Review Topical Sections

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

  • 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.

    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

    Related Papers:

  • 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.


    加载中


    [1] R. A. Adams, J. J. Fournier, Sobolev Spaces, Academic Press, 2003.
    [2] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Commun. Pur. Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405
    [3] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. Pur. Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104
    [4] M. Amara, D. C. Papaghiuc, E. Chaćon-Vera, et al. Vorticity-velocity-pressure formulation for Navier-Stokes equations, Comput. Visual. Sci., 6 (2004), 47-52. doi: 10.1007/s00791-003-0107-y
    [5] C. Amrouche, P. Penel, N. Seloula, Some remarks on the boundary conditions in the theory of Navier-Stokes equations, Anna. Math. Blais. Pasc., 20 (2013), 37-73. doi: 10.5802/ambp.321
    [6] I. Babuška, Finite element method for domains with corners, Computing, 6 (1970), 264-273. doi: 10.1007/BF02238811
    [7] G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
    [8] J. M. Bernard, Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure, existence and regularity, Nonlinear Anal. Real., 4 (2003), 805-839. doi: 10.1016/S1468-1218(03)00016-6
    [9] L. Bers, Survey of local properties of solutions of elliptic partial differential equations, Commun. Pur. Appl. Math., 9 (1956), 339-350. doi: 10.1002/cpa.3160090306
    [10] H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing, 28 (1982), 53-63. doi: 10.1007/BF02237995
    [11] G. A. Brés, T. Colonius, Three-dimensional instabilities in compressible flow over open cavities, J. Fluid Mech., 599 (2008), 309-339. doi: 10.1017/S0022112007009925
    [12] Z. Cai, S. Kim, B. C. Shin, Solution methods for the Poisson equation with corner singularities: Numerical results, SIAM J. Sci. Comput., 23 (2001), 672-682. doi: 10.1137/S1064827500372778
    [13] G. F. Carrier, M. Krook, C. E. Pearson, Functions of a Complex Variable: Theory and Technique, Cambridge University Press, 2005.
    [14] S. E. Chen, R. B. Kellogg, An interior discontinuity of a nonlinear elliptic-hyperbolic system, SIAM J. Math. Anal., 22 (1991), 602-622. doi: 10.1137/0522038
    [15] X. F. Chen, W. Q. Xie, Discontinuous solutions of steady state, viscous compressible NavierStokes equations, J. Differ. Equations, 115 (1995), 99-119. doi: 10.1006/jdeq.1995.1006
    [16] Y. Chen, T. Jiang, The pressure boundary conditions for the incompressible navier-stokes equations computation, Commun. Nonlinear. Sci., 1 (1996), 70-72. doi: 10.1016/S1007-5704(96)90042-8
    [17] Y. Z. Chen, L. C. Wu, Second Order Elliptic Equations and Elliptic Systems, American Mathematical Society, 2004.
    [18] H. J. Choi, J. R. Kweon, For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon, J. Differ. Equation, 250 (2011), 2440-2461. doi: 10.1016/j.jde.2010.12.018
    [19] H. J. Choi, J. R. Kweon, The stationary Navier-Stokes system with no-slip boundary condition on polygons: Corner singularity and regularity, Commun. Part. Diff. Eq., 38 (2013), 1235-1255. doi: 10.1080/03605302.2012.752386
    [20] H. J. Choi, J. R. Kweon, A finite element method for singular solutions of the Navier-Stokes equations on a non-convex polygon, J. Comput. Appl. Math., 292 (2016), 342-362. doi: 10.1016/j.cam.2015.07.006
    [21] N. Chorfi, Geometric singularities of the Stokes problem, Abstr. Appl. Anal., 2014 (2014), 1-8.
    [22] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, 2002.
    [23] M. G. Crandall, P. H. Rabinowitz, L. Tartar, On a dirichlet problem with a singular nonlinearity, Commun. Part. Diff. Eq., 2 (1977), 193-222. doi: 10.1080/03605307708820029
