| Title:
|
An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions (English) |
| Author:
|
Skalák, Zdeněk |
| Author:
|
Kučera, Petr |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
45 |
| Issue:
|
2 |
| Year:
|
2000 |
| Pages:
|
81-98 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends only on certain norms of the given data. In the proof we use a fixed point theorem method in Sobolev spaces with non-integer order derivatives. The proof is performed for Lipschitz domains and a wide class of data. (English) |
| Keyword:
|
Boussinesq equations |
| Keyword:
|
non-Dirichlet boundary conditions |
| Keyword:
|
Sobolev space with non-integer order derivatives |
| Keyword:
|
Schauder principle |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| idZBL:
|
Zbl 1067.35080 |
| idMR:
|
MR1745614 |
| DOI:
|
10.1023/A:1022224328555 |
| . |
| Date available:
|
2009-09-22T18:02:49Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134430 |
| . |
| Reference:
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[1] S. Acharya, R. Jetli: Heat transfer due to buoyancy in a partially divided square box.Int. J. Heat Mass Transfer 33 (1990), 931–942. 10.1016/0017-9310(90)90075-6 |
| Reference:
|
[2] C. R. Doering, J. D. Gibbon: Applied Analysis of the Navier-Stokes Equations.Cambridge University Press, Cambridge, 1995. MR 1325465 |
| Reference:
|
[3] J. Cronin: Fixed Points and Topological Degree in Nonlinear Analysis.Amer. Math. Soc., Providence, Rhode Island, 1964. Zbl 0117.34803, MR 0164101 |
| Reference:
|
[4] K. H. Fuh, W. C. Chen and P. W. Liang: Temperature rise in twist drills with a finite element approach.Int. Com. Heat Mass Transfer 21 (1994), 345–358. 10.1016/0735-1933(94)90003-5 |
| Reference:
|
[5] H. Gajewski, K. Gröger and K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Akademie-Verlag, Berlin, 1974. MR 0636412 |
| Reference:
|
[6] P. Grisvard: Elliptic Problems in Nonsmooth Domains.Pitman Publishing Inc., Boston-London-Melbourne, 1985. Zbl 0695.35060, MR 0775683 |
| Reference:
|
[7] P. M. Gresho, R. L. Sani: Résumé and remarks on the open boundary condition minisymposium.Int. J. Numer. Methods Fluids 18 (1994), 983–1008. MR 1277046, 10.1002/fld.1650181006 |
| Reference:
|
[8] J. G. Heywood, R. Rannacher and S. Turek: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations.Int. J. Numer. Methods Fluids 22 (1996), 325–352. MR 1380844, 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y |
| Reference:
|
[9] S. Kračmar, J. Neustupa: Global existence of weak solutions of a nonsteady variational inequality of the Navier-Stokes type with mixed boundary conditions.In: Proceedings of Int. Symp. on Num. Anal. ISNA’92, Part III, 1992, pp. 156–177. |
| Reference:
|
[10] S. Kračmar, J. Neustupa: Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities.Z. Angew. Math. Mech. 74 (1994), T637–T639. |
| Reference:
|
[11] P. Kučera: Some properties of solutions to the nonsteady Navier-Stokes equations with mixed boundary conditions on the infinite time interval.In: Proceedings of the 3rd Summer Conference “Numerical Modelling in Continuum Mechanics”, Part 2, M. Feistauer, R. Rannacher and K. Kozel (eds.), Matfyzpress, Publishing House of the Faculty of Mathematics and Physics, Charles University, Prague 1997, pp. 375–383. |
| Reference:
|
[12] A. Kufner, O. John and S. Fučík: Function Spaces.Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977. MR 0482102 |
| Reference:
|
[13] J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications, Vol. I and II.Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR 0350177 |
| Reference:
|
[14] T. H. Mai, N. El Wakil and J. Padet: Transient mixed convection in an upward vertical pipe flow: temporal evolution following an inlet temperature step.Int. Com. Heat Mass Transfer 21 (1994), 755–764. 10.1016/0735-1933(94)90076-0 |
| Reference:
|
[15] J. Málek, M. Růžička and G. Thäter: Fractal dimension, attractors, and the Boussinesq approximation in three dimensions.Acta Applicandae Mathematicae 37 (1994), 83–97. MR 1308748, 10.1007/BF00995132 |
| Reference:
|
[16] M. P. Reddy, L. G. Reifschneider, J. N. Reddy and H. U. Akay: Accuracy and convergence of element-by-element iterative solvers for incompressible fluid flows using penalty finite element model.Int. J. Numer. Methods Fluids 17 (1993), 1019–1033. 10.1002/fld.1650171202 |
| Reference:
|
[17] G. Ren, T. Utnes: A finite element solution of the time-dependent incompressible Navier-Stokes equations using a modified velocity correction method.Int. J. Numer. Methods Fluids 17 (1993), 349–364. 10.1002/fld.1650170502 |
| Reference:
|
[18] S. A. M. Said and R. J. Krane: An analytical and experimental investigation of natural convection heat transfer in vertical channels with a single obstruction.Int. J. Heat Mass Transfer 33 (1990), 1121–1134. 10.1016/0017-9310(90)90245-P |
| Reference:
|
[19] Li Q. Tang, Tate T. H. Tsang: A least-squares finite element method for time-dependent incompressible flows with thermal convection.Int. J. Numer. Methods Fluids 17 (1993), 271–289. 10.1002/fld.1650170402 |
| Reference:
|
[20] R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis.North-Holland Publishing Company, Amsterdam-New York-Oxford, 1979. Zbl 0426.35003, MR 0603444 |
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