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Research article Special Issues

Dirichlet characters of the rational polynomials

  • Received: 27 July 2021 Accepted: 22 November 2021 Published: 02 December 2021
  • MSC : 11L05, 11L10

  • Denote by χ a Dirichlet character modulo q3, and ¯a means a¯a1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form

    qa=1χ(ma+¯a),

    where qa=1 denotes the summation over all 1aq with (a,q)=1. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [6] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.

    Citation: Wenjia Guo, Xiaoge Liu, Tianping Zhang. Dirichlet characters of the rational polynomials[J]. AIMS Mathematics, 2022, 7(3): 3494-3508. doi: 10.3934/math.2022194

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  • Denote by χ a Dirichlet character modulo q3, and ¯a means a¯a1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form

    qa=1χ(ma+¯a),

    where qa=1 denotes the summation over all 1aq with (a,q)=1. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [6] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.



    The objective of this work is to deal with the rotation-Camassa-Holm model (RCH)

    mt+Vmx+2Vxm+kVxα0αVxxx+h1β2V2Vx+h2β3V3Vx=0, (1.1)

    where m=VVxx,

    k=1+ϝ2ϝ,β=k21+k2,α0=k(k4+6k21)6(k2+1)2,α=3k4+8k216(k2+1)2h1=3k(k21)(k22)2(1+k2)3,h2=(k22)(k21)2(8k21)2(1+k2)5, (1.2)

    in which constant ϝ is a parameter to depict the Coriolis effect due to the Earth's rotation. Gui et al. [9] derive the nonlinear RCH equation (1.1) (also see [3,10]), depicting the motion of the fluid associated with the Coriolis effect.

    Recently, many works focus on the study of Eq (1.1). Zhang [23] investigates the well-posedness for Eq (1.1) on the torus in the sense of Hadamard if assuming its initial value in the space Hs with the Sobolev index s>32, and gives a Cauchy-Kowalevski type proposition for Eq (1.1) under certain conditions. It is shown in Gui et al. [9] that Eq (1.1) has similar dynamical features with those of Camassa-Holm and irrotational Euler equations. The travelling wave solutions are found and classified in [10]. The well-posedness, geometrical analysis and a more general classification of travelling wave solution for Eq (1.1) are carried out in Silva and Freire [17]. Tu et al. [20] investigate the well-posedness of the global conservative solutions to Eq (1.1).

    If ϝ=0 (implying h1=h2=0), namely, the Coriolis effects disappear, Eq (1.1) becomes the standard Camassa-Holm (CH) model [2], which has been investigated by many scholars [1,6,7,8,16]. For some dynamical characteristics of the CH, we refer the reader to the references [11,12,13,14,15,22].

    Motivated by the works made in [4,21], in which the H1(R) global weak solution to the CH model is studied without restricting that the initial value obeys the sign condition, we investigate the rotation-Camassa-Holm equation (1.1) and utilize the viscous approximation technique to handle the existence of global weak solution in H1(R). As the term Vxxx appears in Eq (1.1), it yields difficulties to establish estimates of solutions for the viscous approximation of Eq (1.1) (In fact, using a change of coordinates, Silva and Freire [18] eliminate the term Vxxx and discuss other dynamical features of Eq (1.1)). The key contribution of this work is that we overcome these difficulties and establish a high order integrable estimate and prove that V(t,x)x possesses upper bound. These two estimates take key roles in proving the existence of the H1(R) global weak solution for Eq (1.1) without the sign condition.

    This work is structured by the following steps. Definition of the H1(R) global weak solutions and several Lemmas are given in Section 2. The main conclusion and its proof are presented in Section 3.

    We rewrite the initial value problem for the RCH equation (1.1)

    {VtVtxx+3VVx+kVx+h1β2V2Vx+h2β3V3Vx=2VxVxx+(V+α0α)Vxxx,V(0,x)=V0(x),xR. (2.1)

    Employing operator Λ2=(12x)1 to multiply Eq (1.1), we have

    {Vt+(V+α0α)Vx+Ax=0,V(0,x)=V0(x), (2.2)

    where

    Ax=Λ2[(kα0α)V+V2+h13β2V3+h24β3V4+12V2x]x.

    It can be found in [9,10,17,18] that

    R(V2+V2x)dx=R(V20+V20x)dx.

    We cite the definition (see [4,21]).

    Definition 2.1. The solution V(t,x):[0,)×RR is called as a global weak solution to system (2.1) or (2.2) if

    (1) VC([0,)×R)L([0,);H1(R));

    (2) V(t,.)H1(R)V0H1(R);

    (3) V=V(t,x) obeys (2.2) in the sense of distribution.

    Define ϕ(x)=e1x21 if |x|<1 and ϕ(x)=0 if |x|1. Set ϕε(x)=ε14ϕ(ε14x) with 0<ε<14. Assume Vε,0=ϕεV0, where represents the convolution, we see that Vε,0C for V0(x)Hs,s>0. To discuss global weak solutions for Eq (1.1), we handle the following viscous approximation problem:

    {Vεt+(Vε+α0α)Vεx+Aεx=ε2Vεx2,V(0,x)=Vε,0(x), (2.3)

    in which

    Aε(t,x)=Λ2[(kα0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12(Vεx)2].

    Utilizing (2.3) and denoting pε(t,x)=Vε(t,x)x yield

    pεt+(Vε+α0α)pεxε2pεx2+12p2ε=(kα0α)Vε+V2ε+h13β2V3ε+h24β3V4εΛ2(V2ε+(kα0α)Vε+h13β2V3ε+h24β3V4ε+12(Vεx)2)=Bε(t,x). (2.4)

    Simply for writing, let c represent arbitrary positive constants (independent of ε).

    Lemma 2.1. Let V0H1(R). For each number σ2, system (2.3) has a unique solution Vε C([0,);Hσ(R)) and

    R(V2ε+(Vεx)2)dx+2εt0R[(Vεx)2+(2Vεx2)2](s,x)dxds=Vε,02H1(R), (2.5)

    which has the equivalent expression

    Vε(t,.)2H1(R)+2εt0Vεx(s,.)2H1(R)ds=Vε,02H1(R).

    Proof. For parameter σ2, we acquire Vε,0C([0,);H(R)). Employing the conclusion in [5] derives that system (2.3) has a unique solution Vε(t,x)C([0,);Hσ(R)). Using (2.3) arises

    ddtR(V2ε+V2εx)dx=2RVε(VεtVεtxx)dx=2εR(VεVεxxVεVεxxxx)dx=2εR((Vεx)2+(Vεxx)2)dx.

    Integrating about variable t for both sides of the above identity, we obtain (2.5).

    In fact, as ε0, we have

    VεL(R)VεH1(R)Vε,0H1(R)V0H1(R), and Vε,0V0inH1(R). (2.6)

    Lemma 2.2. If V0(x)H1(R), for Aε(t,x) and Bε(t,x), then

    Aε(t,)L(R)c,Aεx(t,)L(R)c, (2.7)
    Aε(t,)L1(R)c,Aεx(t,)L1(R)c, (2.8)
    Aε(t,)L2(R)c,Aεx(t,)L2(R)c, (2.9)

    and

    Bε(t,)L(R)c,Bε(t,)L2(R)c, (2.10)

    where c=c(V0H1(R)).

    Proof. For any function U(x) and the operator Λ2, it holds that

    Λ2U(x)=12Re|xy|U(y)dy for U(x)Lr(R),1r, (2.11)

    and

    |Λ2Ux(x)|=|12Re|xy|U(y)ydy|=|12exxU(y)dy+12exxeyU(y)dy|12e|xy||U(y)|dy. (2.12)

    Utilizing (2.6), (2.11), (2.12), the expression of function Aε(t,x) and the Tonelli theorem, we have

    Λ2((kα0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12V2εx)L(R)c

    and

    Λ2((kα0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12V2εx)xL(R)c,

    which derive that (2.7) and (2.8) hold. Utilizing (2.7) and (2.8) yields

    Aε(t,)2L2(R)Aε(t,)L(R)Aε(t,)L1(R)c

    and

    Aε(t,)x2L2(R)Aε(t,)xL(R)Aε(t,)xL1(R)c,

    which complete the proof of (2.9). Furthermore, using (2.4) and (2.6), we have

    BεL(R)c,BεL2(R)c,

    which finishes the proof of (2.10).

    Lemma 2.3. Provided that 0<α1<1, T>0, constants a<b, then

    T0ba|Vε(t,x)x|2+α1dxdtc1, (2.13)

    where constant c1 depends on a,b,α1,T,k and V0H1(R).

