Abstract
This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.
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References
Brandolese, L., Schonbek, E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364, 5057–5090 (2012)
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of Navier-Stokes equation. Commun. Pure. Appl. Math. 35(6), 771–831 (1982)
Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi- geostrophic equation. Ann. Math. 171(3), 1903–1930 (2010)
Caffarelli, L., Vasseur, A.: The De Giorgi method for nonlocal fluid dynamics, Nonlinear Partial Differential Equations Advanced Courses in Mathematics, pp. 1–38. CRM, Barcelona (2012)
Cannon, R., DiBenedetto, E.: The Initial Value Problem for the Boussinesq Equations with Data in \(L^p\), Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, vol. 771, pp. 129–144. Springer, Berlin (1980)
Dong, H., Gu, X.: Partial regularity of solutions to the four-dimensional Navier-Stokes equations. Dyn. Partial Differ. Equ. 11, 53C69 (2014)
De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)
Giga, Y., Sohr, H.: Abstract Lp-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)
Guo, B., Yuan, G.: On the suitable weak solutions for the Cauchy problem of the Boussinesq equations. Nonlinear Anal. 26, 1367–1385 (1996)
Gustafson, S., Kang, K., Tsai, T.: Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations. Commun. Math. Phys. 273, 161–176 (2007)
Han, P.: Algebraic \(L^2\) decay for weak solutions of a viscous Boussinesq system in exterior domains. J. Differ. Equ. 252, 6306–6323 (2012)
Hmidi, T., Rousset, F.: Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data. Ann. Inst. H. Poincaré 27, 1227–1246 (2010)
Karch, G., Prioux, N.: Self-similarity in viscous Boussinesq equation. Proc. Am. Math. Soc. 136, 879–888 (2008)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg Theorem. Commun. Pure Appl. Math. 51(3), 241–257 (1998)
Miao, C., Zheng, X.: On the global well-posedness for the Boussinesq system with horizontal dissipation. Commun. Math. Phys. 321, 33–67 (2013)
Miao, C., Zheng, X.: Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity. J. Math. Pures Appl. 101, 842C–872 (2014)
Nazarov, A.I., Ural’tseva, N.N.: The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients. (Russian) Algebra i Analiz 23, 136–168 (2011); translation in St. Petersburg Math. J. 23, 93–115 (2012)
Pedlosky, J.: Geophysical Fluid Dyanmics. Springer, New York (1987)
Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)
Seregin, G., Silvestre, L., Šverák, V., Zlatoš, A.: On divergence-free drifts. J. Differ. Equ. 252, 505–540 (2012)
Scheffer, V.: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66, 535–552 (1976)
Scheffer, V.: Hausdorff measure and the Navier-Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)
Scheffer, V.: The Navier-Stokes equations in space dimension four. Commun. Math. Phys. 61, 41–68 (1978)
Tian, G., Xin, Z.: Gradient estimation on Navier-Stokes equations. Commun. Anal. Geom. 7, 221–257 (1999)
Vasseur, A.: A new proof of partial regularity of solutions to Navier Stokes equations. Nonlinear Differ. Equ. Appl. 14(5–6), 753–785 (2007)
Wang, X.: A note on long time behavior of solutions to the Boussinesq system at large Prandtl number. Nonlinear partial differential equations and related analysis. Contemp. Math. 371, 315–323 (2005)
Wang, Y., Wu, G.: A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations. J. Differ. Equ. 256(3), 1224–1249 (2014)
Acknowledgments
We are deeply grateful to the anonymous referee and the associated editor for the invaluable comments and suggestions on our paper. The first author is supported in part by National Natural Sciences Foundation of China (Nos.11171229, 11231006 and 11228102) and Project of Beijing Chang Cheng Xue Zhe. The third author is supported in part by the National Natural Science Foundation of China under Grant No.11101405 and the President Fund of UCAS.
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Jiu, Q., Wang, Y. & Wu, G. Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations. J Dyn Diff Equat 28, 567–591 (2016). https://doi.org/10.1007/s10884-016-9536-4
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DOI: https://doi.org/10.1007/s10884-016-9536-4