Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation

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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

Dominique Blanchard ^1, , Nicolas Bruyère ^1, and Olivier Guibé ^2,

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

Abstract
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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.

Keywords: Navier-Stokes equations, parabolic equations., Nonlinear systems of PDE.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35D05, 35D30, 76D0.

Citation: Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

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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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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