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November  2018, 17(6): 2423-2439. doi: 10.3934/cpaa.2018115

On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations

1. 

Universidade Estadual de Campinas, IMECC-Departamento de Matemática, CEP 13083-859, Campinas-SP, Brazil

2. 

Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia

* Corresponding author: Élder J. Villamizar-Roa

Received  August 2017 Revised  February 2018 Published  June 2018

Fund Project: The first author has been partially supported by CNPq and FAPESP, Brazil. The second author has been supported by CNPq, Brazil, and by Universidad Industrial de Santander. The third autor has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016.

This paper is devoted to the Boussinesq equations that models natural convection in a viscous fluid by coupling Navier-Stokes and heat equations via a zero order approximation. We consider the problem in $ \mathbb{R}^{n}$ and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.

Citation: Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, Journal of Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.  Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976.  Google Scholar

[3]

A. P. Blozinski, On a convolution theorem for $ L(p,q)$ spaces, Transactions of the American Mathematical Society, 164 (1972), 255-265.  doi: 10.2307/1995972.  Google Scholar

[4]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Normale. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[5]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.   Google Scholar

[6]

L. Brandolese and M. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Transactions of the American Mathematical Society, 364 (2012), 5057-5090.  doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

[7]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in Lp. Approximation Methods for Navier-Stokes Problems, Proc. Sympos., Univ. Paderborn, Paderborn, 1979,129–144, Lecture Notes in Math., 771, Springer, Berlin, 1980.  Google Scholar

[8]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system?, Journal of Differential Equations, 197 (2004), 247-274.  doi: 10.1016/j.jde.2003.10.003.  Google Scholar

[9]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1961.  Google Scholar

[10]

L. C. F. Ferreira, On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées, 105 (2016), 288-247.  doi: 10.1016/j.matpur.2015.10.004.  Google Scholar

[11]

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

[12]

L. C. F. Ferreira and E. J. Villamizar-Roa, On the stability problem for the Boussinesq equations in weak-Lp spaces, Commun. Pure Appl. Anal., 9 (2010), 667-684.  doi: 10.3934/cpaa.2010.9.667.  Google Scholar

[13]

P. G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, 2nd edition, Monographs in Mathematics. Springer, New York, 2011.  Google Scholar

[14]

T. Hishida, On a class of stable steady flow to the exterior convection problem, Journal of Differential Equations, 141 (1997), 54-85.  doi: 10.1006/jdeq.1997.3323.  Google Scholar

[15]

R. A. Hunt, On $ L(p,q)$ spaces, Enseignement Math., 12 (1966), 249-276.   Google Scholar

[16]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier--Stokes equation with distributions in new function spaces as initial data, Commun. Partial Differ. Eqns., 15 (1994), 959-1014.  doi: 10.1080/03605309408821042.  Google Scholar

[17]

H. Kozono and M. Yamazaki, The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation, Indiana Univ. Math. J., 4 (1995), 1307-1336.   Google Scholar

[18]

H. Kozono and M. Yamazaki, On a larger class of stable solutions to the Navier--Stokes equations in exterior domains, Math. Z., 228 (1998), 751-785.  doi: 10.1007/PL00004644.  Google Scholar

[19]

P. G. Lemarie-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, pp. xiv+395, 2002.  Google Scholar

[20]

S. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, Journal of Differential equations, 124 (1996), 389-406.  doi: 10.1006/jdeq.1996.0016.  Google Scholar

[21]

A. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE., Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[22]

H. Morimoto, On the existence and uniqueness of the stationary solution to the equations of natural convection, Tokyo J. Math., 14 (1991), 217-226.  doi: 10.3836/tjm/1270130501.  Google Scholar

[23]

R. O'Neil, Convolution operators and $ L(p,q)$ spaces, Duke Mathematical Journal, 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[24]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Advanced Texts: Basler Lehrbücher, Birkhuser Verlag, Basel, 2001.  Google Scholar

show all references

References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, Journal of Differential Equations, 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.  Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976.  Google Scholar

[3]

A. P. Blozinski, On a convolution theorem for $ L(p,q)$ spaces, Transactions of the American Mathematical Society, 164 (1972), 255-265.  doi: 10.2307/1995972.  Google Scholar

[4]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Normale. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.  Google Scholar

[5]

W. Borchers and T. Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math., 174 (1995), 311-382.   Google Scholar

[6]

L. Brandolese and M. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Transactions of the American Mathematical Society, 364 (2012), 5057-5090.  doi: 10.1090/S0002-9947-2012-05432-8.  Google Scholar

[7]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in Lp. Approximation Methods for Navier-Stokes Problems, Proc. Sympos., Univ. Paderborn, Paderborn, 1979,129–144, Lecture Notes in Math., 771, Springer, Berlin, 1980.  Google Scholar

[8]

M. Cannone and G. Karch, Smooth or singular solutions to the Navier-Stokes system?, Journal of Differential Equations, 197 (2004), 247-274.  doi: 10.1016/j.jde.2003.10.003.  Google Scholar

[9]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1961.  Google Scholar

[10]

L. C. F. Ferreira, On a bilinear estimate in weak-Morrey spaces and uniqueness for Navier-Stokes equations, Journal de Mathématiques Pures et Appliquées, 105 (2016), 288-247.  doi: 10.1016/j.matpur.2015.10.004.  Google Scholar

[11]

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

[12]

L. C. F. Ferreira and E. J. Villamizar-Roa, On the stability problem for the Boussinesq equations in weak-Lp spaces, Commun. Pure Appl. Anal., 9 (2010), 667-684.  doi: 10.3934/cpaa.2010.9.667.  Google Scholar

[13]

P. G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, 2nd edition, Monographs in Mathematics. Springer, New York, 2011.  Google Scholar

[14]

T. Hishida, On a class of stable steady flow to the exterior convection problem, Journal of Differential Equations, 141 (1997), 54-85.  doi: 10.1006/jdeq.1997.3323.  Google Scholar

[15]

R. A. Hunt, On $ L(p,q)$ spaces, Enseignement Math., 12 (1966), 249-276.   Google Scholar

[16]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier--Stokes equation with distributions in new function spaces as initial data, Commun. Partial Differ. Eqns., 15 (1994), 959-1014.  doi: 10.1080/03605309408821042.  Google Scholar

[17]

H. Kozono and M. Yamazaki, The stability of small stationary solutions in Morrey spaces of the Navier-Stokes equation, Indiana Univ. Math. J., 4 (1995), 1307-1336.   Google Scholar

[18]

H. Kozono and M. Yamazaki, On a larger class of stable solutions to the Navier--Stokes equations in exterior domains, Math. Z., 228 (1998), 751-785.  doi: 10.1007/PL00004644.  Google Scholar

[19]

P. G. Lemarie-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, pp. xiv+395, 2002.  Google Scholar

[20]

S. Lorca and J. L. Boldrini, Stationary solutions for generalized Boussinesq models, Journal of Differential equations, 124 (1996), 389-406.  doi: 10.1006/jdeq.1996.0016.  Google Scholar

[21]

A. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE., Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[22]

H. Morimoto, On the existence and uniqueness of the stationary solution to the equations of natural convection, Tokyo J. Math., 14 (1991), 217-226.  doi: 10.3836/tjm/1270130501.  Google Scholar

[23]

R. O'Neil, Convolution operators and $ L(p,q)$ spaces, Duke Mathematical Journal, 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[24]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhuser Advanced Texts: Basler Lehrbücher, Birkhuser Verlag, Basel, 2001.  Google Scholar

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