Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 79, pp. 1–25.

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

DOI: 10.58997/ejde.2025.79

STABILITY OF LERAY WEAK SOLUTIONS TO 3D NAVIER-STOKES

EQUATIONS

ZUJIN ZHANG, WEIJUN YUAN, ZHENGAN YAO

Abstract. In this article, we show that if the Leray weak solution u of the three-dimensional
Navier-Stokes system satisfies

∇u ∈ Lp(0,∞; Ḃ0
q,∞(R3)),

2

p
+

3

q
= 2,

3

2
< q < ∞,

or

∇u ∈ L
2

2−r (0,∞; Ḃ−r
∞,∞(R3)), 0 < r < 1,

then u is uniformly stable, under small perturbation of initial data and external force, is asymp-

totically stable in the L2 sense, is unique amongst all the Leray weak solutions, and satisfies
some energy type equalities. Also under spectral condition on the initial perturbation, we obtain

optimal upper and lower bounds of convergence rates. Our results extend the results in [6, 11].

1. Introduction

In this article, we study the incompressible Navier-Stokes equations (NSE) in R3 ,

∂tu+ (u · ∇)u−∆u+∇Πu = f ,

∇ · u = 0,

u|t=0 = u0,

(1.1)

where u = (u1, u2, u3) is fluid velocity field, Πu presents the pressure, f is an external force, and
u0 is the prescribed initial data satisfying the compatibility condition ∇ · u0 = 0 . Hereafter, we
use the following notation:

∂t =
∂

∂t
, ∂i =

∂

∂xi
, ∇ = (∂1, ∂2, ∂3), ∆ =

3∑
i=1

∂2
i , (u · ∇) =

3∑
i=1

ui∂i, ∇ · u =

3∑
i=1

∂iui.

For finite energy initial data u0, the existence of a global weak solution satisfying the basic
energy estimate

∥u(t)∥2L2 + 2

∫ t

0

∥∇u(s)∥2L2ds ≤ ∥u0∥2L2 (∀t ≥ 0) (1.2)

of (1.1) has been established in the pioneer works of Leray [14] and Hopf [10] (for the case of
bounded domains). However, the problem of regularity and uniqueness of such a weak solution
remains open. To understand the above problem, Leray [14] posed the following time decay
problem: whether or not the weak solution u of (1.1) with f = 0 satisfies limt→∞ ∥u(t)∥L2 = 0
. This was confirmed in [18] (and references therein). Notice that the time decay problem can
be renamed as the asymptotically stable problem of the trivial solution u = 0 . It is natural to
investigate the stability issue of nontrivial solution of (1.1) with f not small for large t . Precisely,

2020 Mathematics Subject Classification. 35Q30, 76D03.

Key words and phrases. Navier-Stokes equations; stability; energy equality; weak-strong uniqueness.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 25, 2025 Published July 24, 2025.

1



2 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

we consider the following 3D perturbed Navier-Stokes equations

∂tv + (v · ∇)v −∆v +∇Πv = f + g,

∇ · v = 0,

v|t=0 = u0 +w0,

(1.3)

where g(x, t) is a perturbed force and w0(x) is a perturbed initial velocity field. The study of
stability behaviors (uniform stability and asymptotic stability) is beneficial to the understanding
of regularity and uniqueness of Leray weak solutions of (1.1).

For the uniform stability, Ponce-Racke-Sideris-Titi [17] showed that if the Leray weak solution
u of (1.3) with f = 0 satisfies

∇u ∈ L4(0,∞;L2(R3)), (1.4)

then condition

∥w0∥H1 +

∫ ∞

0

(
∥g(t)∥L2 + ∥g(t)∥2L2

)
dt ≤ δ

for sufficiently small δ > 0 implies that (1.3) has a unique global solution v(t) with the property

sup
t>0

∥u(t)− v(t)∥H1 ≤ M(δ), lim
δ→0

M(δ) = 0.

