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

Vladimir Angulo-Castillo; Lucas C. F. Ferreira

doi:10.3934/dcdsb.2020112

Previous Article
Differentiable solutions of the Feigenbaum-Kadanoff-Shenker equation
DCDS-B Home
This Issue
Next Article
Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain

December 2020, 25(12): 4553-4573. 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 December 2020 Early access 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, 2020, 25 (12) : 4553-4573. 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
[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

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

[1]	Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[2]	Jeongho Kim, Weiyuan Zou. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain. Kinetic & Related Models, 2020, 13 (3) : 623-651. doi: 10.3934/krm.2020021
[3]	María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic & Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024
[4]	Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
[5]	Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
[6]	Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[7]	Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[8]	Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[9]	Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[10]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[11]	Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[12]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[13]	Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[14]	Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[15]	Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143
[16]	Vladimir Angulo-Castillo, Lucas C. F. Ferreira, Leonardo Kosloff. Long-time solvability for the 2D dispersive SQG equation with improved regularity. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1411-1433. doi: 10.3934/dcds.2020082
[17]	Min Chen, Olivier Goubet. Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 37-53. doi: 10.3934/dcdss.2009.2.37
[18]	Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[19]	Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[20]	Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors