High-speed trains (HSTs) positioning is a critical technology that affects the safety and operational efficiency of trains. The unique operating environment of HSTs, coupled with the limitations of real data collection, poses challenges in obtaining large-scale and diverse positioning data. To tackle this problem, we introduce a comprehensive method for generating virtual position data for HSTs. Utilizing virtual simulation technology and expert expertise, this method constructs a HST operating simulation environment on the Unity 3D platform, effectively simulating a range of operating scenarios and complex scenes. Positioning data is collected using virtual sensors, while error characteristics are incorporated to emulate real data collection behavior. The contribution of this paper lies in providing abundant, reliable, controllable and diverse positioning data for HSTs, thereby offering novel insights and data support for the evaluation and optimization of positioning algorithms. This method is not only applicable to various routes and scenarios, but also delivers fresh perspectives on data generation for research in other domains, boasting a broad scope of application.
Citation: Xiaoyu Zheng, Dewang Chen, Liping Zhuang. Empowering high-speed train positioning: Innovative paradigm for generating universal virtual positioning big data[J]. Electronic Research Archive, 2023, 31(10): 6197-6215. doi: 10.3934/era.2023314
[1] | Ailing Xiang, Liangchen Wang . Boundedness of a predator-prey model with density-dependent motilities and stage structure for the predator. Electronic Research Archive, 2022, 30(5): 1954-1972. doi: 10.3934/era.2022099 |
[2] | Jialu Tian, Ping Liu . Global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis. Electronic Research Archive, 2022, 30(3): 929-942. doi: 10.3934/era.2022048 |
[3] | Jiani Jin, Haokun Qi, Bing Liu . Hopf bifurcation induced by fear: A Leslie-Gower reaction-diffusion predator-prey model. Electronic Research Archive, 2024, 32(12): 6503-6534. doi: 10.3934/era.2024304 |
[4] | Xuemin Fan, Wenjie Zhang, Lu Xu . Global dynamics of a predator-prey model with prey-taxis and hunting cooperation. Electronic Research Archive, 2025, 33(3): 1610-1632. doi: 10.3934/era.2025076 |
[5] | Shuxia Pan . Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle. Electronic Research Archive, 2019, 27(0): 89-99. doi: 10.3934/era.2019011 |
[6] | Yuan Tian, Yang Liu, Kaibiao Sun . Complex dynamics of a predator-prey fishery model: The impact of the Allee effect and bilateral intervention. Electronic Research Archive, 2024, 32(11): 6379-6404. doi: 10.3934/era.2024297 |
[7] | Érika Diz-Pita . Global dynamics of a predator-prey system with immigration in both species. Electronic Research Archive, 2024, 32(2): 762-778. doi: 10.3934/era.2024036 |
[8] | Miao Peng, Rui Lin, Zhengdi Zhang, Lei Huang . The dynamics of a delayed predator-prey model with square root functional response and stage structure. Electronic Research Archive, 2024, 32(5): 3275-3298. doi: 10.3934/era.2024150 |
[9] | Pinglan Wan . Dynamic behavior of stochastic predator-prey system. Electronic Research Archive, 2023, 31(5): 2925-2939. doi: 10.3934/era.2023147 |
[10] | Ling Xue, Min Zhang, Kun Zhao, Xiaoming Zheng . Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions. Electronic Research Archive, 2022, 30(12): 4530-4552. doi: 10.3934/era.2022230 |
High-speed trains (HSTs) positioning is a critical technology that affects the safety and operational efficiency of trains. The unique operating environment of HSTs, coupled with the limitations of real data collection, poses challenges in obtaining large-scale and diverse positioning data. To tackle this problem, we introduce a comprehensive method for generating virtual position data for HSTs. Utilizing virtual simulation technology and expert expertise, this method constructs a HST operating simulation environment on the Unity 3D platform, effectively simulating a range of operating scenarios and complex scenes. Positioning data is collected using virtual sensors, while error characteristics are incorporated to emulate real data collection behavior. The contribution of this paper lies in providing abundant, reliable, controllable and diverse positioning data for HSTs, thereby offering novel insights and data support for the evaluation and optimization of positioning algorithms. This method is not only applicable to various routes and scenarios, but also delivers fresh perspectives on data generation for research in other domains, boasting a broad scope of application.
This paper is devoted to the existence of weak solutions to the Cauchy problem for the two-component Novikov equation [18]
{mt+uvmx+(2vux+uvx)m=0,m=u−uxx,t>0,nt+uvnx+(2uvx+vux)n=0,n=v−vxx. | (1) |
Note that this system reduces respectively to the Novikov equation [23]
mt+3uuxm+u2mx=0, | (2) |
when
mt+2uxm+umx=0, | (3) |
when
The CH equation was proposed as a nonlinear model describing the unidirectional propagation of the shallow water waves over a flat bottom [1]. Based on the Hamiltonian theory of integrable systems, it was found earlier by using the method of recursion operator due to Fuchssteiner and Fokas [10]. It can also be obtained by using the tri-Hamiltonian duality approach related to the bi-Hamiltonian representation of the Korteweg-de Vries (KdV) equation [9,25]. The CH equation exhibits several remarkable properties. One is the the existence of the multi-peaked solitons on the line
The Novikov equation (2) can be viewed as a cubic generalization of the CH equation, which was introduced by Novikov [23,24] in the classification for a class of equations while they possesses higher-order generalized symmetries. Eq. (2) was proved to be integrable since it enjoys Lax-pair and bi-Hamiltonian structure [14], and is equivalent to the first equation in the negative flow of the Sawada-Kotera hierarchy via Liouville transformation [16]. The Novikov equation (2) also admits peaked solitons over the line
As the two-component generalization of Novikov equation (2), the so-called Geng-Xue system [11]
mt+3vuxm+uvmx=0,nt+3uvxn+uvnx=0, | (4) |
has been studied extensively [11,13]. The integrability [11,19], dynamics and structure of the peaked solitons of (4) [21] were discussed. In [13], well-posedness and wave breaking phenomena of the Cauchy problem of (4) were discussed. The single peakons and multi-peakons of system (4) were constructed in [21] by using compatibility of Lax-pair, which are not the weak solutions in the sense of distribution. Furthermore, the Geng-Xue system does not have the
The main object in this work is to investigate the existence of weak solutions to system (1). It is of great interest to understand the effect from interactions among the two-components, nonlinear dispersion and various nonlinear terms. More specifically, we shall consider the Cauchy problem of (1) and aim to leverage ideas from previous works on CH and Novikov equations. The weak solution of the Cauchy problem associated with (1) is established in Theorem 3.1.
