Denote by χ a Dirichlet character modulo q≥3, and ¯a means a⋅¯a≡1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form
q∑a=1′χ(ma+¯a),
where q∑a=1′ denotes the summation over all 1≤a≤q 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 [
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 q≥3, and ¯a means a⋅¯a≡1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form
q∑a=1′χ(ma+¯a),
where q∑a=1′ denotes the summation over all 1≤a≤q 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 [
The objective of this work is to deal with the rotation-Camassa-Holm model (RCH)
mt+Vmx+2Vxm+kVx−α0αVxxx+h1β2V2Vx+h2β3V3Vx=0, | (1.1) |
where m=V−Vxx,
k=√1+ϝ2−ϝ,β=k21+k2,α0=k(k4+6k2−1)6(k2+1)2,α=3k4+8k2−16(k2+1)2h1=−3k(k2−1)(k2−2)2(1+k2)3,h2=(k2−2)(k2−1)2(8k2−1)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)
{Vt−Vtxx+3VVx+kVx+h1β2V2Vx+h2β3V3Vx=2VxVxx+(V+α0α)Vxxx,V(0,x)=V0(x),x∈R. | (2.1) |
Employing operator Λ−2=(1−∂2x)−1 to multiply Eq (1.1), we have
{Vt+(V+α0α)Vx+∂A∂x=0,V(0,x)=V0(x), | (2.2) |
where
∂A∂x=Λ−2[(k−α0α)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,∞)×R→R is called as a global weak solution to system (2.1) or (2.2) if
(1) V∈C([0,∞)×R)∩L∞([0,∞);H1(R));
(2) ‖V(t,.)‖H1(R)≤‖V0‖H1(R);
(3) V=V(t,x) obeys (2.2) in the sense of distribution.
Define ϕ(x)=e1x2−1 if |x|<1 and ϕ(x)=0 if |x|≥1. Set ϕε(x)=ε−14ϕ(ε−14x) with 0<ε<14. Assume Vε,0=ϕε⋆V0, where ⋆ represents the convolution, we see that Vε,0∈C∞ for V0(x)∈Hs,s>0. To discuss global weak solutions for Eq (1.1), we handle the following viscous approximation problem:
{∂Vε∂t+(Vε+α0α)∂Vε∂x+∂Aε∂x=ε∂2Vε∂x2,V(0,x)=Vε,0(x), | (2.3) |
in which
Aε(t,x)=Λ−2[(k−α0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12(∂Vε∂x)2]. |
Utilizing (2.3) and denoting pε(t,x)=∂Vε(t,x)∂x yield
∂pε∂t+(Vε+α0α)∂pε∂x−ε∂2pε∂x2+12p2ε=(k−α0α)Vε+V2ε+h13β2V3ε+h24β3V4ε−Λ−2(V2ε+(k−α0α)Vε+h13β2V3ε+h24β3V4ε+12(∂Vε∂x)2)=Bε(t,x). | (2.4) |
Simply for writing, let c represent arbitrary positive constants (independent of ε).
Lemma 2.1. Let V0∈H1(R). For each number σ≥2, system (2.3) has a unique solution Vε∈ C([0,∞);Hσ(R)) and
∫R(V2ε+(∂Vε∂x)2)dx+2ε∫t0∫R[(∂Vε∂x)2+(∂2Vε∂x2)2](s,x)dxds=‖Vε,0‖2H1(R), | (2.5) |
which has the equivalent expression
‖Vε(t,.)‖2H1(R)+2ε∫t0‖∂Vε∂x(s,.)‖2H1(R)ds=‖Vε,0‖2H1(R). |
Proof. For parameter σ≥2, we acquire Vε,0∈C([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
ddt∫R(V2ε+V2εx)dx=2∫RVε(Vεt−Vεtxx)dx=2ε∫R(VεVεxx−VεVεxxxx)dx=−2ε∫R((Vεx)2+(Vεxx)2)dx. |
Integrating about variable t for both sides of the above identity, we obtain (2.5).
In fact, as ε→0, we have
‖Vε‖L∞(R)≤‖Vε‖H1(R)≤‖Vε,0‖H1(R)≤‖V0‖H1(R), and Vε,0→V0inH1(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,‖∂Aε∂x(t,⋅)‖L∞(R)≤c, | (2.7) |
‖Aε(t,⋅)‖L1(R)≤c,‖∂Aε∂x(t,⋅)‖L1(R)≤c, | (2.8) |
‖Aε(t,⋅)‖L2(R)≤c,‖∂Aε∂x(t,⋅)‖L2(R)≤c, | (2.9) |
and
‖Bε(t,⋅)‖L∞(R)≤c,‖Bε(t,⋅)‖L2(R)≤c, | (2.10) |
where c=c(‖V0‖H1(R)).
