
Objectives: The aim of this narrative review of the literature was to synthesize and comment the mechanisms of action of avocado/soybean unsaponifiable mixture (ASU-E, Piascledine®300) on articular tissues involved in the OA pathogenesis. Materials and methods: The search was performed in Pubmed and Scopus between January 1981 and December 2016. Keywords used were—any field—(Cartilage OR Bone OR Synovium) AND Avocado AND Soybean. 32 articles out-off 35 found have been considered. The review has included eleven in vitro and animal studies investigating Avocado Soybean Unsaponifiables (ASU) from Laboratoires Expanscience (Piascledine®300) used separately or in combination. Only research articles published in English and French have been taken into account. Results: ASU-E stimulated proteoglycans synthesis in chondrocytes cultures and counteracted the effects of IL-1 on metalloproteases and inflammatory mediators. Some of these effects were associated with inhibition of NF-kB nuclear translocation and stimulation of TGF-synthesis. ASU-E also positively modulated the altered phenotype of OA subchondral bone osteoblasts and reduced the production of collagenases by synovial cells. Conclusions: ASU-E has positive effects on the metabolic changes of synovium, subchondral bone and cartilage which are the main tissues involved in the pathophysiology of OA. These findings contribute to explain the beneficial effects of ASU-E in clinical trials.
Citation: Yves Edgard Henrotin. Avocado/Soybean Unsaponifiables (Piacledine®300) show beneficial effect on the metabolism of osteoarthritic cartilage, synovium and subchondral bone: An overview of the mechanisms[J]. AIMS Medical Science, 2018, 5(1): 33-52. doi: 10.3934/medsci.2018.1.33
[1] | Paola Goatin, Chiara Daini, Maria Laura Delle Monache, Antonella Ferrara . Interacting moving bottlenecks in traffic flow. Networks and Heterogeneous Media, 2023, 18(2): 930-945. doi: 10.3934/nhm.2023040 |
[2] | Felisia Angela Chiarello, Paola Goatin . Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14(2): 371-387. doi: 10.3934/nhm.2019015 |
[3] | Jan Friedrich, Oliver Kolb, Simone Göttlich . A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024 |
[4] | Abraham Sylla . Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks and Heterogeneous Media, 2021, 16(2): 221-256. doi: 10.3934/nhm.2021005 |
[5] | Christophe Chalons, Paola Goatin, Nicolas Seguin . General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433 |
[6] | Caterina Balzotti, Simone Göttlich . A two-dimensional multi-class traffic flow model. Networks and Heterogeneous Media, 2021, 16(1): 69-90. doi: 10.3934/nhm.2020034 |
[7] | Dong Li, Tong Li . Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2011, 6(4): 681-694. doi: 10.3934/nhm.2011.6.681 |
[8] | Raimund Bürger, Kenneth H. Karlsen, John D. Towers . On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(3): 461-485. doi: 10.3934/nhm.2010.5.461 |
[9] | Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada . A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, 2021, 16(2): 187-219. doi: 10.3934/nhm.2021004 |
[10] | Alexander Kurganov, Anthony Polizzi . Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks and Heterogeneous Media, 2009, 4(3): 431-451. doi: 10.3934/nhm.2009.4.431 |
Objectives: The aim of this narrative review of the literature was to synthesize and comment the mechanisms of action of avocado/soybean unsaponifiable mixture (ASU-E, Piascledine®300) on articular tissues involved in the OA pathogenesis. Materials and methods: The search was performed in Pubmed and Scopus between January 1981 and December 2016. Keywords used were—any field—(Cartilage OR Bone OR Synovium) AND Avocado AND Soybean. 32 articles out-off 35 found have been considered. The review has included eleven in vitro and animal studies investigating Avocado Soybean Unsaponifiables (ASU) from Laboratoires Expanscience (Piascledine®300) used separately or in combination. Only research articles published in English and French have been taken into account. Results: ASU-E stimulated proteoglycans synthesis in chondrocytes cultures and counteracted the effects of IL-1 on metalloproteases and inflammatory mediators. Some of these effects were associated with inhibition of NF-kB nuclear translocation and stimulation of TGF-synthesis. ASU-E also positively modulated the altered phenotype of OA subchondral bone osteoblasts and reduced the production of collagenases by synovial cells. Conclusions: ASU-E has positive effects on the metabolic changes of synovium, subchondral bone and cartilage which are the main tissues involved in the pathophysiology of OA. These findings contribute to explain the beneficial effects of ASU-E in clinical trials.
Macroscopic traffic flow models based on fluid-dynamics equations have been introduced in the transport engineering literature since the mid-fifties of last century, with the celebrated Lighthill, Whitham [11] and Richards [13] (LWR) model. Since then, the engineering and applied mathematical literature on the subject has considerably grown, addressing the need for more sophisticated models better capturing traffic flow characteristics. Indeed, the LWR model is based on the assumption that the mean traffic speed is a function of the traffic density, which is not experimentally verified in congested regimes. To overcome this issue, the so-called "second order" models (e.g. Payne-Whitham [12,15] and Aw-Rascle-Zhang [3,16]) consist of a mass conservation equation for the density and an acceleration balance law for the speed, thus considering the two quantities as independent.
More recently, "non-local" versions of the LWR model have been proposed in [5,14], where the speed function depends on a weighted mean of the downstream vehicle density to better represent the reaction of drivers to downstream traffic conditions.
Another limitation of the standard LWR model is the first-in first-out rule, not allowing faster vehicles to overtake slower ones. To address this and other traffic heterogeneities, "multi-class" models consist of a system of conservation equations, one for each vehicle class, coupled in the speed terms, see [4] and references therein for more details.
In this paper, we consider the following class of non-local systems of
$ ∂tρi(t,x)+∂x(ρi(t,x)vi((r∗ωi)(t,x)))=0,i=1,...,M, $ | (1) |
where
$ r(t,x):=M∑i=1ρi(t,x), $ | (2) |
$ vi(ξ):=vmaxiψ(ξ), $ | (3) |
$ (r∗ωi)(t,x):=∫x+ηixr(t,y)ωi(y−x)dy, $ | (4) |
and we assume:
We couple (1) with an initial datum
$ ρi(0,x)=ρ0i(x),i=1,…,M. $ | (5) |
Model (1) is obtained generalizing the
Due to the possible presence of jump discontinuities, solutions to (1), (5) are intended in the following weak sense.