The remainder of this paper is organized as follows. In the next section 2, we review some basic results and lemmas as well as invariant properties of momentum densities
In this section, we recall the local well-posedness, some properties of strong and weak solutions to equation (1) and several approximation results.
First, we introduce some notations. Throughout the paper, we denote the convolution by
With
{mt+uvmx+(2vux+uvx)m=0,m=u−uxx,t>0,x∈R,nt+uvnx+(2uvx+vux)n=0,n=v−vxx,u(0,x)=u0(x),v(0,x)=v0(x),x∈R. | (5) |
Note that if
{ut+uvux+Px∗(12u2xv+uuxvx+u2v)+12P∗(u2xvx)=0,t>0,x∈R,vt+uvvx+Px∗(12v2xu+vvxux+v2u)+12P∗(v2xux)=0,u(0,x)=u0(x),v(0,x)=v0(x),x∈R. | (6) |
Next we recall the local well-posedness and the conservation laws.
Lemma 2.1. [12] Let
u,v∈C([0,T);Hs(R))∩C1([0,T);Hs−1(R)) |
Moreover, the solution depends continuously on the initial data, i.e. the mapping
Lemma 2.2. [12] Let
∫R(u2(t,x)+u2x(t,x))dx=∫R(u20+u20x)dx,∫R(v2(t,x)+v2x(t,x))dx=∫R(v20+v20x)dx,∫R(u(t,x)v(t,x)+ux(t,x)vx(t,x))dx=∫R(u0v0+u0xv0x)dx. |
Moreover, we have
|u(t,x)|≤√22‖u0‖1,|v(t,x)|≤√22‖v0‖1. |
Note that equation (1) has the solitary waves with corner at their peaks. Obviously, such solitons are not strong solutions to equation (6). In order to provide a mathematical framework for the study of these solitons, we define the notion of weak solutions to equation (6). Let
Fu(u,v)=uvux+Px∗(12u2xv+uuxvx+u2v)+12P∗(u2xvx),Fv(u,v)=uvvx+Px∗(12v2xu+vvxux+v2u)+12P∗(v2xux). |
Then equation (6) can be written as
{ut+Fu(u,v)=0,vt+Fv(u,v)=0,u(0,x)=u0(x),v(0,x)=v0(x). | (7) |
Lemma 2.3. [22] Let
f,g∈L2((0,T);H1(R))anddfdt,dgdt∈L2((0,T);H−1(R)), |
then
⟨f(t),g(t)⟩−⟨f(s),g(s)⟩=∫ts⟨df(τ)dτ,g(τ)⟩dτ+∫ts⟨dg(τ)dτ,f(τ)⟩dτ |
for all
Throughout this paper, let
ρn=(∫Rρ(ξ)dξ)−1nρ(nx),x∈R,n≥1, |
where
ρ(x)={e1x2−1,for|x|<1,0,for|x|≥1. |
Next, we recall two crucial approximation results and two identities.
Lemma 2.4. [7] Let
[ρn∗(fμ)−(ρn∗f)(ρn∗μ)]→0,asn→∞inL1(R). |
Lemma 2.5. [7] Let
ρn∗(fg)−(ρn∗f)(ρn∗g)→0,asn→∞inL∞(R). |
Lemma 2.6. [7] Assume that
ddt∫R|ρn∗u|dx=∫R(ρn∗ut)sgn(ρn∗u)dx |
and
ddt∫R|ρn∗ux|dx=∫R(ρn∗uxt)sgn(ρn∗ux)dx. |
Consider the flow governed by
{dq(t,x)dt=(uv)(t,q),t>0,x∈R,q(0,x)=x,x∈R. | (8) |
Applying classical results in the theory of ODEs, one can obtain the following useful result on the above initial value problem.
