Nonhomogeneous Boundary Value Problem for the Steady Navier-Stokes Equations in 2D Symmetric Domains with Several Outlets to Infinity

TL;DR

This paper investigates the existence of weak solutions to the steady Navier-Stokes equations in 2D symmetric domains with multiple outlets to infinity, without restrictions on flux sizes and considering various domain geometries.

Contribution

It constructs symmetric solenoidal extensions satisfying Leray-Hopf inequality, enabling the reduction of boundary problems to homogeneous cases in complex geometries.

Findings

Existence of at least one weak solution in various symmetric domain types.

No restrictions on flux sizes over boundaries.

Dirichlet integral can be finite or infinite depending on domain geometry.

Abstract

In this paper we study the nonhomongeneous boundary value problem for the stationary Navier-Stokes equations in two dimensional symmetric domains with finitely many outlets to infinity. The domains may have no self-symmetric outlet (V-type domain), one self-symmetric outlet (Y-type domain) or two self-symmetric outlets (I-type domain). We construct a symmetric solenoidal extension of the boundary value satisfying the Leray-Hopf inequality. After having such an extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak solution follows. Notice that we do not impose any restrictions on the size of the fluxes over the inner and outer boundaries. Moreover, the Dirichlet integral of the solution can be either finite or infinite depending on the geometry of the domains.