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A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale
The nonstationary flows of micropolar fluids with thermal convection: An iterative approach
| 1. | Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, Brazil |
| 2. | Departamento de Matemática, Universidad de Tarapacá, Arica, Chile |
We consider a problem that describes the motion of a viscous incompressible and heat-conducting micropolar fluids in a bounded domain $ \Omega \subset \mathbb{R}^3 $. We use an iterative method to analyze the existence, uniqueness, and regularity of the solutions. We also determine the convergence rates in several norms.
References:
| [1] |
C. Amrouche and V. Girault,
On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.
doi: 10.3792/pjaa.67.171. |
| [2] |
J. Boussinesq, Théorie Analytique de la Chaleur II, Gauthier-Villars, 1903. |
| [3] |
A. C. Eringen,
Simple microfluids, Internat. J. Engrg. Sci., 2 (1964), 205-217.
doi: 10.1016/0020-7225(64)90005-9. |
| [4] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
| [5] |
D. D. Joseph, Stability of Fluid Motions. I, Springer Tracts in Natural Philosophy, 27, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-80991-0. |
| [6] |
Y. Kagei and M. Skowron,
Nonstationary flows of nonsymmetric fluids with thermal convection, Hiroshima Math. J., 23 (1993), 343-363.
doi: 10.32917/hmj/1206128257. |
| [7] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. |
| [8] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968. |
| [9] |
J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéares, Dunod; Gauthier-Villars, Paris, 1969. |
| [10] |
G. Łukaszewicz,
On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 13 (1989), 105-120.
|
| [11] |
G. Łukaszewicz and W. Waluś,
On stationary flows of asymmetric fluids with heat convection, Math. Methods Appl. Sci., 11 (1989), 343-351.
doi: 10.1002/mma.1670110304. |
| [12] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
| [13] |
L. G. Petrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Izd. Erevan Univ., (1984). |
| [14] |
M. A. Rojas-Medar and E. E. Ortega-Torres,
The equations of a viscous asymmetric fluid: An interactive approach, ZAMM Z. Angew. Math. Mech., 85 (2005), 471-489.
doi: 10.1002/zamm.199910189. |
| [15] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
doi: 10.1002/mana.19971880116. |
| [16] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
| [17] |
A. G. Zarubin,
An iterative method for the approximate solution of an initial-boundary value problem for heat convection equations, Comput. Math. Math. Phys., 33 (1993), 1077-1085.
|
show all references
References:
| [1] |
C. Amrouche and V. Girault,
On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.
doi: 10.3792/pjaa.67.171. |
| [2] |
J. Boussinesq, Théorie Analytique de la Chaleur II, Gauthier-Villars, 1903. |
| [3] |
A. C. Eringen,
Simple microfluids, Internat. J. Engrg. Sci., 2 (1964), 205-217.
doi: 10.1016/0020-7225(64)90005-9. |
| [4] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
| [5] |
D. D. Joseph, Stability of Fluid Motions. I, Springer Tracts in Natural Philosophy, 27, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-80991-0. |
| [6] |
Y. Kagei and M. Skowron,
Nonstationary flows of nonsymmetric fluids with thermal convection, Hiroshima Math. J., 23 (1993), 343-363.
doi: 10.32917/hmj/1206128257. |
| [7] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and its Applications, 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. |
| [8] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968. |
| [9] |
J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéares, Dunod; Gauthier-Villars, Paris, 1969. |
| [10] |
G. Łukaszewicz,
On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 13 (1989), 105-120.
|
| [11] |
G. Łukaszewicz and W. Waluś,
On stationary flows of asymmetric fluids with heat convection, Math. Methods Appl. Sci., 11 (1989), 343-351.
doi: 10.1002/mma.1670110304. |
| [12] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
| [13] |
L. G. Petrosyan, Some problems of mechanics of fluids with antisymmetric stress tensor, Izd. Erevan Univ., (1984). |
| [14] |
M. A. Rojas-Medar and E. E. Ortega-Torres,
The equations of a viscous asymmetric fluid: An interactive approach, ZAMM Z. Angew. Math. Mech., 85 (2005), 471-489.
doi: 10.1002/zamm.199910189. |
| [15] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
doi: 10.1002/mana.19971880116. |
| [16] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
| [17] |
A. G. Zarubin,
An iterative method for the approximate solution of an initial-boundary value problem for heat convection equations, Comput. Math. Math. Phys., 33 (1993), 1077-1085.
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