Problem of second grade fluids in convex polyhedrons (Q2902748)

The stationary problem of a grade-two fluid is studied in convex 3D polyhedron \(\Omega\). The velocity \(v\) and the pressure \(p\) satisfy to the equations NEWLINE\[NEWLINE\begin{aligned} & -\nu\Delta v+\text{curl}(v-\alpha\Delta v)\times v+\nabla p=f,\quad \\ &\text{div}\,v=0\;\text{in}\;\Omega, v=g\quad \text{on}\;\partial\Omega,\end{aligned}NEWLINE\]NEWLINE where \(\nu>0\) is the kinematic viscosity coefficient, \(\alpha\neq 0\) is the normal stress module, \(g\cdot n=0\). The problem is reformulated in an equivalent form using a transport equation.NEWLINENEWLINEThe solvability of the problem is proved for small data \((f,g)\). The Galerkin method is the base of the proof. Uniqueness is established for inner angles of a polyhedron smaller than \(\frac{3\pi}{4}\).