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Problem of second grade fluids in convex polyhedrons. (English) Zbl 1251.35078
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 $$\frac{3\pi}{4}$$.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76A05 Non-Newtonian fluids 35G30 Boundary value problems for nonlinear higher-order PDEs
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