Lemma 2.7. [12] Let
qx=exp(∫t0(uv)x(s,q(s,x))ds),∀(t,x)∈[0,T)×R. |
Furthermore, setting
m(t,q)=exp(−∫t0(2vux+uvx)(s,q(s,x))ds)m0,n(t,q)=exp(−∫t0(2uvx+vux)(s,q(s,x))ds)n0,∀(t,x)∈[0,T)×R. |
Theorem 2.8. Let
u,v∈C([0,T);Hs(R))∩C1([0,T);Hs−1(R)). |
Set
(i).m(t,⋅)≥0,n(t,⋅)≥0,u(t,⋅)≥0,v(t,⋅)≥0and|ux(t,⋅)|≤u(t,⋅),|vx(t,⋅)|≤v(t,⋅)onR;(ii).‖u(t,⋅)‖L1≤‖m(t,⋅)‖L1,‖u(t,⋅)‖L∞≤√22‖u(t,⋅)‖1=√22‖u0‖1,and‖v(t,⋅)‖L1≤‖n(t,⋅)‖L1,‖v(t,⋅)‖L∞≤√22‖v(t,⋅)‖1=√22‖v0‖1;(iii).‖ux(t,⋅)‖L1≤‖m(t,⋅)‖L1and‖vx(t,⋅)‖L1≤‖n(t,⋅)‖L1. |
Moreover, if
‖m(t,⋅)‖L1≤e‖u0‖1‖v0‖1t‖m0‖L1and‖n(t,⋅)‖L1≤e‖u0‖1‖v0‖1t‖n0‖L1. |
Proof. Let
u(t,x)=e−x2∫x−∞eym(t,y)dy+ex2∫∞xe−ym(t,y)dy, | (9) |
and
ux(t,x)=−e−x2∫x−∞eym(t,y)dy+ex2∫∞xe−ym(t,y)dy. | (10) |
From the above two relations and
|ux(t,x)|≤u(t,x)≤√22‖u(t,x)‖1. |
In view of Lemma 2.2, we obtain that
u(t,x)≤√22‖u0‖1,∀(t,x)∈R+×R. |
Since
((mn)13)t+((mn)13uv)x=0, |
it immediately follows that
ddt∫Rm(t,x)dx=−∫∞−∞(uvmx+(2vux+uvx)m)dx=∫∞−∞(vuxm−(uvm)x)dx≤‖u‖L∞‖v‖L∞∫∞−∞m(t,x)dx≤‖u0‖1‖v0‖1∫∞−∞m(t,x)dx. |
Since
‖m(t,⋅)‖L1≤e‖u0‖1‖v0‖1t‖m0‖L1. |
Similarly, we find
‖n(t,⋅)‖L1≤e‖u0‖1‖v0‖1t‖n0‖L1. |
This completes the proof of Theorem 2.8.
In this section, we will prove that there exists a unique global weak solution to equation (6), provided the initial data
Theorem 3.1. Let
u,v∈W1,∞(Rx×R)∩L∞(R+;H1(R)) |
with the initial data
Proof. First, we shall prove
‖u0‖L1=‖P∗m0‖L1=sup‖f‖L∞≤1∫Rf(x)(P∗m0)(x)dx=sup‖f‖L∞≤1∫Rf(x)∫RP(x−y)dm0(y)dx=sup‖f‖L∞≤1∫R(P∗f)(y)dm0(y)≤sup‖f‖L∞≤1‖P‖L1‖f‖L∞‖m0‖M(R)=‖m0‖M(R). | (11) |
Similarly, we have
‖v0‖L1≤‖n0‖M(R). | (12) |
We first prove that there exists a corresponding
Let us define
un0⟶u0H1(R),n→∞,vn0⟶v0H1(R),n→∞, | (13) |
and for all
‖un0‖1=‖ρn∗u0‖1≤‖u0‖1,‖vn0‖1≤‖v0‖1,‖un0‖L1=‖ρn∗u0‖L1≤‖u0‖L1,‖vn0‖L1≤‖v0‖L1, | (14) |
in view of Young's inequality. Note that for all
mn0=un0−un0,xx=ρn∗m0≥0,andnn0=vn0−vn0,xx=ρn∗v0≥0. |
Comparing with the proof of relation (11) and (12), we get
‖mn0‖L1≤‖m0‖M(R),and‖nn0‖L1≤‖n0‖M(R),n≥1. | (15) |
By Theorem 2.8, we obtain that there exists a global strong solution
un=un(⋅,un0),vn=vn(⋅,vn0)∈C([0,T);Hs(R))∩C1([0,T);Hs−1(R)) |
for every
‖unx(t,⋅)‖L∞≤‖un(t,⋅)‖L∞≤‖un(t,⋅)‖1=‖un0‖1≤‖u0‖1,‖vnx(t,⋅)‖L∞≤‖vn(t,⋅)‖L∞≤‖vn(t,⋅)‖1=‖vn0‖1≤‖v0‖1. | (16) |
By the above inequality, we have
‖un(t,⋅)vn(t,⋅)unx(t,⋅)‖L2≤‖un(t,⋅)‖L∞‖vn(t,⋅)‖L∞‖unx(t,⋅)‖L2≤‖un(t,⋅)‖21‖vn(t,⋅)‖1≤‖u0‖21‖v0‖1. | (17) |
Similarly, we have
‖vn(t,⋅)un(t,⋅)vnx(t,⋅)‖L2≤‖v0‖21‖u0‖1. | (18) |
By Young's inequality and (16), for all
‖Px∗(12(unx)2vn+ununxvnx+(un)2vn)+12P∗((unx)2vnx)‖L2≤‖Px‖L2‖12(unx)2vn+ununxvnx+(un)2vn‖L1+12‖P‖L2‖(unx)2vnx‖L1≤12‖unx‖2L2‖vn‖L∞+12‖un‖L∞‖unx‖L2‖vnx‖L2+‖un‖2L2‖vn‖L∞+12‖vnx‖L∞‖unx‖2L2≤52‖un‖21‖vn‖1≤52‖u0‖21‖v0‖1. | (19) |
Similarly, we get
‖Px∗(12(vnx)2un+vnunxvnx+(vn)2un)+12P∗((vnx)2unx)‖L2≤52‖v0‖21‖u0‖1. | (20) |
Combining (17)-(20) with equation (6) for all
‖ddtun(t,⋅)‖L2≤72‖u0‖21‖v0‖1,and‖ddtvn(t,⋅)‖L2≤72‖v0‖21‖u0‖1. | (21) |
For fixed
∫T0∫R([un(t,x)]2+[unx(t,x)]2+[unt(t,x)]2)dxdt≤(‖u0‖21+494‖u0‖41‖v0‖21)T,∫T0∫R([vn(t,x)]2+[vnx(t,x)]2+[vnt(t,x)]2)dxdt≤(‖v0‖21+494‖v0‖41‖u0‖21)T. | (22) |
It follows that the sequence
unk⇀uweaklyinH1(0,T)×R)fornk→∞ | (23) |
and
unk⟶u,a.e.on(0,T)×Rfornk→∞, | (24) |
for some
V[unkx(t,⋅)]=‖unkxx(t,⋅)‖L1≤‖unk(t,⋅)‖L1+‖mnk(t,⋅)‖L1≤2‖mnk(t,⋅)‖L1≤2e‖unk0‖1‖vnk0‖1t‖mnk0‖L1≤2e‖u0‖1‖v0‖1t‖m0‖M(R) |
and
‖unkx(t,⋅)‖L∞≤‖unk(t,⋅)‖1=‖unk0(t,⋅)‖1≤‖u0‖1. |
Applying Helly's theorem, we obtain that there exists a subsequence, denoted again by
V[ˆu(t,⋅)]≤2e‖u0‖1‖v0‖1t‖m0‖M(R). |
Since for almost all
unkx⟶uxa.e.on(0,T)×Rfornk→∞, | (25) |
and for a.e.
