concerning the regularity with respect to ω, and we finally arrive at the desired regularity of ϱ.

    Before proceeding to the uniqueness part, we mention that the solution that was guaranteed to exist in the previous process also belongs to L(1)1(T) thanks to Lemma 4.3, and consequently, to V2m(T). This is directly obtained from Lemma 4.3.

    Finally, we discuss the uniqueness part of the statement. Assume that there exist two solutions ϱi(i=1,2) to (2.3) on (0,T) with the same initial data, and let us define ˜˜ϱϱ1ϱ2.

    Then, it satisfies:

    {˜˜ϱt+ω˜˜ϱθD2˜˜ϱθ2+θ(F[˜˜ϱ,ϱ1])+θ(F[ϱ2,˜˜ϱ])=0                                  θΩ,t(0,T),(x,ω)R2,i˜˜ϱθi|θ=0=i˜˜ϱθi|θ=2π  (i=0,1),t(0,T),(x,ω)R,˜˜ϱ|t=0=0  θΩ,(x,ω)R2. (4.10)

    We multiply (4.10)1 by ˜˜ϱ. Then, integration by parts yields

    Ω˜˜ϱ(θ,t;x,ω)θ(F[˜˜ϱ,ϱ1])dθ=12ΩF(1)[(˜˜ϱ)2,ϱ]dθ                     C1M02|||˜˜ϱ(t)|||2,Ω˜˜ϱ(θ,t;x,ω)θ(F[ϱ2,˜˜ϱ])dθ                         =Ω˜˜ϱ(θ,t;x,ω)F[ϱ2θ,˜˜ϱ]dθ+Ω˜˜ϱ(θ,t;x,ω)F(1)[ϱ2,˜˜ϱ]dθ                         2πM0{˜C0|||ϱ(1,0)2(t)|||+˜C1|||ϱ2(t)|||}|||˜˜ϱ(t)|||2.

    These yield

    12ddt|||˜˜ϱ(t)|||2+D|||˜˜ϱ(1,0)(t)|||2C1M0|||˜˜ϱ(t)|||2  t(0,T).

    By virtue of the Gronwall's inequality and the initial condition (4.10)3, we have

    |||˜˜ϱ(t)|||=0  t(0,T),

    which indicates the uniqueness of the solution in the desired function space.

    Now we discuss the global-in-time solvability of (2.3). Let ϱ0 be provided, which satisfies the assumption (ⅵ) in Theorem 4.1. In accordance with Theorem 4.1, we first construct the local-in-time solution ϱ on t(0,T), where T is the time provided in that theorem. Then, we have the a-priori estimate.

    Lemma 4.8. Let T>0 be an arbitrary number. If there exists a solution to 2.3 on (0,T), estimates of the form

    |||ϱ(k,0)(t)|||c4(k)  (k=1,2,,2m) (4.11)

    hold with certain constants c4(k) independent of t.

    Proof. For the sake of simplicity, we introduce the notation ˜ϱϱˉϱ and derive the estimate of its norm which leads to the desired estimates. From (2.3), it is obvious that ˜ϱ satisfies

    {˜ϱt+ω˜ϱθ+θ(F[˜ϱ+ˉϱ,˜ϱ+ˉϱ])D2˜ϱθ2=0                                        θΩ,t(0,T),(x,ω)R2,i˜ϱθi|θ=0=i˜ϱθi|θ=2π  (i=0,1),t(0,T),(x,ω)R2,˜ϱ|t=0=˜ϱ0ϱ0ˉϱ  θΩ,(x,ω)R2. (4.12)

    Multiply (4.12)1 by ˜ϱ, and making use of Lemma 4.2 and the periodicity of F[˜ϱ+ˉϱ,˜ϱ+ˉϱ] with respect to θ yields

    Ω˜ϱ(θ,t;x,ω)θ(F[˜ϱ+ˉϱ,˜ϱ+ˉϱ])dθ=Ωϱ(θ,t;x,ω)θ(F[˜ϱ+ˉϱ,˜ϱ+ˉϱ])dθ=12ΩF(1)[ϱ2,ϱ]dθC1M02ϱ(,t;x,ω)2.

    On the other hand, in the same line as the arguments by Lavrentiev[24], we have

    ϱ(,t;x,ω)2Ωϱ(θ,t;x,ω)(12π+2πϱθ(,t;x,ω))dθ=12π+2πϱθ(,t;x,ω)12π+Cε+ε˜ϱθ(,t;x,ω)2,

    where we have applied the Young's inequality in the last inequality.

    Thus, after taking the supremum with respect to (x,ω), we have the estimate of the form

    12ddt|||˜ϱ(t)|||2+D|||˜ϱ(1,0)(t)|||2c410+ε|||˜ϱ(1,0)(t)|||2. (4.13)

    Therefore, if we take ε so small that ε<D holds, then by virtue of the classical Gronwall's inequality, we have the estimate of the form (see, for instance, p.85 of[35])

    |||˜ϱ(t)|||2|||˜ϱ0|||2exp(2(Dε)t)+c411Dε(1exp(2(Dε)t))c412  t(0,T). (4.14)

    Next, we show the estimate of ˜ϱ(1,0), which satisfies

    ˜ϱ(1,0)t+ω˜ϱ(1,0)θD2˜ϱ(1,0)θ2+θ(F(1)[˜ϱ+ˉϱ,˜ϱ]+F[˜ϱ(1,0),˜ϱ])=0.

    Then, due to the estimates

    Ω˜ϱ(1,0)(θ,t;x,ω)θ(F(1)[˜ϱ+ˉϱ,˜ϱ])dθ  =ΩF(1)[(˜ϱ(1,0)(θ,t;x,ω))2,˜ϱ]dθ+12ΩF(2)[θ(˜ϱ(θ,t;x,ω))2,˜ϱ]dθ                    +12πΩF(2)[˜ϱ(1,0),˜ϱ]dθ  C1M0˜ϱ(1,0)(,t;x,ω)2+C3M02˜ϱ(,t;x,ω)2                    +˜C2M0˜ϱ(1,0)(,t;x,ω)|||˜ϱ(t)|||,Ω˜ϱ(1,0)(θ,t;x,ω)θ(F[˜ϱ(1,0),˜ϱ])dθ=12ΩF[θ(˜ϱ(1,0)(θ,t;x,ω))2,˜ϱ]dθ                                        C1M02˜ϱ(1,0)(,t;x,ω)2,

    and the Young's inequality, and taking the supremum with respect to x and ω, we have the estimate of the form

    12ddt|||˜ϱ(1,0)(t)|||2+D|||˜ϱ(2,0)(t)|||2χ(0,0)1|||˜ϱ(t)|||2+χ(1,0)1|||˜ϱ(1,0)(t)|||2. (4.15)

    with constants χ(i,0)1(i=0,1). Now we divide the second term in the left-hand side of (4.13) into two terms by using a small constant ε>0, and apply the Poincaré's inequality

    ˜ϱ(,t;x,ω)2π˜ϱ(1,0)(,t;x,ω)

    to the first term:

    (Dε)|||˜ϱ(1,0)(t)|||2+ε|||˜ϱ(1,0)(t)|||2Dε4π2|||˜ϱ(t)|||2+ε|||˜ϱ(1,0)(t)|||2.

    Then, we obtain

    12ddt|||˜ϱ(t)|||2+Dε4π2|||˜ϱ(t)|||2+ε|||˜ϱ(1,0)(t)|||2c410+ε|||˜ϱ(1,0)(t)|||2.

    Summing up this and (4.15) multiplied by a positive constant m(1,0), which will be specified later, we have

    12ddt(|||˜ϱ(t)|||2+m(1,0)|||˜ϱ(1,0)(t)|||2)+{(Dε)4π2m(1,0)χ(0,0)1}|||˜ϱ(t)|||2                    +{εεm(1,0)χ(1,0)1}|||˜ϱ(1,0)(t)|||2+m(1,0)D|||˜ϱ(2,0)(t)|||2              c413.

    Therefore, we take ε, ε and m(1,0) in the following manner:

    (ⅰ) Take ε and ε so that ε<ε<D holds;

    (ⅱ) Then, take m(1,0)>0 so small that

    {Dε4π2χ(0,0)1m(1,0)>0,εεχ(1,0)1m(1,0)>0

    hold.

    Then, in the same line with the deduction of (4.14), we have

    |||˜ϱ(1,0)|||2c414  t>0. (4.16)

    Similarly, for k=2,3,,(2m2), we have the estimate of the form:

    12ddt|||˜ϱ(i,0)(t)|||2+D|||˜ϱ(i+1,0)(t)|||2ij=0χ(j,0)i|||˜ϱ(j,0)(t)|||2(i=2,3,,(2m2)). (4.17)

    For estimates of ˜ϱ(l,0)(l=2m1,2m), we introduce the Friedrichs mollifier ΦC0(Ω) with respect to θ[28], and define

    Φδδ1Φ(θδ1)

    with a constant δ>0. We also define

    f1f2Rf1(θθ)f2(θ)dθ

    for functions f1,f2L2(R) in general. Note that when they are defined on Ω, we extend them onto R preserving the regularity[23].

    Now, by operating Φδ to (4.12)1, we have

    ˜ϱ(δ)t+ω˜ϱ(δ)θD2˜ϱ(δ)θ2+θ(F[(˜ϱ(δ)+ˉϱ),˜ϱ])=˜H(δ), (4.18)

    where

                            ˜ϱ(δ)=Φδ˜ϱ,˜H(δ)θ(F[(˜ϱ(δ)+ˉϱ),˜ϱ])Φδθ(F[(˜ϱ+ˉϱ),˜ϱ]).

    Operating (θ)l to (4.18) yields

    ˜ϱ(l,0)(δ)t+ω˜ϱ(l,0)(δ)θD2˜ϱ(l,0)(δ)θ2+li=0lCiθ(F(li)[˜ϱ(i,0)(δ),˜ϱ])=˜H(l,0)(δ). (4.19)

    Hereafter we use the notation lCi to denote the binomial coefficient of l choose i.

    Then, we multiply (4.19) by ˜ϱ(l,0)(δ), and with the similar process as above, obtain the estimate

    12t|||˜ϱ(l,0)(δ)(t)|||2+D|||˜ϱ(l+1,0)(δ)(t)|||2lj=0χ(j,0)l|||˜ϱ(j,0)(δ)(t)|||2+|||˜H(l,0)(δ)(t)||||||˜ϱ(i,0)(δ)(t)||| (4.20)

    with constants {χ(j,0)l}lj=0. By noting limδ0|||˜H(l,0)(δ)|||=0, we obtain the following estimate from (4.20) by letting δ tend to zero:

    12ddt|||˜ϱ(l,0)(t)|||2+D|||˜ϱ(l+1,0)(t)|||2lj=0χ(j,0)l|||˜ϱ(j,0)(t)|||2(l=2m1,2m). (4.21)

    We now multiply each estimate for ˜ϱ(j,0)(j=1,2,,2m) in (4.15), (4.17) and (4.21) by a positive constant m(j,0), which will be specified later, and sum up them. We also introduce m(0,0)=1 for simplicity. These yield

    12ddt(2mj=0m(j,0)|||˜ϱ(j,0)(t)|||2)+D2mi=0m(i,0)|||˜ϱ(i+1,0)(t)|||2c410+ε|||˜ϱ(1,0)(t)|||2+2mi=1m(i,0)(ij=0χ(j,0)i|||˜ϱ(j,0)(t)|||2).

    It should be noted that a straightforward estimate, like that by Ha and Xiao[11] will require D to be monotonically and increasingly dependent on m. Instead, we partially apply the Poincaré's inequality as before:

    12ddt(2mj=0m(j,0)|||˜ϱ(j,0)(t)|||2)+Dε4π22m1i=0m(i,0)|||˜ϱ(i,0)(t)|||2+ε2m1i=0m(i,0)|||˜ϱ(i+1,0)(t)|||2+Dm(2m,0)|||˜ϱ(2m+1,0)(t)|||2        c410+ε|||˜ϱ(1,0)(t)|||2+2mi=1m(i,0)(ij=0χ(j,0)i|||˜ϱ(j,0)(t)|||2) (4.22)

    We take ε, ε and {m(i,0)}2mi=1 as follows.

    (ⅰ) Take ε and ε so that ε<ε<D hold;

    (ⅱ) Take m(1,0)>0 so small that

    {Dε4π2χ(0,0)1m(1,0)>0,(Dε4π2χ(1,0)1)m(1,0)<εε

    hold;

    (ⅲ) Take m(2,0)>0 so small that

    {Dε4π22i=1χ(0,0)im(i,0)>0,(Dε4π2χ(1,0)1)m(1,0)+(εε)χ(1,0)2m(2,0)>0,(Dε4π2χ(2,0)2)m(2,0)+εm(1,0)>0

    hold;

    (ⅳ) As for {m(i,0)}2m1i=3, take them so small inductively that:

    {Dε4π2ip=1χ(0,0)pm(p,0)>0,(Dε4π2χ(1,0)1)m(1,0)+(εε)ip=1χ(1,0)pm(p,0)>0,(Dε)m(q,0)4π2+εm(q1,0)is=qχ(q,0)sm(s,0)>0  (q=2,3,,i)

    hold;

    (ⅴ) Finally, take m(2m,0) so small that

    {Dε4π22mp=1χ(0,0)pm(p,0)>0,(Dε4π2χ(1,0)1)m(1,0)+(εε)2mp=1χ(1,0)pm(p,0)>0,(Dε)m(q,0)4π2+εm(q1,0)2ms=qχ(q,0)sm(s,0)>0  (q=2,3,,2m1),εm(2m1,0)χ(2m,0)2mm(2m,0)>0

    hold.

    Thus, (4.22) becomes

    12ddt(2mi=0m(i,0)|||˜ϱ(i,0)(t)|||2)+{Dε4π22mp=1χ(0,0)pm(p,0)}|||˜ϱ(t)|||2+{Dε4π2m(1,0)+(εε)2mp=1χ(1,0)pm(p,0)}|||˜ϱ(1,0)(t)|||2+2m1q=2{Dε4π2m(q,0)+εm(q1,0)2ms=qχ(q,0)sm(s,0)}|||˜ϱ(q,0)(t)|||2+(εm(2m1,0)χ(2m,0)2mm(2m,0))|||˜ϱ(2m,0)(t)|||2+Dm(2m,0)|||˜ϱ(2m+1,0)(t)|||2c410,

    where the coefficients of each term in the left-hand side are all positive. Thus, as we have obtained (4.14) and (4.16), the Gronwall's inequality again yields

    (2mi=0m(i,0)|||˜ϱ(i,0)(t)|||2)c415.

