| Abstract: |
| We consider the following system of equations
\begin{equation}
\begin{array}{c}
{\rm div}\, (\varrho \mathbf{u}) = 0,\
{\rm div}\, (\varrho \mathbf{u} \otimes \mathbf{u}) - {\rm div} {\mathbf{S}}+ \nabla {\pi} =\varrho\mathbf{f},\
{\rm div}\, (\varrho E \mathbf{u}) + {\rm div}\, (\mathbf {Q}+\pi\mathbf{u} -\mathbf{S}\mathbf{u})= \varrho\mathbf{f}\cdot \mathbf{u},\
{\rm div}\, (\varrho Y_k \mathbf{u})+ {\rm div}\, \mathbf {F}_k = m_k\omega_{k},\quad k=1,\dots, L
\end{array}
\end{equation}
which describes the steady flow of a compressible heat conducting mixture of gases whose component may chemically react, where only the barycentric velocity is taken into account. We consider either the no-slip or the Navier boundary conditions for the velocity and the Newton boundary condition for the temperature. Based on papers [1]--[3] we present existence results of weak and variational entropy solutions in different situations.
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{\bf References}
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[1] Giovangigli, V., Pokorn\`y, M., Zatorska, E.: \emph{On the steady flow of reactive gaseous mixture},
Analysis (Berlin) 35, no. 4, 319--341 (2015).
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[2] Piasecki, T., Pokorn\`y, M.: \emph{Weak and variational entropy solutions to the system describing steady flow
of a compressible reactive mixture}, Nonlinear Anal. (TMA) 159 (2017), 365--392 (2017).
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[3] Piasecki, T., Pokorn\`y, M.: \emph{On steady solutions to a model of chemically reacting heat conducting compressible mixture with slip boundary conditions}, to appear in Contemporary Mathematics (2018). |
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