Mixed Initial-Boundary Value Problem for the Three-Dimensional Navier-Stokes Equations in Polyhedral Domains

TL;DR

This paper investigates the existence, uniqueness, and smoothness of solutions to a complex 3D Navier-Stokes problem with mixed boundary conditions in polyhedral domains, advancing understanding of fluid dynamics in irregular geometries.

Contribution

It establishes the well-posedness of the Navier-Stokes equations with mixed boundary conditions in polyhedral domains, a significant extension of prior results.

Findings

Proves existence of solutions for the mixed boundary value problem.

Demonstrates uniqueness of solutions within a certain time interval.

Shows solutions are smooth on the interval (0,T^*).

Abstract

We study a mixed initial-boundary value problem for the Navier-Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval $(0,T^*)$, where $0<T^*\leq T$.