Definition 1.1. A function
$ \int_0^T\!\!\int_{-\infty}^\infty \left(\rho_i \partial_t \varphi +\rho_i v_i(r\ast\omega_i) \partial_x \varphi \right)(t, x) \mathinner{{\rm{d}}{x}} \mathinner{{\rm{d}}{t}} +\int_{-\infty}^\infty \rho_i^0 (x)\varphi(0, x) \mathinner{{\rm{d}}{x}} = 0 $ |
for all
The main result of this paper is the proof of existence of weak solutions to (1), (5), locally in time. We remark that, since the convolution kernels
Theorem 1.2. Let
In this work, we do not address the question of uniqueness of the solutions to (1). Indeed, even if discrete entropy inequalities can be derived as in [5,Proposition 3], in the case of systems this is in general not sufficient to single out a unique solution.
The paper is organized as follows. Section 2 is devoted to prove uniform
First of all, we extend
To this end, we approximate the initial datum
$ ρ0i,j=1Δx∫xj+1/2xj−1/2ρ0i(x)dx,j∈Z. $ |
Similarly, for the kernel, we set
$ ωki:=1Δx∫(k+1)ΔxkΔxω0i(x)dx,k∈N, $ |
so that
$ Vni,j:=vmaxiψ(Δx+∞∑k=0ωkirnj+k),i=1,…,M,j∈Z. $ | (6) |
We consider the following Godunov-type scheme adapted to (1), which was introduced in [8] in the scalar case:
$ ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j) $ | (7) |
where we have set
We provide here the necessary estimates to prove the convergence of the sequence of approximate solutions constructed via the Godunov scheme (7).
Lemma 2.1. (Positivity) For any
$ λ≤1vmaxM‖ψ‖∞, $ | (8) |
the scheme (7) is positivity preserving on
Proof. Let us assume that
$ ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≥0 $ | (9) |
under assumption (8).
Corollary 1. (
$ ‖ρni‖1=‖ρ0i‖1,i=1,…,M, $ | (10) |
where
Proof. Thanks to Lemma 2.1, for all
$ ‖ρn+1i‖1=Δx∑jρn+1i,j=Δx∑j(ρni,j−λρni,jVni,j+1+λρni,j−1Vni,j)=Δx∑jρni,j, $ |
proving (10).
Lemma 2.2. (
$ T < \left( M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0\right)^{-1}. $ |
Proof. Let
$ ρn+1i,j=ρni,j(1−λVni,j+1)+λρni,j−1Vni,j≤ˉρ(1+λ(Vni,j−Vni,j+1)) $ | (11) |
and
$ |Vni,j−Vni,j+1|=vmaxi|ψ(Δx+∞∑k=0ωkirnj+k)−ψ(Δx+∞∑k=0ωkirnj+k+1)|≤vmaxi‖ψ′‖∞Δx|+∞∑k=0ωki(rnj+k+1−rnj+k)|=vmaxi‖ψ′‖∞Δx|−ω0irnj++∞∑k=1(ωk−1i−ωki)rnj+k|≤vmaxi‖ψ′‖∞ΔxM‖ρn‖∞ωi(0) $ | (12) |
where
$ {\left\|{{{ \boldsymbol{\rho}}}^{n+1}}\right\|}_\infty \leq {\left\|{{{ \boldsymbol{\rho}}}^{n}}\right\|}_\infty \left( 1+ M K v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 \Delta t \right), $ |
which implies
$ ‖ρn‖∞≤‖ρ0‖∞eCnΔt, $ |
with
$ t\leq \frac{1}{ M K v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}\ln \left(\frac{K}{{\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty}\right) \leq \frac{1}{ M e {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}\, , $ |
where the maximum is attained for
Iterating the procedure, at time
$ t^{m+1}\leq t^m + \frac{m}{ M e^m {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0}. $ |
Therefore, the approximate solution remains bounded, uniformly in
$ T\leq \frac{1}{ M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0} \sum\limits_{m = 1}^{+\infty} \frac{m}{e^m} \leq \frac{1}{ M {\left\|{{ \boldsymbol{\rho}}^0}\right\|}_\infty v_M^{\max} {\left\|{\psi'}\right\|}_\infty W_0 }\, . $ |
Remark 1. Figure 1 shows that the simplex
$ \mathcal{S}: = \left\{ \boldsymbol{\rho}\in \mathbb{R}^M \colon \sum\limits_{i = 1}^M \rho_i \leq 1, \; \rho_i \geq 0\; \mbox{for}\; i = 1, \ldots, M\right\} $ |
is not an invariant domain for (1), unlike the classical multi-population model [4]. Indeed, let us consider the system
$ ∂tρi(t,x)+∂x(ρi(t,x)vi(r(t,x)))=0,i=1,...,M, $ | (13) |
where
Lemma 2.3. Under the CFL condition
$ λ≤1vmaxM(‖ψ‖∞+‖ψ′‖∞), $ |
for any initial datum
$ ρn+1j=ρnj−λ[F(ρnj,ρnj+1)−F(ρnj−1,ρnj)], $ | (14) |
with
$ ρnj∈S∀j∈Z,n∈N. $ | (15) |
Proof. Assuming that
$ ρn+1i,j=ρni,j−λ[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]. $ |
Summing on the index
$ rn+1j=M∑i=1ρn+1i,j=M∑i=1ρni,j−λM∑i=1[vmaxiρni,jψ(rnj+1)−vmaxiρni,j−1ψ(rnj)]=rnj+λψ(rnj)M∑i=1vmaxiρni,j−1−λψ(rnj+1)M∑i=1vmaxiρni,j. $ |
Defining the following function of
$ \Phi(\rho^n_{1, j}, \ldots, \rho^n_{M, j}) = r^n_j+\lambda {\psi(r^n_j)\sum\limits_{i = 1}^{M}} v_i^{\max}\rho^n_{i, j-1}-\lambda {\psi(r^n_{j+1})\sum\limits_{i = 1}^{M}} v_i^{\max}\rho^n_{i, j}, $ |
we observe that
$ \Phi(0, \ldots, 0) = \lambda \psi(0)\sum\limits_{i}^M v_i^{\max}\rho^n_{i, j-1}\leq \lambda {\left\|{\psi}\right\|}_\infty v_{M}^{\max} \leq 1 $ |
if
$ \Phi(\rho^n_{1, j}, ..., \rho^n_{M, j}) = 1-\lambda \psi(r^n_{j+1})\sum\limits_{i = 1}^{M}v_i^{\max}\rho^n_{i, j}\leq 1 $ |
for
$ \frac{ \partial\Phi}{ \partial\rho^n_{i, j}} ( \boldsymbol{\rho}_j^n) = 1 + \lambda \psi'(r^n_j) \sum\limits_{i = 1}^M v_i^{\max} \rho^n_{i, j-1} - \lambda\psi(r^n_{j+1}) v_i^{\max} \geq 0 $ |
if
$ ρn+1i,j=ρni,j(1−λvmaxiψ(rnj+1))+λvmaxiρni,j−1ψ(rnj)≥0 $ |
if
Lemma 2.4. (Spatial
$ T≤mini=1,…,M 1H(TV(ρ0i)+1), $ | (16) |
where
Proof. Subtracting the identities
$ ρn+1i,j+1=ρni,j+1−λ(ρni,j+1Vni,j+2−ρni,jVni,j+1), $ | (17) |
$ ρn+1i,j=ρni,j−λ(ρni,jVni,j+1−ρni,j−1Vni,j), $ | (18) |
and setting
$ Δn+1i,j+1/2=Δni,j+1/2−λ(ρni,j+1Vni,j+2−2ρni,jVni,j+1+ρni,j−1Vni,j). $ |
Now, we can write
$ Δn+1i,j+1/2=(1−λVni,j+2)Δni,j+1 $ | (19) |
$ +λVni,jΔni,j−1/2−λρni,j(Vni,j+2−2Vni,j+1+Vni,j). $ | (20) |
Observe that assumption (8) guarantees the positivity of (19). The term (20) can be estimated as
$ Vni,j+2−2Vni,j+1+Vni,j==vmaxi(ψ(Δx+∞∑k=0ωkirnj+k+2)−2ψ(Δx+∞∑k=0ωkirnj+k+1)+ψ(Δx+∞∑k=0ωkirnj+k))=vmaxiψ′(ξj+1)Δx(+∞∑k=0ωkirnj+k+2−+∞∑k=0ωkirnj+k+1)+vmaxiψ′(ξj)Δx(+∞∑k=0ωkirnj+k−+∞∑k=0ωkirnj+k+1)=vmaxiψ′(ξj+1)Δx(+∞∑k=1(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωki−ωk−1i)rnj+k+ω0irnj)=vmaxi(ψ′(ξj+1)−ψ′(ξj))Δx(+∞∑k=1(ωk−1i−ωki)rnj+k+1−ω0irnj+1)+vmaxiψ′(ξj)Δx(+∞∑k=1(ωk−1i−ωki)(rnj+k+1−rnj+k)+ω0i(rnj−rnj+1))=vmaxiψ″(˜ξj+1/2)(ξj+1−ξj)Δx(+∞∑k=1M∑β=1ωkiΔnβ,j+k+3/2)+vmaxiψ′(ξj)Δx(M∑β=1N−1∑k=1(ωk−1i−ωki)Δnβ,j+k+1/2−ω0iΔnβ,j+1/2), $ |
with
$ ξj+1−ξj=ϑΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+2+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k=ϑΔx+∞∑k=1ωk−1iM∑β=1ρnβ,j+k+1+(1−ϑ)Δx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−μΔx+∞∑k=0ωkiM∑β=1ρnβ,j+k+1−(1−μ)Δx+∞∑k=−1ωk+1iM∑β=1ρnβ,j+k+1=Δx+∞∑k=1[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]M∑β=1ρnβ,j+k+1+(1−ϑ)Δxω0iM∑β=1ρnβ,j+1−μΔxω0iM∑β=1ρnβ,j+1−(1−μ)Δx(ω0iM∑β=1ρnβ,j+ω1iM∑β=1ρnβ,j+1). $ |
By monotonicity of
$ \vartheta \omega_i^{k-1} + (1- \vartheta)\omega_i^{k} -\mu \omega_i^{k} -(1-\mu)\omega_i^{k+1} \geq 0\, . $ |
Taking the absolute values we get
$ |ξj+1−ξj|≤Δx{+∞∑k=2[ϑωk−1i+(1−ϑ)ωki−μωki−(1−μ)ωk+1i]+4ω0i}M‖ρn‖∞≤Δx{+∞∑k=2[ωk−1i−ωk+1i]+4ω0i}M‖ρn‖∞≤Δx6W0M‖ρn‖∞. $ |
Let now
$ ∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1−λ(Vni,j+2−Vni,j+1))+ΔtHK1, $ |
where
$ ∑j|Δn+1i,j+1/2|≤∑j|Δni,j+1/2|(1+ΔtG)+ΔtHK1, $ |
with
$ ∑j|Δni,j+1/2|≤eGnΔt∑j|Δ0i,j+1/2|+eHK1nΔt−1, $ |
that we can rewrite as
$ TV(ρΔxi)(nΔt,⋅)≤eGnΔtTV(ρ0i)+eHK1nΔt−1≤eHK1nΔt(TV(ρ0i)+1)−1, $ |
since
$ t≤1HK1ln(K1+1TV(ρ0i)+1), $ |
where the maximum is attained for some