    [24] D. G. Crowdy, S. J. Brzezicki, Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle, Proc. R. Soc. A., 473 (2017), 20170134.
    [25] D. G. Crowdy, A. M. J. Davis, Stokes flow singularities in a two-dimensional channel: A novel transform approach with application to microswimming, Proc. R. Soc. A., 469 (2013), 20130198.
    [26] M. Dauge, Singularities along the edges, In: Elliptic Boundary Value Problems on Corner Domains, Berlin: Springer, 1988, 128-152.
    [27] M. Dauge, Stationary Stokes and Navier-Stokes systems on two or three-dimensional domains with corners. Part I. Linearized equations, SIAM J. Math. Anal., 20 (1989), 74-97. doi: 10.1137/0520006
    [28] M. Dauge, Singularities of corner problems and problems of corner singularities, ESAIM: Proc., 6 (1999), 19-40. doi: 10.1051/proc:1999044
    [29] M. Dauge, Elliptic boundary value problems on corner domains: Smoothness and asymptotics of solutions, Springer, 2006.
    [30] W. R. Dean, P. E. Montagnon, On the steady motion of viscous liquid in a corner, Math. Proc. Cambridge, 45 (1949), 389-394. doi: 10.1017/S0305004100025019
    [31] F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. Appl. Sci., 25 (2002), 1091-1119. doi: 10.1002/mma.328
    [32] F. Dubois, M. Saläun, S. Salmon, Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem, J. Math. Pure. Appl., 82 (2003), 1395-1451. doi: 10.1016/j.matpur.2003.09.002
    [33] M. Durand, Singularities in elliptic problems, In: Singularities and Constructive Methods for Their Treatment, Berlin: Springer, 1985, 104-112.
    [34] M. Elliotis, G. Georgiou, C. Xenophontos, The solution of Laplacian problems over L-shaped domains with a singular function boundary integral method, Commun. Num. Meth. Eng., 18 (2002), 213-222. doi: 10.1002/cnm.489
    [35] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 2010.
    [36] M. Feistauer, Mathematical Methods in Fluid Dynamics, Chapman and Hall/CRC, 1993.
    [37] G. Georgiou, A. Boudouvis, A. Poullikkas, Comparison of two methods for the computation of singular solutions in elliptic problems, J. Comput. Appl. Maths., 79 (1997), 277-287. doi: 10.1016/S0377-0427(96)00173-2
    [38] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2015.
    [39] V. Girault, P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Science and Business Media, 2012.
    [40] S. Gontara, H. Mâagli, S. Masmoudi, et al. Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J. Math. Anal. Appl., 369 (2010), 719-729. doi: 10.1016/j.jmaa.2010.04.008
    [41] P. Grisvard, Edge behavior of the solution of an elliptic problem, Math. Nachr., 132 (1987), 281-299. doi: 10.1002/mana.19871320119
    [42] P. Grisvard, Singularities in Boundary Value Problems, Springer, 1992.
    [43] P. Grisvard, Singular behavior of elliptic problems in non-Hilbertian Sobolev spaces, J. Math. Pure. Appl., 74 (1995), 3-33.
    [44] P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, In: Numerical solution of partial differential equations-III, Elsevier, 1976, 207-274.
    [45] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Advanced Pub, Program, Boston, 2 (1985), 2-2.
    [46] W. Hackbusch, Elliptic Differential Equations Theory and Numerical Treatment: The Poisson Equation, Springer, 2010.
    [47] J. H. Han, J. R. Kweon, M. Park, Interior discontinuity for a stationary compressible Stokes system with inflow datum, Comput. Math. Appl., 74 (2017), 2321-2329. doi: 10.1016/j.camwa.2017.07.002
    [48] J. Hernández, F. J. Mancebo, J. M. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Ann. I. H. Poincaré-AN, 19 (2002), 777-813.
    [49] D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with non-smooth initial data, P. Roy. Soc. Edinb. A., 103 (1986), 301-315.
    [50] D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions, Commun. Pur. Appl. Math., 55 (2002), 1365-1407.
