In this paper, we establish local-in-time existence and uniqueness of strong solutions in Hs for to the viscous, zero thermal-diffusive Boussinesq equations in . Beale-Kato-Majda type blow-up criterion has been established in three dimensions with respect to the BMO-norm of the vorticity. We further prove the local-in-time existence for nonviscous and fully ideal Boussinesq systems in . Moreover, we establish blow-up criterion for nonviscous Boussinesq system in three dimensions and for fully ideal Boussinesq system in both two and three dimensions. Commutator estimates from the work of Kato and Ponce [Comm. Pure Appl. Math. 41, 891 (1988)] and Fefferman et al. [J. Funct. Anal. 267, 1035 (2014)] play important roles in the calculations.
REFERENCES
1.
Abidi
, H.
and Hmidi
, T.
, “On the global well-posedness for Boussinesq system
,” J. Differ. Equations
233
(1
), 199
–220
(2007
).2.
Abidi
, H.
, Hmidi
, T.
, and Keraani
, S.
, “On the global regularity of axisymmetric Navier-Stokes-Boussinesq system
,” Discrete Contin. Dyn. Syst.
29
(3
), 737
–756
(2011
).3.
Adams
, R. A.
and Fournier
, J. J. F.
, Sobolev Spaces
, Pure and Applied Mathematics (Academic Press
, Amsterdam
, 1975
), Vol. 140.4.
Beale
, J. T.
, Kato
, T.
, and Majda
, A.
, “Remarks on the breakdown of smooth solutions for the 3-D Euler equations
,” Commun. Math. Phys.
94
, 61
–66
(1984
).5.
Bessaih
, H.
and Ferrario
, B.
, “The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
,” J. Differ. Equations
262
(3
), 1822
–1849
(2017
).6.
Brezis
, H.
and Gallouet
, T.
, “Nonlinear Schrödinger evolution equations
,” Nonlinear Anal. TMA
4
, 677
–681
(1980
).7.
Brezis
, H.
and Wainger
, S.
, “A note on limiting cases of Sobolev embeddings and convolution inequalities
,” Commun. Partial Differ. Equations
5
, 773
–789
(1980
).8.
Cannon
, J. R.
and DiBenedetto
, E.
, “The initial value problem for the Boussinesq equations with data in Lp
,” in Approximation Methods for Navier-Stokes Problems
, Lecture Note in Mathematics, edited by Rautmann
, R.
(Springer
, Berlin
, 1980
), Vol. 771, pp. 129
–144
.9.
Chae
, D.
, “Global regularity for the 2D Boussinesq equations with partial viscosity terms
,” Adv. Math.
203
, 497
–513
(2006
).10.
Chae
, D.
, Kim
, S.-K.
, and Nam
, H.-S.
, “Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations
,” Nagoya Math. J.
155
, 55
–88
(1999
).11.
Chae
, D.
and Nam
, H.-S.
, “Local existence and blow-up criterion for the Boussinesq equations
,” Proc. R. Soc. Edinburgh, Sect. A
127
(5
), 935
–946
(1997
).12.
Danchin
, R.
and Paicu
, M.
, “Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
,” Phys. D
237
(10-12
), 1444
–1460
(2008
).13.
Danchin
, R.
and Paicu
, M.
, “Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux
,” Bull. Soc. Math. France
136
(2
), 261
–309
(2008
).14.
Danchin
, R.
and Paicu
, M.
, “Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data
,” Commun. Math. Phys.
290
(1
), 1
–14
(2009
).15.
Dhongade
, U. D.
and Deo
, S. G.
, “A nonlinear generalization of Bihari’s inequality
,” Proc. Am. Math. Soc.
54
(1
), 211
–216
(1976
).16.
Evans
, L. C.
, Partial Differential Equations
, Graduate Studies in Mathematics, 2nd ed. (American Mathematical Society
, Providence, RI
, 2010
), Vol. 19.17.
Fefferman
, C. L.
, “Characterizations of bounded mean oscillation
,” Bull. Am. Math. Soc.
77
(4
), 587
–588
(1971
).18.
Fefferman
, C. L.
, McCormick
, D. S.
, Robinson
, J. C.
, and Rodrigo
, J. L.
, “Higher order commutator estimates and local existence for the non-resistive MHD equations and related models
,” J. Funct. Anal.
267
, 1035
–1056
(2014
).19.
Geng
, J.
and Fan
, J.
, “A note on regularity criterion for the 3D Boussinesq system with zero thermal conductivity
,” Appl. Math. Lett.
25
(1
), 63
–66
(2012
).20.
Hmidi
, T.
and Keraani
, S.
, “On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity
,” Adv. Differ. Equations
12
(4
), 461
–480
(2007
).21.
Hou
, T. Y.
and Li
, C.
, “Global well-posedness of the viscous Boussinesq equations
,” Discrete Cont. Dyn. Syst.
12
(1
), 1
–12
(2005
).22.
Ishimura
, N.
and Morimoto
, H.
, “Remarks on the blow-up criterion for the 3-D Boussinesq equations
,” Math. Models Methods Appl. Sci.
9
(9
), 1323
–1332
(1999
).23.
Kato
, T.
and Ponce
, G.
, “Commutator estimates and the Euler and Navier-Stokes equations
,” Commun. Pure Appl. Math.
41
, 891
–907
(1988
).24.
Kesavan
, S.
, Topics in Functional Analysis and Applications
, 2nd ed. (New Age International Publishers
, 2015
).25.
Kozono
, H.
and Taniuchi
, Y.
, “Limiting case of the Sobolev inequality in BMO, with application to the Euler equations
,” Commun. Math. Phys.
214
, 191
–200
(2000
).26.
Lemarié-Rieusset
, P. G.
, Recent Developments in the Navier-Stokes Problem
, CRC Research in Mathematics Series (Chapman & Hall
, 2002
), Vol. 431.27.
Lions
, J. L.
and Magenes
, E.
, Non-Homogeneous Boundary Value Problems and Applications
(Springer-Verlag
, New York
, 1972
), Vol. 1.28.
Robinson
, J. C.
, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors
(Cambridge University Press
, UK
, 2001
).29.
Temam
, R.
, “Navier-Stokes equations
.” in Theory and Numerical Analysis
, Studies in Mathematics and its Applications, (North-Holland Publishing
, Amsterdam, New York
, 1979
), Vol. 2.30.
Qiu
, H.
, Du
, Y.
, and Yao
, Z.
, “A blow-up criterion for 3D Boussinesq equations in Besov spaces
,” Nonlinear Anal.
73
(3
), 806
–815
(2010
).31.
Ye
, Z.
and Xu
, X.
, “Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation
,” J. Differ. Equations
260
(8
), 6716
–6744
(2016
).32.
Ye
, Z.
, “Regularity criteria for 3D Boussinesq equations with zero thermal diffusion
,” Electron. J. Differ. Equations
2015
(97
), 1
–7
(2015
).33.
Yosida
, K.
, Functional Analysis
, 6th ed. (Springer-Verlag
, Berlin Heidelberg New York
, 1980
).© 2019 Author(s).
2019
Author(s)
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