    Now, by virtue of Lemma 4.8, ϱ|t=T satisfies the assumption (ⅵ) in Theorem 4.1 imposed on ϱ0. Thereby, we are able to extend ϱ over the time interval (T,2T) as a solution to (2.3). This solution, which is now defined over (0,2T), again satisfies the estimate (4.11), and we then extend it over the region (2T,3T). Iterating this procedure sufficiently many times, we are able to obtain the solution over the time interval (0,T) for arbitrary T>0. We summarize these arguments as follows.

    Theorem 4.9. Let T be an arbitrary positive number. Then, under the same assumptions as in Theorem 4.1, there exists a solution ϱ(θ,t;x,ω)V2m(T) to 2.3 on (0,T).

    Remark 3. From these considerations, it is obvious that Theorem 4.9 holds when g is compactly supported. we can also extend these arguments when g is the Dirac's delta function, i.e. g(ω)=δ(0), since Lemma 4.2 holds in that case.

    Corollary 4.10. Under the assumptions in Theorem 4.1 with (ⅳ) replaced by g(ω)=δ(0), the same statement as in Theorem 4.9 holds.

    In this section, we discuss the nonlinear stability of the trivial stationary solution ˉϱ12π. As in the proof of Lemma 4.8, we introduce the notation ˜ϱϱˉϱ, and derive the estimate of its norm with respect to time.

    The asymptotic stability of ˜ϱ reads

    Theorem 5.1. In addition to the assumptions in Theorem 4.1, we assume

    D>2π2M0(C1+˜C1).

    Then, ˜ϱ is asymptotically stable in ¯H2m and satisfies the inequality

    ˜ϱ(t)¯H2mc51˜ϱ0¯H2mec52t

    with certain positive constants c5i(i=1,2).

    Proof. The line of the argument is similar to that of Lemma 4.8, but this time we have to confirm the non-positiveness of the left-hand side of the energy type inequalities. First, let us multiply (4.12)1 by ˜ϱ. By making use of Lemma 4.2, we have the estimate:

    Ω˜ϱ(θ,t;x,ω)θ(F[(˜ϱ+ˉϱ,˜ϱ+ˉϱ])dθ=12ΩF[θ(˜ϱ(θ,t;x,ω))2,(˜ϱ+ˉϱ)]dθ+12πΩF(1)[˜ϱ,(˜ϱ+ˉϱ)]dθM02(C1+˜C1)˜ϱ(t)2.

    Thus, we have the energy estimate

    12ddt|||˜ϱ(t)|||2+D|||˜ϱ(1,0)(t)|||2M02(C1+˜C1)|||˜ϱ(t)|||2. (5.1)

    For the estimate of |||˜ϱ(t)||| only, if D>2π2M0(C1+˜C1) holds, with the aid of the Poincaré's inequality, we have

    12ddt|||˜ϱ(t)|||2+{D4π2M02(C1+˜C1)}|||˜ϱ(t)|||20,

    which leads to the estimate of the form

    |||˜ϱ(t)|||c53exp(c54t) (5.2)

    by virtue of the Gronwall's inequality. Estimate (5.2) implies the asymptotic stability of ˉϱ in ¯H0.

    Next, we show the estimate up to the first-order spatial derivative. First, by applying the Poincaré's inequality to the second term of the left-hand side in (5.1) partially, as in the previous section, we have

    12ddt|||˜ϱ(t)|||2+Dε4π2|||˜ϱ(t)|||2+ε|||˜ϱ(1,0)(t)|||2M02(C1+˜C1)|||˜ϱ(t)|||2. (5.3)

    Then, let us sum up (5.3) and (4.15), and we have

    12ddt(|||˜ϱ(t)|||2+m(1,0)|||˜ϱ(1,0)(t)|||2)+{Dε4π2M02(C1+˜C1)χ(0,0)1m(1,0)}|||˜ϱ(t)|||2+(εχ(1,0)1m(1,0))|||˜ϱ(1,0)(t)|||2+m(1,0)D|||˜ϱ(2,0)(t)|||20. (5.4)

    As in the previous section, we take ε and m(1,0) in the following manner:

    (ⅰ) Take ε so small that Dε4π2M02(C1+˜C1)>0 holds;

    (ⅱ) Then, take m(1,0)>0 so small that

    {Dε4π2M02(C1+˜C1)χ(0,0)1m(1,0)>0,εχ(1,0)1m(1,0)>0

    hold.

    By applying the Gronwall's inequality to (5.4), these lead to the estimate of the form

    (|||˜ϱ(t)|||2+m(1,0)|||˜ϱ(1,0)(t)|||2)c55exp(c56t),

    that is, the asymptotic stability of ˉϱ holds in ¯H1.

    Similarly, we make use of (4.17) and (4.21) to deduce

    12ddt|||˜ϱ(i,0)(t)|||2+Dε4π2|||˜ϱ(i,0)(t)|||2+ε|||˜ϱ(i+1,0)(t)|||2ij=0χ(j,0)i|||˜ϱ(j,0)(t)|||2(i=1,2,,(2m1)), (5.5)
    12ddt|||˜ϱ(2m,0)(t)|||2+D|||˜ϱ(2m+1,0)(t)|||22mj=0χ(j,0)2m|||˜ϱ(j,0)(t)|||2. (5.6)

    Summing up (5.3), (5.5) and (5.6) multiplied by constants {m(i,0)}2mi=0 with m(0,0)=1, we arrive at

    12ddt(2mi=0m(i,0)|||˜ϱ(i,0)(t)|||2)+{Dε4π2M02(C1+˜C1)(2mp=1χ(0,0)pm(p,0))}|||˜ϱ(t)|||2+2m1i=1{(Dε)m(i,0)4π2+εm(i1,0)(2ms=iχ(i,0)sm(s,0))}|||˜ϱ(i,0)(t)|||2+(εm(2m1,0)m(2m,0)χ(2m,0)2m|||˜ϱ(2m,0)(t)|||2)+m(2m,0)D|||˜ϱ(2m+1,0)(t)|||20.

    Now, we determine ε and {m(i,0)}2mi=1 as follows.

    (ⅰ) Take ε so small that Dε4π2M02(C1+˜C1)>0;

    (ⅱ) Take m(1,0)>0 so small that

    {(Dε)m(i,0)4π2M02(C1+˜C1)χ(0,0)1m(1,0)>0,(Dε)m(1,0)4π2+εχ(1,0)1m(1,0)>0

    hold;

    (ⅲ) Take m(2,0)>0 so small that

    {(Dε)m(i,0)4π2M02(C1+˜C1)2i=1χ(0,0)im(i,0)>0,(Dε)m(1,0)4π2+εχ(1,0)1m1,0χ(1,0)2m(2,0)>0,(Dε)m(2,0)4π2+εm(1,0)χ(2,0)2m(2,0)>0

    hold;

    (ⅵ) As for {m(q,0)}2m1q=3, take them so small inductively that:

    {Dε4π2M02(C1+˜C1)qp=1χ(0,0)pm(p,0)>0,(Dε)m(i,0)4π2+εm(i1,0)qs=iχ(i,0)sm(s,0)>0  (i=1,2,,q)

    hold;

    (ⅴ) Finally, take m(2m,0) so small that

    {Dε4π2M02(C1+˜C1)2mi=1χ(0,0)im(i,0)>0,Dε4π2m(i,0)+εm(i1,0)2ms=iχ(i,0)sm(s,0)>0  (i=1,2,,2m),εm(2m1,0)χ(2m,0)2mm(2m,0)>0

    hold.

