  As a continuum model for compressible fluid flows, Howard Brenner proposed
the so-called Brenner-Navier-Stokes-Fourier(BNSF) system that improves some
flaws of the Navier-Stokes-Fourier(NSF) system. For BNSF system, the volume
velocity concept is introduced and is far different from the mass velocity of
NSF, since the density of a compressible fluid is inhomogeneous. Although BNSF
was introduced more than ten years ago, the mathematical study on BNSF is still
in its infancy. We consider the BNSF system in the Lagrangian mass coordinates.
We prove the existence and uniqueness of monotone traveling wave solutions to
the BNSF system. We also present some quantitative estimates for them.
