On linearization principle for Navier-Stokes system with dynamic slip boundary condition

Dalibor Pražák; Michael Zelina

doi:10.3934/cpaa.2023034

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2023, Volume 22, Issue 5: 1456-1476. Doi: 10.3934/cpaa.2023034

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On linearization principle for Navier-Stokes system with dynamic slip boundary condition

Dalibor Pražák^, and
Michael Zelina

Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83,186 75 Prague 8, Czech Republic

^*Corresponding author: Dalibor Pražák
^*Corresponding author: Dalibor Pražák

Received: December 2021

Revised: August 2022

Early access: February 27, 2023

Published online: February 27, 2023

The authors thank the Czech Science Foundation, Grant Number 20-11027X, for its support.

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Abstract

We consider incompressible Navier-Stokes equations in a bounded 3D regular domain, coupled with the so-called dynamic boundary condition. We rigorously establish the principle of linearized stability. Namely, given a smooth stationary state, we prove that the linearized equation has a complete basis of generalized eigenfunctions, and that the non-linear stability depends on the supremum of the real part of spectrum in the usual way. We work in the class of weak solutions satisfying the energy inequality, for which the global existence (but not uniqueness) is known.

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Mathematics Subject Classification: Primary: 37L15; Secondary: 35B35, 76D05.

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