March  2020, 12(1): 107-140. doi: 10.3934/jgm.2020006

On the degenerate boussinesq equations on surfaces

1. 

Department of Mathematics, Rice University, MS 136 P.O. Box 1892, Houston, Texas, 77251, USA

2. 

Department of Mathematics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 0B9, Canada

3. 

Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, Oklahoma, 74078, USA

4. 

Department of Mathematics, Tulane University, 6823 Saint Charles Avenue, New Orleans, LA 70118, USA

* Corresponding author: Kun Zhao

Received  May 2019 Revised  November 2019 Published  January 2020

In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface $ \Sigma $. When $ \Sigma $ has intrinsic curvature of finite Lipschitz norm, we prove the existence of global strong solutions to the Cauchy problem of the Boussinesq equations with full or partial dissipations. The issues of uniqueness and singular limits (vanishing viscosity/vanishing thermal diffusivity) are also addressed. In addition, we establish a breakdown criterion for the strong solutions for the case of zero viscosity and zero thermal diffusivity. These appear to be among the first results for Boussinesq systems on Riemannian manifolds.

Citation: Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006
References:
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H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Diff. Equ., 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.

[2]

D. AdhikariC. CaoH. ShangJ. WuX. Xu and Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Diff. Equ., 260 (2016), 1893-1917.  doi: 10.1016/j.jde.2015.09.049.

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D. AdhikariC. Cao and J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Diff. Equ., 249 (2010), 1078-1088.  doi: 10.1016/j.jde.2010.03.021.

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D. AdhikariC. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Diff. Equ., 251 (2011), 1637-1655.  doi: 10.1016/j.jde.2011.05.027.

[5]

D. AdhikariC. CaoJ. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Diff. Equ., 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.

[6]

D. Alonso–OránA. Córdoba and Á. D. Martínez, Continuity of weak solutions of the critical surface quasigeostrophic equation on $\mathbb{S}^2$, Adv. Math., 328 (2018), 264-299.  doi: 10.1016/j.aim.2018.01.015.

[7]

D. Alonso–OránA. Córdoba and Á. D. Martínez, Global well-posedness of critical surface quasigeostrophic equation on the sphere, Adv. Math., 328 (2018), 248-263.  doi: 10.1016/j.aim.2018.01.016.

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T. Aubin, Nonlinear Analysis on Manifolds. Monge–Amperè Equations, Grundlehern der Mathematischen Wissenschaften, Springer–Verlag, 252, 1982. doi: 10.1007/978-1-4612-5734-9.

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J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$–$D$ Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

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A. BiswasC. Foias and A. Larios, On the attractor for the semi-dissipative Boussinesq equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 381-405.  doi: 10.1016/j.anihpc.2015.12.006.

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

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H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. PDE, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

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L. CaffarelliR. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.

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J.R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, Approximation Methods for Navier-Stokes Problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), Lecture Notes in Math., 771, Springer, Berlin, 1980,129–144. doi: 10.1007/BFb0086903.

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D. ChaeS. Kim and H. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155 (1999), 55-80.  doi: 10.1017/S0027763000006991.

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D. Chae and O. Y. Imanuvilov, Generic solvability of the axisymmetric $3$-D Euler equations and the $2$-D Boussinesq equations, J. Diff. Equ., 156 (1999), 1-17.  doi: 10.1006/jdeq.1998.3607.

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D. Chae and J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230 (2012), 1618-1645.  doi: 10.1016/j.aim.2012.04.004.

[22]

D. CórdobaC. Fefferman and R. De La Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2004), 204-213.  doi: 10.1137/S0036141003424095.

[23]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Phys. D, 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.

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

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R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21 (2011), 421-457.  doi: 10.1142/S0218202511005106.

[26]

C. DoeringJ. WuK. Zhao and X. Zheng, Long-time behavior of two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, 376/377 (2018), 144-159.  doi: 10.1016/j.physd.2017.12.013.

[27]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.

