| Title:
|
Steady-state buoyancy-driven viscous flow with measure data (English) |
| Author:
|
Roubíček, Tomáš |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
493-504 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Steady-state system of equations for incompressible, possibly non-Newtonean of the $p$-power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain $\Omega \subset \mathbb{R}^n$, $n=2$ or 3, with heat sources allowed to have a natural $L^1$-structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if $p>3/2$ (for $n=2$) or if $p>9/5$ (for $n=3$). (English) |
| Keyword:
|
non-Newtonean fluids |
| Keyword:
|
heat equation |
| Keyword:
|
dissipative heat |
| Keyword:
|
adiabatic heat |
| MSC:
|
35J60 |
| MSC:
|
35Q35 |
| MSC:
|
76A05 |
| MSC:
|
76D03 |
| MSC:
|
80A20 |
| idZBL:
|
Zbl 0981.35054 |
| idMR:
|
MR1844286 |
| DOI:
|
10.21136/MB.2001.134009 |
| . |
| Date available:
|
2009-09-24T21:53:13Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134009 |
| . |
| Reference:
|
[1] Alibert, J. J., Raymond, J. P.: Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls.Numer. Funct. Anal. Optim. 18 (1997), 235–250. MR 1448889, 10.1080/01630569708816758 |
| Reference:
|
[2] Baranger, J., Mikelič, A.: Stationary solutions to a quasi-Newtonean flow with viscous heating.Math. Models Methods Appl. Sci. 5 (1995), 725–738. MR 1348583, 10.1142/S0218202595000401 |
| Reference:
|
[3] Bayly, B. J., Levermore, C. D., Passot T.: Density variations in weakly compressible flows.Phys. Fluids A 4 (1992), 945–954. MR 1160287, 10.1063/1.858275 |
| Reference:
|
[4] Frehse, J., Málek, J., Steinhauer, M.: An existence result for fluids with shear dependent viscosity-steady flows.Nonlinear Anal., Theory Methods Appl. 30 (1997), 3041–3049. MR 1602949, 10.1016/S0362-546X(97)00392-1 |
| Reference:
|
[5] Gebhart, B., Jaluria, Y., Mahajan, R. L., Sammakia, B.: Buoyancy-Induced Flows and Transport.Hemisphere Publ., Washington, 1988. |
| Reference:
|
[6] Kagei, Y.: Attractors for two-dimensional equations of thermal convection in the presence of the dissipation function.Hiroshima Math. J. 25 (1995), 251–311. Zbl 0843.35074, MR 1336900, 10.32917/hmj/1206127712 |
| Reference:
|
[7] Kagei, Y., Růžička, M., Thäter, G.: Natural convection with dissipative heating.(to appear). MR 1796023 |
| Reference:
|
[8] Kaplický, P., Málek, J., Stará, J.: $C^{1,\alpha }$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem.Zapisky nauchnych seminarov POMI (Sankt Peterburg) 259 (1999), 89–121. |
| Reference:
|
[9] Landau, L. D., Lifshitz, E. M.: Fluid Mechanics.Pergamon Press, London, 1959. MR 0108121 |
| Reference:
|
[10] Lions, J. L.: Quelques méthodes de résolution des problémes aux limites non linéaires.Dunod, Paris, 1969. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[11] Lions, J. L., Magenes, E.: Problèmes aux limites non homogènes.Dunod, Paris, 1968. |
| Reference:
|
[12] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to the Evolutionary PDE’s.Chapman & Hall, London, 1996. MR 1409366 |
| Reference:
|
[13] Málek, J., Růžička, M., Thäter, G.: Fractal dimension, attractors and Boussinesq approximation in three dimensions.Act. Appl. Math. 37 (1994), 83–98. 10.1007/BF00995132 |
| Reference:
|
[14] Moseenkov, V. B.: Kachestvenyje metody issledovaniya zadach konvekciĭ slabo szhimaemoĭ zhidkosti.Inst. Mat. NAN Ukraïni, Kiïv, 1998. MR 1742952 |
| Reference:
|
[15] Nečas, J.: Les méthodes directes dans la théorie des équations elliptiques.Academia, Prague, 1967. |
| Reference:
|
[16] Nečas, J., Roubíček, T.: Buoyancy-driven viscous flow with $L^1$-data.(to appear). |
| Reference:
|
[17] Rabinowitz, P.: Existence and nonuniqueness of rectangular solutions of the Bénard problems.Arch. Rational Mech. Anal. 29 (1968), 32–57. MR 0233557, 10.1007/BF00256457 |
| Reference:
|
[18] Rajagopal, K. R., Růžička, M., Srinivasa, A. R.: On the Oberbeck-Boussinesq Approximation.Math. Models Methods Appl. Sci. 6 (1996), 1157–1167. MR 1428150, 10.1142/S0218202596000481 |
| Reference:
|
[19] Rodriguez, J. F.: A steady-state Boussinesq-Stefan problem with continuous extraction.Annali Mat. Pura Appl. IV 144 (1986), 203–218. MR 0870877 |
| Reference:
|
[20] Rodriguez, J. F., Urbano, J. M.: On a three-dimensional convective Stefan problem for a non-Newtonian fluid.Nonlinear Applied Analysis, A. Sequiera et al. (eds.), Plenum Press, 1999, pp. 457–468. MR 1727466 |
| Reference:
|
[21] Roubíček, T.: Nonlinear heat equation with $L^1$-data.Nonlinear Diff. Eq. Appl. 5 (1998), 517–527. |
| Reference:
|
[22] Růžička, M.: A note on steady flow fluids with shear dependent viscosity.Nonlinear Anal., Theory Methods Appl. 30 (1997), 3029–3039. MR 1602945, 10.1016/S0362-546X(97)00391-X |
| Reference:
|
[23] Turcotte, D. L., Hsui, A. T., Torrance, K. E., Schubert, G.: Influence of viscous dissipation on Bénard convection.J. Fluid Mech. 64 (1974), 369–374. 10.1017/S0022112074002448 |
| . |
