Abstract
The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.
Highlights
In the paper we consider the boundary value problem for the steady NavierStokes system [2]−ν∆u + (u · ∇)u + ∇p = f, x ∈ Ω, div u = 0, u|∂Ω = a(x), (1.1)in a two-dimensional1 bounded domain Ω = Ω0 ∪ GH, where GH = x ∈ R2 : |x1| < γ0xλ2, x2 ∈ (0, H] for some γ0 > 0 and λ > 1, γ0 =Copyright c 2021 The Author(s)
We mention the papers [7, 8, 9] where the existence of a solution to the Navier–Stokes problem with a sink or source in the cusp point O was proved for arbitrary data and the papers [12, 13, 14] where the asymptotics of a solution to the nonstationary Stokes problem is studied in domains with conical points and conical outlets to infinity
In order to prove the existence of the solution to problem (1.1), we first construct the formal asymptotic decomposition of it such that discrepancies belong to L2-space
Summary
In the paper we consider the boundary value problem for the steady NavierStokes system [2]. We mention the papers [7, 8, 9] where the existence of a solution (with an infinite Dirichlet integral) to the Navier–Stokes problem with a sink or source in the cusp point O was proved for arbitrary data and the papers [12, 13, 14] where the asymptotics of a solution to the nonstationary Stokes problem is studied in domains with conical points and conical outlets to infinity.
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