    [51] S. Itoh, N. Tanaka, A. Tani, On some boundary value problem for the stokes equations in an infinite sector, Anal. Appl., 4 (2006), 357-375. doi: 10.1142/S0219530506000826
    [52] C. Johnson, Streamline diffusion finite element methods for incompressible and compressible fluid flow, In: Computational Fluid Dynamics and Reacting Gas Flows, Springer, 1988, 87-106.
    [53] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009.
    [54] T. Jonsson, M. G. Larson, K. Larsson, Graded parametric CutFEM and CutIGA for elliptic boundary value problems in domains with corners, Comput. Meth. Appl. Mech. Eng., 354 (2019), 331-350. doi: 10.1016/j.cma.2019.05.024
    [55] V. V. Katrakhov, S. V. Kiselevskaya, A singular elliptic boundary value problem in domains with corner points. I. Function spaces, Diff. Equat., 42 (2006), 395-403.
    [56] B. Kellogg, Some simple boundary value problems with corner singularities and boundary layers, Comput. Math. Appl., 51 (2006), 783-792. doi: 10.1016/j.camwa.2006.03.010
    [57] R. B. Kellogg, J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal., 21 (1976), 397-431. doi: 10.1016/0022-1236(76)90035-5
    [58] R. B. Kellogg, Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations, SIAM J. Math. Anal., 19 (1988), 567-579.
    [59] R. B. Kellogg, Corner singularities and singular perturbations, Ann. Univ. Ferrara, 47 (2001), 177-206.
    [60] V. A. Kondratíev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mos. Matem. Obsh., 16 (1967), 209-292.
    [61] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, American Mathematical Society, 1997.
    [62] V. A. Kozlov, V. G. Maz'ya, J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, American Mathematical Society, 2001.
    [63] M. Kumar, G. Mishra, A review on nonlinear elliptic partial differential equations and approaches for solution, Int. J. Nonlin. Sci., 13 (2012), 401-418.
    [64] J. R. Kweon, A regularity result of solution to the compressible Stokes equations on a convex polygon, Z. Angew. Math. Phys., 55 (2004), 435-450. doi: 10.1007/s00033-003-2042-7
    [65] J. R. Kweon, Singularities of a compressible Stokes system in a domain with concave edge in R3, J. Differ. Equations, 229 (2006), 24-48.
    [66] J. R. Kweon, Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon, J. Differ. Equations, 235 (2007), 166-198. doi: 10.1016/j.jde.2006.12.008
    [67] J. R. Kweon, Edge singular behavior for the heat equation on polyhedral cylinders in R3, Potential Anal., 38 (2013), 589-610.
    [68] J. R. Kweon, Corner singularity dynamics and regularity of compressible viscous Navier-Stokes flows, SIAM J. Math. Anal., 44 (2012), 3127-3161. doi: 10.1137/120867937
    [69] J. R. Kweon, A jump discontinuity of compressible viscous flows grazing a nonconvex corner, J. Math. Pure. Appl., 100 (2013), 410-432. doi: 10.1016/j.matpur.2013.01.007
    [70] J. R. Kweon, Jump dynamics due to jump datum of compressible viscous NavierStokes flows in a bounded plane domain, J. Differ. Equations, 261 (2016), 3463-3492. doi: 10.1016/j.jde.2016.05.031
    [71] J. R. Kweon, The compressible Stokes flows with no-slip boundary condition on non-convex polygons, J. Math. Fluid Mech., 19 (2017), 47-57. doi: 10.1007/s00021-016-0264-7
    [72] J. R. Kweon, R. B. Kellogg, Compressible Navier-Stokes equations in a bounded domain with inflow boundary condition, SIAM J. Math. Anal., 28 (1997), 94-108. doi: 10.1137/S0036141095284254
    [73] J. R. Kweon, R. B. Kellogg, Compressible Stokes problem on nonconvex polygonal domains, J. Differ. Equations, 176 (2001), 290-314. doi: 10.1006/jdeq.2000.3964
    [74] J. R. Kweon, R. B. Kellogg, Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon, Arch. Ration. Mech. Anal., 163 (2002), 35-64. doi: 10.1007/s002050200191
    [75] J. R. Kweon, R. B. Kellogg, The pressure singularity for compressible Stokes flows in a concave polygon, J. Math. Fluid Mech., 11 (2009), 1-21. doi: 10.1007/s00021-007-0245-y
