# American Institute of Mathematical Sciences

September  2013, 12(5): 2213-2227. doi: 10.3934/cpaa.2013.12.2213

## 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

Received  August 2011 Revised  December 2012 Published  January 2013

We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq's systems. The main tools rely on regularity results for the Navier-Stokes equations with precise estimates on the solution with respect to the data in dimension $2$ and on the techniques involved for renormalized solutions of parabolic problems.
Citation: Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
##### References:
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##### References:
 [1] A. Attaoui, D. Blanchard and O. Guibé, Weak-renormalized solution for a nonlinear Boussinesq system,, Differential Integral Equations, 22 (2009), 465. Google Scholar [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 Mod\'el. Math. Anal. Num\'er, 29 (1995), 871. Google Scholar [3] D. Blanchard, A few result on coupled systems of thermomechanics,, In, 23 (2009), 145. Google Scholar [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. doi: 10.1017/S0308210500026986. Google Scholar [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. doi: 10.1006/jdeq.2000.4013. Google Scholar [6] D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection,, J. Differential Equations, 210 (2005), 383. doi: 10.1016/j.jde.2004.06.012. Google Scholar [7] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, J. Funct. Anal., 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar [8] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149. doi: 10.1016/0022-1236(89)90005-0. Google Scholar [9] J. Boussinesq, "Thèorie analytique de la chaleur," volume 2., Gauthier-Villars, (1903). Google Scholar [10] N. Bruyère, "Comportement asymptotique de problèmes posés dans des cylindres. Problèmes d'unicité pour les systèmes de Boussinesq,", PhD thesis, (2007). Google Scholar [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. doi: 10.1016/j.crma.2008.03.005. Google Scholar [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. doi: 10.1142/S0218202598000275. Google Scholar [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. Google Scholar [14] C. Gerhardt, $L^p$-estimates for solutions to the instationary Navier-Stokes equations in dimension two,, Pacific J. Math., 79 (1978), 375. Google Scholar [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, (1996). Google Scholar [16] L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733. Google Scholar [17] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [18] R. Temam, "Navier-Stokes Equations,", AMS Chelsea Publishing, (2001). Google Scholar [19] R. Temam and A. Miranville, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005). doi: 10.1017/CBO9780511755422. Google Scholar
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