$ \ln\left(\frac{K_1+1}{ \mathinner{{\rm{TV}}}(\rho^0_i)+1}\right) = \frac{K_1}{K_1+1}\, . $ |
Therefore the total variation is uniformly bounded for
$ t≤1He(TV(ρ0i)+1). $ |
Iterating the procedure, at time
$ tm+1≤tm+mHem(TV(ρ0i)+1). $ | (21) |
Therefore, the approximate solution has bounded total variation for
$ T≤1H(TV(ρ0i)+1). $ |
Corollary 2. Let
Proof. If
$ TV(ρΔxi;[0,T]×R)=nT−1∑n=0∑j∈ZΔt|ρni,j+1−ρni,j|+(T−nTΔt)∑j∈Z|ρnTi,j+1−ρnTi,j|⏟≤Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|. $ |
We then need to bound the term
$ \sum\limits_{n = 0}^{n_T-1}\sum\limits_{j\in \mathbb{Z}}\Delta x {\left|{\rho_{i, j}^{n+1}-\rho_{i, j}^n}\right|}. $ |
From the definition of the numerical scheme (7), we obtain
$ ρn+1i,j−ρni,j=λ(ρni,j−1Vni,j−ρni,jVni,j+1)=λ(ρni,j−1(Vni,j−Vni,j+1)+Vni,j+1(ρni,j−1−ρni,j)). $ |
Taking the absolute values and using (12) we obtain
$ |ρn+1i,j−ρni,j|≤λ(vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δx|ρni,j−1|+vmaxi‖ψ‖∞|ρni,j−1−ρni,j|). $ |
Summing on
$ ∑j∈ZΔx|ρn+1i,j−ρni,j|=vmaxi‖ψ′‖∞M‖ρn‖∞ωi(0)Δt∑j∈ZΔx|ρni,j−1|+vmaxi‖ψ‖∞Δt∑j∈Z|ρni,j−1−ρni,j|, $ |
which yields
$ nT−1∑n=0∑j∈ZΔx|ρn+1i,j−ρni,j|≤vmaxM‖ψ‖∞Tsupt∈[0,T]TV(ρΔxi)(t,⋅)+vmaxM‖ψ′‖∞MW0Tsupt∈[0,T]‖ρΔxi(t,⋅)‖1‖ρΔxi(t,⋅)‖∞ $ |
that is bounded by Corollary 1, Lemma 2.2 and Lemma 2.4.
To complete the proof of the existence of solutions to the problem (1), (5), we follow a Lax-Wendroff type argument as in [5], see also [10], to show that the approximate solutions constructed by scheme (7) converge to a weak solution of (1). By Lemma 2.2, Lemma 2.4 and Corollary 2, we can apply Helly's theorem, stating that for
$ nT−1∑n=0∑jφ(tn,xj)(ρn+1i,j−ρni,j)=−λnT−1∑n=0∑jφ(tn,xj)(ρni,jVni,j+1−ρni,j−1Vni,j). $ |
Summing by parts we obtain
$ −∑jφ((nT−1)Δt,xj)ρnTi,j+∑jφ(0,xj)ρ0i,j+nT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))ρni,j+λnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Vni,j+1ρni,j=0. $ | (22) |
Multiplying by
$ −Δx∑jφ((nT−1)Δt,xj)ρnTi,j+Δx∑jφ(0,xj)ρ0i,j $ | (23) |
$ +ΔxΔtnT−1∑n=1∑j(φ(tn,xj)−φ(tn−1,xj))Δtρni,j $ | (24) |
$ +ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))ΔxVni,j+1ρni,j=0. $ | (25) |
By
$ ∫R(ρ0i(x)φ(0,x)−ρi(T,x)φ(T,x))dx+∫T0∫Rρi(t,x)∂tφ(t,x)dxdt, $ | (26) |
as
$ ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)ΔxVni,j+1ρni,j=ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)+ΔxΔtnT−1∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δxρni,jVni,j. $ | (27) |
By (12) we get the estimate
$ ρni,jVni,j+1−ρni,jVni,j≤vmaxi‖ψ′‖∞ΔxM‖ρ‖2∞ωi(0). $ |
Set
$ ΔxΔtnT∑n=0∑jφ(tn,xj+1)−φ(tn,xj)Δx(ρni,jVni,j+1−ρni,jVni,j)≤ΔxΔt‖∂xφ‖∞nT∑n=0j1∑j=j0vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx≤‖∂xφ‖∞vmaxi‖ψ′‖∞M‖ρ‖2∞ωi(0)Δx2RT, $ |
which goes to zero as
Finally, again by the
$ ΔxΔtnT−1∑n=0∑j(φ(tn,xj+1)−φ(tn,xj))Δxρni,jVni,j−12→∫T0∫R∂xφ(t,x)ρi(t,x)vi(r∗ωi)dxdt. $ |
In this section we perform some numerical simulations to illustrate the behaviour of solutions to (1) for
In this example, we consider a stretch of road populated by cars and trucks. The space domain is given by the interval
$ {∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0, $ | (28) |
with
$ ω1(x)=2η1(1−xη1),η1=0.3,ω2(x)=2η2(1−xη2),η2=0.1,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=0.8,vmax2=1.3. $ |
In this setting,
$ {ρ1(0,x)=0.5χ[−1.1,−1.6],ρ2(0,x)=0.5χ[−1.6,−1.9], $ |
in which a platoon of trucks precedes a group of cars. Due to their higher speed, cars overtake trucks, in accordance with what observed in the local case [4].
