ORIGINAL ARTICLE
Regularity criteria for the three-dimensional axisymmetric Boussinesq equations in anisotropic Lorentz spaces
Zhouyu Li,
Corresponding Author
Zhouyu Li
School of Sciences, Xi'an University of Technology, Xi'an, China
Correspondence
Zhouyu Li, School of Sciences, Xi'an University of Technology, Xi'an 710054, China.
Email: [email protected]
Search for more papers by this authorZhouyu Li,
Corresponding Author
Zhouyu Li
School of Sciences, Xi'an University of Technology, Xi'an, China
Correspondence
Zhouyu Li, School of Sciences, Xi'an University of Technology, Xi'an 710054, China.
Email: [email protected]
Search for more papers by this authorFirst published: 19 March 2025
Abstract
We establish regularity criteria of the three-dimensional axisymmetric Boussinesq equations in anisotropic Lorentz spaces. In particular, no a priori assumptions on the temperature are imposed, and our result improves and extends the related result of Chae and Lee.
REFERENCES
- 1H. Abidi and G. Gui, Inviscid limit for axisymmetric Navier–Stokes–Boussinesq system, Commun. Math. Sci. 17 (2019), 1625–1652.
- 2H. Abidi, T. Hmidi, and S. Keraani, On the global regularity of axisymmetric Navier–Stokes–Boussinesq system, Discrete Contin. Dyn. Syst. 29 (2011), 737–756.
- 3A. P. Blozinski, Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc. 263 (1981), 149–167.
- 4D. Chae and J. Lee, On the regularity of axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239 (2002), 645–671.
10.1007/s002090100317 Google Scholar
- 5D. Chae and H. S. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. R. Soc. Edinb. A: Math. 127 (1997), 935–946.
- 6N. Chikami, On Gagliardo–Nirenberg type inequalities in Fourier–Herz spaces, J. Funct. Anal. 275 (2018), 1138–1172.
- 7R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Physica D. 237 (2008), 1444–1460.
- 8D. Fang, W. Le, and T. Zhang, Global solutions of 3D axisymmetric Boussinesq equations with nonzero swirl, Nonlinear Anal. 166 (2018), 48–86.
- 9H. Hajaiej, L. Molinet, T. Ozawa, and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized Boson equations, in: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kokyuroku Bessatsu, vol. B26, Res. Inst. Math. Sci. , Kyoto, 2011, pp. 159–175.
- 10T. Hmidi and F. Rousset, Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré Anal. Non Linéaire. 27 (2010), 1227–1246.
- 11H. Houamed and M. Zerguine, On the global solvability of the axisymmetric Boussinesq system with critical regularity, Nonlinear Anal. 200 (2020), 112003.
- 12Q. Jiu and H. Yu, Global well-posedness for 3D generalized Navier–Stokes–Boussinesq equations, Acta Math. Appl. Sin. Engl. Ser. 32 (2016), 1–16.
- 13D. Q. Khai and N. M. Tri, Solutions in mixed-norm Sobolev–Lorentz spaces to the initial value problem for the Navier–Stokes equations, J. Math. Anal. Appl. 417 (2014), 819–833.
- 14Z. Li, Global well-posedness of the 2D Euler–Boussinesq system with stratification effects, Math. Methods Appl. Sci. 40 (2017), 5212–5221.
- 15Z. Li and X. Pan, BKM-type blow-up criterion of the inviscid axially symmetric Boussinesq system involving a single component of velocity, Z. Angew. Math. Phys. 74 (2023), 9.
- 16A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003.
10.1090/cln/009 Google Scholar
- 17C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Commun. Math. Phys. 321 (2013), 33–67.
- 18J. Pedlosky, Geophysical fluid dynamics, Springer-Verlag, New York, 1987.
10.1007/978-1-4612-4650-3 Google Scholar
- 19P. Wang and Z. Guo, A regularity criterion for the 3D axisymmetric Boussinesq equations with non-zero swirl, J. Math. Phys. 64 (2023), 051502.
- 20S. Wang, Y. Wang, and J. Liu, Regularity criteria to the incompressible axisymmetric Boussinesq equations, Appl. Math. Lett. 112 (2021), 106800.
- 21W. Wei, Y. Wang, and Y. Ye, Calderón–Zygmund theory in Lorentz mixed-norm spaces and its application to compressible fluids, Math. Nachr. 296 (2023), 1–17.