[28]

H. Engler, An alternative proof of the Brezis–Wainger inequality, Comm. PDE, 14 (1989), 541-544. 

[29]

P. Górka, Brézis–Wainger inequality on Riemannian manifolds, J. Ineq. Appl., (2008), ID 715961, 1–6. doi: 10.1155/2008/715961.

[30]

B. Guo and G. Yuan, On the suitable weak solutions to the Boussinesq equations in a bounded domain, Acta Math. Sinica, 12 (1996), 205-216.  doi: 10.1007/BF02108163.

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E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.

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T. Hmidi and S. Keraani, On the global well-posedness of the 2D Boussinesq system with a zero diffusivity, Adv. Diff. Equ., 12 (2007), 461-480. 

[33]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.

[34]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Diff. Equ., 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[35]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. PDE, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

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T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Disc. Cont. Dyn. Sys., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.

[37]

L. Hu and H. Jian, Blow-up criterion for 2-D Boussinesq equations in bounded domain, Front. Math. China, 2 (2007), 559-581.  doi: 10.1007/s11464-007-0034-1.

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W. HuI. Kukavica and M. Ziane, Persistence of regularity for a viscous Boussinesq equations with zero diffusivity, Asymptot. Anal., 91 (2015), 111-124.  doi: 10.3233/ASY-141261.

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Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454.  doi: 10.1137/140958256.

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Q. JiuJ. Wu and W. Yang, Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Science, 25 (2015), 37-58.  doi: 10.1007/s00332-014-9220-y.

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M. LaiR. Pan and K. Zhao, Initial boundary value problem for 2D viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.

[45]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Diff. Equ., 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.

[46]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Par. Diff. Equ., 10 (2013), 255-265.  doi: 10.4310/DPDE.2013.v10.n3.a2.

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show all references

References:
[1]

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

[2]

D. AdhikariC. CaoH. ShangJ. WuX. Xu and Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Diff. Equ., 260 (2016), 1893-1917.  doi: 10.1016/j.jde.2015.09.049.

[3]

D. AdhikariC. Cao and J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Diff. Equ., 249 (2010), 1078-1088.  doi: 10.1016/j.jde.2010.03.021.

[4]

D. AdhikariC. Cao and J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Diff. Equ., 251 (2011), 1637-1655.  doi: 10.1016/j.jde.2011.05.027.

[5]

D. AdhikariC. CaoJ. Wu and X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Diff. Equ., 256 (2014), 3594-3613.  doi: 10.1016/j.jde.2014.02.012.

[6]

D. Alonso–OránA. Córdoba and Á. D. Martínez, Continuity of weak solutions of the critical surface quasigeostrophic equation on $\mathbb{S}^2$, Adv. Math., 328 (2018), 264-299.  doi: 10.1016/j.aim.2018.01.015.

[7]

D. Alonso–OránA. Córdoba and Á. D. Martínez, Global well-posedness of critical surface quasigeostrophic equation on the sphere, Adv. Math., 328 (2018), 248-263.  doi: 10.1016/j.aim.2018.01.016.

[8]

T. Aubin, Nonlinear Analysis on Manifolds. Monge–Amperè Equations, Grundlehern der Mathematischen Wissenschaften, Springer–Verlag, 252, 1982. doi: 10.1007/978-1-4612-5734-9.

[9]

J. T. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$–$D$ Euler equations, Comm. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.

[10]

A. BiswasC. Foias and A. Larios, On the attractor for the semi-dissipative Boussinesq equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 381-405.  doi: 10.1016/j.anihpc.2015.12.006.

[11]

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

[12]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. PDE, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[13]

L. CaffarelliR. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.

[14]

J.R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^p$, Approximation Methods for Navier-Stokes Problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), Lecture Notes in Math., 771, Springer, Berlin, 1980,129–144. doi: 10.1007/BFb0086903.

[15]

C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.

[16]

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.

[17]

D. ChaeP. Constantin and J. Wu, An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations, J. Math. Fluid Mech., 16 (2014), 473-480.  doi: 10.1007/s00021-014-0166-5.