Gui-Zhang [9] made an important improvement in the sense that they studied the uniform
stability of weak solution of horizontal viscous Navier-Stokes equations in the anisotropic Sobolev
spaces C(0,∞;H0,s(R3)) . Gallagher-Planchon [8] considered the n-dimensional perturbed Navier-
Stokes equations, and showed the uniform stability under the assumption

u ∈ Lp(0, T ; Ḃ
2
p+

n
q −1

q,p (Rn)),
2

p
+

n

q
= 1, 2 < q, p < ∞. (1.5)

Here, Ḃ
2
p+

n
q −1

q,p (Rn) is the homogenous Besov space, see Section 2 for details. Recently, Dong-Jia
[6] covered the limiting case p = ∞ in (1.5), and their assumption ensuring the uniform stability
of Leray weak solution is

∇u ∈ Lp(0, T ; Ḃ0
q,∞(R3)),

2

p
+

3

q
= 2, 1 < p < ∞, 2 ≤ q < ∞. (1.6)

On the other hand, it is also interesting and important to investigate the asymptotic stability.
In the case f = g = 0 and w0 ∈ L1(R3) ∩ Lr(R3) (r > 3) with ∥w0∥Lr sufficiently small,
Beirão da Veiga-Secchi [2] derived that there exists a unique global solution v of (1.3) converging
asymptotically to a weak solution u of (1.1) with f = 0 in the sense that

∥v(t)− u(t)∥Lr ≤ C(1 + t)−
3
4 ,

under the subcritical assumption that

u ∈ L∞(0,∞;Lr+2(R3)). (1.7)

Later, Kozono [13] removed the smallness assumption on w0 and g . Precisely, he showed that
the weak solution v of (1.3) with f ∈ L2

loc(0,∞;L2(R3)) and g ∈ L1(0,∞;L2(R3)) converges
asymptotically to the solution u of (1.1), provided that

u ∈ Lp(0,∞;Lq(R3)),
2

p
+

3

q
= 1, 3 < q ≤ ∞. (1.8)

Then Zhou [21] extended (1.8) as

∇u ∈ Lp(0,∞;Lq(R3)),
2

p
+

3

q
= 2,

3

2
< q ≤ ∞. (1.9)

Recently, Dong-Jia [6] refined (1.9) to be (1.6).
The purpose of this article is to explore the uniform stability and asymptotic stability for weak

solutions of 3D Navier-Stokes equations in the critical Besov spaces under large perturbation of
initial data and external force. Roughly, we shall extend the range of q in (1.6) to be of full range
3
2 < q < ∞; and we can even take the regularity index to be negative. Moreover, under some



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 3

spectral condition on the initial perturbation, the optimal upper and lower bounds of convergence
rates are derived.

Before stating the main results, let us first recall the definition of Leray weak solution of the
Navier-Stokes system (1.1) (see [19] for instance).

Definition 1.1. Let u0 ∈ L2(R3) and f ∈ L2
loc(0,∞;L2(R3)) . A measurable function u(x, t)

is called a Leray weak solution of the Navier-Stokes system (1.1) if the following four conditions
hold:

(1) u ∈ L∞
loc(0,∞;L2(R3)) ∩ L2

loc(0,∞; Ḣ1(R3));
(2) u is weakly continuous from [0,∞) to L2(R3) ;

(3) u satisfies (1.1) in the weak sense, that is, for all ϕ ∈ C1([s, t]; Ḣ1(R3)) ,∫
R3

u(t) · ϕ(t)dx+

∫ t

s

∫
R3

{∇u : ∇ϕ+ [(u · ∇)u] · ϕ} dxdτ

=

∫ t

s

∫
R3

u · ∂τϕdxdτ +

∫
R3

u(s)ϕ(s)dx+

∫ t

s

∫
R3

f · ϕdxdτ (t ≥ s ≥ 0) ;

(1.10)

(4) u satisfies the energy inequality:∫
R3

|u(t)|2dx+2

∫ t

s

∫
R3

|∇u|2dxdτ ≤
∫
R3

|u(s)|2dx+2

∫ t

s

∫
R3

f ·udxdτ (0 ≤ s < t ≤ ∞) . (1.11)

Our main results now read as follows.

