A Nonhomogeneous Boundary Value Problem for Steady State Navier-Stokes Equations in a Multiply-Connected Cusp Domain

Kristina Kaulakytė,Konstantinas Pileckas

doi:10.3390/math9172022

Abstract

The boundary value problem for the steady Navier–Stokes system is considered in a 2D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with nonzero flow rates over connected components of the boundary is studied. It is also supposed that there is a source/sink in O. In this case the solution necessarily has an infinite Dirichlet integral. The existence of a solution to this problem is proved assuming that the flow rates are “sufficiently small”. This condition does not require the norm of the boundary data to be small. The solution is constructed as the sum of a function with the finite Dirichlet integral and a singular part coinciding with the asymptotic decomposition near the cusp point.

Highlights

In the paper we study the nonhomogeneous stationary boundary value problem for the Navier-Stokes equations
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In this paper we extend the results of [1] in two directions: first, we study the case of domains with multiply-connected boundaries and, second, we prove the existence of the solution coinciding near the cusp point with the formal asymptotic decomposition assuming only that the flow rates F0, F1, . . . , FN of the boundary value a are sufficiently small

Summary

Introduction

In this paper we extend the results of [1] in two directions: first, we study the case of domains with multiply-connected boundaries and, second, we prove the existence of the solution coinciding near the cusp point with the formal asymptotic decomposition assuming only that the flow rates F0 , F1 , . The proof is based on the construction of an extension of the boundary value which coincides near the cusp point with the asymptotic decomposition and allows to obtain needed a priori estimates assuming only that flow rates are sufficiently small. Note that in this case the norm of a is not obliged to be small.

Function Spaces

Formal Asymptotic Decomposition

Flux Carrier from Inner Boundaries

Flux Carrier from the Outer Boundary

Extension of a2

Existence and Uniqueness of Weak Solution