Unsteady 3D-Navier–Stokes system with Tresca’s friction law

Boukrouche, Mahdi; Boussetouan, Imane; Paoli, Laetitia

doi:10.1090/qam/1563

Authors: Mahdi Boukrouche, Imane Boussetouan and Laetitia Paoli
Journal: Quart. Appl. Math. 78 (2020), 525-543
MSC (2010): Primary 76D05, 35K86, 76D03, 35A35, 49J40
DOI: https://doi.org/10.1090/qam/1563
Published electronically: November 22, 2019
MathSciNet review: 4100292
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Abstract: We consider unsteady incompressible 3D fluid flow with nonconstant viscosity subjected to nonlinear boundary conditions of Tresca’s type: when the modulus of the shear stress at the boundary is smaller than a given threshold, the classical no-slip condition holds; otherwise, the modulus of the shear stress at the boundary equals the given threshold and the fluid is allowed to slip. The problem is thus described by a nonlinear parabolic variational inequality. We construct a sequence of approximate solutions by using both a regularization of the free boundary condition due to friction and a penalty method, reminiscent of the “incompressibility limit” of compressible fluids, allowing us to get a better insight into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.

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