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

• Main Authors: Kristina Kaulakytė, Konstantinas Pileckas
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: stationary Navier–Stokes equations; multi-connected domain; power cusp; singular solutions; asymptotic expansion; regularity
• Online Access: https://www.mdpi.com/2227-7390/9/17/2022

The boundary value problem for the steady Navier–Stokes system is considered in a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>D</mi></mrow></semantics></math></inline-formula> multiply-connected bounded domain with the boundary having a power cusp singularity at the point <i>O</i>. 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 <i>O</i>. 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.