    [76] J. R. Kweon, M. Song, A discontinuous solution for an evolution compressible Stokes system in a bounded domain, J. Differ. Equations, 219 (2005), 202-220. doi: 10.1016/j.jde.2004.10.001
    [77] O. S. Kwon, J. R. Kweon, For the vorticity-velocity-pressure form of the Navier-Stokes equations on a bounded plane domain with corners, Nonlinear. Anal. Theor., 75 (2012), 2936-2956. doi: 10.1016/j.na.2011.11.037
    [78] O. S. Kwon, J. R. Kweon, Interior jump and regularity of compressible viscous Navier-Stokes flows through a cut, SIAM J. Math. Anal., 49 (2017), 1982-2008. doi: 10.1137/15M1042826
    [79] O. S. Kwon, J. R. Kweon, Compressible Navier-Stokes equations in a polyhedral cylinder with inflow boundary condition, J. Math. Fluid Mech., 20 (2018), 581-601. doi: 10.1007/s00021-017-0336-3
    [80] L. Larchevêque, P. Sagaut, I. Mary, et al. Large-eddy simulation of a compressible flow past a deep cavity, Phys. Fluids, 15 (2003), 193-210. doi: 10.1063/1.1522379
    [81] Z. C. Li, Y. L. Chan, G. C. Georgiou, et al. Special boundary approximation methods for laplace equation problems with boundary singularities-applications to the motz problem, Comput. Math. Appl., 51 (2006), 115-142. doi: 10.1016/j.camwa.2005.01.030
    [82] Z. C. Li, T. T. Lu, Singularities and treatments of elliptic boundary value problems, Math. Comput. Model., 31 (2000), 97-145. doi: 10.1016/S0895-7177(00)00062-5
    [83] Z. C. Li, The method of fundamental solutions for annular shaped domains, J. Comput. Appl. Math., 228 (2009), 355-372. doi: 10.1016/j.cam.2008.09.027
    [84] Z. C. Li, Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, Springer, 2011.
    [85] V. Maz'ya, J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Rati. Mech. Anal., 194 (2009), 669-712. doi: 10.1007/s00205-008-0171-z
    [86] W. McLean, Corner singularities and boundary integral equations, In: Contributions of Mathematical Analysis to the Numerical Solution of Partial Differential Equations, Centre Math. Anal. Austral. Nat. Univ., 7 (1984), 197-213.
    [87] S. A. Nazarov, A. Novotny, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners, Math. Annal., 304 (1996), 121-150. doi: 10.1007/BF01446288
    [88] B. Nkemzi, S. Tanekou, Predictor-corrector p-and hp-versions of the finite element method for Poisson's equation in polygonal domains, Comput. Meth. Appl. Mech. Eng., 333 (2018), 74-93. doi: 10.1016/j.cma.2018.01.027
    [89] B. Nkemzi, M. Jung. Flux intensity functions for the Laplacian at polyhedral edges, Int. J. Fracture, 175 (2012), 167-185.
    [90] B. Nkemzi, M. Jung, Flux intensity functions for the Laplacian at axisymmetric edges, Math. Meth. Appl. Sci., 36 (2013), 154-168. doi: 10.1002/mma.2578
    [91] M. Renardy, Corner singularities between free surfaces and open boundaries, Z. Angew. Math. Phys., 41(1990), 419-425. doi: 10.1007/BF00959988
    [92] P. N. Shankar, M. D. Deshpande, Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., 32 (2000), 93-136. doi: 10.1146/annurev.fluid.32.1.93
    [93] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland: Elsevier, 1979.
    [94] H. B. D. Veiga, An Lp-theory for three-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Commun. Math. Phys., 109 (1987), 229-248. doi: 10.1007/BF01215222
    [95] J. R. Whiteman, N. Papamichael, Treatment of harmonic mixed boundary problems by conformal transformation methods, Z. Angew. Math. Phys., 23 (1972), 655-664. doi: 10.1007/BF01593987
    [96] Z. Zhang, The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term, P. Roy. Soc. Edinb. A., 136 (2006), 209-222. doi: 10.1017/S0308210500004522
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4081) PDF downloads(541) Cited by(3)

Article outline

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog