On the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity

TL;DR

This paper investigates the local well-posedness of a 3D incompressible Navier-Stokes model with partial viscosity, focusing on the challenges posed by non-dissipative Robin boundary conditions and establishing well-posedness through a Picard iteration approach.

Contribution

It provides a novel analysis of local well-posedness for a Navier-Stokes model with partial viscosity under non-standard boundary conditions, using a Picard iteration method.

Findings

Established local well-posedness for the model with Robin boundary conditions.

Analyzed the heat equation with Robin boundary conditions to support the well-posedness proof.

Identified the subtlety of non-dissipative boundary conditions in the analysis.

Abstract

In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition.