On singular solutions of time-periodic and steady Stokes problems in a power cusp domain

2021

Mathematics

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.

Section: Existence and Uniqueness Of Weak Solutionmentioning

confidence: 99%

Section: Existence and Uniqueness Of Weak Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kaulakytė

2021

Mathematics

“…The existence of singular solutions to the time-periodic and the non-stationary Stokes problem in the domain with a cusp point was studied in [2,3]. We can also mention the recent paper [4] where the Dirichlet problem for the non-stationary Stokes system is studied in a three-dimensional cone.…”

Section: Introductionmentioning

confidence: 99%

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

Advances in Nonlinear Analysis

Raciene

2021

Self Cite

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.

“…In recent papers the authors have studied existence of singular solutions to the time-periodic and initial boundary value problems for the linear Stokes equations [3,4] and an initial boundary value problem for the Navier-Stokes equations [28,29] in domains having a power-cusp (peak type) singular point on the boundary. The case when the flux of the boundary value is nonzero was considered.…”

Section: Introductionmentioning

confidence: 99%

On Singular Solutions of the Stationary Navier-Stokes System in Power Cusp Domains

Mathematical Modelling and Analysis

Raciene

2021

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.