We consider the Poisson equation by collocation method with linear barycentric rational function. The discrete form of the Poisson equation was changed to matrix form. For the basis of barycentric rational function, we present the convergence rate of the linear barycentric rational collocation method for the Poisson equation. Domain decomposition method of the barycentric rational collocation method (BRCM) is also presented. Several numerical examples are provided to validate the algorithm.
Citation: Jin Li, Yongling Cheng, Zongcheng Li, Zhikang Tian. Linear barycentric rational collocation method for solving generalized Poisson equations[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4782-4797. doi: 10.3934/mbe.2023221
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We consider the Poisson equation by collocation method with linear barycentric rational function. The discrete form of the Poisson equation was changed to matrix form. For the basis of barycentric rational function, we present the convergence rate of the linear barycentric rational collocation method for the Poisson equation. Domain decomposition method of the barycentric rational collocation method (BRCM) is also presented. Several numerical examples are provided to validate the algorithm.
In this paper, we investigate the Cauchy problem of the modified periodic coupled Camassa-Holm system that has the following form:
{mt+2mux+mxu+(mv)x+nvx=0,t>0, x∈R,nt+2nvx+nxv+(nu)x+mux=0,t>0, x∈R,m(0,x)=m0(x), n(0,x)=n0(x),t=0, x∈R,m(t,x)=m(t,x+1), n(t,x)=n(t,x+1),t>0, x∈R, | (1) |
where
It is known to us, when
mt+2mux+mxu=0,m=u−uxx. | (2) |
The Camassa-Holm equation (2) was first implicitly contained in a bi-Hamiltonian generalization of the Korteweg-de Vries equation by Fuchssteiner and Fokas [15], and later deduced as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm [4]. Similar to the KdV equation, the Camassa-Holm equation has a bi-Hamilton structure [15,24], and is completely integrable [4,5,23]. The equation (2) not only holds an infinity of conservative laws, but also can be solved by its corresponding inverse scattering transform [2,9]. The solitary waves of equation (2) are solitons (i.e., it can keep their shape and velocity after interacted by the same type nonlinear wave). Compared to the KdV equation, the Camassa-Holm equation has many advantages, such as, it has both the finite time wave-breaking solutions (i.e. the solution keeps bounded but the slope becomes unbounded in finite time) and the global strong solutions [6,7,11,25]. The solitary wave solutions are peaked waves and a specific case was given in [4], it is not a classical solution because it has a peak at their crest. The local well-posedness of equation (2) with the inial data
Of course, the Camassa-Holm equation has many generalizations such as the modified two-component Camassa-Holm equation (M2CH) and the coupled Camassa-Holm equation. The M2CH equation was firstly introduced by Holm et al. in [21] as a modified version of the two-component Camassa-Holm equation (2CH) that was proposed by Constantin and Ivanov [10] in the context of shallow water theory. The M2CH equation is integrable and its form as follow;
{mt+2mux+mxu+ρˉρx=0,ρt+(ρu)x=0, | (3) |
where
Inspired by [3,22,26,34,35], this paper mainly discusses the global conservative solutions of the modified periodic Camassa-Holm system. As is known to us, equations (1) is a system, so it's more difficult than the single one. Moreover, the interactions between
The rest of this paper is organized as follows. Section 2 is the basic equation. In section 3, we get a equivalent semilinear system and the global solutions of the semilinear system. In section 4, we obtain the global conservative solution of the system (1) and construct a solution semigroup.
