×

Local in time existence of solution of the Navier-Stokes equations with various types of boundary conditions. (English) Zbl 1479.35601

Summary: In this paper we deal with the two-dimensional Navier-Stokes system with three types of boundary conditions, including the so called “do-nothing” boundary condition. We prove the local in time existence and uniqueness of a solution for the initial velocity, which can belong to a class of functions that can be at least a little stronger than \(\boldsymbol{L}^2 (\Omega)\).

MSC:

35Q30 Navier-Stokes equations
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
76D05 Navier-Stokes equations for incompressible viscous fluids
76E09 Stability and instability of nonparallel flows in hydrodynamic stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arndt, R.; Ceretani, A.; Rautenberg, CN, On existence and uniqueness of solutions to a Boussinesq system with nonlinear and mixed boundary conditions, J. Math. Anal. Appl, 490, 29 (2019) · Zbl 1442.35320
[2] Beneš, M., Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains, DCDS Suppl., 1, 135-144 (2011) · Zbl 1306.35081
[3] Beneš, M.; Kučera, P., Solutions to the Navier-Stokes equations with mixed boundary conditions in Two-Dimensional Bounded Domains, Mathematische Nachrichten, 289, 2-3, 194-212 (2016) · Zbl 1381.35116 · doi:10.1002/mana.201400046
[4] Beneš, M., Kučera, P., Vacková, P.: Existence and regularity of the Stokes system with the do-nothing and Navier’s boundary conditions. In: Proceedings of 19th Conference of Applied Mathematics APLIMAT 2020, 47-56 (2020)
[5] Ceretani, A.; Rautenberg, CN, The Boussinesq with mixed non-smooth boundary conditions and do/nothing boundary flow, Z. Angew. Math. Phys., 70, 24 (2019) · Zbl 1409.76022 · doi:10.1007/s00033-018-1058-y
[6] Day, MM, Normed linear spaces (1958), Berlin: Springer, Berlin · Zbl 0082.10603 · doi:10.1007/978-3-662-25249-9
[7] Kračmar, S.; Neustupa, J., Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities, ZAMM, 74, 637-639 (1994) · Zbl 0836.35121
[8] Kračmar, S.; Neustupa, J., A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions, Nonlinear Anal., 47, 4169-4180 (2001) · Zbl 1042.35605 · doi:10.1016/S0362-546X(01)00534-X
[9] Kučera, P., Basic properties of solution of the non-steady Navier-Stokes equations with mixed boundary conditions in a bounded domain, Ann. Univ. Ferrara, 55, 289-308 (2009) · Zbl 1205.35198 · doi:10.1007/s11565-009-0082-4
[10] Kučera, P.; Skalák, Z., Solutions to the Navier-Stokes equations with mixed boundary conditions, Acta Appl. Math., 54, 275-288 (1998) · Zbl 0924.35097 · doi:10.1023/A:1006185601807
[11] Kufner, A.; John, O.; Fučík, S., Function spaces (1977), San Francisco: Academia, San Francisco · Zbl 0364.46022
[12] Neustupa, T.: The analysis of stationary viscous incompressible flow through a rotating radial blade machine, existence of a weak solution. In: Conference: 5th Symposium on Numerical Analysis of Fluid Flow and Heat. Location: Rhodes, Greece, Date 2010. Appl. Math. Comput. 219(7), 3316-3322 (2011) · Zbl 1309.76054
[13] 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, part 3, 687-691 (2016) · Zbl 1410.35098
[14] Neustupa, T., The weak solvability of the steady problem modelling the flow of a viscous incompressible heat-conductive fluid through the profile cascade, Int. J. Numer. Methods Heat Fluid Flow, 27, 7, 1451-1466 (2017) · doi:10.1108/HFF-03-2016-0104
[15] Orlt, M.; Sändig, A-M, Regularity of viscous Navier-Stokes flows in Nonsmooth domains. Boundary value problems and integral equations in Nonsmooth domains, Lecture Notes Pure Appl. Math., 167, 185-201 (1993) · Zbl 0826.35095
[16] Sohr, H., The Navier-Stokes equations. An elementaryfunctional analytic approach (2001), Basel-Boston-Berlin: Birkhäuser advanced texts, Basel-Boston-Berlin · Zbl 0983.35004 · doi:10.1007/978-3-0348-8255-2
[17] Temam, R.: Navier-Stokes equations, theory and numerical analysis. North-Holland Publishing Company (1977) · 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.