×

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade. (English) Zbl 1471.35220

Let \(A_0A_1B_1B_0\) denote a curvilinear rectangle with two opposite sides lying on the lines \(x_1=0\) and \(x_1=d>0\). Let \(P_0\) denote a smooth simply connected domain which belongs to the rectangle, \(\Omega\) be the complement of \(P_0\) to the rectangle. The domains \(P_0\) is periodically continued in the \(x_2\)-direction. The authors consider the periodic two-dimensional boundary value problem for the stationary Stokes equations with the given velocity \(\mathbf u\) on \(A_0A_1\) and \(B_0B_1\) and \(\mathbf u = \mathbf 0\) on \(\partial P_0\). The existence of weak and strong solutions is verified.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
35B65 Smoothness and regularity of solutions to PDEs
35D35 Strong solutions to PDEs
35D30 Weak solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agmon, S., Lectures on Elliptic Boundary Value Problems (1965), New York: Van Nostrand Comp, New York · Zbl 0142.37401
[2] Al Baba, H.; Amrouche, Ch.; Escobedo, M., Semi-group theory for the Stokes operator with Navier-type boundary condition in \(L^p\)-spaces, Arch. Ration. Mech. Anal., 223, 2, 881-940 (2017) · Zbl 1359.35143 · doi:10.1007/s00205-016-1048-1
[3] Amrouche, Ch., Escobedo, M., Ghosh, A.: Semigroup theory for the Stokes operator with Navier boundary condition in \(L^p\) spaces. arXiv:1808.02001v1 [math.AP] (6 Aug 2018)
[4] Bruneau, C. H.; Fabrie, P., New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result, Math. Model. Numer. Anal., 30, 7, 815-840 (1996) · Zbl 0865.76016 · doi:10.1051/m2an/1996300708151
[5] Chen, G. Q.; Qian, Z., A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions, Indiana Univ. Math. J., 59, 2, 721-760 (2010) · Zbl 1206.35193 · doi:10.1512/iumj.2010.59.3898
[6] Chen, G. Q.; Osborne, D.; Qian, Z., The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries, Acta Math. Sci., 29B, 4, 919-948 (2009) · Zbl 1212.35346 · doi:10.1016/S0252-9602(09)60078-3
[7] Dauge, M., Stationary Stokes and Navier-Stokes systems on two– or three-dimensional domains with corners. Part I: linearized equations, SIAM J. Math. Anal., 20, 74-97 (1989) · Zbl 0681.35071 · doi:10.1137/0520006
[8] Dolejší, V.; Feistauer, M.; Felcman, J., Numerical simulation of compressible viscous flow through cascades of profiles, Z. Angew. Math. Mech., 76, 301-304 (1996) · Zbl 0925.76443
[9] Feistauer, M.; Neustupa, T., On some aspects of analysis of incompressible flow through cascades of profiles, Operator Theory, Advances and Applications, 257-276 (2004), Basel: Birkhäuser, Basel · Zbl 1054.35051
[10] Feistauer, M.; Neustupa, T., On non-stationary viscous incompressible flow through a cascade of profiles, Math. Methods Appl. Sci., 29, 16, 1907-1941 (2006) · Zbl 1124.35054 · doi:10.1002/mma.755
[11] Feistauer, M.; Neustupa, T., On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow, J. Math. Fluid Mech., 15, 701-715 (2013) · Zbl 1293.35204 · doi:10.1007/s00021-013-0135-4
[12] Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problems (2011), New York: Springer, New York · Zbl 1245.35002
[13] Geissert, M.; Heck, H.; Hieber, M., On the equation \(\text{div}\,u = g\) and Bogovskii’s operator in Sobolev spaces of negative order, Partial Differential Equations and Functional Analysis, 113-121 (2006), Basel: Birkhüser, Basel · Zbl 1218.35073 · doi:10.1007/3-7643-7601-5_7
[14] Glowinski, R., Numerical Methods for Nonlinear Variational Problems (1984), New York: Springer, New York · Zbl 0575.65123 · doi:10.1007/978-3-662-12613-4
[15] Grisvard, P.: Singularités des solutions du probléme de Stokes dans un polygone. Université de Nice (1979)
[16] Grisvard, P., Elliptic Problems in Non-smooth Domains (1985), Boston: Pitman Advanced Publishing Program, Boston · Zbl 0695.35060
