Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 74, pp. 1–10.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.74

WELL-POSEDNESS OF SOLUTIONS FOR THE 2D STOCHASTIC

QUASI-GEOSTROPHIC EQUATION IN CRITICAL

FOURIER-BESOV-MORREY SPACES

HASSAN KHAIDER, ACHRAF AZANZAL, ABDERRAHMANE RAJI

Abstract. In this article, we apply the Itô integral to obtain the global

solutions for stochastic quasi-geostrophic equations in Fourier-Besov-Morrey
spaces. For comparison we also give the corresponding results of the determin-

istic quasi-geostrophic equations. We assume the initial data is F0 measurable
and the right-hand side is a random function in a Morrey space, to obtain the

well posedness of stochastic quasi-geostrophic equations.

1. Introduction:

In this article we study the two-dimensional dissipative stochastic quasi-geostro-
phic (SQG) equation

∂tθ + Vθ · ∇θ + µΛ2αθ = gẆ , x ∈ R2, t > 0,

Vθ = (−R2θ,R1θ),

θ(0, x) = θ0(x).

(1.1)

Where µ > 0 is the dissipative coefficient, α ≥ 1/2 is a real number, θ(t, x) is a
real-valued function of two space variables t and x. The function θ represents the
potential temperature, g Ẇ is a random external force and W is a standard infinite
dimensional Wiener process. The velocity Vθ is incompressible and determined
from θ by a stream function ζ,

Vθ =
(
− ∂ζ

∂x2
,
∂ζ

∂x1

)
, (1.2)

where the ζ function is satisfied

Λζ = −θ. (1.3)

We define the operator Λ by the fractional power of −∆ :

Λv = (−∆)1/2v, F(Λv) = F((−∆)
1
2 v) = |ξ|F(v),

and more generally

F(Λ2αv) = F((−∆)αv) = |ξ|2αF(v),

2020 Mathematics Subject Classification. 35Q35, 42B37, 35Q85, 35R60.
Key words and phrases. Itô integral; stochastic quasi-geostrophic equations;

Fourier-Besov-Morrey spaces; partial differential equations.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted June 26, 2024. Published November 20, 2024.

1



2 H. KHAIDER, A. AZANZAL, A. RAJI EJDE-2024/74

where F is the Fourier transform. The relation between (1.2) and (1.3) can be
determined by using the Riesz transform as follows

Vθ = (∂x2
Λ−1θ,−∂x1

Λ−1θ) = (−R2θ,R1θ),

where Rj , j = 1, 2, is the Riesz transform defined by

Rj = ∂xj
(−∆)−1/2.

The Quasi-geostrophic equations are frequently used to study the large-scale
motions of the ocean and atmosphere at mid-latitudes, for the modelling of marine
and atmospheric circulation, as well as for stability, frontogenesis and turbulence
studies, because they are much simpler than the basic equations These equations
have been the subject of several researchers. The solutions depend discontinuously
on the initial data, the approximate finite element solutions converge in regions of
free flow [12], and the wavenumber energy spectra of the solutions asymptotically
approach the statistical equilibrium spectra of the spectrally truncated equations
(Kraichnan 1967).

When µ = 0 and g = 0, the equation (1.1) will be called the two-dimensional
non-dissipative quasigeostrophic equation, it was introduced by Constantin, Majda,
and Tabak in 1994 [9].

When g = 0. It is clear that the small case is mathematically very interesting to
understand, or equivalently when the dissipation tends to 0, the quasi-geostrophic
equation converges to the three dimensional Navier-Stokes equations. Although,
this model is physically interesting in the case α = 1

2 . Mathematically, the power
α = 1/2 corresponds to the index for which the nonlinear term and the dissipation
are of the same order (in the sense that Vθ and Λ are two operators deriving once).
The previous remarks motivate the following definition.

Definition 1.1 ([2]). We distinguish the following 3 cases:

• If 0 ≤ α < 1/2, then (1.1) is called supercritical.
• If α = 1/2, then (1.1) is called critical.
• If 1/2 < α ≤ 1, then (1.1) is called subcritical.

There is a copious literature on well-posedness for fluid dynamics PDEs with
singular data in different spaces, where the conditions are taken in norms of critical
spaces. For instance, for Navier-Stokes equations and related models, we have well-
posedness results in the critical case of the following spaces: Lebesgue space Lp [6],
Marcinkiewicz space Lp,∞ [10], Morrey spaces Mµ

p [7, 13, 14, 17], Besov-Morrey
spacesN s

p,λ,q [8, 22], Fourier-Besov spaces FBs
p,q [5, 9], Fourier-Besov-Morrey spaces

FN s
p,λ,q [1, 3].

The small perturbations (numerical, empirical, and physical errors) or thermo-
dynamic fluctuations present in fluid flows are frequently modeled using stochastic
components in the equations of motion. Additionally, they are employed in order to
comprehend turbulence better. Consequently, stochastic partial differential equa-
tions (SDE) such as quasi-geostrophic equations (QG), stochastic Navier-Stokes
equations are gaining more and more interest in fluid mechanics research.

The well-posedness of partial differential equations (PDEs) in Fourier-Besov-
Morrey spaces is crucial for several reasons related to both the theoretical and
practical aspects of solving PDEs. These spaces allow for the description of func-
tions that may not be smooth but still satisfy certain integrability conditions. They
are typically used when analyzing the regularity of solutions to PDEs with less



EJDE-2024/74 2D STOCHASTIC QUASI-GEOSTROPHIC EQUATION 3

regular data or in the context of singular integrals. They are useful for dealing
with solutions that are not necessarily in classical Sobolev spaces but still exhibit
sufficient regularity for solving certain types of PDEs. In this research Fourier-
Besov-Morrey spaces are particularly suited for handling the scaling properties of
nonlinear operators that arise in stochastic quasi-geostrophic equations.

Biswas [5] established the global-in-time well-posedness of (1.1) in the space
FBs

p,q (wich is a particular case of Fourier-Besov-Morrey space FN s
p,λ,q by taking

λ = 0). In 2003, Chae and Lee [8] obtained the global well-posedness in the
super-critical dissipative quasigeostrophic equations in N s

p,λ,q. Recently, Azanzal

et al. [2] obtained the existence in Fourier-Besov-Morrey spaces FN s
p,λ,q but in the

deterministic case. Consequently, our research extends the previous works.

Definition 1.2 ([23]). Let (Ω, F, P, {Ft}t∈[0,T ]) be a filtered probability space with

the expectation E and T > 0, we designate by MT the smallest σ-algebra of Ft

adapted distribution processes f defined on Ω× [0, T ]×R3, which are progressively
measurable, more precisely f(ω, t, ·) ∈ Ft × B[0, T ] for all t ∈ [0, T ].

Definition 1.3. Let
(
Ω, F, P, {Ft}t≥0 ,W

)
be a fixed probability basis, the diver-

gence free process θ is a mild-solution of (1.1), if θ(ω, ·) ∈ L̃4(0, t;FṄ 3−2α− 2
P

p,λ,q )∩Mt

for all t ≥ 0 and

θ(t) = Tα(t)θ0 −
∫ t

0

Tα(t− s)(Vθ · ∇θ)(s) ds+

∫ t

0

Tα(t− s)g dW. (1.4)

To examine how stochastic forces affect quasi-geostrophic equations, we first
present the outcome of the deterministic quasi-geostrophic equations, or the case
g = 0 in (1.1). So to speak

∂tθ + Vθ · ∇θ + µΛ2αθ = 0, x ∈ R2, t > 0,

Vθ = (−R2θ,R1θ),

θ(0, x) = θ0(x).

(1.5)

Theorem 1.4 ([2] Deterministic case). Let g = 0, 1 ≤ p < ∞, 1 ≤ λ < 2,

1 ≤ q ≤ ∞, 1
2 ≤ α ≤ 2 + λ−2

2p and θ0 ∈ FṄ 3−2α− 2−λ
P

p,λq . Then, there exists a

constant β > 0 such that if θ0 satisfies ∥θ0∥
FṄ

3−2α− 2−λ
P

p,λ,q

≤ β, then (1.5) admits

a unique global solution θ ∈ C(R+,FṄ 3−2α− 2−λ
P

p,λ,q ) ∩ L1(R+,FṄ 3− 2−λ
P

p,λ,q ), such that

∥θ∥X ≲ ∥θ0∥
FṄ

3−2α− 2−λ
P

p,λ,q

, where

X = L∞(R+,FṄ 3−2α− 2−λ
P

p,λ,q ) ∩ L1(R+,FṄ 3− 2−λ
P

p,λ,q ).

Furthermore, the solution θ depends continuously on the initial data θ0.

Our main result reads as follows.

Theorem 1.5 (Stochastic ase). Assume 1 ≤ q ≤ ∞ and 1
2 ≤ α ≤ 3

2 + λ
4 . Let

(Ω, F, P, {Ft}t≥0,W ) be a probability basis and θ0 be F0 measurable, g ∈ MT .
Assume that for any positive T ,

(1 + T )∥g∥
L̃4

ΩL̃4
T (FṄ

2−2α+λ
2

2,λ,q )
+ ∥u0∥

L̃4
Ω(FṄ

2−2α+λ
2

2,λ,q )
< +∞.



4 H. KHAIDER, A. AZANZAL, A. RAJI EJDE-2024/74

Then, there exists a unique global mild solution of (1.1) in L̃4(0, T ;FṄ 2−2α+λ
2

2,λ,q ),

for all ω in Ω̃, with Ω̃ is a random set with positive probability.

The structure of this article is as follows. In the second section, we review
Bernstein’s lemma, the fixed point lemma, the definition of the Fourier-Besov-
Morrey space, and the Littlewood-Paley theory. In the third section, we present
the proof of the Theorem 1.5 .

2. Preliminaries

Here we provide notation and review the fundamental characteristics of Fourier-
Besov-Morrey spaces, which will be utilized throughout the article. First, we should
review the Fourier transform, Littlewood-Paley theory, and the definitions of our
spaces. For more details we refer to [15, 4]. The Fourier transform is described as

f̂(ξ) = Ff(ξ) = (2π)−n/2

∫
Rn

e−ix·ξf(x) dx.

And its inverse Fourier transform as

f̆(x) = F−1f(x) = (2π)−n/2

∫
Rn

eix·ξf(ξ) dξ.

Let φ ≥ 0 be a C∞
0 function with suppφ ⊂ {3/4 ≤ |ξ| ≤ 8/3} and nonnegative

χ ∈ C∞
0 (B(0, 4

3 )) such that

χ(ξ) +
∑
j≥0

φj(ξ) = 1, ξ ∈ Rn,

∑
j∈Z

φj(ξ) = 1, ξ ∈ Rn\{0},

where φj(ξ) = φ(2−jξ).
Let S ′

h = S ′/P, where P is the set of polynomials,

hj = F−1φj , h̃j = F−1χj = F−1χ(2−j .),

∆̇j = F−1φj∗, Ṡju = F−1χj ∗ u,

where

χj = χ(2−j ·).
Now we give some definitions concerning Fourrier-Besov-Morrey space.

Definition 2.1. (i) Let 1 ≤ p ≤ ∞ and 0 ≤ λ < d. The homogeneous Morrey
space Mλ

p is the set of all functions f ∈ Lp(B(x, r)) such that

∥f∥Mλ
p
= sup

x∈Rd

sup
r>0

r−
λ
p ∥f∥Lp(B(x,r)) < ∞, (2.1)

where B(x, r) is the open ball in Rd centered at x and with radius r > 0.
(ii) Let 1 ≤ p, q ≤ ∞ and s ∈ R. We can define the homogeneous Fourier-Besov

space FBs
p,q as the set of all distributions f ∈ S ′\P is the set of all polynomials

such that the norm ∥f∥FBs
p,q

is finite, where

∥f∥FBs
p,q

:=


(∑

j∈Z 2
jsq∥φj f̂∥qLp

)1/q

for q < ∞
supj∈Z 2

js∥φj f̂∥Lp for q = ∞.
(2.2)



EJDE-2024/74 2D STOCHASTIC QUASI-GEOSTROPHIC EQUATION 5

(iii) Let 1 ≤ p, q ≤ ∞, 0 ≤ λ < n and s ∈ R. The homogeneous Fourier-Besov-
Morrey space FN s

p,λ,q is defined as the set of all distributions f ∈ S ′\P, such that

the norm ∥f∥FN s
p,λ,q

is finite, where

∥f∥FN s
p,λ,q

:=

{
(
∑

j∈Z 2
jsq∥φj f̂∥qMλ

p
)1/q for q < ∞

supj∈Z 2
js∥φj f̂∥Mλ

p
for q = ∞.

(2.3)

Note that the space FN s
p,λ,q(Rn) equipped with the norm (2.2) is a Banach space.

Since M0
p = Lp, we have FN s

p,0,q = FḂ
s

p,q , and FN s
1,0,1 = FḂ

s

1,1 = X s where X s

is the Lei-Lin space [4].

Definition 2.2. Let 1 ≤ ρ ≤ +∞ and T ∈ (0,+∞], the space L̃ρ
T (FṄ s

p,λ,q(Rd))
called Chemin–Lerner type space is defined as the set of tempered distributions in

S ′
(
R× Rd

)
/P with respect to the norm

∥f∥L̃ρ
T (FṄ s

p,λ,q(Rd)) := ∥
{
2js∥φj f̂∥Lρ([0,T ];Mλ

p (Rd))

}
∥lq(j∈Z) < +∞.

Definition 2.3. Let p, r, σ ∈ (1,+∞] , 1 ≤ q < +∞ , s ∈ R and T > 0. We

define the space (L̃σ
ΩL̃

r
TFṄ s

p,λ,q(Rd)) called Chemin–Lerner type space of Bochner

(CLBFB) as the space of distribution process g ∈ MT such that g(ω, t) ∈ S ′
h(Rd)

and the quasi-norm

∥g∥L̃σ
ΩL̃r

TFṄ s
p,λ,q

= ∥
{
2js

[
E(∥φj ĝ(t)∥Lr

TMλ
p
)σ
]1/σ}∥lq(j∈Z) < +∞.

Lemma 2.4 ([16]). Let 1 ≤ p1, p2, p3 < ∞ and 0 ≤ λ1, λ2, λ3 < d.
(i) (Hölder’s inequality) If 1

p3
= 1

p1
+ 1

p2
and λ3

p3
= λ1

p1
+ λ2

p2
, then

∥fg∥
M

λ3
p3

≤ ∥f∥
M

λ1
p1

∥g∥
M

λ2
p2

. (2.4)

(ii) (Young’s inequality) If ϕ ∈ L1 and h ∈ Mλ1
p1
, then

∥ϕ ∗ h∥
M

λ1
p1

≤ ∥ϕ∥L1∥h∥
M

λ1
p1

, (2.5)

where ∗ denotes the standard convolution operator.

Now, we recall the Bernstein-type lemma in Fourier variables.

Lemma 2.5. Let 1 ≤ p2 ≤ p1 < ∞, 0 ≤ λ1, λ2 < d, d−λ1

p1
≤ d−λ2

p2
and let β be a

multi-index. If supp(φ̂) ⊂ {|ξ| ≤ A2j}, then there is a constant C > 0 independent
of φ and j such that

∥(iξ)βφ̂∥
M

λ2
p2

≤ C2j|β|+j(
d−λ2
p2

− d−λ1
p1

)∥φ̂∥
M

λ1
p1

. (2.6)

Definition 2.6 ([18]). Let V and W be finite-dimensional inner product spaces,
and let m = (m(t))t≥0 be a continuous V -valued semimartingale. Itô’s formula says
that if F ∈ C2(V ;W ), then

dF (m(t)) = DF (m(t))[dm(t)] +
1

2
D2F (m(t))[dm(t), dm(t)],

where DkF is the kth derivative of F .



6 H. KHAIDER, A. AZANZAL, A. RAJI EJDE-2024/74

Theorem 2.7 (Burkholder-Davis-Gundy inequalities [19]). For each p > 0 there
exist two constants cp and Cp such that for any continuous local martingale Λ
vanishing at 0 we have

cpE
[〈
m,m

〉p/2
∞

]
≤ E

(
m∗

∞
)p ≤ CpE

[〈
m,m

〉p/2
∞

]
.

Stopping at a time T , Theorem 2.7 leads to the following result, which is never-
theless very important in applications.

Corollary 2.8 ([19]). For a stopping time T one has

cpE
[
⟨m,m⟩p/2T

]
≤ E

(
m∗

T

)p ≤ CpE
[〈
m,m

〉p/2
T

]
.

In general for a bounded predictable process H we have

cpE
[( ∫ T

0

H2
sd⟨m,Λ⟩s

)p/2]
≤ E

[
sup
t≤T

∣∣∣ ∫ t

0

Hsdms

∣∣∣p]
≤ CpE

[( ∫ T

0

H2
sd⟨m,m⟩s

)p/2]
.

We shall review briefly the existence and uniqueness of an abstract operator
equation in a Banach space at the end of this section. The purpose of this is to
illustrate Theorem 1.5.

Lemma 2.9. . Let B be a Banach space with norm ∥ · ∥B and L : B× B → B be a
bounded bilinear operator satisfying

∥L(θ1, θ2)∥B ≤ β∥θ1∥B∥θ2∥B,

for all θ1, θ2 ∈ B and a constant β > 0. Then, if 0 < ε < 1
4β and if y ∈ B such that

∥y∥B ≤ ε, the equation x := y + L(x, x) has a solution x in B such that ∥x∥B ≤ 2ε.
This solution is the only one in the ball B(0, 2ε). Moreover, the solution depends
continuously on y in the sense: if ∥y′∥B < ε, x′ = y′ + L(x′, x′), and ∥x′∥B ≤ 2ε,
then

∥x− x′∥B ≤ 1

1− 4εβ
∥y − y′∥B .

3. Proof of Theorem 1.5

The random term in equation (1.4) must also be addressed in order to find
a solution, so using the superposition principle, we take the following auxiliary
Cauchy problem

du+ µΛ2αu dt = g dW in Ω× (0,+∞)× R2,

ut=0 = u0 on Ω× R2.
(3.1)

Applying the Fourier transform of (3.1) with regard to the spatial variable produces,
we obtain

dû+ |ξ|2αû dt = ĝ dW in Ω× (0,+∞)× R2,

ût=0 = û0 on Ω× R2.
(3.2)

we can conclude that this linear stochastic ODE has a unique solution, So by the
Fourier transformation we can also obtain the solution of (3.1).We must estimate
the solution of (3.1) in order to use the fixed point theory to find the solution of
(1.1).



EJDE-2024/74 2D STOCHASTIC QUASI-GEOSTROPHIC EQUATION 7

Lemma 3.1. Let θ0 be F0 measurable and f progressively measurable on Ω ×
[0, T ] × R2, and for any q ∈ [2,+∞), θ0 ∈ L̃4

ΩFṄ 2−2α+λ
2

2,λ,q , f ∈ L̃4
ΩL̃

4
TFṄ 2−2α+λ

2

2,λ,q ,

the solution u of (3.1) is in the space L̃4
ΩL̃

4
TFṄ 2−2α+λ

2

2,λ,q , and

∥u∥L̃4
ΩL̃4

TFṄ 2−2α
2,λ,q

≤ C(1 + T )∥f∥
L̃4

ΩL̃4
TFṄ

2−2α+λ
2

2,λ,q

+ ∥θ0∥
L̃4

ΩFṄ
2−2α+λ

2
2,λ,q

. (3.3)

Proof. By multiplying the first equation in (3.2) by φj on both sides, we obtain

dφj û = −|ξ|2αφj û+ φj ĝ dW.

We apply Itô formula to ∥φj û∥Mλ
2
to obtain

d∥φj û∥2Mλ
2

= d( sup
x∈R2

sup
r>0

r−
λ
2 ∥φj û∥L2(B(x,r)))

2

= sup
x∈R2

sup
r>0

r−λd∥φj û∥2L2(B(x,r))

= sup
x∈R2

sup
r>0

r−λ
[
2⟨φj û,−|ξ|2α(φj û) + (φj ĝ) dW ⟩+ ∥φj ĝ∥2L2(B(x,r)) dt

]
= sup

x∈R2

sup
r>0

r−λ
[
(−2∥|ξ|2αφj û∥2L2(B(x,r)) + ∥φj ĝ∥2L2(B(x,r))) dt+ 2⟨φj û, φj ĝ⟩ dW

]
≲

(
− 2∥|ξ|2αφj û∥2Mλ

2
+ ∥φj ĝ∥2Mλ

2

)
+ 2 sup

x∈R2

sup
r>0

r−λ⟨φj û, φj ĝ⟩ dW.

(3.4)
where ⟨·, ·⟩, denotes the inner product in L2(R2).

Again applying Itô formula to (∥φj û∥2Mλ
2
+ ε)2 for ε > 0. As a result of (3.3),

we have

d(∥φj û∥2Mλ
2
+ ε)2

= 2(∥φj û∥2Mλ
2
+ ε)

[
(−2∥|ξ|2αφj û∥2Mλ

2
+ ∥φj ĝ∥2Mλ

2
) dt

+ 2 sup
x∈R2

sup
r>0

r−λ⟨φj û, φj ĝ⟩ dW
]
+ 4 sup

x∈R2

sup
r>0

r−λ⟨φj ĝ, φj û⟩2 dt.

(3.5)

Considering the sequence of stopping times

τN =

{
inf{t ≥ 0 : ∥φj û∥ > N}, if {t : ∥φj û∥ > N} ≠ ∅,
T, if {t : ∥φj û∥ > N} = ∅.

for N = 1, 2, . . .. Integrating (3.5) [0, t] for t ≤ min {T, τN} over interval and taking
the expectation of the resulting term, we obtain

E(∥φj û∥2Mλ
2
+ ε)2 − E(∥φj θ̂0∥2Mλ

2
+ ε)2

= 4E
∫ t

0

sup
x∈R2

sup
r>0

r−λ⟨φj ĝ, φj û⟩2 ds− 4E
∫ t

0

(
∥φj û∥2Mλ

2
+ ε

)
∥|ξ|2αφj û∥2Mλ

2
ds

+ 2E
∫ t

0

(
∥φj û∥2Mλ

2
+ ε

)
∥φj ĝ∥2Mλ

2
ds

+ 4E
∫ t

0

(∥φj û∥2Mλ
2
+ ε) sup

x∈R2

sup
r>0

r−λ⟨φj û, φj ĝ⟩ dW

:= A1 +A2 +A3 +A4.



8 H. KHAIDER, A. AZANZAL, A. RAJI EJDE-2024/74

Next we estimate A1, A3 and A4 since A2 is already the form needed. For that we
apply Hölder’s and Young’s inequalities.

A1 ≲ E
∫ t

0

∥φj ĝ∥2Mλ
2
∥φj û∥2Mλ

2
ds

≲ E sup
s∈[0,t]

∥φj û∥2Mλ
2

∫ t

0

∥φj ĝ∥2Mλ
2
ds

≲ εE sup
s∈[0,t]

∥φj û∥4Mλ
2
+ CεtE

∫ t

0

∥φj ĝ∥4Mλ
2
ds.

A3 ≲ εE sup
s∈[0,t]

(∥φj û∥2Mλ
2
+ ε)2 + CεtE

∫ t

0

∥φj ĝ∥4Mλ
2
ds.

Fore estimating the random integral A4, we apply the Burkholder-Davis-Gundy
inequality and Young’s inequality.

A4 ≲ E sup
s′∈[0,t]

∣∣ ∫ s′

0

(∥φj û∥2Mλ
2
+ ε)⟨φj û, φj ĝ⟩dW

∣∣
≲ E sup

s∈[0,t]

(∥φj û∥2Mλ
2
+ ε)∥φj û∥Mλ

2

(∫ t

0

∥φj ĝ∥2Mλ
2
ds
)1/2

≲ εE sup
s∈[0,t]

[(∥φj û∥2Mλ
2
+ ε)∥φj û∥Mλ

2
]
4
3 + CεEt

∫ t

0

∥φj ĝ∥4Mλ
2
ds.

Combining the estimates of A1, A2, A3 and A4, assuming ε > 0 to be sufficiently
small and passing to the limit as ε → 0, we obtain

E sup
t∈[0,T∧τN ]

∥φj û∥4Mλ
2
+ E

∫ T∧τN

0

∥φj û∥2Mλ
2
∥|ξ|2αφj û∥2Mλ

2
ds

≲ E∥φj θ̂0∥4Mλ
2
+ [1 + (T ∧ τN )]E

∫ T∧τN

0

∥φj ĝ∥4Mλ
2
ds.

(3.6)

Considering the conditions on θ0 and g, thus E
(
supt∈[0,T∧τ ] ∥φj v̂∥4

)
is bounded

by a constant independent of N by (3.6). Hence, let N → ∞ and consider
limN→∞ τN = T , almost certainly. Applying (2.5), we have

E sup
t∈[0,T ]

∥φj û∥4Mλ
2
+ 22αjE

∫ T

0

∥φj û∥4Mλ
2
ds

≲ E∥φj θ̂0∥4Mλ
2
+ (1 + T )E

∫ T

0

∥φj ĝ∥4Mλ
2
ds.