    Proof. We utilize the methods in Xin and Zhang [21] to prove this lemma. Let function g(x)C(R) and satisfy

    0g(x)1,g(x)={0,x(,a1][b+1,),1,x[a,b].

    Define function f(η):=η(|η|+1)α1,ηR. We note that the function f belongs to C1(R) except η=0. Here we give the expressions of its first and second derivatives as follows:

    f(η)=((α1+1)|η|+1)(|η|+1)α11,f(η)=α1sign(η)(|η|+1)α12((α1+1)|η|+2)=α1(α1+1)sign(η)(|η|+1)α11+(1α1)α1sign(η)(|η|+1)α12,

    from which we have

    |f(η)||η|α1+1+|η|,|f(η)|(α1+1)|η|+1,|f(η)|2α1, (2.14)

    and

    ηf(η)12η2f(η)=1α12η2(|η|+1)α1+α12η2(|η|+1)α111α12η2(|η|+1)α1. (2.15)

    Note that

    T0Rg(x)f(pε)pεtdxdt=Rg(x)dxT0df(pε)=R[f(pε(T,x))f(pε(0,x))]g(x)dx, (2.16)
    T0Rg(x)f(pε)(Vε+α0α)pεxdxdt=T0dtRg(x)(Vε+α0α)df(pε)=Rf(pε)[g(x)(Vε+α0α)+g(x)pε]dx. (2.17)

    Making use of g(x)f(pε) to multiply (2.4), from (2.16) and (2.17), integrating over ([0,)×R) by parts, we obtain

    T0Rg(x)pεf(pε)dtdx12T0Rp2εg(x)f(pε)dtdx=R[f(pε(T,x))f(pε(0,x))]g(x)dx+T0R(Vε+α0α)g(x)f(pε)dtdx+εT0Rg(x)f(pε)pεxdtdx+εT0Rg(x)f(pε)(pεx)2dtdxT0RBεf(pε)g(x)dtdx. (2.18)

    Applying (2.15) yields

    T0Rg(x)pεf(pε)dtdx12T0Rp2εg(x)f(pε)dtdx=T0Rg(x)(pεf(pε)12p2εf(pε))dtdx(1α1)2T0Rg(x)p2ε(|pε|+1)α1dtdx. (2.19)

    For t0, using 0<α1<1, (2.14) and the Hölder inequality gives rise to

    |Rg(x)f(pε)dx|Rg(x)(|pε|α1+1+|pε|)dxg(x)L2/(1α1)(R)pε(t,)α1+1L2(R)+g(x)L2(R)pε(t,)L2(R)(b+2a)(1α1)/2V0α1+1H1(R)+(b+2a)1/2V0H1(R), (2.20)

    and

    |T0RVεg(x)f(pε)dtdx|T0R|Vεg(x)|(|pε|α1+1+|pε|)dtdxT0RVε(t,)L(R)|g(x)|(|pε|α1+1+|pε|)dtdxcT0(g(x)L2/(1α1)(R)pε(t,)α1+1L2(R)+g(x)L2(R)pε(t,)L2(R))dtcT0(g(x)L2/(1α1)(R)V0α1+1L2(R)+g(x)L2(R)V0L2(R))dt. (2.21)

    Moreover, we have

    εT0Rpεxg(x)f(pε)dtdx=εT0Rf(pε)g(x)dtdx. (2.22)

    Utilizing the Hölder inequality and (2.14) leads to

    |εT0Rg(x)pεxf(pε)dtdx|εT0R|f(pε)g(x)|dtdxεT0R(|pε|α1+1+|pε|)| g(x)|dtdxεT0( gL2/(1α1)(R)pε(t,)α1+1L2(R)+gL2(R)pε(t,)L2(R))dtεT(gL2/(1α1)(R)V0α1+1H1(R)+gL2(R)V0H1(R)). (2.23)

    Using the last part of (2.14), we have

    ε|ΠT(pεx)2g(x)f(pε)dtdx|2α1εΠT(pεx)2dtdxα1V02H1(R). (2.24)

    As shown in Lemma 2.2, there exists a constant c0>0 to ensure that

    BεL(R)c0. (2.25)

    Utilizing the second part in (2.14) arises

    |T0RBεg(x)f(pε)dtdx|c0T0R g(x)[(α1+1)|pε|+1]dtdxc0T0((α1+1) g(x)L2(R)pε(t,)L2(R)+g(x)L1(R))dtc0T. (2.26)

    Applying (2.18)–(2.26) yields

    1α12T0R|pε|2f(x)(1+|pε|α1)dtdxc,

    where c>0 relies only on T>0,a,b,α1 and V0H1(R). Furthermore, we have

    T0ba|Vεx(t,x)|2+α1dxdtT0R|pε|g(x)(|pε|+1)α1+1dtdx2c(1α1).

