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
[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 L^{p}-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 |