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

Abstract

We prove existence of L²-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 L²-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 L², allowing us to use L² 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 $p\in [1,2),$ 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.

Keywords:

• Hyperbolic conservation laws
• quasi-linear wave equation
• boundary conditions
• weak solutions
• vanishing viscosity
• compensated compactness
• entropy solutions
• Clausius inequality

MATHEMATICS SUBJECT CLASSIFICATION:

• 35L40
• 35D40

Acknowledgments

We thank Olivier Glass for very helpful discussions. We also thank an anonymous referee, whose critics and comments helped us to improve the first version of this article.

Additional information

Funding

This work has been partially supported by the grants ANR-15-CE40-0020-01 LSD of the French National Research Agency.