    The proof of (2.13) is completed.

    Lemma 2.4. For (t,x)(0,)×R, provided that Vε=Vε(t,x) satisfy problem (2.3), then

    Vε(t,x)x2t+c, (2.27)

    in which positive constant c=c(V0H1(R)).

    Proof. Using Lemma 2.2 gives rise to

    pεt+(Vε+α0α)pεxε2pεx2+12p2ε=Bε(t,x)c. (2.28)

    Assume that H=H(t) satisfies the problem

    dHdt+12H2=c,t>0,H(0)=Vε,0xL.

    Due to (2.28), we know that H=H(t) is a supersolution* of parabolic equation (2.4) associated with initial value Vε,0x. Utilizing the comparison principle for parabolic equations arises

    *The supersolution is defined by supxRpε(t,x). If there exists a point (t,x0) such that supxRpε(t,x))=pε(t,x0), then pε(t,x0)x=0 and 2pε(t,x0)x2<0.

    pε(t,x)H(t).

    We choose the function F(t):=2t+2c,t>0. Since dFdt(t)+12F2(t)c= 22ct>0, we conclude

    H(t)F(t),

    which finishes the proof of (2.27).

    Lemma 2.5. There exists a subsequence {εj}jN,εj0 and V(t,x)L([0,);H1(R))H1([0,T]×R), for every T0, such that

    VεjV in H1([0,T]×R),VεjV in L loc ([0,)×R).

    The proof of Lemma 2.5 can be found in Coclite el al. [4].

    Lemma 2.6. Assume V0H1(R). Then {Bε(t,x)}ε is uniformly bounded in W1,1loc([0,)×R). Moreover, there has a sequence {εj}jN, εj0 to guarantee that

    BεjB strongly in Lr loc ([0,T)×R),

    where function BL([0,T);W1,(R)) and 1<r<.

    The standard proof of Lemma 2.6 can be found in [4]. We omit its proof here.

    For conciseness, we use overbars to denote weak limits which are taken in the space Lr[(0,)×R) with 1<r<3.

    Lemma 2.7. There exist a sequence {εj}jN tending to zero and two functions pLrloc([0,)×R),¯p2Lr1loc([0,)×R) such that

    pεjp in Lrloc([0,)×R),pεjp in Lloc([0,);L2(R)), (2.29)
    p2εj¯p2 in Lr1loc([0,)×R) (2.30)

    for each 1<r<3 and 1<r1<32. In addition, it holds that

    p2(t,x)¯p2(t,x), (2.31)
    Vx=p in the sense of distribution . (2.32)

    Proof. Lemmas 2.1 and 2.2 validate (2.29) and (2.30). The weak convergence in (2.30) ensures the reasonableness of (2.31). Using Lemma 2.5 and (2.29) derives that (2.32) holds.

    For conciseness in the following discussion, we denote {pεj}jN, {Vεj}jN and {Bεj}jN by {pε}ε>0, {Vε}ε>0 and {Bε}ε>0. Assume that FC1(R) is an arbitrary convex function with F being bounded, Lipschitz continuous on R. Using (2.29) derives that

    F(pε)¯F(p) in Lrloc([0,)×R),F(pε)¯F(p) in Lloc([0,);L2(R)).

    Multiplying (2.4) by F(pε) yields

    tF(pε)+x((Vε+α0α)F(pε))ε2x2F(pε)+εF(pε)(pεx)2=pεF(pε)12F(pε)p2ε+BεF(pε). (2.33)

    Lemma 2.8. Suppose that FC1(R) is a convex function with F being bounded, Lipschitz continuous on R. In the sense of distribution, then

    ¯F(p)t+x((Vε+α0α)¯F(p))¯pF(p)12¯F(p)p2+B¯F(p), (2.34)

    where ¯pF(p) and ¯F(p)p2 represent the weak limits of pεF(pε) and F(pε)p2ε in Lr1 loc ([0,)×R),1<r1<32, respectively.