Now, we reformulate the system (1). Let
{ut+(u+v)ux+P1+P2,x=0,t>0, x∈R,vt+(u+v)vx+Q1+Q2,x=0,t>0, x∈R,u(0,x)=u0(x), v(0,x)=v0(x),t=0, x∈R,u(t,x)=u(t,x+1), v(t,x)=v(t,x+1),t>0, x∈R, | (4) |
where
{P1=G∗(uvx),P2=G∗(u2+12u2x+uxvx+12v2−12v2x),Q1=G∗(vux),Q2=G∗(v2+12v2x+uxvx+12u2−12u2x). |
According to the fact that the above representations,
In fact, for smooth solutions, differentiating the first and the second equations in (4) with respect to
{uxt+u2x+uxvx+uuxx+vuxx+P1,x+P2−(u2+12u2x+uxvx+12v2−12v2x)=0,vxt+v2x+uxvx+vvxx+uvxx+Q1,x+Q2−(v2+12v2x+uxvx+12u2−12u2x)=0. | (5) |
Multiplying the first and the second equations in (4) by
{(u22)t+(u2x2)t+(u2v+uu2x+vu2x+2uP22)x− ux(v2+v2x)+vx(u2−u2x)2+uP1+uxP1,x=0,(v22)t+(v2x2)t+(v2u+vv2x+uv2x+2vQ22)x − vx(u2+u2x)+ux(v2−v2x)2+vQ1+vxQ1,x=0. | (6) |
For regular solutions, using (6) and integrating by parts, it is clear that the total energy
E(t)=∫S(u2+u2x+v2+v2x)dx |
is a constant in time. If
(u2+u2x+v2+v2x)t+((u+v)(u2+u2x+v2+v2x))x=(u3−2uP2−2uP1,x+v3−2vQ2−2vQ1,x)x. | (7) |
Firstly, we introduce the space
E1={f∈H1loc(R)|f(θ+1)=f(θ)+1}, |
and define the characteristics
yt(t,θ)=(u+v)(t,y(t,θ)). | (8) |
In addition, we denote
{U(t,θ)=u(t,y(t,θ)),V(t,θ)=v(t,y(t,θ),M(t,θ)=ux(t,y(t,θ),N(t,θ)=vx(t,y(t,θ), | (9) |
and
H(t,θ)=∫y(t,θ)y(t,0)(u2+u2x+v2+v2x)dx. | (10) |
By (7) and (8), we get
{dHdt=[(u3−2uP1,x−2uP2+v3−2vQ1,x−2vQ2)∘y]θ0,Hθ=[(u2+u2x+v2+v2x)∘y]yθ. | (11) |
Using (10), the periodicity of
H(t,θ+1)−H(t,θ)=H(t,1)−H(t,0). |
According to (11), it is very easy to prove that
E={f∈H1loc(R)|f(θ+1)−f(θ)=f(1)−f(0)}. |
We define the norm
We derive formally a system equivalent to system (4). From the definition of the characteristics, it follows that
{Ut(t,θ)=(−P1−P2,x)∘y(t,θ),Vt(t,θ)=(−Q1−Q2,x)∘y(t,θ),Mt(t,θ)=(−M22−N22+U2+V22−P1,x−P2)∘y(t,θ),Nt(t,θ)=(−N22−M22+V2+U22−Q1,x−Q2)∘y(t,θ). | (12) |
Then, we get the explicit expression for
{P1(u,v)=12sinh12∫10cosh(x−y−[x−y]−12)(u(t,y)vx(t,y))dy,P1,x(u,v)=12sinh12∫10sinh(x−y−[x−y]−12)(u(t,y)vx(t,y))dy,P2(u,v)=12sinh12∫10cosh(x−y−[x−y]−12)×(u2(t,y)+12u2x(t,y)+ux(t,y)vx(t,y)+12v2(t,y)−12v2x(t,y))dy,P2,x(u,v)=12sinh12∫10sinh(x−y−[x−y]−12)×(u2(t,y)+12u2x(t,y)+ux(t,y)vx(t,y)+12v2(t,y)−12v2x(t,y))dy, |
and
In the above formulae, we can perform the change of variables
{P1(t,θ)=12(e−1)∫10cosh(y(t,θ)−y(t,θ′))(UNyθ)(t,θ′)dθ′+14∫10exp(−sgn(θ−θ′)(y(θ)−y(θ′)))(UNyθ))(t,θ′)dθ′,P1,x(t,θ)=12(e−1)∫10sinh(y(t,θ)−y(t,θ′))(UN)yθ)(t,θ′)dθ′−14∫10sgn(θ−θ′)exp(−sgn(θ−θ′)(y(θ)−y(θ′)))(UNyθ))(t,θ′)dθ′,P2(t,θ)=12(e−1)∫10cosh(y(t,θ)−y(t,θ′))(Hθ+(U2+2MN−N2)yθ)(t,θ′)dθ′+14∫10exp(−sgn(θ−θ′)(y(θ)−y(θ′)))×(Hθ+(U2+2MN−N2)yθ)(t,θ′)dθ′,P2,x(t,θ)=12(e−1)∫10sinh(y(t,θ)−y(t,θ′))(Hθ+(U2+2MN−N2)yθ)(t,θ′)dθ′ −14∫10sgn(θ−θ′)exp(−sgn(θ−θ′)(y(θ)−y(θ′)))×(Hθ+(U2+2MN−N2)yθ)(t,θ′)dθ′, | (13) |
and
{Q1(u(t,θ),v(t,θ))=P1(v(t,θ),u(t,θ)),Q1,x(u(t,θ),v(t,θ))=P1,x(v(t,θ),u(t,θ)),Q2(u(t,θ),v(t,θ))=P2(v(t,θ),u(t,θ)),Q2,x(u(t,θ),v(t,θ))=P2,x(v(t,θ),u(t,θ)). | (14) |
Straight computation shows that
{P1,θ=P1,xyθ, P1,xθ=−UNyθ+P1yθ,P2,θ=P2,xyθ, P2,xθ=−[Hθ+(U2+2MN−N2)yθ]+P2yθ,Q1,θ=Q1,xyθ, Q1,xθ=−VMyθ+Q1yθ,Q2,θ=Q2,xyθ, Q2,xθ=−[Hθ+(V2+2MN−M2)yθ]+Q2yθ. | (15) |
From (8), (11)-(12) and (13)-(15), we obtain a new system which is equivalent to system (4). And the Cauchy problem of the new system can be rewritten with respect to the variables
{∂y∂t=U+V,∂U∂t=−P1−P2,x,∂V∂t=−Q1−Q2,x,∂M∂t=−M22−N22+U2+V22−P1,x−P2,∂N∂t=−N22−M22+V2+U22−Q1,x−Q2,∂H∂t=(u3−2uP1,x−2uP2+v3−2vQ1,x−2vQ2)|θ0. | (16) |
Differentiating (16) with respect to
{∂yθ∂t=Uθ+Vθ,∂Uθ∂t=Hθ2+(U22+MN−N2−P1,x−P2)yθ, ∂Vθ∂t=Hθ2+(V22+MN−M2−Q1,x−Q2)yθ, ∂Hθ∂t=(3U2−2P2)Uθ+(3V2−2Q2)Vθ−2(UP2,x+VQ2,x)yθ −2(MP1,x+UP1−U2N+NQ1,x+VQ1−V2M)yθ. | (17) |
The system (17) is semilinear for the variables
H1per={f∈H1loc(R)|f(θ+1)=f(θ)}, |
with the norm
Lemma 3.1. The map
It is clear that the space
{∂η∂t=U+V,∂U∂t=−P1−P2,x,∂V∂t=−Q1−Q2,x,∂M∂t=−M22−N22+U2+V22−P1,x−P2,∂N∂t=−N22−M22+V2+U22−Q1,x−Q2,∂σ∂t=(u3−2uP1,x−2uP2+v3−2vQ1,x−2vQ2)|θ0,∂h∂t=0. | (18) |
In the next section, the well-posedness of system (18) will be proved as an ordinary differential equations in the Banach space
W=H1per×H1per×H1per×L∞per×L∞per×H1per×R. |
We have a bijection
Theorem 3.2. Let
Proof. To prove the this theorem, the key step is to prove the Lipchitz continuity of the right side of system (18). We define the map