Proof. For any function U(x) and the operator Λ−2, it holds that
Λ−2U(x)=12∫Re−|x−y|U(y)dy for U(x)∈Lr(R),1≤r≤∞, | (2.11) |
and
|Λ−2Ux(x)|=|12∫Re−|x−y|∂U(y)∂ydy|=|−12e−x∫x−∞U(y)dy+12ex∫∞xe−yU(y)dy|≤12∫∞−∞e−|x−y||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
∥Λ−2((k−α0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12V2εx)∥L∞(R)≤c |
and
∥Λ−2((k−α0α)Vε+V2ε+h13β2V3ε+h24β3V4ε+12V2εx)x∥L∞(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,⋅)∂x‖2L2(R)≤‖∂Aε(t,⋅)∂x‖L∞(R)‖∂Aε(t,⋅)∂x‖L1(R)≤c, |
which complete the proof of (2.9). Furthermore, using (2.4) and (2.6), we have
‖Bε‖L∞(R)≤c,‖Bε‖L2(R)≤c, |
which finishes the proof of (2.10).
Lemma 2.3. Provided that 0<α1<1, T>0, constants a<b, then
∫T0∫ba|∂Vε(t,x)∂x|2+α1dxdt≤c1, | (2.13) |
where constant c1 depends on a,b,α1,T,k and ‖V0‖H1(R).
Proof. We utilize the methods in Xin and Zhang [21] to prove this lemma. Let function g(x)∈C∞(R) and satisfy
0≤g(x)≤1,g(x)={0,x∈(−∞,a−1]∪[b+1,∞),1,x∈[a,b]. |
Define function f(η):=η(|η|+1)α1,η∈R. 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)α1−1,f′′(η)=α1sign(η)(|η|+1)α1−2((α1+1)|η|+2)=α1(α1+1)sign(η)(|η|+1)α1−1+(1−α1)α1sign(η)(|η|+1)α1−2, |
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)=1−α12η2(|η|+1)α1+α12η2(|η|+1)α1−1≥1−α12η2(|η|+1)α1. | (2.15) |
Note that
∫T0∫Rg(x)f′(pε)pεtdxdt=∫Rg(x)dx∫T0df(pε)=∫R[f(pε(T,x))−f(pε(0,x))]g(x)dx, | (2.16) |
∫T0∫Rg(x)f′(pε)(Vε+α0α)pεxdxdt=∫T0dt∫Rg(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
∫T0∫Rg(x)pεf(pε)dtdx−12∫T0∫Rp2εg(x)f′(pε)dtdx=∫R[f(pε(T,x))−f(pε(0,x))]g(x)dx+∫T0∫R(Vε+α0α)g′(x)f(pε)dtdx+ε∫T0∫Rg′(x)f′(pε)∂pε∂xdtdx+ε∫T0∫Rg(x)f′′(pε)(∂pε∂x)2dtdx−∫T0∫RBεf′(pε)g(x)dtdx. | (2.18) |
Applying (2.15) yields
∫T0∫Rg(x)pεf(pε)dtdx−12∫T0∫Rp2εg(x)f′(pε)dtdx=∫T0∫Rg(x)(pεf(pε)−12p2εf′(pε))dtdx≥(1−α1)2∫T0∫Rg(x)p2ε(|pε|+1)α1dtdx. | (2.19) |
For t≥0, using 0<α1<1, (2.14) and the Hölder inequality gives rise to
|∫Rg(x)f(pε)dx|≤∫Rg(x)(|pε|α1+1+|pε|)dx≤‖g(x)‖L2/(1−α1)(R)‖pε(t,⋅)‖α1+1L2(R)+‖g(x)‖L2(R)‖pε(t,⋅)‖L2(R)≤(b+2−a)(1−α1)/2‖V0‖α1+1H1(R)+(b+2−a)1/2‖V0‖H1(R), | (2.20) |
and
|∫T0∫RVεg′(x)f(pε)dtdx|≤∫T0∫R|Vε‖g′(x)|(|pε|α1+1+|pε|)dtdx≤∫T0∫R‖Vε(t,⋅)‖L∞(R)|g′(x)|(|pε|α1+1+|pε|)dtdx≤c∫T0(‖g′(x)‖L2/(1−α1)(R)‖pε(t,⋅)‖α1+1L2(R)+‖g′(x)‖L2(R)‖pε(t,⋅)‖L2(R))dt≤c∫T0(‖g(x)′‖L2/(1−α1)(R)‖V0‖α1+1L2(R)+‖g′(x)‖L2(R)‖V0‖L2(R))dt. | (2.21) |
Moreover, we have
ε∫T0∫R∂pε∂xg′(x)f′(pε)dtdx=−ε∫T0∫Rf(pε)g′′(x)dtdx. | (2.22) |
Utilizing the Hölder inequality and (2.14) leads to
|ε∫T0∫Rg′(x)∂pε∂xf(pε)dtdx|≤ε∫T0∫R|f(pε)‖g′′(x)|dtdx≤ε∫T0∫R(|pε|α1+1+|pε|)| g′′(x)|dtdx≤ε∫T0(‖ g′′‖L2/(1−α1)(R)‖pε(t,⋅)‖α1+1L2(R)+‖g′′‖L2(R)‖pε(t,⋅)‖L2(R))dt≤εT(‖g′′‖L2/(1−α1)(R)‖V0‖α1+1H1(R)+‖g′′‖L2(R)‖V0‖H1(R)). | (2.23) |
