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 2021 Early access  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 and 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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

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