ψ(X)=(U+V,−P1−P2,x,−Q1−Q2,x,−M22−N22+U2+V22−P1,x−P2,−N22−M22+V2+U22−Q1,x−Q2,(u3−2uP1,x−2uP2+v3−2vQ1,x−2vQ2)|θ0,0). |
Firstly, we testify that
‖y‖L∞=‖Id+η‖L∞≤1+C‖η‖H1≤1+CM |
and
|cosh(y(θ)−y(θ′))−cosh(ˉy(θ)−ˉy(θ′))|≤C|y(θ)−y(θ′)−ˉy(θ)+ˉy(θ′)|≤C‖η−ˉη‖L∞, |
and
|exp(−sgn(θ−θ′)(y(θ)−y(θ′))−exp(−sgn(θ−θ′)(ˉy(θ)−ˉy(θ′))|≤C‖η−ˉη‖L∞, |
for all
‖cosh(y(θ)−y(θ′))((Hθ+(U2+2MN−N2)yθ))−cosh(ˉy(θ)−ˉy(θ′))(¯Hθ+(ˉU2+2ˉMˉN−ˉN2)ˉyθ)‖L∞≤C(‖η−ˉη‖L∞+‖U−ˉU‖L∞+‖V−ˉV‖L∞+‖M−ˉM‖L2+‖N−ˉN‖L2+‖ηθ−¯ηθ‖L2+‖σθ−ˉσθ‖L2+|h−ˉh|)≤C‖X−ˉX‖W. |
So does
‖P2−ˉP2‖L∞≤C‖X−ˉX‖W. |
The similar computation shows that
‖P2,x−ˉP2,x‖L∞≤C‖X−ˉX‖W. |
Utilizing (15), we have
‖P2,xθ−ˉP2,xθ‖L2≤C(‖X−ˉX‖W+‖P2−ˉP2‖L∞)≤C‖X−ˉX‖W, |
and
‖P2−ˉP2‖L2≤C(‖X−ˉX‖W. |
In conclusion, the local Lipchitz continuity from
Now, we prove the existence of a global solution of system (18). As we all know that initial data is very significant to system (18), but, here, we will only consider a particular initial data that belong to
W1=W1,∞per×W1,∞per×W1,∞per×L∞per×W1,∞per×R. |
{∂∂tα(t,θ)=β(t,θ)+γ(t,θ),∂∂tβ(t,θ)=ˉh+δ(t,θ)2+(U22+MN−N2−P1,x−P2)(1+α(t,θ)),∂∂tγ(t,θ)=ˉh+δ(t,θ)2+(V22+MN−M2−Q1,x−Q2)(1+α(t,θ)),∂∂tδ(t,θ)=(3U2−2P2−2P1,x+2V2)β(t,θ)+(3V2−2Q2−2Q1,x+2U2)γ(t,θ)−2(UP2,x+UP1+VQ2,x+VQ1)(1+δ(t,θ)). | (19) |
which is obtained by substituting
S={θ∈R||ˉUθ(θ)|≤‖ˉUθ(θ)‖L∞,|ˉVθ(θ)|≤‖ˉvθ(θ)‖L∞,|ˉηθ(θ)|≤‖ˉηθ(θ)‖L∞, |
|ˉσθ(θ)|≤‖ˉσθ(θ)‖L∞}. |
It is not very hard to check that
(α(0,θ),β(0,θ),γ(0,θ),δ(0,θ))=(ˉηθ(θ),ˉUθ(θ),ˉvθ(θ),ˉσθ(θ)). |
For
Lemma 3.3. Given initial data
ˉX=(ˉη,ˉU,ˉV,ˉM,ˉN,ˉσ,ˉh)∈[W1,∞per]3×[L∞per]2×W1,∞per×R. |
Let
X∈C1([0,T],[W1,∞per]3×[L∞per]2×W1,∞per×R), |
and the functions
(α(t,θ),β(t,θ),γ(t,θ),δ(t,θ))=(ηθ(t,θ),Uθ(t,θ),Vθ(t,θ),σθ(t,θ)). | (20) |
Proof. Firstly, let's introduce a key space
{ηθ(t,θ)=ˉηθ+∫t0(Uθ(τ,θ)+Vθ(τ,θ))dτ,Uθ(t,θ)=ˉUθ+∫t0(12(ˉh+δ(τ,θ))+(U22+MN−N2−P1,x−P2)(1+α(τ,θ)))dτ,Vθ(t,θ)=ˉVθ+∫t0(12(ˉh+δ(τ,θ))+(V22+MN−M2−Q1,x−Q2)(1+α(τ,θ)))dτ,σθ(t,θ)=ˉσθ+∫t0((3U2−2P2−2P1,x+2V2)β(τ,θ)+(3V2−2Q2−2Q1,x+2U2)γ(τ,θ)−2(UP2,x+UP1+VQ2,x+VQ1)(1+δ(τ,θ)))dτ. | (21) |
Since
(α(t),β(t),γ(t),δ(t))=(ηθ(t),Uθ(t),Vθ(t),σθ(t)) |
in
Now, we give the initial data as follow
{ˉU(θ)=ˉu∘ˉy(θ), ˉV(θ)=ˉv∘ˉy(θ),ˉM(θ)=ˉux∘ˉy(θ), ˉN(θ)=ˉvx∘ˉy(θ),ˉH(θ)=∫ˉy(θ)0(ˉu2+ˉu2x+ˉv2+ˉv2x)dx,∫ˉy(θ)0(ˉu2+ˉu2x+ˉv2+ˉv2x)dx+ˉy(θ)=(1+ˉh)θ, | (22) |
