Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence of $L^2$-valued thermodynamic entropy solutions for a
  hyperbolic system with boundary conditions   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 $p \in [1,2)$, 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.

0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol