# American Institute of Mathematical Sciences

• Previous Article
Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition
• NHM Home
• This Issue
• Next Article
Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
September  2018, 13(3): 423-448. doi: 10.3934/nhm.2018019

## Perturbations of minimizing movements and curves of maximal slope

 Via della Ricerca Scientifica 1, Rome, 00133, Italy

Received  October 2017 Revised  April 2018 Published  July 2018

We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

Citation: Antonio Tribuzio. Perturbations of minimizing movements and curves of maximal slope. Networks & Heterogeneous Media, 2018, 13 (3) : 423-448. doi: 10.3934/nhm.2018019
##### References:

show all references

##### References:
Graphs of the discrete solutions with different values of $\tau$. The smaller jumps of $u^\tau$ correspond to the larger parameter $\beta$
Pinned motion produced by perturbations diverging in the interval $(1, 2)$
The graphs represent two discrete solutions for the same value of $\tau$, corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear
On the left the time chart of $u^\tau$, on the right the plot of $\phi(u^\tau)$, corresponding to perturbations with $\delta = 1$. Note how the motion exits from the lower energy state when $t = 1$
with $\delta = 8$. It is grater than the critical value $e^2$, indeed when $t = 1$ the motion does not exit the first well">Figure 5.  Graphs as in Figure 4 with $\delta = 8$. It is grater than the critical value $e^2$, indeed when $t = 1$ the motion does not exit the first well
, the motion does not exit the well when $t = 1$, but when $t = 2$ it does">Figure 6.  Graphs for $\delta = 8$. As in Figure 5, the motion does not exit the well when $t = 1$, but when $t = 2$ it does
Graph for $\delta = 1$. The motion always passes to the very next potential well
Graphs of a discrete solution passing through two potential wells at every jump discontinuity
 [1] Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657 [2] Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153 [3] Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 373-393. doi: 10.3934/dcdss.2020324 [4] Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $\Lambda$-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 [5] Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879 [6] Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501 [7] Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345 [8] Gilles A. Francfort, Alessandro Giacomini, Alessandro Musesti. On the Fleck and Willis homogenization procedure in strain gradient plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 43-62. doi: 10.3934/dcdss.2013.6.43 [9] Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028 [10] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [11] Vivek Tewary. Combined effects of homogenization and singular perturbations: A bloch wave approach. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021012 [12] Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks & Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233 [13] José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025 [14] Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295 [15] Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157 [16] Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117 [17] Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063 [18] Maroje Marohnić, Igor Velčić. Homogenization of bending theory for plates; the case of oscillations in the direction of thickness. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2151-2168. doi: 10.3934/cpaa.2015.14.2151 [19] Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 [20] Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

2019 Impact Factor: 1.053