On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows

TL;DR

This paper introduces a new class of energy-preserving boundary conditions for incompressible Newtonian flows, proving local well-posedness in various domain types using maximal regularity techniques.

Contribution

It derives and analyzes a novel class of boundary conditions that preserve energy and establishes their well-posedness for the Navier-Stokes equations in different domain geometries.

Findings

Established local-in-time well-posedness for the boundary conditions.

Proved maximal regularity properties of the linearized problems.

Applicable to bounded and unbounded domains with smooth boundaries.

Abstract

We derive a class of energy preserving boundary conditions for incompressible Newtonian flows and prove local-in-time well-posedness of the resulting initial boundary value problems, i.e. the Navier-Stokes equations complemented by one of the derived boundary conditions, in an Lp-setting in domains, which are either bounded or unbounded with almost flat, sufficiently smooth boundary. The results are based on maximal regularity properties of the underlying linearisations, which are also established in the above setting.