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Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain
Long-time solvability in Besov spaces for the inviscid 3D-Boussinesq-Coriolis equations
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 |
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.
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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. Chae,
Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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. |
[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.
|
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. Chae,
Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
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