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Regularization by monotone perturbations of the hydrostatic approximation of Navier-Stokes equations. (English) Zbl 1096.35105
The hydrostatic approximation of Navier-Stokes equations is a general model used in oceanography for the description of the circulation of water in oceans and lakes. First, the author introduces a functional setting of the problem and shows that the set of smooth functions with compact support is dense in the space where the solutions are sought. This allows to demonstrate that the solutions of the perturbed problem are close to those of the original problem when the perturbations are small enough. As a result, after extracting a weakly convergent (in a suitable Banach space) subsequence of perturbed solutions for vanishing perturbation parameter, the author arrives at a new proof of the existence of solutions to the hydrostatic approximation of Navier-Stokes equations. Moreover, the inclusion of monotone perturbations leads to an energy identity for any solution. Finally, this regularization technique has been applied to prove the existence of a solution for a one-equation hydrostatic turbulence model.

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
86A05 Hydrology, hydrography, oceanography
35J60 Nonlinear elliptic equations
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