Homogenization of a stochastic viscous transport equation

Ioana Ciotir; Nicolas Forcadel; Wilfredo Salazar

doi:10.3934/eect.2020070

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2021, Volume 10, Issue 2: 353-364. Doi: 10.3934/eect.2020070

This issue Previous Article Internal feedback stabilization for parabolic systems coupled in zero or first order terms Next Article Stability and dynamics for a nonlinear one-dimensional full compressible non-Newtonian fluids

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
^* Corresponding author: Ioana Ciotir

Received: December 31, 2019

Early access: June 2020

Published: June 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B27, 60H15; Secondary: 76F25.

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

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