Problem of second grade fluids in convex polyhedrons. (English) Zbl 1251.35078

Bernard, Jean-Marie

SIAM J. Math. Anal. 44, No. 3, 2018-2038 (2012).

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 \[ \begin{aligned} & -\nu\Delta v+\text{curl}(v-\alpha\Delta v)\times v+\nabla p=f,\quad \\ &\text{div}\,v=0\;\text{in}\;\Omega, v=g\quad \text{on}\;\partial\Omega,\end{aligned} \] 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.
The 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 \(\frac{3\pi}{4}\).

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35Q35	PDEs in connection with fluid mechanics
76A05	Non-Newtonian fluids
35G30	Boundary value problems for nonlinear higher-order PDEs

Keywords:

grade-two fluids; regularity in convex polyhedron; transport equation