A new regularity criterion for the 3D-Boussinesq equations in homogeneous Besov and Triebel-Lizorkin spaces

Benallia, Mohamed; Dilmi, Mohamed

doi:10.47974/JIM-2181

Open Access Research Article

A new regularity criterion for the 3D-Boussinesq equations in homogeneous Besov and Triebel-Lizorkin spaces

^*Mohamed BenalliaCorresponding authorbenallia.mohamed@ens-bousaada.dz benalliam@yahoo.frAffiliation 1Department of MathematicsÉcole Normale Supérieure de BousaadaBousaada, 28001, AlgeriaAffiliation 2Laboratoire de Mathématiques et Physique AppliquéesÉcole Normale Supérieure de Bousaada, Laboratory of Functional Analysis and Geometry of Spaces, Road Bourdj Bou ArreiridjUniversity of M’silaM’sila, 28000, AlgeriaView full profile → , Mohamed Dilmidilmi_mohamed@univ-blida.dz mohamed77dilmi@gmail.comDepartment of MathematicsLAMDA-RO Laboratory, BP 270 Soumâa Rd-BlidaUniversity Saad Dahlab Blida 1AlgeriaView full profile →

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pp. 1931–1939Vol. 28Issue 5August 2025DOI: 10.47974/JIM-2181XML

Received:: 07 Aug 2024
Published Online:: 13 Aug 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2181
Pages:: 1931–1939

Abstract

This paper aims to establish a new regularity criterion for weak solutions to the three-dimensional (3D) Boussinesq equations in the homogeneous Besov spaces Ḃsp,q (ℝ3) and the homogeneous Triebel–Lizorkin spaces Ḟsp,q (ℝ3). This regularity is proven if the velocity ω satisfies the following condition ∫I ||ω||2Ḃp,q (ℝ3)2 dt<∞ and ∫I ||ω||F ̇p,qs (ℝ3)2 dt<∞; I := ]0,T[ with some conditions on exponents s, p and q.

Keywords

Besov and Triebel-Lizorkin spaces Regularity criterion 3D-Boussinesq equations

Subject Classifications

35Q3035Q3576D0376D0576N10

References

[1] G. K. Batchelor, “An Introduction to Fluid Dynamics,” Cambridge university press (2000).
[2] J. R. Cannon and E. Dibenedetto, “The initial problem for the Boussinesq equation with data in Lp,” Lect. Notes Math., Springer, Berlin, vol. 771, pp. 129-144 (1980).
[3] A. Majda, “Introduction to PDEs and Waves for the Atmosphere and Ocean,” American Mathematical Soc., vol. 9 (2003).
[4] E. A. Spiegel and G. Veronis, “On the Boussinesq approximation for a compressible fluid,” Astrophysical Journal, vol. 131, pp. 442-447 (1960).
[5] D. J. Tritton, “Physical fluid dynamics,” Springer Science Business Media (2012).
[6] N. Ishimura and H. Morimoto, “Remarks on the blow-up criterion for the 3D-Boussinesq equations,” Math. Models Methods Appl. Sci., vol. 9, no. 09, pp. 1323-1332 (1999).
[7] H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations,” Math. Z., vol. 242, no. 2, pp. 251-278 (2002).
[8] J. S. Fan and T. Ozawa, “Regularity criteria for the 3D density-dependent Boussinesq equations,” Nonlinearity, vol. 22, no. 3, pp. 553-568 (2009).
[9] J. S. Fan and Y. Zhou, “A note on regularity criterion for the 3D Boussinesq systems with partial viscosity,” Appl. Math. Lett., vol. 22, no. 5, pp. 802-805 (2009).
[10] H. Qiu, Y. Du, and Z. A. Yao, “Blow-up criteria for 3D Boussinesq equations in the multiplier space,” Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 4, pp. 1820-1804 (2011).
[11] J. Geng and J. Fan, “A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity,” Appl. Math. Lett., vol. 25, no. 1, pp. 63-66 (2012).
[12] Z. Zhang, “Some regularity criteria for the 3D Boussinesq equations in the class L2 (0,T; Ḃ–1∞,∞(ℝ3)),” ISRN Appl. Math., Art. ID 564758 (2014).
[13] A. Barbagallo, S. Gala, M. A. Ragusa, and M. Théra, “On the regularity of weak solutions of the Boussinesq equations in Besov spaces,” Vietnam J. Math., vol. 49, no. 3, pp. 637-649 (2021).
[14] Z. Ye, “Regularity criteria for 3D Boussinesq equations with zero thermal diffusion,” Electron. J. Differ. Equ., vol. 2015, no. 97, pp. 1-7 (2015).
[15] A. M. Alghamdi, S. Gala, and M. A. Ragusa, “A regularity criterion of weak solutions to the 3D Boussinesq equations,” AIMS Math., vol. 2, no. 3, pp. 451-457 (2017).
[16] A. M. Alghamdi, I. B. Omrane, S. Gala, and M. A. Ragusa, “A regularity criterion to the 3D Boussinesq equations,” Sib. Elektron. Mat. Izv., vol. 16, pp. 1795-1804 (2019).
[17] S. Gala and M. A. Ragusa, “A regularity criterion of weak solutions to the 3D Boussinesq equations,” Bull. Braz. Math. Soc. New Series, vol. 51, no. 2, pp. 513-525 (2019).
[18] I. B. Omrane, S. Gala, and M. Théra, “A blowup criteria of smooth solutions to the 3D Boussinesq equations,” Bull. Braz. Math. Soc. New Series, vol. 55, no. 9 (2024).
[19] ML. Zou and Q. Li, “Some New Regularity Criteria for the 3D Boussinesq Equations in homogeneous Besov Spaces,” Chin. Quart. J. Math., vol. 39, no. 1, pp. 73-81 (2024).
[20] M. Saadi, “On the functions acting on Besov-type and Lizorkin-Triebel-type spaces,” Journal of Interdisciplinary Mathematics, vol. 23, no. 8, pp. 1497-1506 (2020).
[21] J. Peetre, “New Thoughts on Besov Spaces,” Duke Univ. Math. Series I, Durham, N.C. (1976).
[22] H. Triebel, “Theory of Function Spaces,” Birkhäuser, Basel (1983).
[23] M. Moussai, “Realizations of homogeneous Besov and Triebel-Lizorkin spaces and an application to pointwise multipliers,” Anal. Appl. (Singap.), vol. 13, no. 2, pp. 149-183 (2015).
[24] B. Jawerth, “Some observations on Besov and Lizorkin-Triebel spaces,” Math. Scand., vol. 40, no. 1, pp. 94-104 (1977).
[25] G. Bourdaud, “Ce qu’il faut savoir sur les espaces de Besov,” Preprint, Paris (2009).
[26] G. Bourdaud, M. Moussai, and W. Sickel, “Composition operators in Lizorkin-Triebel spaces,” J. Funct. Anal., vol. 259, pp. 1098-1128 (2010).

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