Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments

Yasir Nadeem Anjam; Yasir Nadeem Anjam

doi:10.3934/math.2020030

AIMS Mathematics

2020, Volume 5, Issue 1: 440-466. doi: 10.3934/math.2020030

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Review Topical Sections

Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments

Yasir Nadeem Anjam ^{1,2
,
,}

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P. R. China
2.
Department of Applied Sciences, National Textile University, Faisalabad 37610, Pakistan

Received: 06 October 2019 Accepted: 27 November 2019 Published: 05 December 2019
MSC : 35M32, 35Q30, 76D05, 76N10

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. The corner singularity theory for the solutions of elliptic boundary value problems on domains with corners or edges has been well established in the past century and also in recent years. The corner singularity functions provide an appropriate mathematical structure to understand the physical trajectories of the fluid particles. It has been investigated for general elliptic boundary value problems and also extended to some non-elliptic problems. Currently, the theory has been constructed for compressible viscous Stokes and NavierStokes systems on polygonal and polyhedral domains to analyze the structure of the solution near the corners and edges. Several interesting results about the regularity of the solution cannot be extended if one of the following situations appears: The domain has corners, edges and cusp, etc. On the boundary, change of boundary conditions at some points, discontinuities of the solutions, and singularities of the coefficients. This article reviewed the structure of the solution and regularity results of the stationary Stokes and Navier-Stokes equations on polygonal domains with convex or non-convex corners.
- corner singularities,
- Stokes problem,
- Navier-Stokes system,
- regularity,
- non-convex corners
Citation: Yasir Nadeem Anjam. Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments[J]. AIMS Mathematics, 2020, 5(1): 440-466. doi: 10.3934/math.2020030

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Abstract

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. The corner singularity theory for the solutions of elliptic boundary value problems on domains with corners or edges has been well established in the past century and also in recent years. The corner singularity functions provide an appropriate mathematical structure to understand the physical trajectories of the fluid particles. It has been investigated for general elliptic boundary value problems and also extended to some non-elliptic problems. Currently, the theory has been constructed for compressible viscous Stokes and NavierStokes systems on polygonal and polyhedral domains to analyze the structure of the solution near the corners and edges. Several interesting results about the regularity of the solution cannot be extended if one of the following situations appears: The domain has corners, edges and cusp, etc. On the boundary, change of boundary conditions at some points, discontinuities of the solutions, and singularities of the coefficients. This article reviewed the structure of the solution and regularity results of the stationary Stokes and Navier-Stokes equations on polygonal domains with convex or non-convex corners.

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