where
Definition 3.4. The set
{(η,U,V,M,N,σ,h)∈[W1,∞per]3×[L∞per]2×W1,∞per×R,yθ≥0,Hθ≥0,yθ+Hθ≥0almost everywhere,yθHθ=U2y2θ+U2θ+V2y2θ+V2θalmost everywhere, | (23) |
where
We know that the initial data
Lemma 3.5. Given the initial data
X(t)=(η(t),U(t),V(t),M(t),N(t),σ(t),h(t)) |
be the local solution of system (18) in
(i)
(ii) almost every
Proof. (ⅰ) We continue to use
(yθHθ)t=yθtHθ+yθHθt=(Uθ+Vθ)Hθ+yθ[(3U2−2P2−2P1,x+2V2)Uθ+(3V2−2Q2−2Q1,x+2U2)Vθ−2(UP2,x+UP1+VQ2,x+VQ1)yθ], |
and on the other hand,
(U2y2θ+U2θ+V2y2θ+V2θ)t=2UUtyθ+2U2yθyθt+2UθUθt+2VVtyθ+2V2yθyθt+2VθVθt=(Uθ+Vθ)Hθ+yθ[(3U2−2P2−2P1,x+2V2)Uθ+(3V2−2Q2−2Q1,x+2U2)Vθ−2(UP2,x+UP1+VQ2,x+VQ1)yθ]. |
It is not very difficult to check that
t∗=sup{t∈[0.T]|yθ(t′)≥0 for all t′∈[0,t]}. |
If
yθ(t∗)=0 |
for the continuity of
yθt(t∗)=Uθ(t∗)+Vθ(t∗)=0. |
From system (17) and the fact
yθtt(t∗)=Uθt(t∗)+Vθt(t∗)=Hθ(t∗). |
If
yθ(t∗)=Uθ(t∗)=Vθ(t∗)=Hθ(t∗)=0. |
This is a contradiction to the uniqueness of system (17). If
(ⅱ) Let
A={(t,θ)∈[0,T]×R|yθ(t,θ)=0}. |
Fubini's theorem infers that
meas(A)=∫Rmeas(Aθ)dθ=∫T0meas(At)dθ, | (24) |
where
Aθ={t∈[0,T]|yθ(t,θ)=0}, At={θ∈R|yθ(t,θ)=0}. |
From the above proof, we know that for
meas(At)=0 for almost every t∈[0,T]. |
We denote by
K={t∈R+|meas((A)t)>0}. |
Then,
Theorem 3.6. Given any
Dt(ˉX)=X(t) |
is a continuous semigroup.
Proof. Let us write
supt∈[0,T)‖(η(t,.),U(t,.),V(t,.),M(t,.),N(t,.),σ(t,.),h(t,.))‖W<∞. | (25) |
It is clear that
U2yθ≤Hθ, UθM≤Hθ, V2yθ≤Hθ, VθN≤Hθ. | (26) |
Indeed, using (23), we obtain that
U2+U2θyθ+V2+V2θyθ=Hθ | (27) |
and
{UθM≤U2θyθ+M2yθ, UθN≤V2θyθ+N2yθ,VθN≤V2θyθ+N2yθ, VθM≤U2θyθ+M2yθ. | (28) |
Utilizing (27)-(28), we get (26). From (13)-(22), we have that
supt∈[0,T]‖ρi‖L∞≤Cˉh, supt∈[0,T]‖ρi,x‖L∞≤Cˉh, |
where the constant
supt∈[0,T]‖U(t)‖L∞≤∞,supt∈[0,T]‖V(t)‖L∞≤∞,supt∈[0,T]‖M(t)‖L∞≤∞,supt∈[0,T]‖N(t)‖L∞≤∞. |
For
supt∈[0,T]‖η(t)‖L∞≤∞. |
Now, we have certified that
C1=supt∈[0,T]{‖U(t)‖L∞+‖V(t)‖L∞+‖M(t)‖L∞+‖N(t)‖L∞+‖ρ1‖L∞+‖ρ1,x‖L∞+‖ρ2‖L∞+‖ρ2,x‖L∞+‖ρ3‖L∞+‖ρ3,x‖L∞+‖ρ4‖L∞+‖ρ4,x‖L∞} |
is finite. Let
Z(t)=‖yθ(t)‖L2+‖Uθ(t)‖L2+‖Vθ(t)‖L2+‖Hθ(t)‖L2. |
Since the system (17) is semilinear, we have that
Z(t)=Z(0)+C∫t0Z(τ)dτ, |
where
In this section, we will investigate how to obtain a global conservative solution of system (4) from the global solution of system (17) with the initial variables
u(t,x)≐u(t,θ), v(t,θ)≐v(t,θ), if y(t,θ)=x. | (29) |
Theorem 4.1. Let
‖u(t,⋅)‖2H1+‖v(t,⋅)‖2H1=‖u(0,⋅)‖2H1+‖v(0,⋅)‖2H1, a.e. t≥0. | (30) |
Furthermore, let
‖ˉun−ˉu‖H1→0, ‖ˉvn−ˉv‖H1→0. |
Then the corresponding solutions
Proof. Firstly, what we have to do is to show that the definition of
yθ(θ)=0 in [θ1,θ2]. |
We can obtain that
∫10u2xdx=∫y−1(1)y−1(0)u2x(t,y(θ))yθdθ=∫10u2x(t,y(θ))yθdθ=∫θ∈[0,1]|yθUθyθdθ≤(H(1)−H(0))=ˉh. |
It is similar to
∫R+×R[−(u+v)ϕt+(u+v)(ux+vx)ϕ(t,x)]dxdt =∫R+×R[−(U+V)yθϕt+(U+V)(Uθ+Vθ)ϕ(t,Y)]dθdt. | (31) |
Utilizing
[(U+V)yθϕ∘y]t−[(U+V)2]θ=(Ut+Vt)yθϕ∘y+(U+V)yθϕ−(U+V)(Uθ+Vθ)ϕ. | (32) |
Integrating (32) over
∫R+×R[−(U+V)yθϕt+(U+V)(Uθ+Vθ)ϕ(t,Y)]dθdt=∫R+×R(Ut+Vt)yθ∘yθϕdθdt=∫R+×R(−P1−P2,x−Q1−Q2,x)ϕ(t,x)dxdt=∫R+×R(−P1−P2,x−Q1−Q2,x)yθ(t,θ)ϕ(t,y(t,θ))dθdt. | (33) |
By (31)-(33), we obtain the first two equation of system (4). When
Hθ=U2yθ+Uθyθ+V2yθ+Vθyθ |
holds almost everywhere. By taking
∫10(u2(t,x)+u2x(t,x)+v2(t,x)+v2x(t,x))dx =∫10(u2(0,x)+u2x(0,x)+v2(0,x)+v2x(0,x))dx. |
Therefore, we get (30).
Finally, let
‖ˉyn−ˉy‖L∞→0, ‖ˉUn−ˉU‖L∞→0, ‖ˉVn−ˉV‖L∞→0,‖ˉHn−ˉH‖L∞→0, ‖ˉhn−ˉh‖L∞→0. |
Now, we certify that
‖ˉynθ−ˉyθ‖L2→0, ‖ˉUnθ−ˉUθ‖L2→0, ‖ˉVnθ−ˉVθ‖L2→0,‖ˉMn−ˉM‖L2→0, ‖ˉNn−ˉN‖L2→0. |
Let
(1+h)(ˉyθ−ˉynθ)=(gn∘yn−g∘y)ynθyθ+(h−hn)yθ. | (34) |
The first item on the right side of (34) can be written as follow
(gn∘yn−g∘y)ynθyθ=(gn∘yn−g∘yn)ynθyθ+(g∘yn−g∘y)ynθyθ. |
Because
∫10|(gn∘yn−g∘y)ynθyθ|dθ≤(1+h)‖gn−g‖L1. |
Note that
∫10|g∘y−g∘yn|ynθyθdθ≤∫10(|g∘y−r∘y|+|r∘y−r∘yn|)ynθyθdθ+∫10|r∘yn−g∘yn|ynθyθdθ. |
The first and third item tend to zero for the boundedness of
ˉMn−ˉM=ˉunx∘yn−ˉux∘y=ˉunx∘yn−ˉux∘yn+ˉux∘yn−ˉux∘y. |
For
∫10|ˉunx∘yn−ˉux∘yn|2dθ≤‖ˉunx−ˉux‖2L∞→0. |
Given any
∫10|ˉux∘y−r|2dθ≤ε. | (35) |
Then
ˉunx∘yn−ˉux∘y=ˉunx∘y∘y−1∘yn−ˉux∘y=ˉunx∘y∘y−1∘yn−r∘y−1∘yn+r∘y−1∘yn−r+r−ˉux∘y. |
Utilizing (35), we have
∫10|ˉux∘y∘y−1∘yn−r∘y−1∘yn|2dθ≤ε, |
and
∫10|r−ˉux∘y|2dθ≤ε. |
According to dominated convergence theorem, we get
∫10|r∘y−1∘yn−r|2dθ→0. |