The aim of this test is to study the possible impact of the presence of Connected Autonomous Vehicles (CAVs) on road traffic performances. Let us consider a circular road modeled by the space interval
$ {∂tρ1(t,x)+∂x(ρ1(t,x)vmax1ψ((r∗ω1)(t,x)))=0,∂tρ2(t,x)+∂x(ρ2(t,x)vmax2ψ((r∗ω2)(t,x)))=0,ρ1(0,x)=β(0.5+0.3sin(5πx)),ρ2(0,x)=(1−β)(0.5+0.3sin(5πx)), $ | (29) |
with
$ ω1(x)=1η1,η1=1,ω2(x)=2η2(1−xη2),η2=0.01,ψ(ξ)=max{1−ξ,0},ξ≥0,vmax1=vmax2=1. $ |
Above
As a metric of traffic congestion, given a time horizon
$ J(β)=∫T0d|∂xr|dt, $ | (30) |
$ Ψ(β)=∫T0[ρ1(t,ˉx)vmax1ψ((r∗ω1)(t,ˉx))+ρ2(t,ˉx)vmax2ψ((r∗ω2)(t,ˉx))]dt, $ | (31) |
where
The authors are grateful to Luis M. Villada for suggesting the non-local multi-class traffic model studied in this paper.
We provide here alternative estimates for (1), based on approximate solutions constructed via the following adapted Lax-Friedrichs scheme:
$ ρn+1i,j=ρni,j−λ(Fni,j+1/2−Fni,j−1/2), $ | (32) |
with
$ Fni,j+1/2:=12ρni,jVni,j+12ρni,j+1Vni,j+1+α2(ρni,j−ρni,j+1), $ | (33) |
where
Lemma A.1. For any
$ λα<1, $ | (34) |
$ α≥vmaxM‖ψ‖∞, $ | (35) |
the scheme (33)-(32) is positivity preserving on
Lemma A.2. (
$ T<(M‖ρ0‖∞vmaxM‖ψ′‖∞W0)−1. $ | (36) |
Lemma A.3. (
$ \begin{align} \Delta t &\leq \frac{2}{2 \alpha +\Delta x \, {\left\|{\psi'}\right\|}_\infty \, W_0 \, v_M^{\max}{\left\|{ \boldsymbol{\rho}}\right\|}_\infty} \, \Delta x , \end{align} $ | (37) |
then the solution constructed by the algorithm (33)-(32) has uniformly bounded total variation for any
$ \begin{equation} T \leq {\min\limits_{i = 1, ..., M}}\, \frac{1}{\mathcal{D} \left( \mathinner{{\rm{TV}}}(\rho^0_i)+1\right)\, }, \end{equation} $ | (38) |
where
[1] |
Kraus V, Blanco F, Englund M, et al. (2015) Call for standardized definitions of osteoarthritis and risk stratification for clinical trials and clinical use. Osteoarthr Cartil 23: 1233–1241. doi: 10.1016/j.joca.2015.03.036
![]() |
[2] |
Pesesse L, Sanchez C, Delcour JP, et al. (2013) Consequences of chondrocyte hypertrophy on osteoarthritic cartilage: potential effect on angiogenesis. Osteoarthr Cartil 21: 1913–1923. doi: 10.1016/j.joca.2013.08.018
![]() |
[3] |
Berenbaum F (2013) Osteoarthritis as an inflammatory disease (osteoarthritis is not osteoarthrosis!). Osteoarthr Cartil 21: 16–21. doi: 10.1016/j.joca.2012.11.012
![]() |
[4] |
Henrotin Y, Pesesse L, Lambert C (2014) Targeting the synovial angiogenesis as a novel treatment approach to osteoarthritis. Ther Adv Musculoskeletal Dis 6: 20–34. doi: 10.1177/1759720X13514669
![]() |
[5] | Henrotin Y, Pesesse L, Sanchez C (2012) Subchondral bone and osteoarthritis: biological and cellular aspects. Osteoporosis Int 23: 47–51. |
[6] |
Sanchez C, Pesesse L, Gabay O, et al. (2012) Regulation of subchondral bone osteoblast metabolism by cyclic compression. Arthritis Rheumatol 64: 1193–1203. doi: 10.1002/art.33445
![]() |
[7] |
Sanchez C, Deberg M, Bellahcène A,et al. (2008) Phenotypic characterization of osteoblasts from the sclerotic zones of osteoarthritic subchondral bone. Arthritis Rheumatol 58: 442–455. doi: 10.1002/art.23159
![]() |
[8] |
Pesesse L, Sanchez C, Henrotin Y (2011) Osteochondral plate angiogenesis: a new treatment target in osteoarthritis. Jt Bone Spine 78: 144–149. doi: 10.1016/j.jbspin.2010.07.001
![]() |
[9] | Rahmati M, Mobasheri A (2016) Mozafari M. Inflammatory mediators in osteoarthritis: A critical review of the state-of-the-art, current prospects, and future challenges. Bone 85: 81–90. |
[10] |
Bertuglia A, Lacourt M, Girard C, et al. (2016) Osteoclasts are recruited to the subchondral bone in naturally occurring post-traumatic equine carpal osteoarthritis and may contribute to cartilage degradation. Osteoarthr Cartilage 24: 555–566. doi: 10.1016/j.joca.2015.10.008
![]() |
[11] | Sanchez C, Gabay O, Salvat C, et al. (2009) Mechanical loading highly increases IL-6 production and decreases OPG expression by osteoblasts. Osteoarthritis Cartilage 7: 473–481. |
[12] | Henrotin Y, Pesesse L, Sanchez C (2009) Subchondral bone in osteoarthritis physiopathology: state-of-the art and perspectives. Biomed Mater Eng 19: 311–316. |
[13] | Hügle T, Geurts J (2017) What drives osteoarthritis?-synovial versus subchondral bone pathology. Rheumatology (Oxford) 56: 1461–1471. |
[14] | Veronese N, Trevisan C, De Rui M, et al. (2015) Osteoarthritis increases the risk of cardiovascular diseases in the elderly: The progetto veneto anziano study. Arthritis Rheumatol 68: 1136–1144. |
[15] |
Eymard F, Parsons C, Edwards MH, et al. (2015) Diabetes is a risk factor for knee osteoarthritis progression. Osteoarthr Cartilage 23: 851–859. doi: 10.1016/j.joca.2015.01.013
![]() |
[16] | da Costa BR, Reichenbach S, Keller N, et, al. (2017) Effectiveness of non-steroidal anti-inflammatory drugs for the treatment of pain in knee and hip osteoarthritis: a network meta-analysis. Lancet 387: 2093–2105. |
