October  2020, 40(10): 5711-5728. doi: 10.3934/dcds.2020242

On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity

1. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux, F-33405 Talence Cedex, France

2. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author: Ning Zhu

Received  April 2019 Revised  February 2020 Published  June 2020

Fund Project: M. Paicu is partially supported by the Agence Nationale de la Recherche, Project IFSMACS, grant ANR-15-CE40-0010. N. Zhu was partially supported by the National Natural Science Foundation of China (No. 11771045, No. 11771043)

The goal of this paper is to study the two-dimensional inviscid Boussinesq equations with temperature-dependent thermal diffusivity. Firstly we establish the global existence theory and regularity estimates for this system with Yudovich's type initial data. Then we investigate the vortex patch problem, and proving that the patch remains in Hölder class $ C^{1+s}\; (0<s<1) $ for all the time.

Citation: Marius Paicu, Ning Zhu. On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5711-5728. doi: 10.3934/dcds.2020242
References:
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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]

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J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup., 26 (1993), 517-542. doi: 10.24033/asens.1679.  Google Scholar

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R. Danchin, Évolution temporelle d'une poche de tourbillon singulière, Comm. Partial Differential Equations, 22 (1997), 685-721. doi: 10.1080/03605309708821280.  Google Scholar

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R. Danchin, Poches de tourbillon visqueuses, J. Math. Pures Appl., 76 (1997), 609-647. doi: 10.1016/S0021-7824(97)89964-3.  Google Scholar

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R. Danchin, F. Fanelli and M. Paicu, A well-posedness result for viscous compressible fluids with only bounded density, Anal. PDE, 13 (2020), 275-316. doi: 10.2140/apde.2020.13.275.  Google Scholar

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R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385. doi: 10.1002/cpa.21806.  Google Scholar

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R. Danchin and X. Zhang, On the persistence of hölder regular patches of density for the inhomogeneous Navier-Stokes equations, J. Éc. polytech. Math., 4 (2017), 781-811. doi: 10.5802/jep.56.  Google Scholar

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F. Gancedo and E. García-Juárez, Global regularity for 2D Boussinesq temperature patches with no diffusion, Ann. PDE, 3 (2017), Article number 14. doi: 10.1007/s40818-017-0031-y.  Google Scholar

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T. Hmidi, Régularité höldérienne des poches de tourbillon visqueuses, J. Math. Pures Appl., 84 (2005), 1455-1495. doi: 10.1016/j.matpur.2005.01.004.  Google Scholar

[23]

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

[24]

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

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T. Hmidi and M. Zerguine, Vortex patch problem for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  Google Scholar

[26]

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

[27]

M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.  Google Scholar

[28]

D. Li and X. Xu, Global well-posedness of an inviscid $2D$ Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265. doi: 10.4310/DPDE.2013.v10.n3.a2.  Google Scholar

[29]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.  Google Scholar

[30]

X. Liao and P. Zhang, On the global regularity of the two-dimensional density patch for inhomogeneous incompressible viscous flow, Arch. Ration. Mech. Anal., 220 (2016), 937-981. doi: 10.1007/s00205-015-0945-z.  Google Scholar

[31]

X. Liao and P. Zhang, Global regularity of 2D density patches for viscous inhomogeneous incompressible flow with general density: Low regularity case, Comm. Pure Appl. Math., 72 (2019), 835-884. doi: 10.1002/cpa.21782.  Google Scholar

[32]

M. Paicu and P. Zhang, Striated regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity, Commun. Math. Phys., 376 (2020), 385-439. doi: 10.1007/s00220-019-03446-z.  Google Scholar

[33]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. Google Scholar

[34]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[35]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[36]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[37]

X. Zhai and Z.-M. Chen, Global well-posedness for $N$-dimensional Boussinesq system with viscosity depending on temperature, Commun. Math. Sci., 16 (2018), 1427-1449.  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]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28. doi: 10.1007/BF02097055.  Google Scholar

[4]

J. R. Cannon and E. DiBenedetto, The initial value problem for the Boussinesq equations with data in ${{L}^{p}}$, in: Approximation Methods for Navier-Stokes problems, Lecture Notes in Mathematics, Springer, Berlin, 771 (1980), 129-144.  Google Scholar

[5]

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

[6]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels, Ann. Sci. École Norm. Sup., 26 (1993), 517-542. doi: 10.24033/asens.1679.  Google Scholar

[7] J.-Y. Chemin, Perfect Incompressible Fluids, Vol. 14, Oxford University Press, New York, 1998.   Google Scholar
[8]

