June  2021, 10(2): 353-364. doi: 10.3934/eect.2020070

Homogenization of a stochastic viscous transport equation

Normandie Univ, INSA de Rouen Normandie, LMI (EA 3226 - FR CNRS 3335), 76000 Rouen, France, 685 Avenue de l'Université, 76801 St Etienne du Rouvray cedex

* Corresponding author: Ioana Ciotir

Received  January 2020 Published  June 2020

Fund Project: This work was partially supported by the European Union with the European regional development fund (ERDF, HN0002137 and 18P03390/18E01750/18P02733) and by the Normandie Regional Council (via the M2NUM and M2SiNum projects). The first author was partially supported by the ANR Project QUTE-HPC Quantum Turbulence Exploration by High-Performance Computing (ANR-18-CE46-0013)

In the present paper we prove an homogenisation result for a locally perturbed transport stochastic equation. The model is similar to the stochastic Burgers' equation and it is inspired by the LWR model. Therefore, the interest in studying this equation comes from it's application for traffic flow modelling. In the first part of paper we study the inhomogeneous equation. More precisely we give an existence and uniqueness result for the solution. The technical difficulties of this part come from the presence of the function $ \varphi $ under assumptions coherent for the model, which is giving the inhomogeneity with respect to the space variable, not present in the classical results. The second part of the paper is the homogenisation result in space.

Citation: Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations & Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070
References:
[1]

A. Amosov and G. Panasenko, Homogenization of the integro-differential burgers equation, in Integral Methods in Science and Engineering, Vol. 1, Birkhäuser Boston, Boston, MA, 2010, 1–8. doi: 10.1007/978-0-8176-4899-2_1.  Google Scholar

[2]

G. Da Prato and D. Gatarek, Stochastic burgers equation with correlated noise, Stochastics and Stochastic Reports, 52 (1995), 29-41.  doi: 10.1080/17442509508833962.  Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2008. https://books.google.fr/books?id=JYiL8zz_nC8C. Google Scholar

[4]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers' equation, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[5]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[6]

M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[7]

I. Gyoengy and D. Nualart, On the stochastic Burgers' equation in the real line, Ann. Probab., 27 (1999), 782-802.  doi: 10.1214/aop/1022677386.  Google Scholar

[8]

I. Hosokawa and K. Yamamoto, Turbulence in the randomly forced, one-dimensional Burgers flow, J. Stat. Phys., 13 (1975), 245-272.  doi: 10.1007/BF01012841.  Google Scholar

[9]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[10]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[11]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

show all references

References:
[1]

A. Amosov and G. Panasenko, Homogenization of the integro-differential burgers equation, in Integral Methods in Science and Engineering, Vol. 1, Birkhäuser Boston, Boston, MA, 2010, 1–8. doi: 10.1007/978-0-8176-4899-2_1.  Google Scholar

[2]

G. Da Prato and D. Gatarek, Stochastic burgers equation with correlated noise, Stochastics and Stochastic Reports, 52 (1995), 29-41.  doi: 10.1080/17442509508833962.  Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2008. https://books.google.fr/books?id=JYiL8zz_nC8C. Google Scholar

[4]

G. Da PratoA. Debussche and R. Temam, Stochastic Burgers' equation, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 389-402.  doi: 10.1007/BF01194987.  Google Scholar

[5]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[6]

M. Garavello and B. Piccoli, Traffic Flow on Networks, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[7]

I. Gyoengy and D. Nualart, On the stochastic Burgers' equation in the real line, Ann. Probab., 27 (1999), 782-802.  doi: 10.1214/aop/1022677386.  Google Scholar

[8]

I. Hosokawa and K. Yamamoto, Turbulence in the randomly forced, one-dimensional Burgers flow, J. Stat. Phys., 13 (1975), 245-272.  doi: 10.1007/BF01012841.  Google Scholar

[9]

M. J. Lighthill and G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[10]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[11]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

Figure 1.  Schematic representation of the function $ \varphi $
[1]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[2]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[3]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[4]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[5]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[6]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006

[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[8]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[9]

Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021010

[10]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[11]

Jingni Guo, Junxiang Xu, Zhenggang He, Wei Liao. Research on cascading failure modes and attack strategies of multimodal transport network. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2020159

[12]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[13]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[14]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[15]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[16]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[17]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[18]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

[19]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[20]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (60)
  • HTML views (276)
  • Cited by (0)

[Back to Top]