[17] Heywood, J. G.; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 22, 325-352 (1996) · Zbl 0863.76016 · doi:10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
[18] Kato, T., Perturbation Theory for Linear Operators (1966), Berlin: Springer, Berlin · Zbl 0148.12601 · doi:10.1007/978-3-642-53393-8
[19] Kato, T.; Mitrea, M.; Ponce, G.; Taylor, M., Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 7, 643-650 (2000) · Zbl 0980.53022 · doi:10.4310/MRL.2000.v7.n5.a10
[20] Kellog, R. B.; Osborn, J. E., A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal., 21, 397-431 (1976) · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5
[21] Kozel, K.; Louda, P.; Příhoda, J., Numerical solution of turbulent flow in a turbine cascade, Proc. Appl. Math. Mech., 6, 743-744 (2006) · doi:10.1002/pamm.200610352
[22] Kračmar, S.; Neustupa, J., Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities, Z. Angew. Math. Mech., 74, 6, 637-639 (1994) · Zbl 0836.35121
[23] Kračmar, S.; Neustupa, J., A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions, Nonlinear Anal., 47, 6, 4169-4180 (2001) · Zbl 1042.35605 · doi:10.1016/S0362-546X(01)00534-X
[24] Kračmar, S.; Neustupa, J., Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier-Stokes variational inequality, Math. Nachr., 291, 11-12, 1-14 (2018) · Zbl 1401.35239
[25] Kučera, P., Basic properties of the non-steady Navier-Stokes equations with mixed boundary conditions ina bounded domain, Ann. Univ. Ferrara, 55, 289-308 (2009) · Zbl 1205.35198 · doi:10.1007/s11565-009-0082-4
[26] Kučera, P.; Beneš, M., Solution to the Navier-Stokes equatons with mixed boundary conditions in two-dimensional bounded domains, Math. Nachr., 289, 2-3, 194-212 (2016) · Zbl 1381.35116
[27] Kučera, P.; Skalák, Z., Solutions to the Navier-Stokes equations with mixed boundary conditions, Acta Appl. Math., 54, 3, 275-288 (1998) · Zbl 0924.35097 · doi:10.1023/A:1006185601807
[28] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompresible Flow (1969), New York: Gordon and Breach Science Publishers, New York · Zbl 0184.52603
[29] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications (1968), Paris: Dunod, Paris · Zbl 0165.10801
[30] Medková, D., The Neumann problem for the planar Stokes system, Ann. Univ. Ferrara, 58, 307-329 (2012) · Zbl 1303.35074 · doi:10.1007/s11565-012-0154-8
[31] Medková, D., One problem of the Navier type for the Stokes system inplanar domains, J. Differ. Equ., 261, 5670-5689 (2016) · Zbl 1356.35181 · doi:10.1016/j.jde.2016.08.007
[32] Nečas, J., Les méthodes directes en théorie des équations elliptiques (1967), Paris: Masson et Cie, Paris · Zbl 1225.35003
[33] Neustupa, T., Question of existence and uniqueness of solution for Navier-Stokes equation with linear “do-nothing” type boundary condition on the outflow, Lecture Notes in Computer Science, 431-438 (2009) · Zbl 1233.35159
[34] Neustupa, T., The analysis of stationary viscous incompressible flow through a rotating radial blade machine, existence of a weak solution, Appl. Math. Comput., 219, 3316-3322 (2012) · Zbl 1309.76054
[35] Neustupa, T., A steady flow through a plane cascade of profiles with an arbitrarily large inflow: the mathematical model, existence of a weak solution, Appl. Math. Comput., 272, 687-691 (2016) · Zbl 1410.35098
[36] Sohr, H., The Navier-Stokes Equations. The Eelementary Functional Analytic Approach (2001), Basel: Birkhäuser, Basel · Zbl 0983.35004 · doi:10.1007/978-3-0348-8255-2
[37] Straka, P.; Příhoda, J.; Kožíšek, M.; Fürst, J., Simulation of transitional flows through a turbine blade cascade with heat transfer for various flow conditions, EPJ Web Conf., 143 (2017) · doi:10.1051/epjconf/201714302118
[38] Temam, R., Navier-Stokes Equations (1977), Amsterdam: North-Holland, Amsterdam · Zbl 0383.35057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.