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doi: 10.3934/dcdsb.2020112

Long-time solvability in Besov spaces for the inviscid 3D-Boussinesq-Coriolis equations

Vladimir Angulo-Castillo 1, and  Lucas C. F. Ferreira 2,,

1. 

National University of Colombia, Campus Orinoquia, Department of Mathematics, Kilómetro 9 vía a Caño Limón, Arauca, Colombia
2. 

University of Campinas, IMECC-Department of Mathematics, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brazil

* Corresponding author: Lucas C. F. Ferreira

Received  April 2019 Published  March 2020

Fund Project: The first author was supported by CAPES and CNPq, Brazil. The second author was supported by FAPESP and CNPq, Brazil

We investigate the long-time solvability in Besov spaces of the initial value problem for the inviscid 3D-Boussinesq equations with Coriolis force. First we prove a local existence and uniqueness result with critical and supercritical regularity and existence-time $ T $ uniform with respect to the rotation speed $ \Omega $. Afterwards, we show a blow-up criterion of BKM type, estimates for arbitrarily large $ T $, and then obtain the long-time existence and uniqueness of solutions for arbitrary initial data, provided that $ \Omega $ is large enough.

Keywords: Boussinesq equations, Coriolis force, long-time solvability, blow-up, Besov spaces.
Mathematics Subject Classification: Primary: 35Q35, 76U05; Secondary: 76B03, 35A01, 42B35.
Citation: Vladimir Angulo-Castillo, Lucas C. F. Ferreira. Long-time solvability in Besov spaces for the inviscid 3D-Boussinesq-Coriolis equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020112
References:
[1]

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

[2]

M. F. de Almeida and L. C. F. Ferreira, On the well posedness and large-time behavior for Boussinesq equations in Morrey spaces, Differential Integral Equations, 24 (2011), 719-742.   Google Scholar

[3]

V. Angulo-Castillo and L. C. F. Ferreira, On the 3D Euler equations with Coriolis force in borderline Besov spaces, Commun. Math. Sci., 16 (2018), 145-164.  doi: 10.4310/CMS.2018.v16.n1.a7.  Google Scholar

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

[5]

A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of $3$D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal., 15 (1997), 103-150.  doi: 10.3233/ASY-1997-15201.  Google Scholar

[6]

A. Babin, A. Mahalov and B. Nicolaenko, On the regularity of three-dimensional rotating Euler-Boussinesq equations, Math. Models Methods Appl. Sci., 9 (1999), 1089-1121.  doi: 10.1142/S021820259900049X.  Google Scholar

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.  Google Scholar

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Q. Bie, Q. Wang and Z.-A. Yao, On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces, Kinet. Relat. Models, 8 (2015), 395-411.  doi: 10.3934/krm.2015.8.395.  Google Scholar

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[10]

D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.   Google Scholar

[11]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar

[12]

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

[13]

F. Charve, Global well-posedness for the primitive equations with less regular initial data, Ann. Fac. Sci. Toulouse Math., 17 (2008), 221-238.  doi: 10.5802/afst.1182.  Google Scholar

[14]

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.  Google Scholar

[15]

X. Cui, C. Dou and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces, J. Partial Differ. Equ., 25 (2012), 220-238.  doi: 10.4208/jpde.v25.n3.3.  Google Scholar

[16]

Z. Dai, X. Wang, L. Zhang and W. Hou, Blow-up criterion of weak solutions for the 3D Boussinesq equations, J. Funct. Spaces, (2015), Art. ID 303025, 6 pp. doi: 10.1155/2015/303025.  Google Scholar

[17]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

[18]

C. Deng and S. Cui, Well-posedness of the viscous Boussinesq system in Besov spaces of negative regular index $s = -1$, J. Math. Phys., 53 (2012), 073101, 15 pp. doi: 10.1063/1.4732521.  Google Scholar

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A. Dutrifoy, Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl., 84 (2005), 331-356.  doi: 10.1016/j.matpur.2004.09.007.  Google Scholar

[20]

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

[21]

M. Fu and C. Cai, Remarks on pressure blow-up criterion of the 3D zero-diffusion Boussinesq equations in margin Besov spaces, Adv. Math. Phys., (2017), Art. ID 6754780, 7 pp. doi: 10.1155/2017/6754780.  Google Scholar

