As a continuation of our previous work, we consider lower regularity global well-posedness for a model of the two-dimensional zero diffusivity Boussinesq equations with variable viscosity. More precisely, based on De Giorgi–Nash–Moser estimates and the refined logarithmic Gronwall-type inequality, we prove that it is globally well-posed, provided that the initial data belong to Hs with s > 1. Finally, we show that it is also valid for the two-dimensional zero diffusivity Boussinesq equations with variable viscosity in the non-divergence form.
REFERENCES
1.
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
, 43
–62
(2011
).2.
S.
Lorca
and J.
Boldrini
, “The initial value problem for a generalized Boussinesq model: Regularity and global existence of strong solutions
,” Mat. Contemp.
11
, 71
–94
(1996
).3.
S. A.
Lorca
and J. L.
Boldrini
, “The initial value problem for a generalized Boussinesq model
,” Nonlinear Anal.
36
, 457
–480
(1999
).4.
D.
Turcotte
and G.
Schubert
, Geodynamics Applications of Continuum Physics to Geological Problems
(John Wiley & Sons
, 1982
).5.
M.
Gaedtke
, S.
Abishek
, R.
Mead-Hunter
, A. J. C.
King
, B. J.
Mullins
, H.
Nirschl
, and M. J.
Krause
, “Total enthalpy-based lattice Boltzmann simulations of melting in paraffin/metal foam composite phase change materials
,” Int. J. Heat Mass Transfer
155
, 119870
(2020
).6.
H.
Abidi
and P.
Zhang
, “On the global well-posedness of 2-D Boussinesq system with variable viscosity
,” Adv. Math.
305
, 1202
–1249
(2017
).7.
Q.
Chen
and L.
Jiang
, “Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity
,” Colloq. Math.
135
, 187
–199
(2014
).8.
B.
Dong
, Z.
Ye
, and X.
Zhai
, “Global regularity for the 2D Boussinesq equations with temperature-dependent viscosity
,” J. Math. Fluid Mech.
22
(1
), 2
(2020
).9.
A.
Huang
, “The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity
,” Nonlinear Anal.
113
, 401
–429
(2015
).10.
Q.
Jiu
and J.
Liu
, “Global-wellposedness of 2D Boussinesq equations with mixed partial temperature-dependent viscosity and thermal diffusivity
,” Nonlinear Anal.
132
, 227
–239
(2016
).11.
D.
Li
and X.
Xu
, “Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity
,” Dyn. PDE
10
, 255
–265
(2013
).12.
H.
Li
, R.
Pan
, and W.
Zhang
, “Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion
,” J. Hyperbolic Differ. Equations
12
, 469
–488
(2015
).13.
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. Differ. Equations
255
, 1069
–1085
(2013
).14.
Z.
Ye
, “Global well-posedness for a model of 2D temperature-dependent Boussinesq equations without diffusivity
,” J. Differ. Equations
271
, 107
–127
(2021
).15.
A.
Majda
and A.
Bertozzi
, Vorticity and Incompressible Flow
(Cambridge University Press
, Cambridge
, 2001
).16.
J.
Pedlosky
, Geophysical Fluid Dynamics
(Springer-Verlag
, New York
, 1987
).17.
D.
Chae
, “Global regularity for the 2D Boussinesq equations with partial viscosity terms
,” Adv. Math.
203
, 497
–513
(2006
).18.
T. Y.
Hou
and C.
Li
, “Global well-posedness of the viscous Boussinesq equations
,” Discrete Contin. Dyn. Syst.
12
, 1
–12
(2005
).19.
H.
Abidi
and T.
Hmidi
, “On the global well-posedness for Boussinesq system
,” J. Differ. Equations
233
, 199
–220
(2007
).20.
R.
Danchin
and X.
Zhang
, “Global persistence of geometrical structures for the Boussinesq equation with no diffusion
,” Commun. Partial Differ. Equations
42
, 68
–99
(2017
).21.
F.
Gancedo
and E.
Garcia-Juarez
, “Global regularity for 2D Boussinesq temperature patches with no diffusion
,” Ann. PDE
3
(2
), 14
(2017
).22.
T.
Hmidi
and S.
Keraani
, “On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity
,” Adv. Differ. Equations
12
, 461
–480
(2007
).23.
M.
He
, “On the blowup criteria and global regularity for the non-diffusive Boussinesq equations with temperature-dependent viscosity coefficient
,” Nonlinear Anal.
144
, 93
–109
(2016
).24.
L.
Caffarelli
and A.
Vasseur
, “Drift diffusion equations with fractional diffusion and the quasi-geostrophic equations
,” Ann. Math.
171
(3
), 1903
–1930
(2010
).25.
J. T.
Beale
, T.
Kato
, and A.
Majda
, “Remarks on the breakdown of smooth solutions for the 3-D Euler equations
,” Commun. Math. Phys.
94
(1
), 61
–66
(1984
).26.
T.
Kato
and G.
Ponce
, “Commutator estimates and the Euler and Navier-Stokes equations
,” Commun. Pure Appl. Math.
41
, 891
–907
(1988
).27.
H.
Bahouri
, J.-Y.
Chemin
, and R.
Danchin
, Fourier Analysis and Nonlinear Partial Differential Equations
, Grundlehren der Mathematischen Wissenschaften Vol. 343 (Springer
(2011
).28.
H.
Triebel
, Theory of Function Spaces
, Monographs in Mathematics (Birkhäuser Verlag
, Basel, Boston
, 1983
).29.
Z.
Ye
, “Some new regularity criteria for the 2D Euler-Boussinesq equations via the temperature
,” Acta Appl. Math.
157
, 141
–169
(2018
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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