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

Marius Paicu; Ning Zhu

doi:10.3934/dcds.2020242

Article Contents

2020, Volume 40, Issue 10: 5711-5728. Doi: 10.3934/dcds.2020242

This issue Previous Article Review of local and global existence results for stochastic pdes with Lévy noise Next Article A gradient flow approach of propagation of chaos

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

Marius Paicu^1, and
Ning Zhu^2, ,

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
^* Corresponding author: Ning Zhu

Received: April 2019

Revised: February 2020

Published: June 2020

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)

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary:35Q35, 35K59;Secondary:76B03.

Citation:

Full Text(HTML)

Related Papers

Cited by

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]	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.
[3]	A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28. doi: 10.1007/BF02097055.
[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.
[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.
[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.
[7]	J.-Y. Chemin, Perfect Incompressible Fluids, Vol. 14, Oxford University Press, New York, 1998.
[8]	R. Danchin, Évolution temporelle d'une poche de tourbillon singulière, Comm. Partial Differential Equations, 22 (1997), 685-721. doi: 10.1080/03605309708821280.
[9]	R. Danchin, Poches de tourbillon visqueuses, J. Math. Pures Appl., 76 (1997), 609-647. doi: 10.1016/S0021-7824(97)89964-3.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	T. Hmidi and M. Zerguine, Vortex patch problem for stratified Euler equations, Commun. Math. Sci., 12 (2014), 1541-1563.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[33]	J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013.
[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.
[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.
[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.
[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.