On the existence of $L^2$-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 ∈ [1, 2), that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy 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 Limits and Clausius inequality for Isothermal Non-linear Elastodynamics with Boundary Tension

“…Notice that the noise introduced by the heat bath respects the boundary conditions V (r N +1 (t)) = τ (t) and p 0 (t) = 0 already present in the Hamiltonian part of the dynamics, while it introduces the Neumann type boundary conditions r 0 (t) = r 1 (t) and p N +1 (t) = p N (t). In this sense these boundary conditions are the microscopic analogous of those taken in the viscous approximation used in reference [6].…”

“…Recently (cf. [6]) we have considered L 2valued weak solution to the isothermal Euler equation in Lagrangian coordinates on [0, 1] (also called in the literature non-linear wave equation or p-system):…”

“…The precise sense an L 2 -valued solution satisfies the boundary condition is given in Definition 2.2. In [6] we consider viscous approximations of (1.1), that requires two extra boundary conditions. We choose these extra boundary conditions to be of Neumann type (i.e.…”

“…conservative). Adapting the L 2 -version of the compensated-compactness argument of Shearer [9] and [8], we prove in [6] the existence of vanishing viscosity solutions to the p-system. Furthermore, these solutions satisfy the usual Lax-entropy production characterisation and the thermodynamic Clausius inequality, which relates the change of the total free energy to the work done by the boundary force (see Section 5).…”

“…In this sense these heat baths act like a stochastic viscosity, vanishing after the space-time hyperbolic rescaling. The conservative nature of these stochastic heat baths also provides at the boundaries the analogue of the extra Neumann conditions as used in [6].…”