R. Danchin, Évolution temporelle d'une poche de tourbillon singulière, Comm. Partial Differential Equations, 22 (1997), 685-721. doi: 10.1080/03605309708821280.  Google Scholar

[9]

R. Danchin, Poches de tourbillon visqueuses, J. Math. Pures Appl., 76 (1997), 609-647. doi: 10.1016/S0021-7824(97)89964-3.  Google Scholar

[10]

R. Danchin, F. Fanelli and M. Paicu, A well-posedness result for viscous compressible fluids with only bounded density, Anal. PDE, 13 (2020), 275-316. doi: 10.2140/apde.2020.13.275.  Google Scholar

[11]

R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385. doi: 10.1002/cpa.21806.  Google Scholar

[12]

R. Danchin and P. B. Mucha, A lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480. doi: 10.1002/cpa.21409.  Google Scholar

[13]

R. Danchin and M. Paicu, Les théorèmes de leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136 (2008), 261-309. doi: 10.24033/bsmf.2557.  Google Scholar

[14]

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

[15]

R. Danchin and X. Zhang, Global persistence of geometrical structures for the Boussinesq equation with no diffusion, Comm. Partial Differential Equations, 42 (2017), 68-99. doi: 10.1080/03605302.2016.1252394.  Google Scholar

[16]

R. Danchin and X. Zhang, On the persistence of hölder regular patches of density for the inhomogeneous Navier-Stokes equations, J. Éc. polytech. Math., 4 (2017), 781-811. doi: 10.5802/jep.56.  Google Scholar

[17]

F. Fanelli, Conservation of geometric structures for non-homogeneous inviscid incompressible fluids, Comm. Partial Differential Equations, 37 (2012), 1553-1595. doi: 10.1080/03605302.2012.698343.  Google Scholar

[18]

P. Gamblin and X. Saint-Raymond, On three-dimensional vortex patches, Bull. Soc. Math. France, 123 (1995), 375-424. doi: 10.24033/bsmf.2265.  Google Scholar

[19]

F. Gancedo and E. García-Juárez, Global regularity for 2D Boussinesq temperature patches with no diffusion, Ann. PDE, 3 (2017), Article number 14. doi: 10.1007/s40818-017-0031-y.  Google Scholar

[20]

B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equations, Acta Math. Appl. Sin. Engl. Ser., 5 (1989), 208-218. doi: 10.1007/BF02006004.  Google Scholar

[21]

Z. Hassainia and T. Hmidi, On the inviscid Boussinesq system with rough initial data, J. Math. Anal. Appl., 430 (2015), 777-809. doi: 10.1016/j.jmaa.2015.04.087.  Google Scholar

[22]

T. Hmidi, Régularité höldérienne des poches de tourbillon visqueuses, J. Math. Pures Appl., 84 (2005), 1455-1495. doi: 10.1016/j.matpur.2005.01.004.  Google Scholar

[23]

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

[24]

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

[25]

T. Hmidi and M. Zerguine, Vortex patch problem for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.  Google Scholar

[26]

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

[27]

M.-J. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760. doi: 10.1007/s00205-010-0357-z.  Google Scholar

[28]

D. Li and X. Xu, Global well-posedness of an inviscid $2D$ Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ., 10 (2013), 255-265. doi: 10.4310/DPDE.2013.v10.n3.a2.  Google Scholar

[29]

H. Li, R. Pan and W. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, J. Hyperbolic Differ. Equ., 12 (2015), 469-488. doi: 10.1142/S0219891615500137.  Google Scholar

[30]

X. Liao and P. Zhang, On the global regularity of the two-dimensional density patch for inhomogeneous incompressible viscous flow, Arch. Ration. Mech. Anal., 220 (2016), 937-981. doi: 10.1007/s00205-015-0945-z.  Google Scholar

[31]

X. Liao and P. Zhang, Global regularity of 2D density patches for viscous inhomogeneous incompressible flow with general density: Low regularity case, Comm. Pure Appl. Math., 72 (2019), 835-884. doi: 10.1002/cpa.21782.  Google Scholar

[32]

M. Paicu and P. Zhang, Striated regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity, Commun. Math. Phys., 376 (2020), 385-439. doi: 10.1007/s00220-019-03446-z.  Google Scholar

[33]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. Google Scholar

[34]

P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674.  Google Scholar

[35]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[36]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[37]

X. Zhai and Z.-M. Chen, Global well-posedness for $N$-dimensional Boussinesq system with viscosity depending on temperature, Commun. Math. Sci., 16 (2018), 1427-1449.  Google Scholar

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