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Room P3.10, Mathematics Building

Lu Xu, Inria Lille

Hydrodynamic limit for asymmetric simple exclusion with open boundaries. I

Exclusion process in contact with boundary reservoirs is one of the simplest open particle systems.For symmetric exclusion, the macroscopic time evolution of the particle density is given by the heat equation with varies types of boundary conditions (Dirichlet, Robin, and Neumann). For asymmetric exclusion, the density evolves with the initial-boundary problem of the nonlinear conservation law:

\begin{align*} & \partial_t u+\partial_x [u(1-u)]=0, \\ & u(t,0)=v_-(t), u(t,1)=v_+(t), \\ & u(0,x)=v_0(x). \end{align*}

Its (entropy) solution exhibits specific discontinuous phenomenon (shock waves, boundary layers), which become the main obstacle of applying classical relative entropy method.

The goal of the mini-course is to introduce the entropy solution to the conservation law with irregular initial and boundary data, and present some new methods and results on the hydrodynamical behaviour of open ASEP.

It will include the following topics:

- introduction to the ASEP with reservoirs, the stationary states, the hydrodynamic equation;
- the concept of entropy solution, the viscous approximate, entropy inequality, boundary entropy;
- main methods used to prove the hydrodynamic limit: the Young measure, the stochastic compensated compactness, the grading technique at boundaries.