[17] | Roberts E, Delgado Nunes V, Buckner S, et al. (2016) Paracetamol: not as safe as we thought? A systematic literature review of observational studies. Ann Rheum Dis 75: 552–559. |
[18] |
Pavelka K, Coste P, Géher P, et al. (2010) Efficacy and safety of Piascledine 300 versus chondroitin sulfate in a 6 months treatment plus 2 months observation in patients with osteoarthritis of the knee. Clin Rheumatol 29: 659–670. doi: 10.1007/s10067-010-1384-8
![]() |
[19] | Appelboom T, Schuermans J, Verbruggen G, et al. (2001) Symptoms modifying effect of avocado/soybean unsaponifiables (ASU) in knee osteoarthritis. A double blind, prospective, placebo-controlled study. Acta Rheumatol Scand 30: 242–247. |
[20] |
Maheu E, Mazières B, Valat JP, et al. (1998) Symptomatic efficacy of avocado/soybean unsaponifiables in the treatment of osteoarthritis of the knee and hip: a prospective, randomized, double-blind, placebo-controlled, multicenter clinical trial with a six-month treatment period and a two-month followup demonstrating a persistent effect. Arthritis Rheumatol 41: 81–91. doi: 10.1002/1529-0131(199801)41:1<81::AID-ART11>3.0.CO;2-9
![]() |
[21] | Blotman F, Maheu E, Wulwik A, et al. (1997) Efficacy and safety of avocado/soybean unsaponifiables in the treatment of symptomatic osteoarthritis of the knee and hip. A prospective, multicenter, three-month, randomized, double-blind, placebo-controlled trial. Rev Rhum Engl Ed 64: 825–834. |
[22] |
Christensen R, Bartels EM, Astrup A, et al. (2008) Symptomatic efficacy of avocado-soybean unsaponifiables (ASU) in osteoarthritis (OA) patients: a meta-analysis of randomized controlled trials. Osteoarthritis Cartilage 16: 399–408. doi: 10.1016/j.joca.2007.10.003
![]() |
[23] |
Maheu E, Cadet C, Marty M, et al. (2014) Randomised, controlled trial of avocado-soybean unsaponifiable (Piascledine) effect on structure modification in hip osteoarthritis: the ERADIAS study. Ann Rheum Dis 73: 376–384. doi: 10.1136/annrheumdis-2012-202485
![]() |
[24] |
Zhang W, Doherty M, Arden N, et al. (2005) EULAR evidence based recommendations for the management of hip osteoarthritis: report of a task force of the EULAR Standing Committee for International Clinical Studies Including Therapeutics (ESCISIT). Ann Rheum Dis 64: 669–681. doi: 10.1136/ard.2004.028886
![]() |
[25] |
McAlindon T, Bannuru R, Sullivan M, et al. (2014) OARSI guidelines for the non-surgical management of knee osteoarthritis. Osteoarthr Cartilage 22: 363–388. doi: 10.1016/j.joca.2014.01.003
![]() |
[26] |
Msika P, Baudoin C, Saunois A, et al. (2008) Avocado/Soybean unsaponifiable, ASU EXPANSCIENCETM, are strictly different from the nutraceutical products claiming ASU appellation. Osteoarthr Cartilage 16: 1275–1276. doi: 10.1016/j.joca.2008.02.017
![]() |
[27] |
Henrotin Y (2008) Avocado/soybean unsaponifiable (ASU) to treat osteoarthritis: a clarification. Osteoarthritis Cartilage 16: 1118–1119. doi: 10.1016/j.joca.2008.01.010
![]() |
[28] | Rancurel A (1985) Parfums, cosmétiques, Arômes 61: 91. |
[29] |
Farines M, Soulier J, Rancurel A, et al. (1995) Influence of avocado oil processing on the nature of some unsaponifiable constituents. J Am Oil Chem Soc 72: 473–476. doi: 10.1007/BF02636092
![]() |
[30] | Baillet A (1995) Pharmaceutical expert report. Courbevoie, France: Pharmascience, 1995 (unpublished data). |
[31] | Mauviel A, Daireaux M, Hartman DJ, et al. (1989) Effets des insaponifiables d'avocat/soja (PIAS) sur la production de collagène par des cultures de synoviocytes, chondrocytes articulaires et fibroblastes dermiques. Rev Rhum 56: 207–213. |
[32] | Mauviel A, Loyau G, Pujol JP (1991) Effets des insaponifiables d'avocat/soja (Piascledine) sur l'activité collagénolytique de cultures de synoviocytes rhumatoides humains et de chondrocytes articulaires de lapin traités par l'interleukine-1. Rev Rhum 58: 241–248. |
[33] | Harmand MF (1985) Etude de fraction des insaponifiables d'avocat et de soja sur les cultures de chondrocytes articulaires. Gaz Med Fr 92: 1–3. |
[34] |
Henrotin Y, Labasse A, Jaspar JM, et al. (1998) Effects of three avocado/soybean unsaponifiable mixtures on metalloproteinases, cytokines and prostaglandin E2 production by human articular chondrocytes. Clin. Rheumatol 17: 31–39. doi: 10.1007/BF01450955
![]() |
[35] | Henrotin Y, Sanchez C, Deberg MA, et al. (2003) Avocado/soybean unsaponifiables increase aggrecan synthesis and reduce catabolic and proinflammatory mediator production by human osteoarthritic chondrocytes. J. Rheumatol 30: 1825–1834 |
[36] |
Gabay O, Gosset M, Levy A, et al. (2008) Stress-induced signaling pathways in hyalin chondrocytes: inhibition by Avocado-Soybean Unsaponifiables (ASU). Osteoarthr Cartilage 16: 373–384. doi: 10.1016/j.joca.2007.06.016
![]() |
[37] |
