Traveling Wave Solutions to a Large Class of Brenner-Navier-Stokes-Fourier Systems

TL;DR

This paper proves the existence and uniqueness of small-amplitude traveling wave solutions in a physically realistic one-dimensional Brenner-Navier-Stokes-Fourier system with temperature-dependent coefficients, using geometric singular perturbation and implicit function theorem.

Contribution

It establishes the existence and uniqueness of monotone traveling wave solutions for the BNSF system with variable coefficients, extending previous methods to nonlinear, temperature-dependent cases.

Findings

Existence of monotone traveling wave solutions for small shock amplitudes.

Uniqueness of these solutions under given conditions.

Quantitative estimates on the solutions relevant for stability analysis.

Abstract

The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Howard Brenner, was developed to address some deficiencies in the classical Navier-Stokes-Fourier system, based on the concept of volume velocity. We consider the one-dimensional BNSF system in Lagrangian mass coordinates, incorporating temperature-dependent transport coefficients, which yields a more physically realistic framework. We establish the existence and uniqueness of monotone traveling wave solutions (or viscous shocks) to the BNSF system with any positive $C^2$ dissipation coefficients, provided that the shock amplitude is sufficiently small. We utilize geometric singular perturbation theory as in the constant coefficient case [13]; however, due to the arbitrary nonlinearities of the coefficients, we employ the implicit function theorem, which grants robustness to our approach. This work is motivated by [12], which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive some quantitative estimates on the traveling wave solutions that play a fundamental role in [12].