Hydrodynamic limit for a diffusive system with boundary conditions

Abstract

We study the hydrodynamic limit for the isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling and with nonvanishing viscosity. The temperature is kept constant by a contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed so it contributes to the macroscopic limit. Dirichlet boundary conditions are also considered: one end of the chain is kept fixed, while a time-varying tension is applied to the other end. Moreover, Neumann boundary conditions are added in such a way that the system produces the correct thermodynamic entropy in the macroscopic limit. We show that the volume stretch and momentum converge (in an appropriate sense) to a smooth solution of a system of parabolic conservation laws (isothermal Navier-Stokes equations in Lagrangian coordinates) with boundary conditions.Finally, changing the external tension allows us to define thermodynamic isothermal transformations between equilibrium states. We use this to deduce the first and the second law of Thermodynamics for our model.

Global existence for a class of viscous systems of conservation laws

“…A prototypical model for our study is given by the following viscous p-system obtained as hydrodynamic limit (under hyperbolic space-time scaling) for the isothermal dynamics of an anharmonic chain subject to an external varying tension: Example 1. As described in [10], a suitable choice of the microscopic model leads to the following viscous p-system…”