On the generalized Forchheimer-Stokes-Fourier systems under the Beavers-Joseph-Saffman boundary condition (Q2866654)

The author considers a fluid moving along a porous medium. Precisely speaking, she considers an incompressible, generalized Newtonian fluid in a domain \(\Omega\subset\mathbb R^n\), which is a disjoint union of \(\Omega_f\), \(\Omega_p\), and \(\Gamma\). The fluid is flowing freely in \(\Omega_f\), and it is flowing through a porous medium in \(\Omega_p\). These two domains are separated by the interface \(\Gamma\). One considers a Beaver-Joseph-Saffman boundary value problem as follows. In the free flow region \(\Omega_f\), one considers the Stokes equation, taking the heat transfer into account. In the porous region \(\Omega_p\), one introduces a general form of the Forchheimer system. It is a constitutive equation for a fluid flowing through a porous medium. They are second order elliptic/parabolic equations for the fluid velocity, the pressure, and the temperature. One considers a natural boundary condition on the interface \(\Gamma\), and defines the notion of a weak solution to this boundary value problem. The author gives an existence theorem of the steady solution (resp. of the time-dependent solution) in this weak sense, using the Galerkin method (resp. the Faedo-Galerkin method).