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T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.   Google Scholar

[23]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[24]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

T. Iwabuchi, A. Mahalov and R. Takada, Global solutions for the incompressible rotating stably stratified fluids, Math. Nachr., 290 (2017), 613-631.  doi: 10.1002/mana.201500385.  Google Scholar

[26]

H. Koba, A. Mahalov and T. Yoneda, Global well-posedness for the rotating Navier-Stokes-Boussinesq equations with stratification effects, Adv. Math. Sci. Appl., 22 (2012), 61-90.   Google Scholar

[27]

Y. Koh, S. Lee and R. Takada, Strichartz estimates for the Euler equations in the rotational framework, J. Differential Equations, 256 (2014), 707-744.  doi: 10.1016/j.jde.2013.09.017.  Google Scholar

[28]

S. Lee and R. Takada, Dispersive estimates for the stably stratified Boussinesq equations, Indiana Univ. Math. J., 66 (2017), 2037-2070.  doi: 10.1512/iumj.2017.66.6179.  Google Scholar

[29]

X. Liu, M. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[30]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar

[31]

J. Sun, C. Liu, and M. Yang, Global solutions to 3D rotating Boussinesq equations in Besov spaces, J. Dyn. Diff. Equat., (2019), 1–15. doi: 10.1007/s10884-019-09747-0.  Google Scholar

[32]

J. Sun and M. Yang, Global well-posedness for the viscous primitive equations of geophysics, Bound. Value Probl., 21 (2016), 16 pp. doi: 10.1186/s13661-016-0526-6.  Google Scholar

[33]

R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725.  doi: 10.1007/s00028-008-0403-6.  Google Scholar

[34]

R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), 22 pp. doi: 10.1007/s00033-016-0697-0.  Google Scholar

[35]

Z. Xiang and W. Yan, On the well-posedness of the Boussinesq equation in the Triebel-Lizorkin-Lorentz spaces, Abstr. Appl. Anal., 2012, Art. ID 573087, 17 pp. doi: 10.1155/2012/573087.  Google Scholar

[36]

Z. Ye, A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 112-120.  doi: 10.1016/S0252-9602(14)60144-2.  Google Scholar

[37]

B. Q. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455-468.   Google Scholar

[38]

Y. Zhou, Local well-posedness for the incompressible Euler equations in the critical Besov spaces, Ann. Inst. Fourier (Grenoble), 54 (2004), 773-786.  doi: 10.5802/aif.2033.  Google Scholar

show all references

References:
[1]

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

[2]

M. F. de Almeida and L. C. F. Ferreira, On the well posedness and large-time behavior for Boussinesq equations in Morrey spaces, Differential Integral Equations, 24 (2011), 719-742.   Google Scholar

[3]

V. Angulo-Castillo and L. C. F. Ferreira, On the 3D Euler equations with Coriolis force in borderline Besov spaces, Commun. Math. Sci., 16 (2018), 145-164.  doi: 10.4310/CMS.2018.v16.n1.a7.  Google Scholar

[4]

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

[5]

A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of $3$D Euler and Navier-Stokes equations for rotating fluids, Asymptot. Anal., 15 (1997), 103-150.  doi: 10.3233/ASY-1997-15201.  Google Scholar

[6]

A. Babin, A. Mahalov and B. Nicolaenko, On the regularity of three-dimensional rotating Euler-Boussinesq equations, Math. Models Methods Appl. Sci., 9 (1999), 1089-1121.  doi: 10.1142/S021820259900049X.  Google Scholar

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.  Google Scholar

[8]

Q. Bie, Q. Wang and Z.-A. Yao, On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces, Kinet. Relat. Models, 8 (2015), 395-411.  doi: 10.3934/krm.2015.8.395.  Google Scholar

[9]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equation with data in $L^{p}$, Lecture Notes in Math., 771 (1980), 129-144.  doi: 10.1007/BFb0086903.  Google Scholar

[10]

D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.   Google Scholar

[11]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar

[12]

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

[13]