    These yield the estimate of the form

    (2mi=0m(i,0)|||˜ϱ(i,0)(t)|||2)c57exp(c58t),

    which directly leads to the desired statement.

    Remark 4. For the original Kuramoto-Sakaguchi equation (1.6), Ha[11] deduced a similar result concerning the asymptotic stability of ˉϱ=1/2π. On the basis of some regularity and decay of ϱ0 with respect to ω, they estimated the H3 norm with weight with respect to ω. Their arguments are based on the energy method, which is similar to the one presented here. The advantage of our method is to make the estimate sharper by dividing the terms of a higher derivative into two terms before applying the Poincaré's inequality. Indeed, as we have mentioned before, we need monotonically increasing D with respect to m with the procedure used by Ha and Xiao[11]. As we mentioned in Section 2, the original Kuramoto model (1.3) (or (1.6)) can be regarded as a specific case of the non-local coupling model (2.2) (resp. (2.3))[20]. Therefore, the stability of the incoherent state in (1.6) is verified through similar arguments and assumptions in Theorem 4.1 (see also[13]).

    Finally, we discuss the vanishing limit of the diffusion coefficient. In order to show the dependency of the solution on the diffusion coefficient clearly, we denote the solution of (2.3) as ϱ(D), whereas that of (2.5) is denoted as ϱ(0). As we have stated, Ha and Xiao[11] held a similar discussion for the original Kuramoto-Sakaguchi equation (1.6). However, they estimated the norm of ϱ(D) by using the polynomial of D, which resulted in the convergence in L(Ω) with respect to θ. In the discussion below, we apply the compactness argument for deriving higher order convergence than their result. This method is also applicable to the original Kuramoto-Sakaguchi equation[13]. We first prepare some lemmas below.

    Lemma 6.1. Let T>0 be an arbitrary number. Then, the sequence {ϱ(D)}D>0 is uniformly bounded with respect to D in V2m(T).

    Proof. What we have to verify are

    supt(0,T)|||ϱ(l,k)(D)(t)|||cl,k(T)  (2k+l2m), (6.1)
    T0|||ϱ(l+1,k)(D)(t)|||2dtcl,k(T)  (2k+l2m) (6.2)

    with some constants {cl,k(T)}2k+l2m and {cl,k(T)}2k+l2m dependent on T. For k=0, we have already verified (6.1) in the previous section.

    Now we provide the estimates of the temporal derivative of ϱ inductively. Let us assume that the boundedness of {ϱ(0,i)(D)}D>0 has been proven for i=1,2,,k1(km1). Then, by noting

    ϱ(0,k)(D)t+ωϱ(0,k)(D)θD2ϱ(0,k)(D)θ2+kj=0kCjθ(F[ϱ(0,j)(D),ϱ(0,kj)(D)])=0

    and Lemma 4.2, we have the estimate of the form

    12ddt|||ϱ(0,k)(D)(t)|||2+D|||ϱ(1,k)(D)(t)|||2                   k1j=1ck,j|||ϱ(0,kj)(D)(t)|||2|||ϱ(0,j)(D)(t)|||2+ε|||ϱ(1,k)(D)(t)|||2                            +(C0|||ϱ(1,0)(D)(t)|||2+˜C1|||ϱ(D)(t)|||2+C12)M0|||ϱ(0,k)(D)(t)|||2, (6.3)

    where {ck,j} are the positive constants. By virtue of the assumption of the induction and the Gronwall's inequality, we then have the estimate of the form

    |||ϱ(0,k)(D)(t)|||2c61(t)exp(t0c62(τ)dτ)(k=1,2,,m1). (6.4)

    As for k=m, we use the mollifier again. Recalling the notation defined in Section 4, we consider

    ϱ(D)(δ)t+ωϱ(D)(δ)θD2ϱ(D)(δ)θ2+θ(F[ϱ(D)(δ),ϱ(D)])=H(D)(δ), (6.5)

    where ϱ(D)(δ)=Φδϱ(D), and

    H(D)(δ)=θ(F[(ϱ(D)(δ)),ϱ(D)])Φδθ(F[ϱ(D),ϱ(D)]).

    We operate the temporal derivative (t)m to (6.5) and obtain

    ϱ(0,m)(D)(δ)t+ωϱ(0,m)(D)(δ)θD2ϱ(0,m)(D)(δ)θ2                      +mj=0mCjθ(F[ϱ(0,j)(D)(δ),ϱ(0,mj)(D)])=H(0,m)(D)(δ). (6.6)

    Then, after the energy type estimate, we make δ tend to zero. By making use of the fact

    limδ0|||H(0,m)(D)(δ)|||=0,

    we obtain (6.4) with k=m with the aid of the assumptions of the induction.

    Next, we estimate the term including both temporal and spatial derivatives. We only show the case l+2k+22m, when the mollifier is necessary again. By applying the l-th order spatial derivative to (6.6) with m replaced with k, we have

    ϱ(l,k)(D)(δ)t+ωϱ(l,k)(D)(δ)θD2ϱ(l,k)(D)(δ)θ2+li=0kj=0lCikCjθ(F(li)[ϱ(i,j)(D)(δ),ϱ(0,kj)(D)])=H(l,k)(D)(δ). (6.7)

    Now we show some examples of the energy type estimates. In case (i,j)=(l,k), we have

    Ωϱ(l,k)(D)(δ)(θ,t;x,ω)θ(F[ϱ(l,k)(D)(δ),ϱ(D)])dθC1M02ϱ(l,k)(D)(δ)(,t;x,ω)2.

    For (i,j)=(l1,k), we have

    Ωϱ(l,k)(D)(δ)(θ,t;x,ω)θ(F(1)[ϱ(l1,k)(D)(δ),ϱ(D)])dθ=ΩF(1)[(ϱ(l,k)(D)(δ))2,ϱ(D)]dθ+ΩF(2)[ϱ(l,k)(D)(δ)ϱ(l1,k)(D)(δ),ϱ(D)]dθC1M0ϱ(l,k)(D)(δ)(,t;x,ω)2+C3M02ϱ(l1,k)(D)(δ)(,t;x,ω)2.

    Otherwise, we have

    Ωϱ(l,k)(D)(δ)(θ,t;x,ω)θ(F(li)[ϱ(i,j)(D)(δ),ϱ(0,kj)(D)])dθ˜CliM0|||ϱ(l+1,k)(D)(δ)(t)||||||ϱ(i,j)(D)(δ)(t)||||||ϱ(0,kj)(D)(δ)(t)|||.