V[ux(t,⋅)]=‖uxx(t,⋅)‖M(R)≤2e‖u0‖1‖v0‖1t‖m0‖M(R). |
We can analogously extract a subsequence of
vnk⟶va.e.on(0,T)×Rfornk→∞andvnkx⟶vxa.e.on(0,T)×Rfornk→∞. | (26) |
By Theorem 2.8
‖12(unx)2vn+ununxvnx+(un)2vn+12(unx)2vnx‖L1≤3‖u0‖21‖v0‖1. |
For fixed
Px∗[12(unx)2vn+ununxvnx+(un)2vn]+P∗(12(unx)2vnx)⟶Px∗[12u2xvn+uuxvx+u2v]+P∗(12u2xvx),asn→∞. | (27) |
We can analogously obtain that
Px∗[12(vnx)2un+vnvnxunx+(vn)2un]+P∗(12(vnx)2unx)⟶Px∗[12v2xun+vvxux+v2u]+P∗(12v2xux),asn→∞. | (28) |
Combining (24)-(26) with (27) and (28), we deduce that
Since
u∈Cw,loc(R+;H1(R)). |
For a.e.
‖u(t,⋅)‖L∞≤‖u(t,⋅)‖1≤lim infnk→∞‖un(t,⋅)‖1=lim infnk→∞‖unk0(t,⋅)‖1≤lim infnk→∞‖P‖1‖mnk0(t,⋅)‖L1≤‖m0‖M(R), | (29) |
for a.e.
u∈L∞(R+×R)∩L∞(R+;H1(R)). |
Note that by Theorem 2.8 and (15), we have
‖unx(t,⋅)‖L∞≤‖un(t,⋅)‖L∞≤‖un(t,⋅)‖1≤‖P‖1‖mn0(t,⋅)‖L1≤‖m0(t,⋅)‖M(R). | (30) |
Combining this with (25), we deduce that
ux∈L∞(R+×R). |
This shows that
u∈W1,∞(R+×R)∩L∞(R+;H1(R)). |
Taking the same way as
v∈W1,∞(R+×R)∩L∞(R+;H1(R)). |
Please note that we use the subsequence of
Now, by a regularization technique, we prove that
{ρn∗ut+ρn∗(uvux)+ρn∗Px∗(12u2xv+uuxvx+u2v)+12ρn∗P∗(u2xvx)=0,ρn∗vt+ρn∗(uvvx)+ρn∗Px∗(12v2xv+vuxvx+v2u)+12ρn∗P∗(v2xux)=0. | (31) |
By differentiation of the first equation of (31), we obtain
ρn∗uxt+ρn∗(uvux)x+ρn∗Px∗(12u2xvx)+ρn∗Pxx∗(12u2xv+uuxvx+u2v)=0. | (32) |
Note that
ρn∗uxt+ρnx∗(uvux)+ρn∗P∗(12u2xv+uuxvx+u2v)−ρn∗(12u2xv+uuxvx+u2v)+ρn∗Px∗(12u2xvx)=0. | (33) |
Take these two equation (32) and (33) into the integration below, we obtain
12ddt∫R(ρn∗u)2+(ρn∗ux)2dx=∫R(ρn∗u)(ρn∗ut)+(ρn∗ux)(ρn∗uxt)dx=−∫R(ρn∗u)(ρn∗(uvux)+ρn∗Px∗(12u2xv+uuxvx+u2v)+ρn∗P∗(12u2xvx))dx−∫R(ρn∗ux)(ρnx∗(uvux)+ρn∗P∗(12u2xv+uuxvx+u2v)−ρn∗(12u2xv+uuxvx+u2v)+ρn∗Px∗(12u2xvx))dx. | (34) |
Note that
limn→∞‖ρn∗u−u‖L2=limn→∞‖ρn∗(uvux)−uvux‖L2=0. |
Therefore, by using H
∫R(ρn∗u)(ρn∗(uvux))dx⟶∫Ru2vuxdx,asn→∞. |
Similarly, for a.e.
