Non-stationary Navier–Stokes equations in 2D power cusp domain

Konstantin Pileckas,Alicija Raciene

doi:10.1515/anona-2020-0165

Abstract

Abstract The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. We consider the case where the boundary value has a nonzero flux over the boundary. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point was constructed. In this, second part, the constructed asymptotic decomposition is justified, i.e., existence of the solution which is represented as the sum of the constructed asymptotic expansion and a term with finite energy norm is proved. Moreover, it is proved that the solution represented in this form is unique.

Highlights

In this paper we continue to study the boundary value problem for the non-stationary Navier–Stokes system ut − ν∆u + (u · ∇)u + ∇p = f, div u = (1.1) u|∂Ω O = a, \u(x, ) = b(x) in a two-dimensional bounded domain Ω with the cusp point O = (, ) at the boundary: Ω = GH ∪ Ω, where GH = x ∈ R : |x | < φ(x ), x ∈ (, H], φ(x ) = γ xλ, γ = const, λ >
The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary
Second part, the constructed asymptotic decomposition is justi ed, i.e., existence of the solution which is represented as the sum of the constructed asymptotic expansion and a term with nite energy norm is proved

Summary

Introduction

∂tl It is proved (see inequality (4.15) in [1]) that the vector eld U[J] satis es the following estimates sup U[J](·, y , t) W , Υ + U[J] L T W , Υ In the subsections we construct the sequence of weak solutions vK to the Navier–Stokes equations in regular domains ΩK and prove the uniform (with respect to K) estimates for them.

Results

Conclusion