Theorem 1.2 (Uniform Stability). Let 0 < T ≤ ∞ , and u(x, t) be a Leray weak solution of (1.1)
with u0 ∈ L2(R3) and f ∈ L2(0, T ;L2(R3)) . If

∇u ∈ Lp(0, T ; Ḃ0
q,∞(R3)),

2

p
+

3

q
= 2,

3

2
< q < ∞, (1.12)

then any Leray weak solution v of (1.3) with w0 ∈ L2(R3) and g ∈ L1(0, T ;L2(R3)) satisfies the
estimate

∥v(t)− u(t)∥2L2 +

∫ t

0

∥∇(v − u)∥2L2dτ

≤
(
∥w0∥2L2 +

∫ T

0

∥g∥L2dτ
){

1 + C

∫ T

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ

× exp
[
C

∫ T

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ

]}
.

(1.13)

In particular, if

∥w0∥2L2 +

∫ T

0

∥g∥L2dτ ≤ ε

for sufficiently small ε > 0 , then we have the uniform stability

∥v(t)− u(t)∥L2 ≤ Cε, ∀0 < t < T. (1.14)

Dong-Jia’ (1.6) considered only 2 ≤ q < ∞, while our Theorem 1.2 can treat all the possible q.
Moreover, we could even improve (1.12) to be Besov spaces of negative regular index.

Theorem 1.3. Under the conditions of Theorem 1.2, if (1.12) is replaced by

∇u ∈ L
2

2−r (0, T ; Ḃ−r
∞,∞(R3)), 0 < r < 1, (1.15)

then

∥v(t)− u(t)∥2L2 +

∫ t

0

∥∇(v − u)∥2L2dτ

≤
(
∥w0∥2L2 +

∫ T

0

∥g∥L2dτ
){

1 + C

∫ T

0

(
∥g∥L2 + ∥∇u∥

2
2−r

Ḃ−r
∞,∞

)
dτ

× exp
[
C

∫ T

0

(
∥g∥L2 + ∥∇u∥

2
2−r

Ḃ−r
∞,∞

)
dτ

]}
,

(1.16)



4 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

and the uniform stability (1.14) still holds.

From the embedding Ḃ0
q,∞(R3) ⊂ Ḃ

− 3
q

∞,∞(R3), we see that Theorem 1.3 is an refinement of
Theorem 1.2 for the case 3 < q < ∞ .

The uniform estimate (1.13)/ (1.16) allows us to derive a weak-strong uniqueness of Leray weak
solutions of the 3D Navier-Stokes system. Indeed, if w0 = g = 0 in (1.13)/(1.16), then v = u .
Precisely, we have

Theorem 1.4. (Weak-Strong Uniqueness). Assume u ∈ L2(R3) and f ∈ L2(0, T ;L2(R3)). Let u
be a Leray weak solution of (1.1) and satisfy (1.12) or (1.15). Then u is unique amongst all the
Leray weak solutions associated to the same initial data u0 and external force f .

Remark 1.5. It is worth mentioning that Chen-Miao-Zhang [4] showed the weak-strong unique-
ness under the assumption that

u ∈ Lp(0, T ;Br
q,∞(R3)),

2

p
+

3

q
= 1+r,

3

1 + r
< q ≤ ∞, 0 < r ≤ 1, (q, r) ̸= (∞, 1). (1.17)

From the equivalence relation ∇f ∈ Ḃ−r
∞,∞(R3) (0 < r < 1) ⇔ f ∈ Ḃs

∞,∞(R3) (0 < s < 1) and the

continuous embedding Bs
p,q(R3) ⊂ Ḃs

p,q(R3) for s > 0, we see our weak-strong uniqueness criterion
(1.15) can be reformulated as

u ∈ L
2

1+s (0, T ; Ḃs
∞,∞(R3)), 0 < s < 1, (1.18)

and is better than (1.17) in many cases. For readers interested in weak- strong uniqueness results,
please refer to [3, 7] and references therein.

Now, our asymptotic stability result reads as follows.

Theorem 1.6 (Asymptotic Stability). Assume that u0,w0 ∈ L2(R3), f ∈ L2
loc(0,∞;L2(R3)) and

g ∈ L1(0,∞;L2(R3)). Assume that u(t) is a Leray weak solution of (1.1) and v(t) is a weak
solution of (1.3). If

∇u ∈ Lp(0,∞; Ḃ0
q,∞(R3)),

2

p
+

3

q
= 2,

3

2
< q < ∞, (1.19)

or
∇u ∈ L

2
2−r (0,∞; Ḃ−r

∞,∞(R3)), 0 < r < 1, (1.20)

holds, then v(t) converges asymptotically to u(t) in the sense that

lim
t→∞

∥v(t)− u(t)∥L2 = 0. (1.21)

Remark 1.7. Because of the continuous inclusion Lq(R3) ⊂ Ḃ0
q,∞(R3) for 1 ≤ q ≤ ∞, we see that

Theorem 1.6 extends Zhou’s result (1.9) to homogeneous Besov spaces in a full range (without
the limiting case q = ∞). Moreover, we extend Dong-Jia [6] in the range of regularity index and
integrability index.

If the initial perturbation w0 satisfies some further spectral property (see (1.22)) and there is
no perturbation of external force, then we can rewrite (1.21) with an explicit convergence rate.
For this purpose, we recall the following optimal upper and lower bounds of heat flow (See [16]).

Lemma 1.8 (Decay rate of the heat flow). Assume that w0 ∈ L2(R3) and for some γ > 0,∫
|ξ|=1

|ŵ0(rω)|2dω = Cr2γ−3 + o(rγ−3), as r → 0. (1.22)

Then the solution W(x, t) = et∆w0 of the heat equation

∂tW −∆W = 0,

W|t=0 = w0
(1.23)

obeys the following upper and lower bounds

C1(1 + t)−
γ+s
2 ≤ ∥W(t)∥Ḣs ≤ C2(1 + t)−

γ+s
2 , (1.24)

where s ≥ 0, C1, and C2 are positive constants depending only on s .



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 5

From this lemma we can develop a Fourier splitting technique (see [18, 12]) to show the following
result.

Theorem 1.9 (Decay rate). Under the hypotheses of Theorem 1.6. If in addition, the perturbed
initial data w0 satisfies (1.22) with 2 < γ < 5

2 , then the perturbed external force g = 0. Then
there exists two positive constants C1 and C2 such that

C1(1 + t)−γ/2 ≤ ∥v(t)− u(t)∥L2 ≤ C2(1 + t)−γ/2, ∀t ≥ 0. (1.25)

Remark 1.10. (1) An example of w0 satisfying (1.22) is

ŵ0(ξ) =

{
C|ξ|γ− 3

2 , |ξ| < 1,

0, |ξ| ≥ 1.

(2) In the framework of Morrey spaces, Jia-Xie-Wang [11] showed (1.25). While our result is
built upon the homogeneous Besov spaces (even with negative regularity index).

(3) The upper and lower bound estimates (1.25) are optimal since they coincide with those of
the linear heat flow (see (1.24).

(4) The upper bound of the decay rates in (1.25) can be improved if we assume γ ≥ 5/2. In
this circumstance, we can show that

∥v(t)− u(t)∥L2 ≤ C2(1 + t)−5/4, ∀t ≥ 0. (1.26)

This will be proved at the end of Section 6.
(5) At this moment, it seems not so easy to consider non-zero perturbed external force g. We

hope we can investigate this issue later.

Remark 1.11. At this moment, we do not know whether the margin case r = 0 and r = 1 is valid
in Theorem 1.3, Theorem 1.4, Theorem 1.6 and Theorem 1.9. However, by using the following
bilinear estimate in Hardy spaces (see [20, Lemma 2.1])

∥∇(fg)∥H1 ≤ C∥∇f∥Lp∥g∥Lq + C∥f∥Lp∥∇g∥Lq , 1 < p, q < ∞,
1

p
+

1

q
= 1, (1.27)

we have the trilinear estimate which could be viewed as the margin case r = 1 in Theorem 2.7,∫ T

0

∫
R3

fg∂ihdxdt

= −
∫ T

0

∫
R3

∂i(fg)hdxdt

≤ C

∫ T

0

∥∂i(fg)∥H1∥h∥BMOdt

≤ C

∫ T

0

(∥∇f∥L2∥g∥L2 + ∥f∥L2∥∇g∥L2) ∥h∥BMOdt

≤ C∥g∥L∞
T (L2)∥∇f∥L2

T (L2)∥h∥L2
T (BMO) + C∥f∥L∞

T (L2)∥∇g∥L2
T (L2)∥h∥L2

T (BMO),

(1.28)

where 1 ≤ i ≤ 3. Hence the assumption on the velocity gradient in Theorem 1.3, Theorem 1.4,
Theorem 1.6 and Theorem 1.9 can be replaced by

u ∈ L2(0, T ;BMO(R3)) or u ∈ L2(0,∞;BMO(R3)). (1.29)