Thus, we have
According to (23),
∫RUnθΨdθ=∫Runx∘ynynθΨdθ=∫RunxΨ∘y−1ndx. |
Thus,
limn→∞∫RUnθΨdθ=∫RuxΨ∘y−1ndx=∫RUθΨdθ. |
Then, we obtain
{yn→y in H1,Un→U in H1,Vn→V in H1,Hn→H in H1,Mn→M in L2,Nn→N in L2. | (36) |
Combining (16)-(17) and (36), we get
ddt(‖Un(t)−U(t)‖L∞+‖Vn(t)−V(t)‖L∞+‖yn(t)−y(t)‖L∞+‖Mn(t)−M(t)‖L2+‖Nn(t)−N(t)‖L2+‖Unθ(t)−Uθ(t)‖L2+‖Vnθ(t)−Vθ(t)‖L2+‖Hnθ(t)−Hθ(t)‖L2+‖ynθ(t)−yθ(t)‖L2)≤C(‖Un(t)−U(t)‖L∞+‖Vn(t)−V(t)‖L∞+‖yn(t)−y(t)‖L∞+‖Mn(t)−M(t)‖L2+‖Nn(t)−N(t)‖L2+‖Unθ(t)−Uθ(t)‖L2+‖Vnθ(t)−Vθ(t)‖L2+‖Hnθ(t)−Hθ(t)‖L2+‖ynθ(t)−yθ(t)‖L2). |
According to Gronwall's inequality, we conclude that
un(t,x)→u(t,x), vn(t,x)→v(t,x), |
are uniformly Hölder continuous on any bounded time interval.
Lastly, we shall certify that the solutions obtained in
Theorem 4.2. Given initial data
Proof. Fix
Ft(Fτ(ˉu,ˉv))=Fτ+t(ˉu,ˉv). |
Let
∫ˆy(θ)0(u2+u2x+v2+v2x)dx+ˆy(θ)=(1+ˉh)θ, | (37) |
and
{ˆH(θ)=∫ˆy(θ)0(u2+u2x+v2+v2x)dx,ˆU(θ)=u(t,ˆy(t,θ)), ˆV(θ)=v(t,ˆy(t,θ)),ˆM(θ)=ux(t,ˆy(t,θ)),ˆN(θ)=vx(t,ˆy(t,θ)). | (38) |
Let
(y(t+τ,θ),U(t+τ,θ),V(t+τ,θ),M(t+τ,θ),N(t+τ,θ),H(t+τ,θ))=(ˆy(t+τ,˜θ),ˆU(t+τ,˜θ),ˆV(t+τ,˜θ),ˆM(t+τ,˜θ),ˆN(t+τ,˜θ),ˆH(t+τ,˜θ)), |
where
Actually, the equality
ˆyθ(τ+t,˜θ)dθ=yθ(τ+t,θ)dθ. | (39) |
Utilizing (39), we claim that
Qi(τ+t,˜θ)=Qi(τ+t,θ(˜θ)), Qi,x(τ+t,˜θ)=Qi,x(τ+t,θ(˜θ)), i=1,2. |
For
(y(τ,θ(ˆθ)),U(τ,θ(ˆθ)),V(τ,θ(ˆθ)),M(τ,θ(ˆθ)),N(τ,θ(ˆθ)),H(t+τ,θ))=(ˆy(τ,˜θ),ˆU(τ,˜θ),ˆV(τ,˜θ),ˆM(τ,˜θ),ˆN(τ,˜θ),ˆH(τ,˜θ)). |
Therefore, we get that
The authors are very grateful to the anonymous reviewers and editors for their careful reading and useful suggestions, which greatly improved the presentation of the paper.
The first author Chen is partially supported by Science and Technology Research Program of Chongqing Municipal Educational Commission. The third author Zhang is partially supported by the National Social Science Fund of China (No.19BJY077, No.18BJY093 and No.17CJY031), the National Natural Science Foud of China (No.71901044), Chongqing Social Science Planning Project (No.2018PY74), Science and Technology Project of Chongqing Education Commission(No.KJQN20180051 and No.KJQN201900537).
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