F. Charve, Global well-posedness for the primitive equations with less regular initial data, Ann. Fac. Sci. Toulouse Math., 17 (2008), 221-238.  doi: 10.5802/afst.1182.  Google Scholar

[14]

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.  Google Scholar

[15]

X. Cui, C. Dou and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces, J. Partial Differ. Equ., 25 (2012), 220-238.  doi: 10.4208/jpde.v25.n3.3.  Google Scholar

[16]

Z. Dai, X. Wang, L. Zhang and W. Hou, Blow-up criterion of weak solutions for the 3D Boussinesq equations, J. Funct. Spaces, (2015), Art. ID 303025, 6 pp. doi: 10.1155/2015/303025.  Google Scholar

[17]

R. Danchin and M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data, Comm. Math. Phys., 290 (2009), 1-14.  doi: 10.1007/s00220-009-0821-5.  Google Scholar

[18]

C. Deng and S. Cui, Well-posedness of the viscous Boussinesq system in Besov spaces of negative regular index $s = -1$, J. Math. Phys., 53 (2012), 073101, 15 pp. doi: 10.1063/1.4732521.  Google Scholar

[19]

A. Dutrifoy, Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl., 84 (2005), 331-356.  doi: 10.1016/j.matpur.2004.09.007.  Google Scholar

[20]

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

[21]

M. Fu and C. Cai, Remarks on pressure blow-up criterion of the 3D zero-diffusion Boussinesq equations in margin Besov spaces, Adv. Math. Phys., (2017), Art. ID 6754780, 7 pp. doi: 10.1155/2017/6754780.  Google Scholar

[22]

T. Hmidi and S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461-480.   Google Scholar

[23]

T. Hmidi and F. Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1227-1246.  doi: 10.1016/j.anihpc.2010.06.001.  Google Scholar

[24]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[25]

T. Iwabuchi, A. Mahalov and R. Takada, Global solutions for the incompressible rotating stably stratified fluids, Math. Nachr., 290 (2017), 613-631.  doi: 10.1002/mana.201500385.  Google Scholar

[26]

H. Koba, A. Mahalov and T. Yoneda, Global well-posedness for the rotating Navier-Stokes-Boussinesq equations with stratification effects, Adv. Math. Sci. Appl., 22 (2012), 61-90.   Google Scholar

[27]

Y. Koh, S. Lee and R. Takada, Strichartz estimates for the Euler equations in the rotational framework, J. Differential Equations, 256 (2014), 707-744.  doi: 10.1016/j.jde.2013.09.017.  Google Scholar

[28]

S. Lee and R. Takada, Dispersive estimates for the stably stratified Boussinesq equations, Indiana Univ. Math. J., 66 (2017), 2037-2070.  doi: 10.1512/iumj.2017.66.6179.  Google Scholar

[29]

X. Liu, M. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12 (2010), 280-292.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[30]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar

[31]

J. Sun, C. Liu, and M. Yang, Global solutions to 3D rotating Boussinesq equations in Besov spaces, J. Dyn. Diff. Equat., (2019), 1–15. doi: 10.1007/s10884-019-09747-0.  Google Scholar

[32]

J. Sun and M. Yang, Global well-posedness for the viscous primitive equations of geophysics, Bound. Value Probl., 21 (2016), 16 pp. doi: 10.1186/s13661-016-0526-6.  Google Scholar

[33]

R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725.  doi: 10.1007/s00028-008-0403-6.  Google Scholar

[34]

R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), 22 pp. doi: 10.1007/s00033-016-0697-0.  Google Scholar

[35]

Z. Xiang and W. Yan, On the well-posedness of the Boussinesq equation in the Triebel-Lizorkin-Lorentz spaces, Abstr. Appl. Anal., 2012, Art. ID 573087, 17 pp. doi: 10.1155/2012/573087.  Google Scholar

[36]

Z. Ye, A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 112-120.  doi: 10.1016/S0252-9602(14)60144-2.  Google Scholar

[37]

B. Q. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455-468.   Google Scholar

[38]

Y. Zhou, Local well-posedness for the incompressible Euler equations in the critical Besov spaces, Ann. Inst. Fourier (Grenoble), 54 (2004), 773-786.  doi: 10.5802/aif.2033.  Google Scholar

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