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On a variational inequality for the Navier-Stokes operator with variable viscosity. (English) Zbl 1167.35432

Summary: We investigate the unilateral problem for the operator \(L\) perturbed of Navier-Stokes operator in a cylindrical case, where
\[ Lu=u'-(\nu_0+\nu_1\|u(t)\|^2)\Delta u+(u\nabla)u-f+\nabla p. \]
The mixed problem for the operator \(L\) was proposed by J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; Paris: Gauthier-Villars (1969; Zbl 0189.40603)]. Using an appropriate penalization, we obtain a variational inequality for the Navier-Stokes perturbed system.

MSC:

35Q30 Navier-Stokes equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0189.40603
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