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Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation
1. | Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen, Avenue de l'université, BP12, 76801 Saint Étienne du Rouvray cedex, France, France |
2. | Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS - Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint-Étienne du Rouvray |
References:
[1] |
A. Attaoui, D. Blanchard and O. Guibé, Weak-renormalized solution for a nonlinear Boussinesq system, Differential Integral Equations, 22 (2009), 465-494. |
[2] |
C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: le modèle et son approximation par élément finis, RAIRO Modél. Math. Anal. Numér, 29 (1995), 871-921. |
[3] |
D. Blanchard, A few result on coupled systems of thermomechanics, In "On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments," Quaderni di Matematica, 23 (2009), 145-182. |
[4] |
D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with $L_1$ data: existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[5] |
D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.
doi: 10.1006/jdeq.2000.4013. |
[6] |
D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428.
doi: 10.1016/j.jde.2004.06.012. |
[7] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[8] |
L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.
doi: 10.1016/0022-1236(89)90005-0. |
[9] |
J. Boussinesq, "Thèorie analytique de la chaleur," volume 2. Gauthier-Villars, Paris, 1903. |
[10] |
N. Bruyère, "Comportement asymptotique de problèmes posés dans des cylindres. Problèmes d'unicité pour les systèmes de Boussinesq," PhD thesis, Université de Rouen, 2007. |
[11] |
N. Bruyère, Existence et unicité de la solutions faible-renormalisée pour un système non linéaire de Boussinesq, C. R. Math. Acad. Sci. Paris, 346 (2008), 521-526.
doi: 10.1016/j.crma.2008.03.005. |
[12] |
B. Climent and E. Fernández-Cara, Some existence and uniqueness results for a time-dependent coupled problem of the Navier-Stokes kind, Math. Models Methods Appl. Sci., 8 (1998), 603-622.
doi: 10.1142/S0218202598000275. |
[13] |
J. I. Díaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82. |
[14] |
C. Gerhardt, $L^p$-estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pacific J. Math., 79 (1978), 375-398. |
[15] |
P-L Lions, "Mathematical Topics in Fluid Mechanics," Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications. |
[16] |
L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[18] |
R. Temam, "Navier-Stokes Equations," AMS Chelsea Publishing, Providence, RI, 2001. Theory and Numerical Analysis. |
[19] |
R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, New York, second edition, 2005.
doi: 10.1017/CBO9780511755422. |
show all references
References:
[1] |
A. Attaoui, D. Blanchard and O. Guibé, Weak-renormalized solution for a nonlinear Boussinesq system, Differential Integral Equations, 22 (2009), 465-494. |
[2] |
C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: le modèle et son approximation par élément finis, RAIRO Modél. Math. Anal. Numér, 29 (1995), 871-921. |
[3] |
D. Blanchard, A few result on coupled systems of thermomechanics, In "On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments," Quaderni di Matematica, 23 (2009), 145-182. |
[4] |
D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with $L_1$ data: existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[5] |
D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.
doi: 10.1006/jdeq.2000.4013. |
[6] |
D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383-428.
doi: 10.1016/j.jde.2004.06.012. |
[7] |
L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237-258.
doi: 10.1006/jfan.1996.3040. |
[8] |
L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169.
doi: 10.1016/0022-1236(89)90005-0. |
[9] |
J. Boussinesq, "Thèorie analytique de la chaleur," volume 2. Gauthier-Villars, Paris, 1903. |
[10] |
N. Bruyère, "Comportement asymptotique de problèmes posés dans des cylindres. Problèmes d'unicité pour les systèmes de Boussinesq," PhD thesis, Université de Rouen, 2007. |
[11] |
N. Bruyère, Existence et unicité de la solutions faible-renormalisée pour un système non linéaire de Boussinesq, C. R. Math. Acad. Sci. Paris, 346 (2008), 521-526.
doi: 10.1016/j.crma.2008.03.005. |
[12] |
B. Climent and E. Fernández-Cara, Some existence and uniqueness results for a time-dependent coupled problem of the Navier-Stokes kind, Math. Models Methods Appl. Sci., 8 (1998), 603-622.
doi: 10.1142/S0218202598000275. |
[13] |
J. I. Díaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82. |
[14] |
C. Gerhardt, $L^p$-estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pacific J. Math., 79 (1978), 375-398. |
[15] |
P-L Lions, "Mathematical Topics in Fluid Mechanics," Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications. |
[16] |
L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[18] |
R. Temam, "Navier-Stokes Equations," AMS Chelsea Publishing, Providence, RI, 2001. Theory and Numerical Analysis. |
[19] |
R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics," Cambridge University Press, New York, second edition, 2005.
doi: 10.1017/CBO9780511755422. |
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