(3.7)

Hence,

22αjE
∫ T

0

∥φj û∥4Mλ
2
ds ≲ E∥φj θ̂0∥4Mλ

2
+ (1 + T )E

∫ T

0

∥φj ĝ∥4Mλ
2
ds.

Multiplying the above estimate by 2(
3
2−2α+λ

2 )j and taking lq norm, we obtain

∥u∥
L̃4

ΩL̃4
TFṄ

2−2α+λ
2

2,λ,q

≤ C(1 + T )∥g∥
L̃4

ΩL̃4
TFṄ

2−2α+λ
2

2,λ,q

+ ∥θ0∥
L̃4

ΩFṄ
2−2α+λ

2
2,q

,

where the condition α ≤ 3
2+

λ
4 (i.e, 3

2−2α+λ
2 ≥ 0). Which yields the conclusion. □



EJDE-2024/74 2D STOCHASTIC QUASI-GEOSTROPHIC EQUATION 9

Lemma 3.2 ([20]). . Under condition 3.1, for the solution u of (3.1), there is a

set Ω̃ with positive probability such that u(ω, ·, ·) ∈ L̃4
TFṄ 2−2α+λ

2

2,λ,q with

∥u(ω, ·, ·)∥
L̃4

TFṄ
2−2α+λ

2
2,λ,q

≤ C1

[
∥θ0∥L̃4

ΩFṄ 2−2α
2,λq +λ

2
+ (1 + T )∥g∥

L̃4
ΩL̃4

TFṄ
2−2α+λ

2
2,λ,q

]
,

for all ω ∈ Ω̃, where C1 is a constant.

Proof of Theorem 1.5. We take the working space Z := L̃4
TFṄ 2−2α+λ

2

2,λ,q and define
mappings

Ψ(θ) = Tα(t)θ0 +

∫ t

0

Tα(t− s)g dW −
∫ t

0

Tα(t− s)(Vθ · ∇θ)(s) ds,

e0 = Tα(t)θ0 +

∫ t

0

Tα(t− s)g dW,

B(θ, ξ) = −
∫ t

0

Tα(t− s)(Vθ · ∇ξ)(s) ds.

Thus, to solve (1.1) it is sufficient to search for the fixed point of the mapping Ψ(θ)
by (2.9). By the bilinear estimate in Theorem 1.4 [2], we have

∥
∫ t

0

Tα(t− s)(Vθ · ∇θ)(s) ds∥Z ≤ C0∥θ∥Z∥θ∥Z .

Thus, Lemma 3.2 leads to

∥Ψ(θ)∥Z ≤ ∥Tα(t)θ0 +

∫ t

0

Tα(t− s)g dW∥Z + ∥
∫ t

0

Tα(t− s)(Vθ · ∇θ)(s) ds∥Z

≤ ∥u(ω, ·, ·)∥
L̃4

TFṄ
2−2α+λ

2
2,q

+ C0∥θ∥2Z .

By (2.9), All that remains is for us to prove that

∥u(ω, ·, ·)∥
L̃4

TFṄ
2−2α+λ

2
2,λ,q

≤ 1

4C1C0
,

with positive probability. Indeed, by Lemma 3.2 we have

∥u(ω, ·, ·)∥
L̃4

TFṄ
2−2α+λ

2
2,λ,q

≤ C1

[
∥θ0∥

L̃4
ΩFṄ

2−2α+λ
2

2,λ,q

+ (1 + T )∥g∥
L̃4

ΩL̃4
TFṄ

2−2α+λ
2

2,λ,q

]
≤ 1

4C0
,

for all ω ∈ Ω̃. This completes the proof. □

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Hassan Khaider

Laboratory LMACS, Faculty of Science and Technology of Beni Mellal, Sultan Moulay
Slimane University, Beni Mellal, BP 523, 23000, Morocco

Email address: hassankhaider1998@gmail.com

Achraf Azanzal
Laboratory LEST. High School of Education and Formation (ESEF), Hassan First Uni-

versity, Settat, Morocco
Email address: achraf0665@gmail.com

Abderrahmane Raji
Laboratory LMACS, Faculty of Science and Technology of Beni Mellal, Sultan Moulay

Slimane University, Beni Mellal, BP 523, 23000, Morocco
Email address: rajiabd2@gmail.com


	1. Introduction:
	2. Preliminaries
	3. Proof of Theorem ??
	References