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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem in $\mathbbR^n$, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. D. Landau and E. M. Lifchitz, "Theorical Physics: Fluid Mechanics," 2nd edition, Pergamon Press, 1987. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem in $\mathbbR^n$, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. D. Landau and E. M. Lifchitz, "Theorical Physics: Fluid Mechanics," 2nd edition, Pergamon Press, 1987. Google Scholar

[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.  Google Scholar

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