The Brenner-Navier-Stokes-Fourier (BNSF) system, introduced by Brenner, was developed to address some deficiencies in the 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 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 Eo-Eun-Kang-Oh, J. Diff. Eqs. , 422: 639-658, 2025; 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 Eo-Eun-Kang, preprint: 2024, which proves a contraction property of any large solutions to the BNSF system around the traveling wave solutions. Thus, we also derive quantitative estimates on the traveling wave solutions that play a fundamental role in Eo-Eun-Kang, preprint: 2024.
Eo et al. (Thu,) studied this question.