On the existence of L²-valued thermodynamic entropy solutions for a hyperbolic system with boundary conditions

Abstract

We prove existence of L 2 -weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L 2 -valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in L 2 , allowing us to use L 2 Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in L p , for any p 2 ½1, 2Þ, that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy, beside the local Lax entropy condition, the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness.

Hydrodynamic limit for a diffusive system with boundary conditions

“…In Marchesani and Olla (2020b) the Clausius inequality has been proven for vanishing viscosity solutions to the hyperbolic system obtained from our system by taking δ 1 = δ 2 = 0. This is done entirely at the macroscopic level and takes into account the fact that shocks might arise as the viscosity vanishes.…”

Hydrodynamic limit for a diffusive system with boundary conditions

“…In [10] the Clausius inequality has been proven for vanishing viscosity solutions to the hyperbolic system obtained from our system by taking δ1 = δ2 = 0. This is done entirely at the macroscopic level and takes into account the fact that shocks might arise as the viscosity vanishes.…”