    By combining these and (6.7), and applying the Young's and Schwarz's inequalities, we derive the energy estimate of the form

    12ddt|||ϱ(l,k)(D)(δ)(t)|||2+D2|||ϱ(l+1,k)(D)(δ)(t)|||2c63|||ϱ(l,k)(D)(δ)(t)|||2+ε|||ϱ(l+1,k)(D)(δ)(t)|||2              +Cε(i,j)(l,k)|||ϱ(i,j)(D)(δ)(t)|||2|||ϱ(0,kj)(D)(t)|||2+C3M02|||ϱ(l1,k)(D)(δ)(t)|||2              +|||ϱ(l,k)(D)(δ)(t)||||||H(l,k)(D)(δ)(t)|||. (6.8)

    After making δ0 and applying the Gronwall's inequality, we have the estimate of the form

    |||ϱ(l,k)(D)(t)|||2c64(t)exp(t0c65(τ)dτ).

    Thus, we have shown the first part of the statement. Estimate (6.2) is derived from (6.3), (6.8), and the estimates we have already obtained. These complete the proof.

    By virtue of Lemma 6.1, we see that the sequence {ϱ(D)}D>0 includes a sub-sequence, denoted as {ϱ(D)} again, which is convergent in the weak-star sense as D tends to zero:

    ϱ(D)ˆϱinL(0,T;¯H2m) weaklystar; (6.9)
    ϱ(D)tˆϱinL(0,T;¯H2m2) weaklystar. (6.10)

    Then, in the relationship

    ϱ(D)=ϱ0+t0ϱ(D)t(τ)dτinL(0,T;¯H2m2),

    if we make D tend to zero, we have

    ˆϱ=ϱ0+t0ˆϱ(τ)dτinL(0,T;¯H2m2),

    which means ˆϱ=ˆϱt.

    The next lemma clarifies the space to which this sequence converges.

    Lemma 6.2. Let T>0 be an arbitrary number. Then, the sequence {ϱ(D)}D>0 forms the Cauchy sequence in V2m(T).

    Proof. Let us define ˘ϱϱ(D)ϱ(D) for arbitrary D, D>0, which satisfies

    ˘ϱt+ω˘ϱθD2˘ϱθ2(DD)2ϱ(D)θ2+θ(F[˘ϱ,ϱ(D)])+θ(F[ϱ(D),˘ϱ])=0.

    Then, with the aid of the estimates

    Ω˘ϱ(θ,t;x,ω)θ(F[˘ϱ,ϱ(D)])dθ=Ω˘ϱθ(θ,t;x,ω)F[˘ϱ,ϱ(D)]dθC1M02˘ϱ(,t;x,ω)2,
    Ω˘ϱ(θ,t;x,ω)θ(F[ϱ(D),˘ϱ])dθ(˜C0|||ϱ(D)θ(t)|||+˜C1|||ϱ(D)(t)|||)M0|||˘ϱ(t)|||2,Ω(DD)˘ϱ(θ,t;x,ω)2ϱ(D)θ2(θ,t;x,ω)dθ                             |DD|22+12˘ϱ(,t;x,ω)22ϱ(D)θ2(,t;x,ω)2,

    we have

    12ddt|||˘ϱ(t)|||2+D|||˘ϱ(1,0)(t)|||2c66|||˘ϱ(t)|||2+|DD|22.

    Thus, by virtue of the Gronwall's inequality and the fact that ˘ϱ|t=0=0, we have the estimate of the form

    |||˘ϱ(t)|||2c67|DD|2exp(c68t).

    This implies that the sequence {ϱ(D)}D>0 is the Cauchy sequence in ¯H0. The estimates of higher derivative terms are obtained in a similar manner. To show this, we subtract (6.7) with D replaced with a certain D>0 from itself and obtain

    ˘ϱ(l,k)(δ)t+ω˘ϱ(l,k)(δ)θD2˘ϱ(l,k)(δ)θ2(DD)2˘ϱ(l,k)(δ)θ2+li=0kj=0lCikCjθ(F(li)[˘ϱ(i,j)(δ),ϱ(0,kj)(D)])+li=0kj=0lCikCjθ(F(li)[ϱ(i,j)(D)(δ),˘ϱ(0,kj)])=˘H(l,k)(δ),

    where ˘ϱ(δ)Φδ˘ϱ, and ˘H(δ)=H(D)(δ)H(D)(δ).

    We show inductively that ˘ϱ(l,k)(δ)0 hold as D and D tend to zero based on the assumption that {ϱ(i,j)(D)}D>0 form the Cauchy sequences with 0il1 and 0jk. Let us show some examples of the estimates. In case (i,j)=(l,k), we have

    Ω˘ϱ(l,k)(δ)(θ,t;x,ω)θ(F(li)[˘ϱ(l,k)(δ),ϱ(D)])dθC1M02˘ϱ(l,k)(δ)(,t;x,ω)2.

    In case (i,j)=(l1,k), we have

    Ω˘ϱ(l,k)(δ)(θ,t;x,ω)θ(F(1)[˘ϱ(l1,k)(δ),ϱ(D)])dθ=ΩF(1)[(˘ϱ(l,k)(δ))2,ϱ(D)]dθ+12ΩF(2)[(˘ϱ(l1,k)(δ))2,ϱ(D)]dθC1M0˘ϱ(l,k)(δ)(,t;x,ω)2+C3M02˘ϱ(l1,k)(δ)(,t;x,ω)2.

    Otherwise, we have

    Ω˘ϱ(l,k)(δ)(θ,t;x,ω)θ(F(li)[˘ϱ(i,j)(δ),ϱ(0,kj)(D)])dθ=ΩF(li)[˘ϱ(l+1,k)(δ)˘ϱ(i,j)(δ),ϱ(kj)(D)]dθ˜CliM0|||˘ϱ(l+1,k)(δ)(t)||||||˘ϱ(i,j)(δ)(t)||||||ϱ(0,kj)(D)(t)|||.

    After applying the Schwarz's inequality, we make δ tend to zero. Then, on the basis of the assumption of the induction, we have the estimate of the form:

    12ddt|||˘ϱ(l,k)(t)|||2+D2|||˘ϱ(l+1,k)(t)|||2c69|||˘ϱ(l,k)(t)|||2+C(D,D),

    where C(D,D)0 as D and D tend to zero.

    From these considerations, the sequence {ϱ(D)}D>0 forms the Cauchy sequence in V2m(T). This completes the proof of Lemma 6.2.

    By Lemma 6.2, we see that ˆϱ belongs to V2m(T). Now, we show that ˆϱ certainly satisfies (2.5). To do this, we take an arbitrary function h(θ,t)C1(0,T;C0(Ω)) satisfying h(θ,t)|t=T=0, h(θ,t)|t=00, and consider

    T0dtΩ{ϱ(D)t+ωϱ(D)θD2ϱ(D)θ2+θ(F[ϱ(D),ϱ(D)])}h(θ,t)dθ=0(x,ω)R2, (6.11)

    In virtue of (6.9)-(6.10), if we make D tend to zero,

    T0dtΩ{ϱ(D)t+ωϱ(D)θD2ϱ(D)θ2}h(θ,t)dθT0dtΩ{ˆϱt+ωˆϱθ}h(θ,t)dθ  (x,ω)R2.