∫R(ρn∗u)(ρn∗Px∗(12u2xv+uuxvx+u2v))dx⟶∫RuPx∗(12u2xv+uuxvx+u2v)dx,asn→∞, |
∫R(ρn∗u)(ρn∗P∗(12u2xvx))dx⟶∫RuP∗(12u2xvx)dx,asn→∞, |
∫R(ρn∗ux)(ρn∗P∗(12u2xv+uuxvx+u2v))dx⟶∫RuxP∗(12u2xv+uuxvx+u2v)dx,asn→∞, |
∫R(ρn∗ux)(ρn∗(12u2xv+uuxvx+u2v))dx⟶∫Rux(12u2xv+uuxvx+u2v)dx,asn→∞, |
∫R(ρn∗ux)(ρn∗Px∗(12u2xvx))dx⟶∫RuxPx∗(12u2xvx)dx,asn→∞, |
as
∫R(ρn∗ux)(ρnx∗(uvux))dx=−∫R(ρn,xx∗u)(ρ∗(uvux))dx+∫R(ρn,xx∗u)(ρn∗uv)(ρn∗ux)dx+12∫R(ρn∗ux)2(ρn∗(uv)x)dx. | (35) |
Observe that
∫R(ρn∗ux)2(ρn∗(uv)x)dx⟶∫Ru2x(uv)xdx,asn→∞. |
On the other hand
‖ρnxx∗u‖L1≤‖uxx‖M(R)≤2e‖u0‖1‖v0‖1t‖m0‖M(R),∀t∈[0,T). |
As
‖(ρn∗uv)(ρn∗ux)−(ρn∗(uvux))‖L∞→0,n→∞. |
Therefore,
∫R(ρn,xx∗u)((ρn∗uv)(ρn∗ux)−ρn∗(uvux))dx→0,n→∞. |
In view of the above relations and (35), we obtain
∫R(ρn∗ux)(ρnx∗(uvux))dx→12∫Ru2x(uv)xdx,n→∞. | (36) |
Let us define
Eun(t)=∫R(ρn∗u)2+(ρn∗ux)2dx, | (37) |
and
Gun(t)=−2∫R(ρn∗u)(ρn∗(uvux)+ρn∗Px∗(12u2xv+uuxvx+u2v)+ρn∗P∗(12u2xvx))dx−2∫R(ρn∗ux)(ρnx∗(uvux)+ρn∗P∗(12u2xv+uuxvx+u2v)−ρn∗(12u2xv+uuxvx+u2v)+ρn∗Px∗(12u2xvx))dx. |
We have proved that for fixed
{ddtEun(t)=Gun(t),n≥1,Gun(t)→0,n→∞. | (38) |
Therefore, we get
Eun(t)−Eun(0)=∫t0Gun(s)ds,t∈[0,T),n≥1. | (39) |
By Young's inequality and H
|Gun(t)|≤Ku(T),n≥1. |
In view of (38) and (39), an application of Lebesgue's dominated convergence theorem yields that for fixed a.e.
limn→∞(Eun(t)−Eun(0))=0. |
By (24) and the above relation, for fixed
Eu(u)=limn→∞Eun(t)=limn→∞Eun(0)=Eu(u0). |
By Theorem 2.8, we infer that for all fixed
Next, we prove that
By differentiation of the second equation of (31), we obtain this relation:
ρn∗vxt+ρnx∗(uvvx)+ρn∗P∗(12v2xu+vuxvx+v2u)−ρn∗(12v2xu+vuxvx+v2u)+ρn∗Px∗(12v2xux)=0. | (40) |
In view of (31), (33) and (40), we obtain
ddt∫R(ρn∗u)(ρn∗v)+(ρn∗ux)(ρn∗vx)dx=∫R(ρn∗u)(ρn∗vt)+(ρn∗ux)(ρn∗vxt)+(ρn∗ut)(ρn∗v)+(ρn∗uxt)(ρn∗vx)dx=−∫R(ρn∗u)(ρn∗(uvvx)+ρn∗Px∗(12v2xu+vuxvx+v2u) |
+ρn∗P∗(12v2xux))dx−∫R(ρn∗ux)(ρnx∗(uvvx)+ρn∗P∗(12v2xu+vuxvx+v2u)−ρn∗(12v2xu+vuxvx+v2u)+ρn∗Px∗(12v2xux))dx−∫R(ρn∗v)(ρn∗(vuux)+ρn∗Px∗(12u2xv+uuxvx+u2v)+ρn∗P∗(12u2xvx))dx−∫R(ρn∗vx)(ρnx∗(uvux)+ρn∗P∗(12u2xv+uuxvx+u2v)−ρn∗(12u2xv+uuxvx+u2v)+ρn∗Px∗(12u2xvx))dx. | (41) |
We can analogously get the similar convergence like the case
It is nature to define
Hn(t)=∫R(ρn∗u)(ρn∗v)+(ρn∗ux)(ρn∗vx)dx, | (42) |
and
Gu,vn(t)=−∫R(ρn∗u)(ρn∗(uvvx)+ρn∗Px∗(12v2xu+vuxvx+v2u)+ρn∗P∗(12v2xux))dx−∫R(ρn∗ux)(ρnx∗(uvvx)+ρn∗P∗(12v2xu+vuxvx+v2u)−ρn∗(12v2xu+vuxvx+v2u)+ρn∗Px∗(12v2xux))dx−∫R(ρn∗v)(ρn∗(vuux)+ρn∗Px∗(12u2xv+uuxvx+u2v)+ρn∗P∗(12u2xvx))dx−∫R(ρn∗vx)(ρnx∗(uvux)+ρn∗P∗(12u2xv+uuxvx+u2v)−ρn∗(12u2xv+uuxvx+u2v)+ρn∗Px∗(12u2xvx))dx. | (43) |
And it is easy to get
Hn(t)−Hn(0)=∫t0Gu,vn(s)ds,t∈[0,T),n≥1. | (44) |
Similarly, we get this estimate by using Young's inequality and Holder's inequality:
|Gu,vn(t)|≤Ku,v(T),n≥1. |
An application of Lebesgue's dominated convergence theorem yields that for fixed a.e.
