| Abstract: |
| We consider the model describing the steady flow of a compressible heat conducting fluid in a bounded three-dimensional domain
\[
\begin{array}{c}
\displaystyle \mbox{div}\, (\varrho \mathbf{u}) = 0, \
\displaystyle
\mbox{div}\, (\varrho \mathbf{u} \otimes \mathbf{u}) - \mbox{div}\, \mathbf{S} + \nabla p = \varrho \mathbf{f}, \
\mbox{div}\, (\varrho E \mathbf{u}) = \varrho \mathbf{f} \cdot \mathbf{u} - \mbox{div}\, (p \mathbf{u}) + \mbox{div}\, (\mathbf{S} \mathbf{u}) -\mbox{div}\, \mathbf{q}
\end{array}
\]
with \(\varrho\) the density, \(\mathbf{u}\) the velocity field, \(\mathbf{S}\) the stress tensor (here we assume the fluid to be Newtonian with temperature dependent viscosity), \(p\) the pressure, \(\mathbf{f}\) the given volume force, \(\mathbf{q}\) the heat flux and the total energy \(E= \frac 12 |\mathbf{u}|^2 + e\) with \(e\) the internal energy. We assume the pressure law of the form \(p(\varrho, \vartheta) \sim \varrho^{\gamma} + \varrho \vartheta\) with \(\gamma>1\) and the viscosities \(\mu(\vartheta), \xi (\vartheta) \sim (1+ \vartheta)^\alpha\), \(\alpha \in [0,1]\).
We show the existence of a weak or variational entropy solution for the above model with internal energy fulfilling the Gibbs relation and the heat flux fulfilling the Fourier law \(\mathbf{q} \sim (1+\vartheta)^m \nabla \vartheta\) with \(\vartheta\) the temperature, \(m = m(\gamma, \alpha)>0\).
We first review the results for the case $\alpha=1$ which has been known before. Then we show extensions for $\alpha \frac 32$ both for Dirichlet and Navier boundary conditions for the velocity and Robin and Dirichlet boundary conditions for the temperature. Finally we briefly touch the situation when $\gamma \leq \frac 32$, here only for Robin boundary conditions for the temperature.
The solutions are constructed for arbitrarily large sufficiently integrable data. |
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