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

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].