Using the last part of (2.14), we have
ε|∫ΠT(∂pε∂x)2g(x)f′′(pε)dtdx|≤2α1ε∫ΠT(∂pε∂x)2dtdx≤α1‖V0‖2H1(R). | (2.24) |
As shown in Lemma 2.2, there exists a constant c0>0 to ensure that
‖Bε‖L∞(R)≤c0. | (2.25) |
Utilizing the second part in (2.14) arises
|∫T0∫RBεg(x)f′(pε)dtdx|≤c0∫T0∫R g(x)[(α1+1)|pε|+1]dtdx≤c0∫T0((α1+1)‖ g(x)‖L2(R)‖pε(t,⋅)‖L2(R)+‖g(x)‖L1(R))dt≤c0T. | (2.26) |
Applying (2.18)–(2.26) yields
1−α12∫T0∫R|pε|2f(x)(1+|pε|α1)dtdx≤c, |
where c>0 relies only on T>0,a,b,α1 and ‖V0‖H1(R). Furthermore, we have
∫T0∫ba|∂Vε∂x(t,x)|2+α1dxdt≤∫T0∫R|pε|g(x)(|pε|+1)α1+1dtdx≤2c(1−α1). |
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)∂x≤2t+c, | (2.27) |
in which positive constant c=c(‖V0‖H1(R)).
Proof. Using Lemma 2.2 gives rise to
∂pε∂t+(Vε+α0α)∂pε∂x−ε∂2pε∂x2+12p2ε=Bε(t,x)≤c. | (2.28) |
Assume that H=H(t) satisfies the problem
dHdt+12H2=c,t>0,H(0)=‖∂Vε,0∂x‖L∞. |
Due to (2.28), we know that H=H(t) is a supersolution* of parabolic equation (2.4) associated with initial value ∂Vε,0∂x. Utilizing the comparison principle for parabolic equations arises
*The supersolution is defined by supx∈Rpε(t,x). If there exists a point (t,x0) such that supx∈Rpε(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= 2√2ct>0, we conclude
H(t)≤F(t), |
which finishes the proof of (2.27).
Lemma 2.5. There exists a subsequence {εj}j∈N,εj→0 and V(t,x)∈L∞([0,∞);H1(R))∩H1([0,T]×R), for every T≥0, such that
Vεj⇀V in H1([0,T]×R),Vεj→V in L∞ loc ([0,∞)×R). |
The proof of Lemma 2.5 can be found in Coclite el al. [4].
Lemma 2.6. Assume V0∈H1(R). Then {Bε(t,x)}ε is uniformly bounded in W1,1loc([0,∞)×R). Moreover, there has a sequence {εj}j∈N, εj→0 to guarantee that
Bεj→B strongly in Lr loc ([0,T)×R), |
where function B∈L∞([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}j∈N tending to zero and two functions p∈Lrloc([0,∞)×R),¯p2∈Lr1loc([0,∞)×R) such that
pεj⇀p in Lrloc([0,∞)×R),pεj⋆⇀p in L∞loc([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) |
∂V∂x=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}j∈N, {Vεj}j∈N and {Bεj}j∈N by {pε}ε>0, {Vε}ε>0 and {Bε}ε>0. Assume that F∈C1(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 L∞loc([0,∞);L2(R)). |
Multiplying (2.4) by F′(pε) yields
∂∂tF(pε)+∂∂x((Vε+α0α)F(pε))−ε∂2∂x2F(pε)+εF′′(pε)(∂pε∂x)2=pεF(pε)−12F′(pε)p2ε+BεF′(pε). | (2.33) |
Lemma 2.8. Suppose that F∈C1(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 ε→0 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), η∈R.
Using Lemmas 2.4 and 2.7 leads to
pε,p≤2t+c,0<t<T. |
Lemma 2.10. For t≥0,x∈R, in the sense of distribution, it holds that
∂p∂t+∂∂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 ε→0.
Lemma 2.11. Provided that F∈C1(R) is a convex function with F′∈L∞(R), for every T>0, in the sense of distribution, then
∂F(p)∂t+∂∂x((Vε+α0α)F(p))=pF(p)+(12¯p2−p2)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)∂pδ∂t=F′(pδ)(−∂∂x((Vε+α0α)p)⋆wδ+12ˉp2⋆wδ+B⋆wδ)=F′(pδ)[(−Vε+α0α)px⋆wδ−Vp2⋆wδ]+F′(pδ)(12V¯q2⋆wδ+B⋆wδ). | (2.36) |
Utilizing the assumptions on F and F′ and letting δ→0 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 V0∈H1(R). Then
limt→0∫Rp2(t,x)dx=limt→0∫R¯p2(t,x)dx=∫R(∂V0∂x)2dx. |
Lemma 2.13. [4] If V0∈H1(R), L>0, then
limt→0∫R(¯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 F−L(ρ)=FL(ρ)χ(−∞,0](ρ), ρ∈(−∞,∞).
Lemma 2.14. [4] Let L>0. For FL(ρ) defined in (2.37), then
{FL(ρ)=12ρ2−12(L−|ρ|)2χ(−∞,−L)∩(L,∞)(ρ),F′L(ρ)=ρ+(L−|ρ|)sign(ρ)χ(−∞,−L)∩(L,∞)(ρ),F+L(ρ)=12(ρ+)2−12(L−ρ)2χ(L,∞)(ρ),(F+L)′(ρ)=ρ++(L−ρ)χ(L,∞)(ρ),F−L(ρ)=12(ρ−)2−12(L+ρ)2χ(−∞,−L)(ρ),(F−L)′(ρ)=ρ−−(L+ρ)χ(−∞,−L)(ρ). |
Lemma 2.15. Assume V0∈H1(R). For almost all t>0, then
12∫R(¯(p+)2−p2+)(t,x)dx≤∫t0∫RB(s,x)[¯p+(s,x)−p+(s,x)]dxds. |
Lemma 2.16. Assume V0∈H1(R). For almost all t>0, then
∫(¯F−L(p)−F−L(p))(t,x)dx≤L22∫t0∫R¯(L+p)χ(−∞,−L)(p)dxds−L22∫t0∫R(L+p)χ(−∞,−L)(p)dxds+L∫t0∫R[¯F−L(p)−F−L(p)]dxds+L2∫t0∫R(¯p2+−p2+)dxds+∫t0∫RB(t,x)(¯(F−L)′(p)−(F−L)′(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 V0∈H1(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]+[¯F−L−F−L])(t,x)dx≤L22∫t0∫R¯(L+p)χ(−∞,−L)(p)dxds−L22∫t0∫R(L+p)χ(−∞,−L)(p)dxds+L∫t0∫R[¯F−L(p)−F−L(p)]dxds+L2∫t0∫R(¯p2+−p2+)dxds+∫t0∫RB(s,x)([¯p+−p+]+[¯(F−L)′(p)−(F−L)′(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++(F−L)′(p)=p−(L+p)χ(−∞,−L),¯p++¯(F−L)′(p)=p−¯(L+p)χ(−∞,−L)(p). | (2.41) |
Since the map ρ→ρ++(F−L)′(ρ) is convex, it holds that
0≤[¯p+−p+]+[¯(F−L)′(p)−(F−L)′(p)]=(L+p)χ(−∞,−L)−¯(L+p)χ(−∞,−L)(p). | (2.42) |
Using (2.40) gives rise to
B(s,x)([¯p+−p+]+[¯(F−L)′(p)−(F−L)′(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+]+[(¯F−L)′(p)−(F−L)′(p)])≤(L22−N)(¯(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.
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