limn→∞[Hn(t)−Hn(0)]=0. |
By these convergence above, for fixed
H(u,v)=limn→∞Hn(t)=limn→∞Hn(0)=H(u0,v0), |
which indicates that
Since
‖m(t,⋅)‖≤3e‖u0‖1‖v0‖1t‖m0‖M(R). |
For any fixed
(u(t,⋅)−uxx(t,⋅))∈M(R). |
Therefore, in view of (24) and (25), we obtain that for all
unk(t,⋅)−unkxx(t,⋅)→u(t,⋅)−uxx(t,⋅)inD′(R). |
Since
u(t,⋅)−uxx(t,⋅)∈M+(R). |
Similarly, we arrive at the conclusion:
v(t,⋅)−vxx(t,⋅)∈M+(R). |
Finally, we show the uniqueness of the weak solutions of equation (6). Let
(f,g)∈W1,∞(R+×R)∩L∞(R+;H1(R))×W1,∞(R+×R)∩L∞(R+;H1(R)) |
Note that
‖u(t,⋅)−uxx(t,⋅)‖M(R)≤3e‖u0‖1‖v0‖1t‖m0‖M(R),‖v(t,⋅)−vxx(t,⋅)‖M(R)≤3e‖u0‖1‖v0‖1t‖n0‖M(R)fora.e.t∈[0,T). |
Define
M(T)=supt∈[0,T){‖u(t,⋅)−uxx(t,⋅)‖M(R)+‖v(t,⋅)−vxx(t,⋅)‖M(R)+‖ˉu(t,⋅)−ˉuxx(t,⋅)‖M(R)+‖ˉv(t,⋅)−ˉvxx(t,⋅)‖M(R)}. |
Then for fixed
‖u(t,⋅)‖L1≤‖P‖L1M(T)=M(T),‖ux(t,⋅)‖L1≤‖Px‖L1M(T)=M(T),‖v(t,⋅)‖L1,‖vx(t,⋅)‖L1,‖ˉu(t,⋅)‖L1,‖ˉux(t,⋅)‖L1,‖ˉv(t,⋅)‖L1and‖ˉvx(t,⋅)‖L1≤M(T). | (45) |
On the other hand, from (29) and (30), we have
‖u(t,⋅)‖L∞≤‖m0‖M(R)≤N,‖ux(t,⋅)‖L∞≤‖m0‖M(R)≤N,‖v(t,⋅)‖L∞≤‖n0‖M(R)≤N,‖vx(t,⋅)‖L∞≤‖n0‖M(R)≤N,‖ˉu(t,⋅)‖L∞,‖ˉux(t,⋅)‖L∞,‖ˉv(t,⋅)‖L∞and‖ˉvx(t,⋅)‖L∞≤N. | (46) |
Let us define
ˆu(t,x)=u(t,x)−ˉu(t,x)andˆv(t,x)=v(t,x)−ˉv(t,x),(t,x)∈[0,T)×R. |
Convoluting equation (6) for
ddt∫R|ρn∗ˆu|dx=∫Rρn∗ˆutsgn(ρn∗ˆu)dx=−∫Rρn∗(ˆuvux+ˉuuxˆv+ˉuˉvˆux)sgn(ρn∗ˆu)dx−∫Rρn∗Pxx∗(12ˆu(ux+ˉux)v+12ˉu2xˆv+ˆuuxvx+ˉuvxˆux+ˉuˉuxˆvx+ˆu(u+ˉu)v+u2ˆv)sgn(ρnx∗ˆu)dx−∫Rρn∗12P∗(ˆux(ux+ˉux)vx+ˉu2xˆvx)sgn(ρn∗ˆu)dx. | (47) |
Using (46) and Young's inequality, we infer that for a.e.
ddt∫R|ρn∗ˆu|dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx). | (48) |
where
ddt∫R|ρn∗ˆux|dx=∫Rρn∗ˆuxtsgn(ρnx∗ˆu)dx=−∫Rρn∗(ˆuvux+ˉuuxˆv+ˉuˉvˆux)xsgn(ρnx∗ˆu)dx−∫Rρn∗Pxx∗(12ˆu(ux+ˉux)v+12ˉu2xˆv+ˆuuxvx+ˉuvxˆux+ˉuˉuxˆvx+ˆu(u+ˉu)v+u2ˆv)sgn(ρnx∗ˆu)dx−∫Rρn∗12Px∗(ˆux(ux+ˉux)vx+ˉu2xˆvx)sgn(ρnx∗ˆu)dx=I1+I2+I3. | (49) |
For the term
I1=−∫Rρn∗(ˆuxvux+ˆuuxvx+ˆuvuxx+ˉuxuxˆv+ˉuuxxˆv+ˉuuxˆvx+ˉuxˉvˆux+ˉuˉvxˆux+ˉuˉvˆuxx)sgn(ρnx∗ˆu)dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)−∫Rρn∗(ˆuvuxx+ˉuuxxˆv+ˉuˉvˆuxx)sgn(ρnx∗ˆu)dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)−∫R(ρn∗ˆuv)(ρn∗uxx)sgn(ρnx∗ˆu)dx−∫R(ρn∗ˉuˆv)(ρn∗uxx)sgn(ρnx∗ˆu)dx−∫R(ρn∗ˉuˉv)(ρn∗ˆuxx)sgn(ρnx∗ˆu)dx+Rn(t) |
≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)+∫R(ρn∗(ˆuv)x)(ρn∗ux)sgn(ρnx∗ˆu)dx+∫R(ρn∗(ˉuˆv)x)(ρn∗ux)sgn(ρnx∗ˆu)dx+∫R(ρn∗(ˉuˉv)x)(ρn∗ˆux)sgn(ρnx∗ˆu)dx+Rn(t)≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)+Rn(t), | (50) |
where C is a constant depending on
{Rn(t)⟶0,n→∞,|Rn(t)|≤κ(T),n≥1,t∈[0,T). | (51) |
For the second term
I2=−∫Rρn∗Pxx∗(12ˆu(ux+ˉux)v+12ˉu2xˆv+ˆuuxvx+ˉuvxˆux+ˉuˉuxˆvx+ˆu(u+ˉu)v+u2ˆv)sgn(ρnx∗ˆu)dx≤2∫Rρn∗|12ˆu(ux+ˉux)v+12ˉu2xˆv+ˆuuxvx+ˉuvxˆux+ˉuˉuxˆvx+ˆu(u+ˉu)v+u2ˆv|dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx). | (52) |
For the final term
I3=−∫Rρn∗12Px∗(ˆux(ux+ˉux)vx+ˉu2xˆvx)sgn(ρnx∗ˆu)dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx). | (53) |
Adding these three terms, we obtain
ddt∫R|ρn∗ˆux|dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)+Rn(t). | (54) |
For these terms
ddt∫R|ρn∗ˆv|dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx),ddt∫R|ρn∗ˆvx|dx≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)+Rn(t). | (55) |
From (48), (54) and (55), we infer that
ddt(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)≤C(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)+Rn(t). | (56) |
If
(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)≤e∫t0C+˜Rn(τ)dτ(|ρn∗ˆu|+|ρn∗ˆux|+|ρn∗ˆv|+|ρn∗ˆvx|)(0,x), | (57) |
where
(∫R|ρn∗ˆu|dx+∫R|ρn∗ˆux|dx+∫R|ρn∗ˆv|dx+∫R|ρn∗ˆvx|dx)≤eCt(|ρn∗ˆu|+|ρn∗ˆux|+|ρn∗ˆv|+|ρn∗ˆvx|)(0,x), | (58) |
As
[1] |