[18]

D. Chae and H. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946.  doi: 10.1017/S0308210500026810.

[19]

D. ChaeS. Kim and H. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155 (1999), 55-80.  doi: 10.1017/S0027763000006991.

[20]

D. Chae and O. Y. Imanuvilov, Generic solvability of the axisymmetric $3$-D Euler equations and the $2$-D Boussinesq equations, J. Diff. Equ., 156 (1999), 1-17.  doi: 10.1006/jdeq.1998.3607.

[21]

D. Chae and J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230 (2012), 1618-1645.  doi: 10.1016/j.aim.2012.04.004.

[22]

D. CórdobaC. Fefferman and R. De La Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2004), 204-213.  doi: 10.1137/S0036141003424095.

[23]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Phys. D, 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.

[24]

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

[25]

R. Danchin and M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21 (2011), 421-457.  doi: 10.1142/S0218202511005106.

[26]

C. DoeringJ. WuK. Zhao and X. Zheng, Long-time behavior of two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, 376/377 (2018), 144-159.  doi: 10.1016/j.physd.2017.12.013.

[27]

W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids, 6 (1994), 49-58.  doi: 10.1063/1.868044.

[28]

H. Engler, An alternative proof of the Brezis–Wainger inequality, Comm. PDE, 14 (1989), 541-544. 

[29]

P. Górka, Brézis–Wainger inequality on Riemannian manifolds, J. Ineq. Appl., (2008), ID 715961, 1–6. doi: 10.1155/2008/715961.

[30]

B. Guo and G. Yuan, On the suitable weak solutions to the Boussinesq equations in a bounded domain, Acta Math. Sinica, 12 (1996), 205-216.  doi: 10.1007/BF02108163.

[31]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics, 5. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.

[32]

T. Hmidi and S. Keraani, On the global well-posedness of the 2D Boussinesq system with a zero diffusivity, Adv. Diff. Equ., 12 (2007), 461-480. 

[33]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.

[34]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Diff. Equ., 249 (2010), 2147-2174.  doi: 10.1016/j.jde.2010.07.008.

[35]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. PDE, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

[36]

T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Disc. Cont. Dyn. Sys., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.

[37]

L. Hu and H. Jian, Blow-up criterion for 2-D Boussinesq equations in bounded domain, Front. Math. China, 2 (2007), 559-581.  doi: 10.1007/s11464-007-0034-1.

[38]

W. HuI. Kukavica and M. Ziane, Persistence of regularity for a viscous Boussinesq equations with zero diffusivity, Asymptot. Anal., 91 (2015), 111-124.  doi: 10.3233/ASY-141261.

[39]

W. Hu, I. Kukavica and M. Ziane, On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys., 54 (2013), 081507, 10 pp. doi: 10.1063/1.4817595.

[40]

A. A. Il'in, The Navier–Stokes equation and Euler euqation on two-dimensional closed manifolds, Mathematics of the USSR–Sbornik, 69 (1991), 559-579. 

[41]

Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454.  doi: 10.1137/140958256.

[42]

Q. JiuJ. Wu and W. Yang, Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Science, 25 (2015), 37-58.  doi: 10.1007/s00332-014-9220-y.

[43]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.I, Interscience, 1963.

[44]

M. LaiR. Pan and K. Zhao, Initial boundary value problem for 2D viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.

[45]

A. LariosE. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Diff. Equ., 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.

[46]

D. Li and X. Xu, Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity, Dyn. Par. Diff. Equ., 10 (2013), 255-265.  doi: 10.4310/DPDE.2013.v10.n3.a2.

[47]

J. LiH. ShangJ. WuX. Xu and Z. Ye, Regularity criteria for the 2D Boussinesq equations with supercritical dissipation, Comm. Math. Sci., 14 (2016), 1999-2022.  doi: 10.4310/CMS.2016.v14.n7.a10.

[48]

J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001.  doi: 10.1007/s00205-015-0946-y.

[49]

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