    On the other hand, thanks to the Rellich's theorem[27], we have

    ϱ(D)ˆϱ  inL2(0,T;¯H0)

    strongly as D0; therefore,

    T0dtΩθ(F[ϱ(D),ϱ(D)])h(θ,t)dθT0dtΩθ(F[ˆϱ,ˆϱ])h(θ,t)dθ

    holds. Thus, we arrive at

    T0dtΩ{ˆϱt+ωˆϱθ+θ(F[ˆϱ,ˆϱ])}h(θ,t)dθ=0, (6.12)

    which means that ˆϱ certainly satisfies (2.5)1. Next, integrate (6.11) and (6.12) by part with respect to t, and the assumptions on h yield

    ϱ0(θ;x,ω)h(θ,0)T0dtΩϱ(D)(θ,t;x,ω)ht(θ,t)dθ    +T0dtΩ{ωϱ(D)θD2ϱ(D)θ2+θ(F[ϱ(D),ϱ(D)])}h(θ,t)dθ=0, (6.13)
    ˆϱ(θ,0;x,ω)h(θ,0)T0dtΩˆϱ(θ,t;x,ω)ht(θ,t)dθ+T0dtΩ{ωˆϱθ+θ(F[ˆϱ,ˆϱ])}h(θ,t)dθ=0, (6.14)

    respectively. Comparing (6.13) and (6.14) with the aid of (6.9)-(6.10) implies \hat{\varrho}|_{t=0} = \varrho_{(0)}, and so the initial condition (2.5)_3 is satisfied. The periodicity (2.5)_2 obviously holds due to the function space to which \hat{\varrho} belongs. Thus, \hat{\varrho}=\varrho_{(0)}. We summarize these arguments as follows.

    Theorem 6.3. Let T>0 be an arbitrary number, and impose the same assumptions as in Theorem 4.1. Then, the solution \varrho_{(D)} of (2.3) converges to that of (2.5) in {\mathcal V}^{2m}(T), which is denoted as \varrho_{(0)}, as D tends to zero.

    In this paper, we discussed the mathematical analysis of the nonlinear Fokker-Planck equation of Kuramoto's non-local coupling model of oscillators. We first showed the local and global-in-time solvability, and then the nonlinear asymptotic stability of the incoherent state. Finally, we verified the existence of the vanishing diffusion limit solution as the diffusion coefficient tends to zero.

    Our future work will be concerned with the mathematical stability analysis of the chimera state of this model and the coupled oscillator model on the complex graph. We will also tackle the bifurcation problem.



