Global Existence of Weak Solutions to the Compressible Navier--Stokes Equations with Temperature-Depending Viscosity Coefficients

Some existence results for the Brinkman-Fourier system

“…Compared to [10][11][12]40] for the Navier-Stokes-Fourier system, there are some differences in our proof. We state them as below.…”

“…Note that the effect of thermal radiation has been taken into account in [12], where the standard 'molecular' heat conductivity coefficient, pressure, internal energy and entropy are augmented by their radiation counterparts, respectively. More recently, Wang and Zuo [40] extended [11] to the case that the viscosity coefficients may be degenerate.…”

“…It should be emphasised that in the above radiative setting, to the best of our knowledge, we cannot deal with the case that the viscosity coefficients may be degenerate. An alternative way to establish a global existence result for the Brinkman-Fourier system with temperaturedependent and degenerate viscosity coefficients, inspired by [10,11,40], is to replace the entropy equation (1.1) 3 by the thermal energy equation…”

“…with p e representing the elastic pressure. In this setting, similarly to [10,11,40], the possible temperature concentrations can be tackled by the renormalisation of the temperature equation (1.17). To be specific, multiplying (1.17) by h(θ) for a suitable function h, we have…”

“…• Benefits. The absence of the nonlinear terms ∂ t (ρu) and div(ρu ⊗ u) in the Brinkman-type equation makes it easier to obtain pressure estimates compared to [10][11][12]40]. In those works, the main idea to obtain the pressure estimates is to 'compute' the pressure p in the momentum equation and using the energy estimates available.…”