December  2017, 37(12): 5979-6034. doi: 10.3934/dcds.2017259

Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system

1. 

BCAM -Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain

2. 

Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33400 Talence, France

Received  February 2016 Revised  July 2017 Published  August 2017

Fund Project: This research was partially supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.

We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under the effects of gravitational forces. We prove that if the Froude number is sufficiently small such system is globally well posed with respect to a $ H^s, \ s>1/2 $ Sobolev regularity. Moreover if the Froude number converges to zero we prove that the solutions of the aforementioned system converge (strongly) to a stratified three-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data.

Citation: Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259
References:
[1]

J.-P. Aubin, Un théoréme de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. 

[2]

A. BabinA. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mech. B Fluids, 15 (1996), 291-300. 

[3]

——, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Appl. Math. Lett., 13 (2000), 51–57. doi: 10.1016/S0893-9659(99)00208-6.

[4]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales scientifiques de l'École Normale Supérieure, 14 (1981), 209-246 (fre).  doi: 10.24033/asens.1404.

[6]

D. BreschD. Gérard-Varet and E. Grenier, Derivation of the planetary geostrophic equations, Arch. Ration. Mech. Anal., 182 (2006), 387-413.  doi: 10.1007/s00205-006-0008-6.

[7]

F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Comm. Partial Differential Equations, 29 (2004), 1919-1940.  doi: 10.1081/PDE-200043510.

[8]

——, Convergence of weak solutions for the primitive system of the quasigeostrophic equations, Asymptot. Anal., 42 (2005), 173–209.

[9]

F. Charve and V.-S. Ngo, Global existence for the primitive equations with small anisotropic viscosity, Rev. Mat. Iberoam., 27 (2011), 1-38.  doi: 10.4171/RMI/629.

[10]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[11]

J.-Y. Chemin, À propos d'un probléme de pénalisation de type antisymétrique, J. Math. Pures Appl.(9), 76 (1997), 739-755.  doi: 10.1016/S0021-7824(97)89967-9.

[12]

——, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[13]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335, Special issue for R.  doi: 10.1051/m2an:2000143.

[14]

——, Anisotropy and dispersion in rotating fluids, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., vol. 31, North-Holland, Amsterdam, 2002,171–192. doi: 10.1016/S0168-2024(02)80010-8.

[15]

——, Mathematical Geophysics, Oxford Lecture Series in Mathematics and its Applications, vol. 32, The Clarendon Press, Oxford University Press, Oxford, 2006, An introduction to rotating fluids and the Navier-Stokes equations.

[16]

B. Cushman-Roisin and J. -M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011.

[17]

H. Fujita and T. Kato, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, 32 (1962), 243-260. 

[18]

——, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269–315. doi: 10.1007/BF00276188.

[19]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl.(9), 77 (1998), 989-1054.  doi: 10.1016/S0021-7824(99)80002-6.

[20]

E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl.(9), 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X.

[21]

D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France, 127 (1999), 473-517.  doi: 10.24033/bsmf.2358.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[23]

M. Paicu, Étude asymptotique pour les fluides anisotropes en rotation rapide dans le cas périodique, J. Math. Pures Appl.(9), 83 (2004), 163-242.  doi: 10.1016/j.matpur.2003.10.001.

[24]

——, Équation periodique de Navier-Stokes sans viscosité dans une direction, Comm. Partial Differential Equations, 30 (2005), 1107–1140. doi: 10.1080/036053005002575529.

[25]

H. Poincaré, Sur la Précession Des Corps Déformables, Bulletin Astronomique, Serie I, 1910.

[26]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.

[27]

S. Scrobogna, Highly Rotating Fluids with Vertical Stratification for Periodic Data and Anisotropic Diffusion, to appear in Revista Matemática Iberoamericana.

[28]

K. Widmayer, Convergence to Stratified Flow for an Inviscid 3d Boussinesq System, http://arxiv.org/abs/1509.09216.

show all references

References:
[1]

J.-P. Aubin, Un théoréme de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. 

[2]

A. BabinA. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, European J. Mech. B Fluids, 15 (1996), 291-300. 

[3]

——, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Appl. Math. Lett., 13 (2000), 51–57. doi: 10.1016/S0893-9659(99)00208-6.

[4]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales scientifiques de l'École Normale Supérieure, 14 (1981), 209-246 (fre).  doi: 10.24033/asens.1404.

[6]

D. BreschD. Gérard-Varet and E. Grenier, Derivation of the planetary geostrophic equations, Arch. Ration. Mech. Anal., 182 (2006), 387-413.  doi: 10.1007/s00205-006-0008-6.

[7]

F. Charve, Global well-posedness and asymptotics for a geophysical fluid system, Comm. Partial Differential Equations, 29 (2004), 1919-1940.  doi: 10.1081/PDE-200043510.

[8]

——, Convergence of weak solutions for the primitive system of the quasigeostrophic equations, Asymptot. Anal., 42 (2005), 173–209.

[9]

F. Charve and V.-S. Ngo, Global existence for the primitive equations with small anisotropic viscosity, Rev. Mat. Iberoam., 27 (2011), 1-38.  doi: 10.4171/RMI/629.

[10]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[11]

J.-Y. Chemin, À propos d'un probléme de pénalisation de type antisymétrique, J. Math. Pures Appl.(9), 76 (1997), 739-755.  doi: 10.1016/S0021-7824(97)89967-9.

[12]

——, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998, Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie.

[13]

J.-Y. CheminB. DesjardinsI. Gallagher and E. Grenier, Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., 34 (2000), 315-335, Special issue for R.  doi: 10.1051/m2an:2000143.

[14]

——, Anisotropy and dispersion in rotating fluids, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., vol. 31, North-Holland, Amsterdam, 2002,171–192. doi: 10.1016/S0168-2024(02)80010-8.

[15]

——, Mathematical Geophysics, Oxford Lecture Series in Mathematics and its Applications, vol. 32, The Clarendon Press, Oxford University Press, Oxford, 2006, An introduction to rotating fluids and the Navier-Stokes equations.

[16]

B. Cushman-Roisin and J. -M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011.

[17]

H. Fujita and T. Kato, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova, 32 (1962), 243-260. 

[18]

——, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269–315. doi: 10.1007/BF00276188.

[19]

I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl.(9), 77 (1998), 989-1054.  doi: 10.1016/S0021-7824(99)80002-6.

[20]

E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl.(9), 76 (1997), 477-498.  doi: 10.1016/S0021-7824(97)89959-X.

[21]

D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France, 127 (1999), 473-517.  doi: 10.24033/bsmf.2358.

[22]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[23]

M. Paicu, Étude asymptotique pour les fluides anisotropes en rotation rapide dans le cas périodique, J. Math. Pures Appl.(9), 83 (2004), 163-242.  doi: 10.1016/j.matpur.2003.10.001.

[24]

——, Équation periodique de Navier-Stokes sans viscosité dans une direction, Comm. Partial Differential Equations, 30 (2005), 1107–1140. doi: 10.1080/036053005002575529.

[25]

H. Poincaré, Sur la Précession Des Corps Déformables, Bulletin Astronomique, Serie I, 1910.

[26]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.  doi: 10.1006/jdeq.1994.1157.

[27]

S. Scrobogna, Highly Rotating Fluids with Vertical Stratification for Periodic Data and Anisotropic Diffusion, to appear in Revista Matemática Iberoamericana.

[28]

K. Widmayer, Convergence to Stratified Flow for an Inviscid 3d Boussinesq System, http://arxiv.org/abs/1509.09216.

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