The rest of this article is organized as follows. In Section 2, we establish the continuity of

the trilinear form
∫ T

0

∫
R3 fghdxdt by the Fourier localization technique. In Section 3 we study

the energy equality of Leray weak solutions of (1.1). In Sections 4 and 5 we discuss the uni-
form stability under small perturbation and the asymptotic stability under large perturbation.
Finally, in Sections 6 and 7 we study the upper and lower bounds of the asymptotic convergence.
Throughout this paper, we denote by C a generic positive constant which may vary from line to
line, by Lp(R3) with 1 ≤ p ≤ ∞ the classical Lebesgue space endowed with the norm ∥ · ∥Lp , and



6 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

by Lp(0, T ;Lq(R3)) the anisotropic (in time and space) Lebesgue space endowed with the norm
∥ · ∥Lp

T (Lq). The Fourier transform of f is

Ff(ξ) = f̂(ξ) =
1

(2π)3/2

∫
R3

f(x)e−ix·ξdx,

The inverse Fourier transform of g is

F−1g(x) =
1

(2π)3/2

∫
R3

g(x)eix·ξdξ.

2. Continuity of the trilinear form

In this section, we establish the continuity of the trilinear form
∫ T

0

∫
R3 fghdxdt using the Fourier

localization technique.
Let χ(ξ) and ϕ(ξ) be radial smooth functions defined on R3, supported in {ξ ∈ R3; |ξ| ≤ 4

3}
and {ξ ∈ R3; 3

4 ≤ |ξ| ≤ 8
3} respectively. Assume that

χ(ξ) +
∑
j≥0

ϕ(2−jξ) = 1, ∀ξ ∈ R3;
∑
j∈Z

ϕ(2−jξ) = 1, ∀ξ ∈ R3\{0} (2.1)

(see [1, Proposition 2.10] for the construction of such χ and ϕ). Then we can define the homoge-

neous dyadic blocks ∆̇j as

∆̇jf = ϕ(2−jD)f = 23j
∫
R3

F−1ϕ(2jy)f(x− y)dy, ∀j ∈ Z,

and the homogeneous low-frequency cut-off operator Ṡj as

Ṡjf = χ(2−jD)f = 23j
∫
R3

F−1χ(2jy)f(x− y)dy, ∀j ∈ Z.

In view of the spectral support, we have

∆̇j∆̇kf = 0 if |j − k| ≥ 2; ∆̇j(Ṡk−1f∆̇kf) = 0 if |j − k| ≥ 5;

∆̇j

( ∑
|k′−k|≤1

∆̇kf∆k′f
)
= 0, if j − k ≥ 4. (2.2)

Also, the Bernstein inequality

∥∆̇jf∥Lq ≤ C23j(
1
p−

1
q )∥∆̇jf∥Lp , ∀1 ≤ p ≤ q ≤ ∞ (2.3)

holds. Moreover, by (2.1), we have the homogeneous Littlewood-Paley decomposition

f =
∑
j∈Z

∆̇jf. (2.4)

With the homogeneous dyadic blocks ∆̇j in hand, we may define the seminorm

∥f∥Ḃs
p,q

=
(∑

j∈Z
2jsr∥∆̇jf∥rLp

)1/r

, s ∈ R, 1 ≤ p, q ≤ ∞.

Then the homogeneous Besov space Ḃs
p,q(R3) is the space of distributions f satisfy ∥f∥Ḃs

p,q
< ∞

and

lim
λ→∞

∥θ(λD)f∥L∞ = 0,

for any smooth function θ with compactly support. It should be pointed out that Ḃs
2,2(R3) =

Ḣs(R3), the homogeneous Sobolev space.
Now, we recall some often-used lemmas. The first one is the characterization of homogeneous

Besov space with negative regularity index by the homogeneous low-frequency cut-off operator,
see [1, Proposition 2.33].



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 7

Lemma 2.1. Let s < 0 and 1 ≤ p, q ≤ ∞. Then f ∈ Ḃs
p,q(R3) if and only if(

2js∥Ṡjf∥Lp

)
j
∈ ℓq.

Moreover, there exists an absolute constant C such that

C−|s|+1∥f∥Ḃs
p,q

≤ ∥
(
2js∥Ṡjf∥Lp

)
j
∥ℓr ≤ C

(
1 +

1

|s|

)
∥f∥Ḃs

p,q
.

The second lemma concerns the continuous embedding properties of homogeneous Besov spaces,
see [1, Proposition 2.20].

Lemma 2.2. Let 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞, Then for any real number s , Ḃs
p1,r1(R

3)

is continuously embedded in Ḃ
s−3( 1

p1
− 1

p2
)

p2,r2 (R3).

The third lemma discusses the interpolation properties of homogeneous Besov spaces, see [1,
Proposition 2.22].

Lemma 2.3. There exists a constant C that satisfies the following properties. If s1 and s2 are
real numbers such that s1 < s2 and θ ∈ (0, 1), then we have, for any pair (p, r) ∈ [1,∞]2 ,

∥u∥
Ḃ

θs1+(1−θ)s2
p,r

≤ ∥u∥θ
Ḃ

s1
p,r

∥u∥1−θ

Ḃ
s2
p,r

,

and

∥u∥
Ḃ

θs1+(1−θ)s2
p,1

≤ C

s2 − s1

(1
θ
+

1

1− θ

)
∥u∥θ

Ḃ
s1
p,∞

∥u∥1−θ

Ḃ
s2
p,∞

.

And the fourth lemma is the duality properties of homogeneous Besov spaces, see [1, Proposition
2.29].

Lemma 2.4. For all 1 ≤ p, r ≤ ∞ and s ∈ R, the mapping

(u, ϕ) 7→
∫
R3

uϕdx =
∑

|j−j′|≤1

∫
R3

∆ju∆j′ϕdx

defines a continuous bilinear functional from Ḃs
p,r × Ḃ−s

p′,r′ to R.

A fine tool in Fourier frequency technique is the following Bony decomposition

uv = Ṫuv + Ṫvu+ Ṙ(u, v), (2.5)

where
Ṫuv =

∑
j

Ṡj−1u∆̇jv, Ṙ(u, v) =
∑

|j′−j|≤1

∆̇ju · ∆̇j′v.

With the Bony decomposition, we have the Hölder type inequality for homogeneous Besov spaces,
see [1, Corollary 2.54] for a special case.

Lemma 2.5. Let (s, p, q, p1, p2, p3, p4) ∈ (0,∞)× [1,∞]6. Then there exists a constant C, depend-
ing on s such that

∥uv∥Ḃs
p,q

≤ C
(
∥u∥Lp1∥v∥Ḃs

p2,q
+ ∥u∥Ḃs

p3,q
∥v∥Lp4

)
(2.6)

with
1

p
=

1

p1
+

1

p2
=

1

p3
+

1

p4
.

Proof. We provide the proof in full detail for convenience of the readers. By (2.5), we have the
the Bony decomposition

uv = Ṫuv + Ṫvu+ Ṙ(u, v),

and we need to estimate these three terms.

Estimates for Ṫuv and Ṫvu. By (2.2),

∆̇j(Ṫuv) =
∑

|j′−j|≤4

∆j(Ṡj′−1u∆̇j′v),



8 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

and thus

∥Ṫuv∥Ḃs
p,q

= ∥
(
2js∥∆j(Ṫuv)∥Lp

)
j
∥ℓq

= ∥
(
2js∥

∑
|j′−j|≤4

∆j(Ṡj′−1u∆̇j′v)∥Lp

)
j
∥ℓq

≤ C∥
(
2j

′s∥Ṡj′−1u∆̇j′v∥Lp

)
j′
∥ℓq

≤ C∥
(
2j

′s∥Ṡj′−1u∥Lp1 ∥∆̇j′v∥Lp2

)
j′
∥ℓq

≤ C∥u∥Lp1 ∥
(
2j

′s∥∆̇j′v∥Lp2

)
j′
∥ℓq

= C∥u∥Lp1 ∥v∥Ḃs
p2,q

.

(2.7)

Similarly,

∥Ṫvu∥Ḃs
p,q

≤ C∥v∥Lp3∥u∥Ḃs
p4,q

. (2.8)

Estimation of Ṙ(u, v). By (2.2) again,

∆̇j′Ṙ(u, v) =
∑

j≥j′−3, |ν|≤1

∆̇j′
(
∆̇j−νu∆̇jv

)
.

Consequently,

2j
′s∥∆j′Ṙ(u, v)∥Lp ≤ 2j

′s
∑

j≥j′−3, |ν|≤1

∥∆̇j′

(
∆̇j−νu∆̇jv

)
∥Lp ≤ C2j

′s
∑

j≥j′−3, |ν|≤1

∥∆̇j−νu∆̇jv∥Lp

≤ C2j
′s

∑
j≥j′−3, |ν|≤1

∥∆̇j−νu∥Lp1 ∥∆̇jv∥Lp2

≤ C∥u∥Lp1

∑
j≥j′−3

2(j
′−j)s · 2js∥∆̇jv∥Lp2

= C∥u∥Lp1

∑
i≤3

2is · 2(j
′−i)s∥∆̇j′−iv∥Lp2 (j′ − j = i)

= C∥u∥Lp1

(
(ai) ∗

(
2is∥∆̇iv∥Lp2

))
j′
,

where

ai =

{
2is, i ≤ 3

0, i > 3
,

and ((ai)∗(bj))j′ denotes the j′-th term of the convolution of these two sequence, namely
∑

i aibj′−i

. Invoking Young’s inequality for series, we find that

∥Ṙ(u, v)∥Ḃs
p,q

= ∥
(
2j

′s∥∆j′Ṙ(u, v)∥Lp

)
j′
∥ℓq

≤ C∥u∥Lp1 ∥
(
(ai) ∗

(
2is∥∆̇iv∥Lp2

))
j′
∥ℓq

≤ C∥u∥Lp1 ∥ (ai)i ∥ℓ1∥
(
2is∥∆̇iv∥Lp2

)
i
∥ℓq

≤ C∥u∥Lp1 ∥v∥Ḃs
p2,q

(since s > 0).

(2.9)

Combining (2.7)-(2.9), we complete the proof of Lemma 2.5. □

Now we are ready to state our trilinear estimate.

Theorem 2.6. Let 0 < T ≤ ∞ and ε > 0. Assume that f, g ∈ L∞(0, T ;L2(R3))∩L2(0, T ; Ḣ1(R3))
and h verifies (1.12), that is,

h ∈ Lp(0, T ; Ḃ0
q,∞(R3)),

2

p
+

3

q
= 2, 1 ≤ p < ∞,

3

2
< q < ∞. (2.10)



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 9

Then we have the following estimates∫ T

0

∫
R3

fghdxdt ≤ C∥f∥2−
3
q

L∞
T (L2)∥∇f∥

3
q−1

L2
T (L2)

∥∇g∥L2
T (L2)∥h∥Lp

T (Ḃ0
q,∞)

+ C∥∇f∥L2
T (L2)∥g∥

2− 3
q

L∞
T (L2)∥∇g∥

3
q−1

L2
T (L2)

∥h∥Lp
T (Ḃ0

q,∞)

+ C∥f∥1−
3
2q

L∞
T (L2)∥∇f∥

3
2q

L2
T (L2)

∥g∥1−
3
2q

L∞
T (L2)∥∇g∥

3
2q

L2
T (L2)

∥h∥Lp
T (Ḃ0

q,∞),

(2.11)

if 3
2 < q ≤ 3;

∣∣ ∫ T

0

∫
R3

fghdxdt
∣∣ ≤ C∥f∥L∞

T (L2)∥g∥
1− 3

q

L∞
T (L2)∥∇g∥

3
q

L2
T (L2)

∥h∥Lp
T (Ḃ0

q,∞)

+ C∥f∥1−
3
q

L∞
T (L2)∥∇f∥

3
q

L2
T (L2)

∥g∥L∞
T (L2)∥h∥Lp

T (Ḃ0
q,∞),

(2.12)

if 3 < q < ∞. Moreover, if f = g, then∫ T

0

∫
R3

g2hdxdt ≤ C

∫ T

0

∥h∥p
Ḃ0

q,∞
∥g∥2L2dτ + ε

∫ T

0

∥∇g∥2L2dτ. (2.13)