    [1] Kanzari F, Syakti AD, Asia L, et al. (2014) Distributions and sources of persistent organic pollutants (aliphatic hydrocarbons, PAHs, PCBs and pesticides) in surface sediments of an industrialized urban river (Huveaune), France. Sci Total Environ 478: 141-151. doi: 10.1016/j.scitotenv.2014.01.065
    [2] Abdel-Shafy HI, Mansour MSM (2016) A review on polycyclic aromatic hydrocarbons: source, environmental impact, effect on human health and remediation. Egypt J Pet 25: 107-123. doi: 10.1016/j.ejpe.2015.03.011
    [3] Lin Y, Qiu X, Ma Y, et al. (2015) Concentrations and spatial distribution of polycyclic aromatic hydrocarbons (PAHs) and nitrated PAHs (NPAHs) in the atmosphere of North China, and the transformation from PAHs to NPAHs. Environ Pollut 196: 164-170. doi: 10.1016/j.envpol.2014.10.005
    [4] Friedman CL, Pierce JR, Selin NE (2014) Assessing the Influence of Secondary Organic versus Primary Carbonaceous Aerosols on Long-Range Atmospheric Polycyclic Aromatic Hydrocarbon Transport. Environ Sci Technol 48: 3293-3302. doi: 10.1021/es405219r
    [5] Basavaiah N, Mohite RD, Singare PU, et al. (2017) Vertical distribution, composition profiles, sources and toxicity assessment of PAH residues in the reclaimed mudflat sediments from the adjacent Thane Creek of Mumbai. Mar Pollut Bull 118: 112-124. doi: 10.1016/j.marpolbul.2017.02.049
    [6] Singare PU (2015) Studies on Polycyclic Aromatic Hydrocarbons in Sediments of Mithi River of Mumbai, India: Assessment of Sources, Toxicity Risk and Biological Impact. Mar Pollut Bull 101: 232-242.
    [7] Tongo I, Ezemonye L, Akpeh K (2017) Levels, distribution and characterization of polycyclic aromatic hydrocarbons (PAHs) in Ovia river, Southern Nigeria. J Environ Chem Eng 5: 504-512. doi: 10.1016/j.jece.2016.12.035
    [8] Hong WJ, Jia H, Li YF et al. (2016) Polycyclic aromatic hydrocarbons (PAHs) and alkylated PAHs in the coastal seawater, surface sediment and oyster from Dalian, Northeast China. Ecotoxicol Environ Saf 128: 11-20. doi: 10.1016/j.ecoenv.2016.02.003
    [9] Qin XB, Sun HW, Wang CP, et al. (2010) Impacts of crab bioturbation on the fate of polycyclic aromatic hydrocarbons in sediment from the Beitang Estuary of Tianjin, China. Environ Toxicol Chem 29: 1248-1255.
    [10] Gu YG, Lin Q, Lu TT, et al. (2013) Levels, composition profiles and sources of polycyclic aromatic hydrocarbons in surface sediments from Nan'ao Island, a representative mariculture base in South China. Mar Pollut Bull 75: 310-316. doi: 10.1016/j.marpolbul.2013.07.039
    [11] Hawliczek A, Nota B, Cenijn P, et al. (2012) Developmental toxicity and endocrine disrupting potency of 4-azapyrene, benzo[b]fluorene and retene in the zebrafish Danio rerio. Reprod Toxicol 33: 213-223. doi: 10.1016/j.reprotox.2011.11.001
    [12] Liu LY, Wang JZ, Wei GL, et al. (2012) Polycyclic aromatic hydrocarbons (PAHs) in continental shelf sediment of China: implications for anthropogenic influences on coastal marine environment. Environ Pollut 167: 155-162. doi: 10.1016/j.envpol.2012.03.038
    [13] Lewis MA, Russel MJ (2015) Contaminant profiles for surface water, sediment, flora and fauna associated with the mangrove fringe along middle and lower eastern Tampa Bay. Mar Pollut Bull 95: 273-282. doi: 10.1016/j.marpolbul.2015.04.001
    [14] Santana JL, Massone CG, Valdes M, et al. (2015) Occurrence and source appraisal of polycyclic aromatic hydrocarbons (PAHs) in surface waters of the Almendares River, Cuba. Arch Environ Contam Toxicol 69: 143-152. doi: 10.1007/s00244-015-0136-9
    [15] Sarria-Villa R, Ocampo-Duque W, Paez M, et al. (2016) Presence of PAHs in water and sediments of the Colombian Cauca River during heavy rain episodes, and implications for risk assessment. Sci Total Environ 540: 455-465. doi: 10.1016/j.scitotenv.2015.07.020
    [16] Singare PU (2016) Carcinogenic and endocrine disrupting PAHs in the aquatic ecosystem of India. Environ Monit Assess 188: 1-25. doi: 10.1007/s10661-015-4999-z
    [17] Yan J, Liu J, Shi X, et al. (2016) Polycyclic aromatic hydrocarbons (PAHs) in water from three estuaries of China: distribution, seasonal variations and ecological risk assessment. Mar Pollut Bull 109: 471-479. doi: 10.1016/j.marpolbul.2016.05.025
    [18] Manoli E, Samara C, Konstantinou I, et al. (2000) Polycyclic aromatic hydrocarbons in the bulk precipitation and surface waters of Northern Greece. Chemosphere 41: 1845-1855. doi: 10.1016/S0045-6535(00)00134-X
    [19] Mane S, Sundaram S (2014) Studies on some aspects on the biology of green mussel Perna viridis (Linnaeus, 1758) from Versova creek, Mumbai, northwest coast of India. Int Res J Sci Eng 2: 47-50.
    [20] Shirke S, Pinto SM, Kushwaha VK, et al. (2016) Object-based image analysis for the impact of sewage pollution in Malad Creek, Mumbai, India. Environ Monit Assess 188: 95-99. doi: 10.1007/s10661-015-4981-9
    [21] Zeng EY, Yu CC, Tran K (1999) In situ measurements of chlorinated hydrocarbons in the water column off the Palos Verdes Peninsula, California. Environ Sci Technol 33: 392-398. doi: 10.1021/es980561e
    [22] WHO (1998) Polynuclear aromatic hydrocarbons. Guidelines for Drinking-Water Quality, 2nd edition. Addendum to Vol. 2 Health Criteria and Other Supporting Information. World Health Organization, Geneva, pp. 123-152.
    [23] Xiang N, Jiang C, Yang T, et al. (2018) Occurrence and distribution of Polycyclic aromatic hydrocarbons (PAHs) in seawater, sediments and corals from Hainan Island, China. Ecotoxicol Environ Saf 152: 8-15. doi: 10.1016/j.ecoenv.2018.01.006
    [24] Santos E, Souza MRR, Vilela Junior AR, et al. (2018) Polycyclic aromatic hydrocarbons (PAH) in superficial water from a tropical estuarine system: Distribution, seasonal variations, sources and ecological risk assessment. Mar Pollut Bull 127: 352-358. doi: 10.1016/j.marpolbul.2017.12.014
    [25] Nwineewii JD, Marcus AC (2015) Polycyclic Aromatic Hydrocarbons (PAHs) In Surface Water and Their Toxicological Effects in Some Creeks of South East Rivers State (Niger Delta) Nigeria. IOSR J Environ Sci Toxicol Food Technol 9: 27-30.
    [26] Yang D, Qi SH, Zhang Y, et al. (2013) Levels, sources and potential risks of polycyclic aromatic hydrocarbons (PAHs) in multimedia environment along the Jinjiang River mainstream to Quanzhou Bay, China. Mar Pollut Bull 76: 298-306. doi: 10.1016/j.marpolbul.2013.08.016
    [27] Adeniji AO, Okoh OO, Okoh AI (2019) Levels of Polycyclic Aromatic Hydrocarbons in the Water and Sediment of Bufalo River Estuary, South Africa and Their Health Risk Assessment. Arch Environ Contam Toxicol 76: 657-669. doi: 10.1007/s00244-019-00617-w
    [28] Edokpayi JN, Odiyo JO, Popoola OE, et al. (2016) Determination and Distribution of Polycyclic Aromatic Hydrocarbons in Rivers, Sediments and Wastewater Effluents in Vhembe District, South Africa. Int J Environ Res Public Health 13: 387, 1-12. doi: 10.3390/ijerph13040387
    [29] Nekhavhambe TJ., van Ree T, Fatoki OS (2014) Determination and distribution of polycyclic aromatic hydrocarbons in rivers, surface runoff, and sediments in and around Thohoyandou, Limpopo Province, South Africa. Water SA 40: 415-424.
    [30] Eganhouse RP, Simoneit BRT, Kaplan IR (1981) Extractable organic matter in urban stormwater runoff. 2. Molecular characterization. Environ Sci Technol 15: 315-326.
    [31] Hoffman EJ, Mills GL, Latimer JS, et al. (1984) Urban runoff as a source of polycyclic aromatic hydrocarbons to coastal waters. Environ Sci Technol 18: 580-587. doi: 10.1021/es00126a003
    [32] Agarwal T, Khillare P, Shridhar V, et al. (2009) Pattern, sources and toxic potential of PAHs in the agricultural soils of Delhi, India. J Hazard Mater 163: 1033-1039. doi: 10.1016/j.jhazmat.2008.07.058
    [33] Xing XL, Qi S, Zhang J, et al. (2011) Spatial distribution and source diagnosis of polycyclic aromatic hydrocarbons in soils from Chengdu Economic Region, Sichuan Province, western China. J Geochem Explor 110: 146-154. doi: 10.1016/j.gexplo.2011.05.001
    [34] Sprovieri M, Feo ML, Prevedello L, et al. (2007) Heavy metals, polycyclic aromatic hydrocarbons and polychlorinated biphenyls in surface sediments of the Naples harbor (southern Italy). Chemosphere 67: 998-1009. doi: 10.1016/j.chemosphere.2006.10.055
    [35] Ravindra K, Wauters E, Grieken RV (2008) Variation in particulate PAHs levels and their relation with the transboundary movement of the air masses. Sci Total Environ 396: 100-110. doi: 10.1016/j.scitotenv.2008.02.018
    [36] Tobiszewski M, Namiesnik J (2012) PAH diagnostic ratios for the identification of pollution emission sources. Environ Pollut 162: 110-119. doi: 10.1016/j.envpol.2011.10.025
    [37] Cao ZH, Wang YQ, Ma YM, et al. (2005) Occurrence and distribution of polycyclic aromatic hydrocarbons in reclaimed water and surface water of Tianjin, China. J Hazard Mater 122: 51-59. doi: 10.1016/j.jhazmat.2005.04.003
    [38] Boonyatumanond R, Wattayakorn G, Togo A, et al. (2006) Distribution and origins of polycyclic aromatic hydrocarbons (PAHs) in riverine, estuarine, and marine sediments in Thailand. Mar Pollut Bull52: 942-956. doi: 10.1016/j.marpolbul.2005.12.015
    [39] Mostert MMR., Ayoko GA, Kokot S (2010) Application of chemometrics to analysis of soil pollutants. Trends Anal Chem 29: 430-435.
    [40] Mai BX, Qi SH, Zeng EY, et al. (2003) Distribution of polycyclic aromatic hydrocarbons in the coastal region off Macao, China: assessment of input sources and transport pathways using compositional analysis. Environ Sci Technol 37: 4855-4863. doi: 10.1021/es034514k
    [41] Rocher V, Azimi S, Moilleron R, et al. (2004) Hydrocarbons and heavy metals in the different sewer deposits in the Le Marais' catchment (Paris, France): stocks, distributions and origins. Sci Total Environ 323: 107-122. doi: 10.1016/j.scitotenv.2003.10.010
    [42] Wang XC, Sun S, Ma HQ, et al. (2006) Sources and distribution of aliphatic and polyaromatic hydrocarbons in sediments of Jiaozhou Bay, Qingdao, China. Mar Pollut Bull 52: 129-138. doi: 10.1016/j.marpolbul.2005.08.010
    [43] Montuori P, Aurino S, Garzonio F, et al. (2016) Distribution, sources and ecological risk assessment of polycyclic aromatic hydrocarbons in water and sediments from Tiber River and estuary, Italy. Sci Total Environ 566-567: 1254-1267. doi: 10.1016/j.scitotenv.2016.05.183
    [44] Zhang W, Zhang S, Wan C, et al. (2008) Source diagnostics of polycyclic aromatic hydrocarbons in urban road runoff, dust, rain and canopy throughfall. Environ Pollut 153: 594-601. doi: 10.1016/j.envpol.2007.09.004
    [45] Chung MK, Hu R, Cheung KC, et al. (2007) Pollutants in Hongkong soils: polycyclicaromatic hydrocarbons. Chemosphere 67: 464-473. doi: 10.1016/j.chemosphere.2006.09.062
    [46] Li G, Xia X, Yang Z, et al. (2006) Distribution and sources of polycyclic aromatic hydrocarbons in the middle and lower reaches of the Yellow River, China. Environ Pollut 144: 985-993. doi: 10.1016/j.envpol.2006.01.047
    [47] De La Torre-Roche RJ, Lee WY, Campos-Diaz SI (2009) Soil-borne polycyclic aromatic hydrocarbons in El Paso, Texas: analysis of a potential problem in the United States/Mexico border region. J Hazard Mater 163: 946-958. doi: 10.1016/j.jhazmat.2008.07.089
    [48] Akyuz M, Cabuk H (2010) Gaseparticle partitioning and seasonal variation of polycyclic aromatic hydrocarbons in the atmosphere of Zonguldak, Turkey. Sci Total Environ 408: 5550-5558. doi: 10.1016/j.scitotenv.2010.07.063
    [49] Dhananjayan V, Muralidharan S, Peter VR (2012) Occurrence and distribution of polycyclic aromatic hydrocarbons in water and sediment collected along the Harbour Line, Mumbai, India. Int J Oceanogr Article ID 403615, 7.
    [50] Katsoyiannis A, Sweetman AJ, Jones KC (2011) PAH molecular diagnostic ratios applied to atmospheric sources: a critical evaluation using two Decades of source Inventory and air concentration data from the UK. Environ Sci Technol 45: 8897-8906. doi: 10.1021/es202277u
    [51] Pozo K, Perra G, Menchi V, et al. (2011) Levels and spatial distribution of polycyclic aromatic hydrocarbons (PAHs) in sediments from Lenga Estuary, central Chile. Mar Pollut Bull 62: 1572-1576. doi: 10.1016/j.marpolbul.2011.04.037
    [52] Law RJ, Dawes VJ, Woodhead RJ, et al. (1997) Polycyclic aromatic hydrocarbons (PAH) in seawater around England and Wales. Mar Pollut Bull 34: 306-322.
    [53] Barron MG, Podrabsky T, Ogle S, et al. (1999) Are aromatic hydrocarbons the primary determinant of petroleum toxicity to aquatic organisms? Aquat Toxicol 46: 253-268. doi: 10.1016/S0166-445X(98)00127-1
    [54] Agroudy NA, Soliman YA, Hamed MA, et al. (2017) Distribution of PAHs in Water, Sediments Samples of Suez Canal During 2011. J Aquat Pol Toxicol 1: 1-10.
    [55] Pohl A, Kostecki M, Jureczko I, et al. (2018) Polycyclic aromatic hydrocarbons in water and bottom sediments of a shallow, lowland dammed reservoir (on the example of the reservoir Blachownia, South Poland). Arch Environ Prot 44: 10-23.
    [56] US Environmental Protection Agency (USEPA), (2012) Regional screening levels for chemical contaminants at superfund sites. Regional screening table. User's guide. (Access date: November 2012). < http://www.epa.gov/reg3hwmd/risk/human/rb-concentration_table/usersguide.htm > .
    [57] Di Toro DM, McGrath JA, Hansen DJ (2000) Technical basis for narcotic chemicals and polycyclic aromatic hydrocarbon criteria. I. Water and tissue. Environ Toxicol Chem 19: 1951-1970.
    [58] Kalf DF, Crommentuijn T, van de Plassche EJ (1997) Environmental quality objectives for 10 polycyclic aromatic hydrocarbons (PAHs). Ecotoxicol Environ Saf 36: 89-97. doi: 10.1006/eesa.1996.1495
    [59] Canadian Council of Ministers of the Environment (CCME) (2010) Canadian Soil Quality Guidelines, Carcinogenic and Other Polycyclic Aromatic Hydrocarbons (PAHs)-Environmental and Human Health Effects. ISBN 978-1-896997-94-0 PDF.
    [60] Yang B, Xue N, Zhou L, et al. (2012) Risk assessment and sources of polycyclic aromatic hydrocarbons in agricultural soils of Huanghuai plain, China. Ecotoxicol Environ Saf 84: 304-310. doi: 10.1016/j.ecoenv.2012.07.027
    [61] Omayma EA, Sawsan AM, El Nady MM (2016) Application of polycyclic aromatic hydrocarbons in identification of organic pollution in seawater around Alexandria coastal area, Egypt. J Environ Life Sci 1: 39-55.
    [62] Daisey JM, Leyko MA, Kneip TJ (1979) Source identification and allocation of polynuclear aromatic hydrocarbon compounds in the New York City aerosol: methods and applications. In: Jones, P.W., Leber, P. (Eds.), Polynuclear Aromatic Hydrocarbons. Ann Arbor Science, Ann Arbor, pp. 201-215.
    [63] Harrison RM, Smith DJT, Luhana L (1996) Source apportionment of atmospheric polycyclic aromatic hydrocarbons collected from an urban location in Birmingham, UK. Environ Sci Technol 30: 825-832. doi: 10.1021/es950252d
    [64] Rogge WF, Hildemann LM, Mazurek MA, et al. (1993) Source of fine organic aerosol 2. Noncatalyst and catalyst-equipped automobiles and heavy-duty diesel trucks. Environ Sci Technol 27: 636-651.
    [65] Gschwend PM, Hites RA (1981) Fluxes of polycyclic aromatic hydrocarbons to marine and lacustrine sediments in the northeastern United States. Geochimica et Cosmochimica Acta 45: 2359-2367. doi: 10.1016/0016-7037(81)90089-2
    [66] Mitra S, Bianchi TS, Mckee BA, et al. (2002) Black carbon from the Mississippi River: quantities, sources and potential implications for the global carbon cycle. Environ Sci Technol 36: 2296-2302. doi: 10.1021/es015834b
    [67] Masclet P, Bresson MA, Mouvier G (1987) Polycyclic aromatic hydrocarbons emitted by power station, and influence of combustion conditions. Fuel 66: 556-562. doi: 10.1016/0016-2361(87)90163-3
  • This article has been cited by:

    1. Hirotada Honda, On Kuramoto-Sakaguchi-type Fokker-Planck equation with delay, 2023, 19, 1556-1801, 1, 10.3934/nhm.2024001
  • Reader Comments
  • © 2021 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(2179) PDF downloads(133) Cited by(0)

Figures and Tables

Figures(2)  /  Tables(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog