On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions

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December 2011, 15(3): 825-847. doi: 10.3934/dcdsb.2011.15.825

On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions

Elder J. Villamizar-Roa ^1, and Elva E. Ortega-Torres ^2,

1.	Universidad Nacional de Colombia-Medellín, Escuela de Matemáticas, Medellìn, A.A. 3840, Colombia
2.	Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta, Chile

Received June 2009 Revised March 2010 Published February 2011

Abstract
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The aim of this work is to prove the existence of strong solutions for a generalized Boussinesq model, with nonlinear diffusion for the equations of velocity and temperature, occupying a domain $\Omega,$ exterior to a rigid body that rotates with constant angular velocity $\omega.$

Keywords: Rotating obstacle, Strong solutions., Boussinesq equations.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03, 76U0.

Citation: Elder J. Villamizar-Roa, Elva E. Ortega-Torres. On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 825-847. doi: 10.3934/dcdsb.2011.15.825

References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
[2]	J. L. Boldrini and S. Lorca, The initial value problem for a generalized Boussinesq model, Nonlinear Analysis, 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.
[3]	P. Braz e Silva, M. A. Rojas-Medar and E. J. Villamizar-Roa, Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains, Math. Methods Appl. Sci., 33 (2010), 358-372.
[4]	L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem in $\mathbb{R}^{N}$, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.
[5]	L. C. F. Ferreira and E. J. Villamizar-Roa, On the stability problem for the Boussinesq equations in weak-$L^p$ spaces, Communication on Pure and Applied Analysis, 9 (2010), 667-684. doi: 10.3934/cpaa.2010.9.667.
[6]	G. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations," Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Nature Philosophy, 39, Springer-Verlag, New York, 1994.
[7]	G. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical fluid dynamics, Vol. I 653-791, North-Holland, Amsterdam (2002), 653-791.
[8]	G. Galdi and A. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rational Mech. Anal., 176 (2005), 331-350. doi: 10.1007/s00205-004-0348-z.
[9]	G. Galdi and A. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific Journal of Mathematics, 223 (2006), 251-267. doi: 10.2140/pjm.2006.223.251.
[10]	J. Heywood, The Navier-Stokes Equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. Journal, 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.
[11]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rat. Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[12]	J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," vol. 1, Travaux et Recherches Mathéatiques, No. 17 Dunod, Paris 1968.
[13]	L. D. Landau and E. M. Lifchitz, "Theorical Physics: Fluid Mechanics," 2nd edition, Pergamon Press, 1987.
[14]	M. A. Rojas-Medar and S. A. Lorca, The equations of a viscous incompressible chemical active fluid II. Regularity of solutions, Rev. Mat. Apl., 16 (1995), 81-95.

show all references

References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
[2]	J. L. Boldrini and S. Lorca, The initial value problem for a generalized Boussinesq model, Nonlinear Analysis, 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4.
[3]	P. Braz e Silva, M. A. Rojas-Medar and E. J. Villamizar-Roa, Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains, Math. Methods Appl. Sci., 33 (2010), 358-372.
[4]	L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem in $\mathbb{R}^{N}$, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.
[5]	L. C. F. Ferreira and E. J. Villamizar-Roa, On the stability problem for the Boussinesq equations in weak-$L^p$ spaces, Communication on Pure and Applied Analysis, 9 (2010), 667-684. doi: 10.3934/cpaa.2010.9.667.
[6]	G. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations," Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Nature Philosophy, 39, Springer-Verlag, New York, 1994.
[7]	G. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical fluid dynamics, Vol. I 653-791, North-Holland, Amsterdam (2002), 653-791.
[8]	G. Galdi and A. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rational Mech. Anal., 176 (2005), 331-350. doi: 10.1007/s00205-004-0348-z.
[9]	G. Galdi and A. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pacific Journal of Mathematics, 223 (2006), 251-267. doi: 10.2140/pjm.2006.223.251.
[10]	J. Heywood, The Navier-Stokes Equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. Journal, 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.
[11]	T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rat. Mech. Anal., 150 (1999), 307-348. doi: 10.1007/s002050050190.
[12]	J. L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," vol. 1, Travaux et Recherches Mathéatiques, No. 17 Dunod, Paris 1968.
[13]	L. D. Landau and E. M. Lifchitz, "Theorical Physics: Fluid Mechanics," 2nd edition, Pergamon Press, 1987.
[14]	M. A. Rojas-Medar and S. A. Lorca, The equations of a viscous incompressible chemical active fluid II. Regularity of solutions, Rev. Mat. Apl., 16 (1995), 81-95.

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