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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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[12]

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[13]

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[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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