Generalized Navier-Stokes equations with nonlinear anisotropic viscosity. (English) Zbl 1428.35354
Summary: The purpose of this work is to study the generalized Navier-Stokes equations with nonlinear viscosity that, in addition, can be fully anisotropic. Existence of very weak solutions is proved for the associated initial and boundary-value problem, supplemented with no-slip boundary conditions. We show that our existence result is optimal in some directions provided there is some compensation in the remaining directions. A particular simplification of the problem studied here, reduces to the Navier-Stokes equations with (linear) anisotropic viscosity used to model either the turbulence or the Ekman layer in atmospheric and oceanic fluid flows.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q30 | Navier-Stokes equations |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35D30 | Weak solutions to PDEs |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
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