Global regularity for a coupled Cahn-Hilliard-Boussinesq system on bounded domains
Author:
Kun Zhao
Journal:
Quart. Appl. Math. 69 (2011), 331-356
MSC (2010):
Primary 35Q35, 35B65, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-2011-01241-5
Published electronically:
March 9, 2011
MathSciNet review:
2729892
Full-text PDF Free Access
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Abstract: We study an initial-boundary value problem (IBVP) for a coupling of the Cahn-Hilliard equation with the 2D inviscid heat-conductive Boussinesq equations. For large initial data with finite energy, we prove global existence and uniqueness of classical solutions to the IBVP, together with some uniform-in-time and decay estimates of the solution.
References
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References
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- Agmon, S., Douglis, A., Nirenberg, L. (1959; 1964). Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I $\&$ II. Comm. Pure. Appl. Math. 12: 623–727; 17: 35–92. MR 0125307 (23:A2610); MR 0162050 (28:5252)
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- Bates, P., Fife, P. (1993). The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53 no. 4: 990–1008. MR 1232163 (94g:82034)
- Bourguignon, J. P., Brezis, H. (1974). Remarks on the Euler equation. J. Funct. Anal. 15: 341–363. MR 0344713 (49:9452)
- Boyer, F. (1999). Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptotic Analysis 20: 175–212. MR 1700669 (2000g:35166)
- Boyer, F. (2001). Nonhomogeneous Cahn-Hilliard fluid. Ann. Inst. Henri Poincaré, Anal. non Linéaire 18 no. 2: 225–259. MR 1808030 (2002d:76032)
- Boyer, F. (2002). A theoretical and numerical model for the study of incompressible mixture flows. Computers and Fluids 31 no. 1: 41–68.
- Cannon, J., DiBenedetto, E. (1980). The initial value problem for the Boussinesq equations with data in $L^ {p}$. Lecture Notes in Math. 771: 129–144. Berlin: Springer. MR 565993 (81f:35101)
- Carr, J., Gurtin, M., Slemrod, M. (1984). Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86: 317–351. MR 759767 (86i:80001)
- Chae, D. H. (2006). Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203 no. 2: 497–513. MR 2227730 (2007e:35223)
- Chae, D. H., Imanuvilov, O. (1999). Generic solvability of the axisymmetric $3$-D Euler equations and the $2$-D Boussinesq equations. J. Differential Equations 156 no. 1: 1–17. MR 1700862 (2000d:76009)
- Chella, R., Vinals, J. (1996). Mixing of a two-phase fluid by a cavity flow. Phys. Rev. E 53: 3832–3840.
- Debussche, A., Dettori, L. (1995). On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Analysis 24 no. 10: 1491–1514. MR 1327930 (96c:35080)
- Elliott, C., Garcke, H. (1996). On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 no. 2: 404–423. MR 1377481 (97c:35081)
- Gunton, J., San Miguel, M., Sahni, P. (1983). in: Domb, Lebowitz (Eds.), Phase transitions and critical phenomena 8. London: Academic Press. MR 794319
- Gurtin, M., Polignone, D., Vinals, J. (1996). Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models and Methods in Appl. Sciences 6: 815–831. MR 1404829 (99e:76123)
- Hmidi, T., Keraani, S. (2007). On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. Adv. Differential Equations 12 no. 4: 461–480. MR 2305876 (2009c:35404)
- Hou, T., Li, C. (2005). Global well-posedness of the viscous Boussinesq equations. Discr. Cont. Dynam. Sys. 12: 1–12. MR 2121245 (2005j:76026)
- Kato, T. (1967). On classical solutions of the two-dimensional non-stationary Euler equations. Arch. Ration. Mech. Anal. 25: 188–200. MR 0211057 (35:1939)
- Ladyzhenskaya, O. A., Solonnikov, V. A., Uraltseva, N. N. (1968). Linear and Quasi-linear Equations of Parabolic Type. AMS.
- Lai, M. J., Pan, R. H., Zhao, K. (2008). Initial boundary value problem for 2D viscous Boussinesq equations. Arch. Ration. Mech. Analysis. To appear.
- Lions, P. L. (1996). Mathematical Topics in Fluid Mechanics, Vol. I. New York: Oxford University Press. MR 1422251 (98b:76001)
- Lions, J. L., Magenes, E. (1961). Problemi ai limiti non omogenei III. Ann. Scuo. Norm. Sup. Pisa XV: 41–101. MR 0146526 (26:4048)
- Lions, J. L., Magenes, E. (1961). Problèmes aux limites non homogènes IV. Ann. Scuo. Norm. Sup. Pisa XV: 311–326. MR 0140938 (25:4351)
- Lorca, S., Boldrini, J. (1999). The initial value problem for a generalized Boussinesq model. Nonlinear Analysis 36: 457–480. MR 1675260 (99m:35210)
- Majda, A., Bertozzi, A. (2002). Vorticity and Incompressible Flow. Cambridge: Cambridge University Press. MR 1867882 (2003a:76002)
- Miranville, A. (1999). A model of Cahn-Hilliard equation based on a microforce balance. C. R. Acad. Sci. Paris Sér. I Math. 328 no. 12: 1247–1252. MR 1701394 (2000e:35091)
- Onuki, A. (1997). Phase transitions of fluids in shear flow. J. Phys.: Condens. Matter 9: 6119–6157.
- Pedlosky, J. (1987). Geophysical Fluid Dynamics. New York: Springer-Verlag.
- Wei, J., Winter, M. (1998). Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. Henri Poincaré, Anal. non Linéaire 15 no. 4: 459–492. MR 1632937 (2000b:35093)
- Zhao, K. (2010). 2D inviscid heat conductive Boussinesq equations on a bounded domain. Michigan Math. J. 59: 329–352. MR 2677625
- Zhao, K. (2009). Initial-boundary value problem for a coupling of Cahn-Hilliard equation with 2D Boussinesq equations. Preprint.
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Additional Information
Kun Zhao
Affiliation:
Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210
Email:
kzhao@mbi.ohio-state.edu.
Keywords:
Cahn-Hilliard equation,
2D Boussinesq equations,
global existence and uniqueness
Received by editor(s):
October 23, 2009
Published electronically:
March 9, 2011
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.