    Proof. Applying Lemmas 2.5 and 2.7, letting ε0 in (2.33) and noticing the convexity of function F, we finish the proof of (2.34).

    Lemma 2.9. [4] Almost everywhere in [0,)×R, it has

    p=p++p=¯p++¯p,p2=(p+)2+(p)2,¯p2=¯(p+)2+¯(p)2,

    where η+:=ηχ[0,+)(η),η:=ηχ(,0](η), ηR.

    Using Lemmas 2.4 and 2.7 leads to

    pε,p2t+c,0<t<T.

    Lemma 2.10. For t0,xR, in the sense of distribution, it holds that

    pt+x((Vε+α0α)p)=12¯p2+B(t,x). (2.35)

    Proof. Making use of (2.4), Lemmas 2.5–2.7, we derive that (2.35) holds by letting ε0.

    Lemma 2.11. Provided that FC1(R) is a convex function with FL(R), for every T>0, in the sense of distribution, then

    F(p)t+x((Vε+α0α)F(p))=pF(p)+(12¯p2p2)F(p)+BF(p).

    Proof. Suppose that {wδ}δ is a kind of mollifiers defined in (,). Let pδ(t,x):=(p(t,)wδ)(x) in which denotes the convolution with respect to variable x. Using (2.35) yields

    F(pδ)t=F(pδ)pδt=F(pδ)(x((Vε+α0α)p)wδ+12ˉp2wδ+Bwδ)=F(pδ)[(Vε+α0α)pxwδVp2wδ]+F(pδ)(12V¯q2wδ+Bwδ). (2.36)

    Utilizing the assumptions on F and F and letting δ0 in (2.36), we complete the proof.

    Following the ideas in [21], we hope that the weak convergence of pε should be strong convergence in (2.30). The strong convergence leads to the existence of global weak solution for system (2.1).

    Lemma 2.12. [4] Assume V0H1(R). Then

    limt0Rp2(t,x)dx=limt0R¯p2(t,x)dx=R(V0x)2dx.

    Lemma 2.13. [4] If V0H1(R), L>0, then

    limt0R(¯F±L(p)(t,x)F±L(p)(t,x))dx=0,

    where

    FL(ρ):={12ρ2, if |ρ|L,L|ρ|12L2, if |ρ|>L, (2.37)

    F+L(ρ)=FL(ρ)χ[0,)(ρ) and FL(ρ)=FL(ρ)χ(,0](ρ), ρ(,).

    Lemma 2.14. [4] Let L>0. For FL(ρ) defined in (2.37), then

    {FL(ρ)=12ρ212(L|ρ|)2χ(,L)(L,)(ρ),FL(ρ)=ρ+(L|ρ|)sign(ρ)χ(,L)(L,)(ρ),F+L(ρ)=12(ρ+)212(Lρ)2χ(L,)(ρ),(F+L)(ρ)=ρ++(Lρ)χ(L,)(ρ),FL(ρ)=12(ρ)212(L+ρ)2χ(,L)(ρ),(FL)(ρ)=ρ(L+ρ)χ(,L)(ρ).

    Lemma 2.15. Assume V0H1(R). For almost all t>0, then

    12R(¯(p+)2p2+)(t,x)dxt0RB(s,x)[¯p+(s,x)p+(s,x)]dxds.

    Lemma 2.16. Assume V0H1(R). For almost all t>0, then

    (¯FL(p)FL(p))(t,x)dxL22t0R¯(L+p)χ(,L)(p)dxdsL22t0R(L+p)χ(,L)(p)dxds+Lt0R[¯FL(p)FL(p)]dxds+L2t0R(¯p2+p2+)dxds+t0RB(t,x)(¯(FL)(p)(FL)(p))dxds.

    Using Lemmas 2.8 and 2.11–2.14, the proofs of Lemmas 2.15 and 2.16 are analogous to those of Lemmas 4.4 and 4.5 in Tang et al. [19]. Here we omit their proofs.

