In this paper, the authors investigate the probability distribution of solutions within the phase space for the non-autonomous tropical climate model in two-dimensional bounded domains. They first prove that the associated process possesses a pullback attractor and a family of invariant Borel probability measures. Then they establish that this family of invariant Borel probability measures satisfies Liouville’s theorem and is a statistical solution of the tropical climate model. Afterwards, they prove that the statistical solution possesses degenerate Lusin’s type regularity provided that the associated Grashof number is small enough.

1.
D. M. W.
Frierson
,
A. J.
Majda
, and
O. M.
Pauluis
, “
Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit
,”
Commun. Math. Sci.
2
,
591
626
(
2004
).
2.
B.-Q.
Dong
,
W.
Wang
,
J.
Wu
et al, “
Global regularity for a class of 2D generalized tropical climate models
,”
J. Differ. Equations
266
,
6346
6382
(
2019
).
3.
B.-Q.
Dong
,
J.
Wu
, and
Z.
Ye
, “
Global regularity for a 2D tropical climate model with fractional dissipation
,”
J. Nonlinear Sci.
29
,
511
550
(
2019
).
4.
B.-Q.
Dong
,
J.
Wu
, and
Z.
Ye
, “
2D tropical climate model with fractional dissipation and without thermal diffusion
,”
Commun. Math. Sci.
18
,
259
292
(
2020
).
5.
B.-Q.
Dong
,
C.
Li
,
X.
Xu
, and
Z.
Ye
, “
Global smooth solution of 2D temperature-dependent tropical climate model
,”
Nonlinearity
34
(
8
),
5662
5686
(
2021
).
6.
J.
Li
and
E.
Titi
, “
Global well-posedness of strong solutions to a tropical climate model
,”
Discrete Contin. Dyn. Syst.
36
,
4495
4516
(
2016
).
7.
R.
Wan
, “
Global small solutions to a tropical climate model without thermal diffusion
,”
J. Math. Phys.
57
,
021507
(
2016
).
8.
X.
Zhai
and
Y.
Chen
, “
Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion
,”
Math. Methods Appl. Sci.
43
,
7022
7039
(
2020
).
9.
C.
Li
,
X.
Xu
, and
Z.
Ye
, “
On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model
,”
Discrete Contin. Dyn. Syst.
42
,
1535
1568
(
2022
).
10.
P.
Han
,
K.
Lei
,
G.
Liu
et al, “
The H1-uniform attractor for the 2D non-autonomous tropical climate model on some unbounded domains
,”
Bull. Korean Math. Soc.
59
,
1439
1470
(
2022
).
11.
P.
Han
,
K.
Lei
,
C.
Liu
, and
X.
Wang
, “
Global attractors for a tropical climate model
,”
Appl. Math.
68
(
3
),
329
356
(
2023
).
12.
B.
Yuan
and
X.
Chen
, “
Global regularity for the 3D tropical climate model with damping
,”
Appl. Math. Lett.
121
,
107439
(
2021
).
13.
B.
Yuan
and
Y.
Zhang
, “
Global strong solution of 3D tropical climate model with damping
,”
Front. Math. China
16
,
889
900
(
2021
).
14.
D.
Berti
,
L.
Bisconti
, and
D.
Catania
, “
A regularity criterion for a 3D tropical climate model with damping
,”
J. Math. Anal. Appl.
518
,
126685
(
2023
).
15.
D.
Berti
,
L.
Bisconti
, and
D.
Catania
, “
Global attractor for the three-dimensional Bardina tropical climate model
,”
Appl. Anal.
102
,
5123
5131
(
2023
).
16.
D.
Niu
and
H.
Wu
, “
Global strong solutions and large-time behavior of 2D tropical climate model with zero thermal diffusion
,”
Math. Methods Appl. Sci.
45
,
9341
9370
(
2022
).
17.
F.
Wang
and
S.
Yin
, “
Global solutions of 3D tropical climate model with finite energy
,”
Commun. Math. Sci.
19
,
1337
1345
(
2021
).
18.
X.
Yin
and
L.
Yan
, “
Local well-posedness for 2D stochastic tropical climate model
,”
Discrete Contin. Dyn. Syst. B
28
(
9
),
5037
5054
(
2023
).
19.
B.
Yuan
and
F.
Wang
, “
The Liouville theorems for 3D stationary tropical climate model in local Morrey spaces
,”
Appl. Math. Lett.
138
,
108533
(
2023
).
20.
M.
Chekroun
and
N. E.
Glatt-Holtz
, “
Invariant measures for dissipative dynamical systems: Abstract results and applications
,”
Commun. Math. Phys.
316
,
723
761
(
2012
).
21.
G.
Łukaszewicz
and
J. C.
Robinson
, “
Invariant measures for non-autonomous dissipative dynamical systems
,”
Discrete Contin. Dyn. Syst.
34
,
4211
4222
(
2014
).
22.
G.
Łukaszewicz
,
J.
Real
, and
J. C.
Robinson
, “
Invariant measures for dissipative dynamical systems and generalised Banach limits
,”
J. Dyn. Differ. Equations
23
,
225
250
(
2011
).
23.
C.
Zhao
and
L.
Yang
, “
Pullback attractors and invariant measures for the non-autonomous globally modified Navier–Stokes equations
,”
Commun. Math. Sci.
15
,
1565
1580
(
2017
).
24.
C.
Zhao
,
G.
Xue
, and
G.
Łukaszewicz
, “
Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
,”
Discrete Contin. Dyn. Syst. B
23
,
4021
4044
(
2018
).
25.
A.
Bronzi
,
C. F.
Mondaini
, and
R.
Rosa
, “
Trajectory statistical solutions for three-dimensional Navier–Stokes-like systems
,”
SIAM J. Math. Anal.
46
,
1893
1921
(
2014
).
26.
A.
Bronzi
and
R.
Rosa
, “
On the convergence of statistical solutions of the 3D Navier-Stokes-α model as α vanishes
,”
Discrete Contin. Dyn. Syst. A
34
,
19
49
(
2014
).
27.
A.
Bronzi
,
C. F.
Mondaini
, and
R.
Rosa
, “
Abstract framework for the theory of statistical solutions
,”
J. Differ. Equations
260
,
8428
8484
(
2016
).
28.
T.
Caraballo
,
P. E.
Kloeden
, and
J.
Real
, “
Invariant measures and statistical solutions of the globally modified Navier-Stokes equations
,”
Discrete Contin. Dyn. Syst. B
10
,
761
781
(
2008
).
29.
C.
Foias
,
R. M. S.
Rosa
, and
R. M.
Temam
, “
Properties of stationary statistical solutions of the three-dimensional Navier–Stokes equations
,”
J. Dyn. Differ. Equations
31
,
1689
1741
(
2019
).
30.
P. E.
Kloeden
,
P.
Marín-Rubio
, and
J.
Real
, “
Equivalence of invariant measures and stationary statistical solutions for theautonomous globally modified Navier-Stokes equations
,”
Commun. Pure Appl. Anal.
8
,
785
802
(
2009
).
31.
Z.
Lin
,
C.
Xu
,
C.
Zhao
et al, “
Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger type equations
,”
Discrete Contin. Dyn. Syst. B
28
,
20
(
2023
).
32.
H.
Yang
,
X.
Han
,
X.
Wang
, and
C.
Zhao
, “
Homogenization of trajectory statistical solutions for the 3D incompressible magneto-micropolar fluids
,”
Discrete Contin. Dyn. Syst. S
16
,
2672
2685
(
2023
).
33.
C.
Zhao
and
T.
Caraballo
, “
Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations
,”
J. Differ. Equations
266
,
7205
7229
(
2019
).
34.
C.
Zhao
,
Y.
Li
, and
T.
Caraballo
, “
Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications
,”
J. Differ. Equations
269
,
467
494
(
2020
).
35.
C.
Zhao
,
Y.
Li
, and
G.
Łukaszewicz
, “
Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids
,”
Z. Angew. Math. Phys.
71
,
141
(
2020
).
36.
C.
Zhao
,
Y.
Li
, and
Z.
Song
, “
Trajectory statistical solutions for the 3D Navier–Stokes equations: The trajectory attractor approach
,”
Nonlinear Anal.: Real World Appl.
53
,
103077
(
2020
).
37.
C.
Zhao
,
Z.
Song
, and
T.
Caraballo
, “
Strong trajectory statistical solutions and Liouville type equation for dissipative Euler equations
,”
Appl. Math. Lett.
99
,
105981
(
2020
).
38.
C.
Zhao
,
T.
Caraballo
, and
G.
Łukaszewicz
, “
Statistical solution and Liouville type theorem for the Klein-Gordon-Schrödinger equations
,”
J. Differ. Equations
281
,
1
32
(
2021
).
39.
C.
Zhao
,
Y.
Zhang
,
T.
Caraballo
, and
G.
Łukaszewicz
, “
Statistical solutions and degenerate regularity for the micropolar fluid with generalized Newton constitutive law
,”
Math. Methods Appl. Sci.
46
,
10311
10331
(
2023
).
40.
R.
Wang
,
T.
Caraballo
, and
N.
Tuan
, “
Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications
,”
Proc. Am. Math. Soc.
151
,
2449
2458
(
2023
).
41.
D.
Yang
,
Z.
Chen
, and
T.
Caraballo
, “
Dynamics of a globally modified Navier–Stokes model with double delay
,”
Z. Angew. Math. Phys.
73
,
216
(
2022
).
42.
C.
Zhao
,
J.
Wang
, and
T.
Caraballo
, “
Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations
,”
J. Differ. Equations
317
,
474
494
(
2022
).
43.
C.
Zhao
and
R.
Zhuang
, “
Statistical solutions and Liouville theorem for the second order lattice systems with varying coefficients
,”
J. Differ. Equations
372
,
194
234
(
2023
).
44.
A.
Cheskidov
and
L.
Kavlie
, “
Degenerate pullback attractors for the 3D Navier–Stokes equations
,”
J. Math. Fluid Mech.
17
,
411
421
(
2015
).
45.
C.
Foias
,
O.
Manley
,
R.
Rosa
et al,
Navier-Stokes Equations and Turbulence
(
Cambridge University Press
,
Cambridge
,
2001
).
46.
J.
García-Luengo
,
P.
Marín-Rubio
, and
J.
Real
, “
Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behavior
,”
J. Differ. Equations
252
,
4333
4356
(
2012
).
47.
W.
Rudin
,
Real and Complex Analysis
(
China Machine Press
,
Beijing
,
2004
).
You do not currently have access to this content.