Steady solutions to the Navier-Stokes-Fourier system for dense compressible fluid

Steady solutions to a model of compressible chemically reacting fluid with high density

“…In contrast to the single-constituted fluid case studied in Axmann et al 4 or Axmann et al, 5 we are not able to deduce here the comparison principle for the temperature. Therefore, inspired by previous works on compressible mixtures and in order to guarantee the nonnegativity of temperature as well as the specific concentrations, we introduce two approximation parameters , > 0.…”

“…[6][7][8]19 Note also that our approach is also based on the method of decomposition, developed and successfully used in nineties to study strong solutions of the compressible Navier-Stokes equations by Novotný and Padula, see previous study. 20 This fact will be hidden here as we can directly use results from Axmann et al 4 and Axmann et al, 5 which are based on this technique, and we do not have to prove them here again.…”

“…for arbitrary 1 < p < ∞. We may replace the condition on geometric properties of Ω by assuming that f > 0 and f ∼ M. The proof follows similar lines, only in order to estimate the velocity, we need to combine the estimates from the entropy and the total energy balance and the proof becomes slightly more complicated, see also Axmann et al 4 and Axmann et al 5 Remark 1. Integrating Equation (4), by virtue of the boundary conditions ( 16) and (18), one can observe that any stationary solution to our system has to satisfy sort of compatibility conditions…”

“…In order to improve the regularity we use the method of decomposition in the same way as for the case of single-component fluid in Axmann et al 4 and Axmann et al 5 Here, we use the fact that the momentum equation is influenced by the chemical reactions only through the temperature involved in the pressure. Otherwise, all the other terms in the momentum equation are identical to the single-component case, and thus can be treated exactly in the same way.…”

“…The existence and uniqueness of fixed point for Equations ( 71) and (72) with boundary conditions (77) and (78) follows from our previous works. For the reader's convenience we recall the corresponding Lemma 3.1 from Axmann et al 5 We use below the present notation.…”