Boumediene K, Felisaz N, Bogdanowicz P, et al. (1999) Avocado/soya unsaponifiables enhance the expression of transforming growth factor beta1 and beta2 in cultured articular chondrocytes. Arthritis Rheumatol 42: 148–156. doi: 10.1002/1529-0131(199901)42:1<148::AID-ANR18>3.0.CO;2-U
![]() |
[38] |
Campbell IK, Wojta J, Novak U, et al. (1994) Cytokine modulation of plasminogen activator inhibitor-1 (PAI-1) production by human articular cartilage and chondrocytes: down-regulation by tumor necrosis factora. Biochim Biophys Acta 1226: 277–285. doi: 10.1016/0925-4439(94)90038-8
![]() |
[39] | Khayyal MT, el-Ghazaly MA (1998) The possible "chondroprotective" effect of the unsaponifiable constituents of avocado and soya in vivo. Drugs Exp Clin Res 24: 41–50. |
[40] |
Boileau C, Martel-Pelletier J, Caron J, et al. (2009) Protective effects of total fraction of avocado/soybean unsaponifiables on the structural changes in experimental dog osteoarthritis: inhibition of nitric oxide synthase and matrix metalloproteinase-13. Arthritis Res Ther 11: 41. doi: 10.1186/ar2649
![]() |
[41] |
Jaberi, F, Tahami, M, Torabinezhad S, et al. (2012) The healing effect of soybean and avocado mixture on knee cartilage defects in a dog animal model. Comp. Clin Pathol 21: 661–666. doi: 10.1007/s00580-010-1152-9
![]() |
[42] |
Altinel L, Saritas ZK, Kose KC, et al. (2007) Treatment with unsaponifiable extracts of avocado and soybean increases TGF-beta1 and TGF-beta2 levels in canine joint fluid. Tohoku J Exp Med 211: 181–186. doi: 10.1620/tjem.211.181
![]() |
[43] | Cake M, Read R, Guillou B, et al. (2008) Modification of articular cartilage and subchondral bone pathology in an ovine meniscectomy model of osteoarthritis by avocado and soya unsaponifiables (ASU). Osteoarthr Cartilage 8: 404–411. |
[44] |
Cinelli M, Guiducci S, Del Rosso A, et al. (2006) Piascledine modulates the production of VEGF and TIMP-1 and reduces the invasiveness of rheumatoid arthritis synoviocytes. Scand J Rheumatol 35: 346–350. doi: 10.1080/03009740600709865
![]() |
[45] | Henrotin Y, Deberg M, Crielaard JM, et al. (2006) Avocado/soybean unsaponifiables prevent the inhibitory effect of osteoarthritic subchondral osteoblasts on aggrecan and type II collagen synthesis by chondrocytes. J Rheumatol 33: 1668–1678. |
[46] | Sanchez C, Deberg M, Piccardi N, et al. (2005) Osteoblasts from the sclerotic subchondral bone downregulate aggrecan but upregulate metalloproteinases expression by chondrocytes. This effect is mimicked by interleukin-6, -1beta and oncostatin M pre-treated non-sclerotic osteoblasts. Osteoarthr Cartilage 13: 979–987. |
[47] | Sanchez C, Deberg M, Piccardi N, et al. (2004) Interleukin-1, interleukin-6 and oncostatin M stimulate normal subchondral osteoblast to induce cartilage degradation. Osteoarthr Cart 12: S98. |
[48] |
Andriamanalijaona R, Benateau H, Barre PE, et al. (2006) Effect of interleukin-1beta on transforming growth factor-beta and bone morphogenetic protein-2 expression in human periodontal ligament and alveolar bone cells in culture: modulation by avocado and soybean unsaponifiables. J Periodontol 77: 1156–1166. doi: 10.1902/jop.2006.050356
![]() |
[49] | Day JS, van der Linden JC, Bank RA, et al. (2004) Adaptation of subchondral bone in osteoarthritis. Biorheology 41: 359–368. |
[50] |
Westacott CI, Webb GR, Warnock MG, et al. (1997) Alteration of cartilage metabolism by cells from osteoarthritic bone. Arthritis Rheumatol 40: 1282–1291. doi: 10.1002/1529-0131(199707)40:7<1282::AID-ART13>3.0.CO;2-E
![]() |
[51] |
Hilal G, Massicotte F, Martel-Pelletier J, et al. (2001) Endogenous prostaglandin E2 and insulin-like growth factor 1 can modulate the levels of parathyroid hormone receptor in human osteoarthritic osteoblasts. J Bone Miner Res 16: 713–721. doi: 10.1359/jbmr.2001.16.4.713
![]() |
[52] |
Hilal G, Martel-Pelletier J, Pelletier JP, et al. (1999) Abnormal regulation of urokinase plasminogen activator by insulin-like growth factor 1 in human osteoarthritic subchondral osteoblasts. Arthritis Rheumatol 42: 2112–2122. doi: 10.1002/1529-0131(199910)42:10<2112::AID-ANR11>3.0.CO;2-N
![]() |
[53] |
Massicotte F, Fernandes JC, Martel-Pelletier J, et al. (2006) Modulation of insulin-like growth factor 1 levels in human osteoarthritic subchondral bone osteoblasts. Bone 38: 333–341. doi: 10.1016/j.bone.2005.09.007
![]() |
[54] |
Hilal G, Martel-Pelletier J, Pelletier JP, et al. (1998) Osteoblast-like cells from human subchondral osteoarthritic bone demonstrate an altered phenotype in vitro: possible role in subchondral bone sclerosis. Arthritis Rheumatol 41: 891–899. doi: 10.1002/1529-0131(199805)41:5<891::AID-ART17>3.0.CO;2-X
![]() |
[55] | Harris SE, Bonewald LF, Harris MA, et al. (1994) Effects of transforming growth factor on bone nodule formation and expression of bone morphogenetic protein 2, osteocalcin, osteopontin, alkaline phosphatase, and type I collagen mRNA in long-term cultures of fetal rat calvarial osteoblasts. J Bone Miner Res 9: 855–863. |
