Acta mathematica scientia, Series B >
EXISTENCE UNIQUENESS AND DECAY OF SOLUTION FOR FRACTIONAL BOUSSINESQ APPROXIMATION
Received date: 2012-04-09
Revised date: 2012-10-18
Online published: 2013-07-20
Supported by
Sponsored by the Fundamental Research Funds for the Central Universities (2010QS04); the National Science Foundation of China (11201475, 11126160, 11201185); Zhejiang Provincial Natural Science Foundation of China under Grant (LQ12A01013).
The Boussinesq approximation finds more and more frequent use in geologi-cal practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of commu-tator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results
obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively.
GUO Chun-Xiao , ZHANG Jing-Jun , GUO Bo-Ling . EXISTENCE UNIQUENESS AND DECAY OF SOLUTION FOR FRACTIONAL BOUSSINESQ APPROXIMATION[J]. Acta mathematica scientia, Series B, 2013 , 33(4) : 883 -900 . DOI: 10.1016/S0252-9602(13)60048-X
[1] Bellout H, Bloom F, Neˇcas J. Weak and measure-valued solutions for non-Newtonian fluids. C R Acad Sci Pairs, 1993, 317: 795–800
[2] Bellout H, Bloom F, Neˇcas J. Young measure-valued solutions for Non-Newtonian incompressible viscous fluids. Commun Part Differ Equ, 1994, 19: 1763–1803
[3] Bleustein J L, Green A E. Dipolar fluids. Int J Engng Sci, 1967, 5: 323–340
[4] Bloom F, Hao W. Regularization of a non-Newtonian system in an unbounded channel: Existence and uniqueness of solutions. Nonlinear Anal, 2001, 44: 281–309
[5] Coifman R, Meyer Y. Nonlinear harmonic analysis, operator theory and P.D.E.//Beijing Lectures in Harmonic Analysis. Princeton University Press, 1986: 3–45
[6] Constantin P, Majda A, Tabak E. Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity, 1994, 7: 1495–1533
[7] Constantin P, Wu J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J Math Anal, 1999, 30: 937–948
[8] Constantin P, Cordoba D, Wu J. On the critical dissipative quasi-geostrophic equations. Indiana Univ Math J, 2001, 50: 97–107
[9] Dong B Q, Li Y S. Large time behavior to the system of incompressible non-Newtonian fluids in R2. J Math Anal Appl, 2004, 298: 667–676
[10] Guo B L, Guo C X. The convergence of non-Newtonian fluids to Navier-Stokes equations. J Math Anal Appl, 2009, 357: 468–478
[11] Guo B L, Shang Y D. The periodic initial value problem and initial value problem for modified Boussinesq approximation. J Part Differ Equ, 2002, 15: 57–71
[12] Guo B L, Shang Y D. The global attractors for the modified Boussinesq approximation. Preprint.
[13] Guo B L, Shang Y D, Lin G G. Non-Newtonian Fluids Dynamical Systems. Changsha: National Defense Industry Press, 2006 (In Chinese)
[14] Guo B L, Zhu P C. Algebraic L2 decay for the solution to a class system of non-Newtonian fluid in Rn. J Math Phys, 2000, 41: 349–356
[15] Hills R , Roberts H. On the motion of a fluid that is incompressible in a generalized sense and its relationship to the Boussinesq approximation. Stab Anal Contin Media, 1991, 1: 205–212
[16] Kening C, Ponce G, Vega L. Well-posedness of the initial value problem for the Korteweg-deVries equation. J Am Math Soc, 1991, 4: 323–347
[17] Ladyzhenskaya O. The mathematical theory of viscous incompressible flow. New York: Gordon and Breach, 1969
[18] Ladyzhenskaya O. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them//Boundary Value Problem of Mathematical Physics, Vol V. Providence, RI: American Mathematical Society, 1970
[19] Lions J. Quelques M´ethodes de R´esolution des Probl´emes aux Limites Nonlin´eaires. Paris: Dunod, 1969
[20] M´alek J, Ruˇziˇcka M, Th¨ater G. Fractal dimension, attractors, and the Boussinesq approximation in three dimensions. Acta Appl Math, 1993, 37: 83–97
[21] Padula M. Mathematical properties of motion of viscous compressible fluids//Galidi G P, Malek J, Necas J, eds. Progress in Theoretical Computational Fluid Mechanics. Pitman Research Notes in Mathematics Series 308. Essex: Longman Scientific Technical, 1994: 128–173
[22] Pokorny M. Cauchy problem for the non-Newtonian incompressible fluids. Appl Math, 1996, 41: 169–201
[23] Robinson J. Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Univ Press, 2001
[24] Pu X K, Guo B L. Global well posedness of the stochastic 2D Boussinesq equations with partial viscosity. Acta Math Sci, 2011, 31B(5): 1968–1984
[25] Pu X K, Guo B L. Existence and decay of solutions to the two-dimensional fractional quasigeostrophic equation. J Math Phys, 2010, 51: 083101-1-083101-15
[26] Schonbek M, Vallis G. Energy decay of solution to the boussinesq, primitive, and planetary geostrophic equationsm. J Math Anal Appl, 1999, 234: 457–481
[27] Stein E. Singular Intetgrals and Differentiability Properties of Functions. Princeton, NJ: Princeton Univ Press, 1970
[28] Temam R. Navier-Stokes Equations: Theory and Numerical Analysis. 3rd ed. North Holland, 1984
[29] Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed. Berlin: Springer, 1997
[30] Zhang L H. Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equation. Comm Part Differ Equ, 1995, 20: 119–127
[31] Zhao C D. Li Y S. A note on the asymptotic smoothing effect of solutions to a non-Newtonian system in 2-D unbounded domains. Nonlinear Analysis, 2005, 60: 475–483
[32] Zhao C D, Zhou S F. Pullback attractors for a non-autonomous incompressible non-Newtonian fluid. J Differ Equ, 2007, 238: 394–425
[33] Zhao C D, Li Y S, Zhou S F. Random attractors for a two-dimensional incompressible non-Newtonian fluid with mulitiplicative noise. Acta Math Sci, 2011, 31B(2): 567–575
/
| 〈 |
|
〉 |