    Lemma 2.17. Assume V0H1(R). Almost everywhere in [0,)×(,), it holds that

    ¯p2=p2. (2.38)

    Proof. Using Lemmas 2.15 and 2.16 arises

    R(12[¯(p+)2(p+)2]+[¯FLFL])(t,x)dxL22t0R¯(L+p)χ(,L)(p)dxdsL22t0R(L+p)χ(,L)(p)dxds+Lt0R[¯FL(p)FL(p)]dxds+L2t0R(¯p2+p2+)dxds+t0RB(s,x)([¯p+p+]+[¯(FL)(p)(FL)(p)])dxds. (2.39)

    Applying Lemma 2.6 drives that there has a constant constant N>0 to ensure

    B(t,x)L([0,T)×R)N. (2.40)

    Using Lemmas 2.9 and 2.14 yields

    {p++(FL)(p)=p(L+p)χ(,L),¯p++¯(FL)(p)=p¯(L+p)χ(,L)(p). (2.41)

    Since the map ρρ++(FL)(ρ) is convex, it holds that

    0[¯p+p+]+[¯(FL)(p)(FL)(p)]=(L+p)χ(,L)¯(L+p)χ(,L)(p). (2.42)

    Using (2.40) gives rise to

    B(s,x)([¯p+p+]+[¯(FL)(p)(FL)(p)])N(¯(L+p)χ(,L)(p)(L+p)χ(,L)(p)). (2.43)

    Since ρ(L+ρ)χ(,L)(ρ) is concave, letting L be sufficiently large, we have

    L22¯(L+p)χ(,L)(p)L22(L+p)χ(,L)(p)+B(s,x)([¯p+p+]+[(¯FL)(p)(FL)(p)])(L22N)(¯(L+p)χ(,L)(p)(L+p)χ(,L)(p))0. (2.44)

    Using (2.39)–(2.44) yields

    \begin{eqnarray} 0 &\leq& \int_{\mathbb{R}}\left(\frac{1}{2}\left[\overline{\left(p_{+}\right)^{2}}-\left(p_{+}\right)^{2}\right]+\left[\overline{F_{L}^{-}(p)}-F_{L}^{-}(p)\right]\right)(t, x) d x \\ &\leq& L \int_{0}^{t} \int_{\mathbb{R}}\left(\frac{1}{2}\left[\overline{\left(p_{+}\right)^{2}}-p_{+}^{2}\right]+\left[\overline{F_{L}^{-}(p)}-F_{L}^{-}(p)\right]\right) d s d x, \end{eqnarray}

    which together with the Gronwall inequality yields

    \begin{eqnarray} 0 \leq \int_{\mathbb{R}}\left(\frac{1}{2}\left[\overline{\left(p_{+}\right)^{2}}-\left(p_{+}\right)^{2}\right]+\left[\overline{F_{R}^{-}(p)}-F_{R}^{-}(p)\right]\right)(t, x) d x \leq 0. \end{eqnarray} (2.45)

    Using the Fatou lemma, Lemma 2.9 and (2.45), sending L \rightarrow \infty , it holds that

    \begin{eqnarray} 0 \leq \int_{\mathbb{R}}\left(\overline{p^{2}}-p^{2}\right)(t, x) d x \leq 0, \quad t > 0, \end{eqnarray}

    which finishes the proof of (2.38).

    Theorem 3.1. Assume that V_{0}(x) \in H^{1}(\mathbb{R}) . Then system (2.1) has at least a global weak solution V(t, x) . Furthermore, this weak solution possesses the features:

    (a) For (t, x) \in[0, \infty)\times\mathbb{R} , there exists a positive constant c = c(\|V_{0}\|_{H^{1}(\mathbb{R})}) such that

    \begin{eqnarray} \frac{\partial V(t, x)}{\partial x} \leq \frac{2}{t}+c. \end{eqnarray} (3.1)

    (b) If a, b \in\mathbb{R}, \; a < b , for any 0 < \alpha_1 < 1 and T > 0 , it holds that

    \begin{eqnarray} \int_{0}^{T} \int_{a}^{b}\left|\frac{\partial V(t, x)}{\partial x}\right|^{2+\alpha_1} d x d t \leq c_0, \end{eqnarray} (3.2)

    where positive constant c_0 relies on \alpha_1, k, T, a, b and \left\|V_{0}\right\|_{H^{1}(\mathbb{R})} .