Y. Ye, P. Huang, Y. Zhang, Deep learning-based fault diagnostic network of high-speed train secondary suspension systems for immunity to track irregularities and wheel wear, Railway Eng. Sci., 30 (2022), 96–116. https://doi.org/10.1007/s40534-021-00252-z doi: 10.1007/s40534-021-00252-z
![]() |
[2] |
V. Lauda, V. Novotný, Role of railway transport in green deal 2050 challenge-situation in the Czech Republic, Promet-Traffic Transp., 34 (2022), 801–812. https://doi.org/10.7307/ptt.v34i5.4117 doi: 10.7307/ptt.v34i5.4117
![]() |
[3] |
L. Zhang, Z. Wang, Q. Wang, J. Mo, J. Feng, K. Wang, The effect of wheel polygonal wear on temperature and vibration characteristics of a high-speed train braking system, Mech. Syst. Signal. Process., 186 (2023), 109864. https://doi.org/10.1016/j.ymssp.2022.109864 doi: 10.1016/j.ymssp.2022.109864
![]() |
[4] |
I. A. Tasiu, Z. Liu, S. Wu, W. Yu, M. Barashi, J. O. Ojo, Review of recent control strategies for the traction converters in high-speed train, IEEE Trans. Transp. Electrif., 8(2022), 2311–2333. https://doi.org/10.1109/TTE.2022.3140470 doi: 10.1109/TTE.2022.3140470
![]() |
[5] |
J. Yang, J. Wang, Y. Zhao, Simulation of nonlinear characteristics of vertical vibration of railway freight wagon varying with train speed, Electron. Res. Arch., 30 (2022), 4382–4400. https://doi.org/10.3934/era.2022222 doi: 10.3934/era.2022222
![]() |
[6] |
H. Song, E. Schnieder, Availability and performance analysis of train-to-train data communication system, IEEE Trans. Intell. Transp. Syst., 20 (2019), 2786–2795. https://doi.org/10.1109/TITS.2019.2914701 doi: 10.1109/TITS.2019.2914701
![]() |
[7] |
W. Li, O. P. Hilmola, J. Wu, Chinese high-speed railway: Efficiency comparison and the future, Promet-Traffic Transp., 31 (2019), 693–702. https://doi.org/10.7307/ptt.v31i6.3220 doi: 10.7307/ptt.v31i6.3220
![]() |
[8] |
F. Bădău, Railway interlockings—A review of the current state of railway safety technology in Europe, Promet-Traffic Transp., 34 (2022), 443–454. https://doi.org/10.7307/ptt.v34i3.3992 doi: 10.7307/ptt.v34i3.3992
![]() |
[9] |
H. Gu, T. Liu, Z. Jiang, Z. Guo, Experimental and simulation research on the aerodynamic effect on a train with a wind barrier in different lengths, J. Wind Eng. Ind. Aerod., 214 (2021), 104644. https://doi.org/10.1016/j.jweia.2021.104644 doi: 10.1016/j.jweia.2021.104644
![]() |
[10] |
H. Song, S. Gao, Y. Li, L. Liu, H. Dong, Train-centric communication based autonomous train control system, IEEE Trans. Intell. Veh., 8 (2022), 721–731. https://doi.org/10.1109/TIV.2022.3192476 doi: 10.1109/TIV.2022.3192476
![]() |
[11] |
R. Kour, A. Patwardhan, A. Thaduri, R. Karim, A review on cybersecurity in railways, Proc. Inst. Mech. Eng., Part F: J. Rail Rapid Transit, 237 (2022), 3–20. https://doi.org/10.1177/09544097221089389 doi: 10.1177/09544097221089389
![]() |
[12] |
L. Zhang, Vibration analysis and multi-state feedback control of maglev vehicle-guideway coupling system, Electron. Res. Arch., 30 (2022), 3887–3901. https://doi.org/10.3934/era.2022198 doi: 10.3934/era.2022198
![]() |
[13] | M. Roth, H. Winter, An open data set for rail vehicle positioning experiments, in 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC), IEEE, (2020), 1–7. https://doi.org/10.1109/ITSC45102.2020.9294594 |
[14] | H. Winter, Rail vehicle positioning data set: Lucy, October 2018, Technische Universität Darmstadt, 2020. https://doi.org/10.25534/tudatalib-360 |
[15] |
Y. Cao, Z. Zhang, F. Cheng, S. Su, Trajectory optimization for high-speed trains via a mixed integer linear programming approach, IEEE Trans. Intell. Transp. Syst., 23 (2022), 17666–17676. https://doi.org/10.1109/TITS.2022.3155628 doi: 10.1109/TITS.2022.3155628
![]() |
[16] |
M. Spiryagin, Q. Wu, O. Polach, J. Thorburn, W. Chua, V. Apiryagin, et al., Problems, assumptions and solutions in locomotive design, traction and operational studies, Railway Eng. Sci., 30 (2022), 265–288. https://doi.org/10.1007/s40534-021-00263-w doi: 10.1007/s40534-021-00263-w
![]() |
[17] |
Y. Feng, C. Zhao, W. Zhai, L. Tong, X. Liang, Y. Shu, Dynamic performance of medium speed maglev train running over girders: field test and numerical simulation, Int. J. Struct. Stab. Dyn., 23 (2023), 2350006. https://doi.org/10.1142/S0219455423500062 doi: 10.1142/S0219455423500062
![]() |
[18] |
W. Yu, D. Huang, Q. Wang, L. Cai, Distributed event-triggered iterative learning control for multiple high-speed trains with switching topologies: A data-driven approach, IEEE Trans. Intell. Transp. Syst., 2023 (2023). https://doi.org/10.1109/TITS.2023.3277452 doi: 10.1109/TITS.2023.3277452
![]() |
[19] |
J. Yin, C. Ning, T. Tang, Data-driven models for train control dynamics in high-speed railways: LAG-LSTM for train trajectory prediction, Inf. Sci., 600 (2022), 377–400. https://doi.org/10.1016/j.ins.2022.04.004 doi: 10.1016/j.ins.2022.04.004
![]() |
[20] |
Q. Wu, C. Cole, Computing schemes for longitudinal train dynamics: sequential, parallel and hybrid, J. Comput. Nonlinear Dyn., 10 (2015), 064502. https://doi.org/10.1115/1.4029716 doi: 10.1115/1.4029716
![]() |
[21] |
Q. Wu, C. Cole, S. Maksym, W. Yucang, W. Ma, C. Wei, Railway air brake model and parallel computing scheme, J. Comput. Nonlinear Dyn., 12 (2017), 051017. https://doi.org/10.1115/1.4036421 doi: 10.1115/1.4036421
![]() |
[22] |
K. Fadhloun, H. Rakha, A. Loulizi, A. Abdelkefi, Vehicle dynamics model for estimating typical vehicle accelerations, Transp. Res. Rec., 2491 (2015), 61–71. https://doi.org/10.3141/2491-07 doi: 10.3141/2491-07
![]() |
[23] |
Y. Cheng, J. Yin, L. Yang, Robust energy-efficient train speed profile optimization in a scenario-based position–Time–Speed network, Front. Eng. Manage., 8 (2021), 595–614. https://doi.org/10.1007/s42524-021-0173-1 doi: 10.1007/s42524-021-0173-1
![]() |
[24] | M. Zhang, Z. Yang, J. Cheng, Speed and distance measurement algorithm of train control onboard equipment based on adaptive federated filter, China Railway Sci., 43 (2022), 144–151. |
[25] |
H. Chen, T. Furuya, S. Fukagai, S. Saga, J. Ikoma, K. Kimura, et al., Wheel slip/slide and low adhesion caused by fallen leaves, Wear, 446 (2020), 203187. https://doi.org/10.1016/j.wear.2020.203187 doi: 10.1016/j.wear.2020.203187
![]() |
[26] |
D. Zhang. Y. Tang, Q. Peng, A novel approach for decreasing driving energy consumption during coasting and cruise for the railway vehicle, Energy, 263 (2023), 125615. https://doi.org/10.1016/j.energy.2022.125615 doi: 10.1016/j.energy.2022.125615
![]() |
[27] |
Y. Peng, Y. Lin, C. Fan, Q, Xu, D. Xu, S. Yi, et al., Passenger overall comfort in high-speed railway environments based on EEG: assessment and degradation mechanism, Build. Environ., 210 (2022), 108711. https://doi.org/10.1016/j.buildenv.2021.108711 doi: 10.1016/j.buildenv.2021.108711
![]() |
[28] | Y. Lu, D. Chen, Z. Zhao, Algorithm for automatically generating a large number of speed curves of subway trains based on AlphaZero, Chin. J. Intell. Sci. Technol., 3 (2021), 179–184. |
1. | Byungsoo Moon, Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation, 2021, 14, 1937-1632, 4409, 10.3934/dcdss.2021123 | |
2. | K.H. Karlsen, Ya. Rybalko, Global semigroup of conservative weak solutions of the two-component Novikov equation, 2025, 86, 14681218, 104393, 10.1016/j.nonrwa.2025.104393 |