REGULARITY FOR THE 3D EVOLUTION NAVIER-STOKES EQUATIONS UNDER NAVIER BOUNDARY CONDITIONS IN SOME LIPSCHITZ DOMAINS

Article Mathematics, Applied

Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions

Alessio Falocchi et al.

Summary: This study investigates the Stokes eigenvalue problem under Navier boundary conditions in C-1,C-1-domains Omega subset of R-3, finding that zero may be the least eigenvalue in this scenario. The analysis fully characterizes the domains where this occurs, and shows that the ball is the only domain where the zero eigenvalue is not simple, having a multiplicity of three. These results are then applied to demonstrate the validity or failure of a suitable Poincare-type inequality, with proofs obtained through a combination of analytic and geometric arguments.

ANNALI DI MATEMATICA PURA ED APPLICATA (2022)

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Article Mathematics, Applied

Uniqueness and Bifurcation Branches for Planar Steady Navier-Stokes Equations Under Navier Boundary Conditions

Gianni Arioli et al.

Summary: The uniqueness of solutions of the stationary Navier-Stokes equations under Navier boundary conditions in a square is studied, showing dependency on the Reynolds number and external force strength. For certain external forcing, uniqueness persists on continuous branches of solutions as these quantities increase. On the other hand, a branch of symmetric solutions is shown to bifurcate for a different forcing, leading to a secondary branch of nonsymmetric solutions. This proof is computer-assisted, based on a local representation of branches as analytic arcs.

JOURNAL OF MATHEMATICAL FLUID MECHANICS (2021)

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Article Mathematics, Applied