    Proof. Utilizing (2.3), (2.5) and Lemma 2.5, we derive (1) and (2) in Definition 2.1. From Lemma 2.17, we have

    \begin{eqnarray} p_\varepsilon^2\rightarrow p^2\; \text{ in }\; L^1_{loc}([0, \infty)\times\mathbb{R}). \end{eqnarray}

    Employing Lemmas 2.5 and 2.6 results in that V is a global weak solution to system (2.2). Making use of Lemmas 2.3 and 2.4 gives rise to inequalities (3.1) and (3.2). The proof is finished.

    In this work, we study the rotation-Camassa-Holm (RCH) model (1.1), a nonlinear equation describing the motion of equatorial water waves with the Coriolis effect due to the Earth's rotation. The presence of the term V_{xxx} in the RCH equation leads to difficulties of establishing estimates of solutions for the viscous approximation. To overcome these difficulties, we establish a high order integrable estimate and show that \frac{\partial V(t, x)}{\partial x} possesses an upper bound. Using these two estimates and the viscous approximation technique, we examine the existence of H^1(\mathbb{R}) global weak solutions to the RCH equation without the sign condition.

    The authors are very grateful to the reviewers for their valuable and meaningful comments of the paper.

    The authors declare no conflicts of interest.



    [1] G. Pólya, Über die Verteilung der quadratischen Reste und Nichtreste, Göttingen Nachr., 167 (1918), 21–29.
    [2] I. M. Vinogradov, On the distribution of residues and non-residues of powers, J. Phys. Math. Soc. Perm., 1 (1918), 94–96.
    [3] A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. U.S.A., 34 (1948), 204–207. doi: 10.1073/pnas.34.5.204.
    [4] W. P. Zhang, Y. Yi, On Dirichlet characters of polynomials, Bull. Lond. Math. Soc., 34 (2002), 469–473. doi: 10.1112/S0024609302001030. doi: 10.1112/S0024609302001030
    [5] W. P. Zhang, W. L. Yao, A note on the Dirichlet characters of polynomials, Acta Arith., 115 (2004), 225–229. doi: 10.4064/aa115-3-3. doi: 10.4064/aa115-3-3
    [6] W. P. Zhang, T. T. Wang, A note on the Dirichlet characters of polynomials, Math. Slovaca, 64 (2014), 301–310. doi: 10.2478/s12175-014-0204-z. doi: 10.2478/s12175-014-0204-z
    [7] D. A. Burgess, Dirichlet characters and polynomials, Tr. Mat. Inst. Steklova, 132 (1973), 203–205.
    [8] E. A. Grechnikov, An estimate for the sum of Legendre symbols, Math. Notes, 88 (2010), 819–826. doi: 10.1134/S0001434610110222. doi: 10.1134/S0001434610110222
    [9] V. Pigno, C. Pinner, Binomial character sums modulo prime powers, J. Théor. Nombres Bordeaux, 28 (2016), 39–53. doi: 10.5802/jtnb.927. doi: 10.5802/jtnb.927
    [10] X. X. Lv, W. P. Zhang, A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums, Lith. Math. J., 57 (2017), 359–366. doi: 10.1007/s10986-017-9366-z. doi: 10.1007/s10986-017-9366-z
    [11] J. F. Zhang, X. X. Lv, On the character sums of polynomials and L-functions, (Chinese), Acta Math. Sinica (Chin. Ser.), 62 (2019), 903–912.
    [12] X. X. Lv, W. P. Zhang, On the character sum of polynomials and the two-term exponential sums, Acta. Math. Sin.-English Ser., 36 (2020), 196–206. doi: 10.1007/s10114-020-9255-y. doi: 10.1007/s10114-020-9255-y
    [13] A. P. Mangerel, Short character sums and the Pólya-Vinogradov inequality, The Quarterly Journal of Mathematics, 71 (2020), 1281–1308. doi: 10.1093/qmath/haaa031. doi: 10.1093/qmath/haaa031
    [14] P. Xi, Moments of certain character sums that are unnamed, 2021, arXiv: 2105.15051.
    [15] A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Scientifiques et Industrielles, 1948.
    [16] A. Weil, Variétés abéliennes et courbes algébriques, Actualités Scientifiques et Industrielles, 1948.
    [17] L. K. Hua, Introduction to number theory, (Chinese), Peking: Science Press, 1964.
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