Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

Alessio Falocchi; Filippo Gazzola

doi:10.3934/dcds.2021151

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2022, Volume 42, Issue 3: 1185-1200. Doi: 10.3934/dcds.2021151

This issue Previous Article Topological mild mixing of all orders along polynomials Next Article Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

Alessio Falocchi^1, and
Filippo Gazzola^2,

1.
Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, Italy

Received: April 2021

Revised: August 2021

Early access: October 2021

Published: March 2022

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Abstract

For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least $ C^{2,1} $). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest, that we call sectors.

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Mathematics Subject Classification: 35Q30, 35K20.

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Figure 1. From left to right: a pipe bifurcation, a joint, a vein, a drop, a tunnel, a bottle

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Figure 2. Some sectors obtained as subdomains of a sphere

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Figure 3. Some Lipschitz domains that are not sectors, according to Definition 3

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Figure 4. Sectors of type $ (B) $

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