1. | Rinaldo M. Colombo, Magali Lecureux-Mercier, Mauro Garavello, 2020, Chapter 5, 978-3-030-50449-6, 83, 10.1007/978-3-030-50450-2_5 | |
2. | Felisia Angela Chiarello, Paola Goatin, Luis Miguel Villada, Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models, 2020, 39, 2238-3603, 10.1007/s40314-020-1097-9 | |
3. | Alexandre Bayen, Jan Friedrich, Alexander Keimer, Lukas Pflug, Tanya Veeravalli, Modeling Multilane Traffic with Moving Obstacles by Nonlocal Balance Laws, 2022, 21, 1536-0040, 1495, 10.1137/20M1366654 | |
4. | Felisia Angela Chiarello, 2021, Chapter 5, 978-3-030-66559-3, 79, 10.1007/978-3-030-66560-9_5 | |
5. | Jan Friedrich, Simone Göttlich, Maximilian Osztfalk, Network models for nonlocal traffic flow, 2022, 56, 2822-7840, 213, 10.1051/m2an/2022002 | |
6. | Zlatinka Dimitrova, Flows of Substances in Networks and Network Channels: Selected Results and Applications, 2022, 24, 1099-4300, 1485, 10.3390/e24101485 | |
7. | Giuseppe Maria Coclite, Lorenzo di Ruvo, On the initial-boundary value problem for a non-local elliptic-hyperbolic system related to the short pulse equation, 2022, 3, 2662-2963, 10.1007/s42985-022-00208-w | |
8. | Kuang Huang, Qiang Du, Stability of a Nonlocal Traffic Flow Model for Connected Vehicles, 2022, 82, 0036-1399, 221, 10.1137/20M1355732 | |
9. | Yanbing Wang, Daniel B. Work, Estimation for heterogeneous traffic using enhanced particle filters, 2022, 18, 2324-9935, 568, 10.1080/23249935.2021.1881186 | |
10. | Felisia Angela Chiarello, Harold Deivi Contreras, Luis Miguel Villada, Nonlocal reaction traffic flow model with on-off ramps, 2022, 17, 1556-1801, 203, 10.3934/nhm.2022003 | |
11. | Ioana Ciotir, Rim Fayad, Nicolas Forcadel, Antoine Tonnoir, A non-local macroscopic model for traffic flow, 2021, 55, 0764-583X, 689, 10.1051/m2an/2021006 | |
12. | Maria Colombo, Gianluca Crippa, Marie Graff, Laura V. Spinolo, On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws, 2021, 55, 0764-583X, 2705, 10.1051/m2an/2021073 | |
13. | Alexander Keimer, Lukas Pflug, 2023, 15708659, 10.1016/bs.hna.2022.11.001 | |
14. | Alexandre Bayen, Jean-Michel Coron, Nicola De Nitti, Alexander Keimer, Lukas Pflug, Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model, 2021, 49, 2305-221X, 957, 10.1007/s10013-021-00506-7 | |
15. | F. A. CHIARELLO, J. FRIEDRICH, P. GOATIN, S. GÖTTLICH, O. KOLB, A non-local traffic flow model for 1-to-1 junctions, 2020, 31, 0956-7925, 1029, 10.1017/S095679251900038X | |
16. | Jan Friedrich, Simone Göttlich, Alexander Keimer, Lukas Pflug, 2024, Chapter 30, 978-3-031-55263-2, 347, 10.1007/978-3-031-55264-9_30 | |
17. | Veerappa Gowda G. D., Sudarshan Kumar Kenettinkara, Nikhil Manoj, Convergence of a second-order scheme for non-local conservation laws, 2023, 57, 2822-7840, 3439, 10.1051/m2an/2023080 | |
18. | Jan Friedrich, Sanjibanee Sudha, Samala Rathan, Numerical schemes for a class of nonlocal conservation laws: a general approach, 2023, 18, 1556-1801, 1335, 10.3934/nhm.2023058 | |
19. | Jan Friedrich, Simone Göttlich, Michael Herty, Lyapunov Stabilization for Nonlocal Traffic Flow Models, 2023, 61, 0363-0129, 2849, 10.1137/22M152181X | |
20. | Felisia A. Chiarello, Harold D. Contreras, 2024, Chapter 26, 978-3-031-55263-2, 303, 10.1007/978-3-031-55264-9_26 | |
21. | Jan Friedrich, Lyapunov stabilization of a nonlocal LWR traffic flow model, 2023, 23, 1617-7061, 10.1002/pamm.202200084 | |
22. | Agatha Joumaa, Paola Goatin, Giovanni De Nunzio, 2023, A Macroscopic Model for Multi-Modal Traffic Flow in Urban Networks, 979-8-3503-9946-2, 411, 10.1109/ITSC57777.2023.10422168 | |
23. | Saeed Mohammadian, Zuduo Zheng, Md. Mazharul Haque, Ashish Bhaskar, Continuum modeling of freeway traffic flows: State-of-the-art, challenges and future directions in the era of connected and automated vehicles, 2023, 3, 27724247, 100107, 10.1016/j.commtr.2023.100107 | |
24. | Harold Deivi Contreras, Paola Goatin, Luis-Miguel Villada, A two-lane bidirectional nonlocal traffic model, 2025, 543, 0022247X, 129027, 10.1016/j.jmaa.2024.129027 | |
25. | Felisia Angela Chiarello, Paola Goatin, 2023, Chapter 3, 978-3-031-29874-5, 49, 10.1007/978-3-031-29875-2_3 | |
26. | Alexander Keimer, Lukas Pflug, Discontinuous nonlocal conservation laws and related discontinuous ODEs – Existence, Uniqueness, Stability and Regularity, 2023, 361, 1778-3569, 1723, 10.5802/crmath.490 | |
27. | Archie J. Huang, Animesh Biswas, Shaurya Agarwal, Incorporating Nonlocal Traffic Flow Model in Physics-Informed Neural Networks, 2024, 25, 1524-9050, 16249, 10.1109/TITS.2024.3429029 | |
28. | Rinaldo M. Colombo, Mauro Garavello, Claudia Nocita, General stability estimates in nonlocal traffic models for several populations, 2025, 32, 1021-9722, 10.1007/s00030-025-01034-w |