Proof. By the Bony decomposition (2.5), the Littlewood-Paley decomposition (2.4), and the van-
ishing property (2.2), it follows that∫ T

0

∫
R3

fghdxdt

=

∫ T

0

∫
R3

[Ṫfg + Ṫgf + Ṙ(f, g)]
∑
j

∆̇jhdxdt

=
∑

|k−j|≤4

∫ T

0

∫
R3

Ṡk−1f∆̇kg∆̇jhdxdt+
∑

|k−j|≤4

∫ T

0

∫
R3

∆̇kfṠk−1g∆̇jhdxdt

+
∑

|k′−k|≤1

∑
k≥j−3

∫ T

0

∫
R3

∆̇kf∆̇k′g∆̇jhdxdt

≡ I1 + I2 + I3.

(2.14)

If 3
2 < q ≤ 3, from the Hölder inequality it follows that

I1 ≤
∑

|k−j|≤4

∫ T

0

∥Ṡk−1f∥
L

2q
q−1

∥∆̇kg∥
L

2q
q−1

∥∆̇jh∥Lqdt

≤
∫ T

0

∑
j

∑
|l|≤4

∥Ṡj+l−1f∥
L

2q
q−1

∥∆̇j+lg∥
L

2q
q−1

∥∆̇jh∥Lqdt (l = k − j)

≤ C
∑
|l|≤4

∫ T

0

∑
j

2−(j+l−1)(1− 3
2q )∥Ṡj+l−1f∥

L
2q

q−1
2(j+l)(1− 3

2q )∥∆̇j+lg∥
L

2q
q−1

∥∆̇jh∥Lqdt

≤ C
∑
|l|≤4

∫ T

0

∥
(
2−(j+l−1)(1− 3

2q )∥Ṡj+l−1f∥
L

2q
q−1

)
j
∥ℓ2∥

(
2(j+l)(1− 3

2q )∥∆̇j+lg∥
L

2q
q−1

)
j
∥ℓ2

× ∥
(
∥∆̇jh∥Lq

)
j
∥ℓ∞dt.

Thanks to Lemma 2.1 and the definition of homogeneous Besov spaces, we have

I1 ≤ C

∫ T

0

∥f∥
Ḃ

−1+ 3
2q

2q
q−1

,2

∥g∥
Ḃ

1− 3
2q

2q
q−1

,2

∥h∥Ḃ0
q,∞

dt.



10 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

Invoking Lemma 2.2 yields

I1 ≤ C

∫ T

0

∥f∥
Ḃ

3
q
−1

2,2

∥g∥Ḃ1
2,2

∥h∥Ḃ0
q,∞

dt.

From the interpolation inequality and the Minkowski inequality, we obtain

I1 = C

∫ T

0

∥f∥
Ḣ

3
q
−1∥g∥Ḣ1∥h∥Ḃ0

q,∞
dt

≤ C

∫ T

0

∥f∥2−
3
q

L2 ∥∇f∥
3
q−1

L2 ∥∇g∥L2∥h∥Ḃ0
q,∞

dt

≤ C∥f∥2−
3
q

L∞
T (L2)∥∇f∥

3
q−1

L2
T (L2)

∥∇g∥L2
T (L2)∥h∥Lp

T (Ḃ0
q,∞).

(2.15)

Interchanging f and g in (2.15) gives the estimate

I2 ≤ C∥g∥2−
3
q

L∞
T (L2)∥∇g∥

3
q−1

L2
T (L2)

∥∇f∥L2
T (L2)∥h∥Lp

T (Ḃ0
q,∞). (2.16)

Now, we treat I3 as follows,

I3 =
∑
|l|≤1

∑
k≥j−3

∫ T

0

∫
R3

∆̇kf∆̇k+lg∆̇jhdxdt (k
′ − k = l)

≤
∑
|l|≤1

∑
k≥j−3

∫ T

0

∥∆̇kf∥L2∥∆̇k+lg∥L2∥∆̇jh∥L∞dt

≤ C
∑
|l|≤1

∑
k≥j−3

∫ T

0

∥∆̇kf∥L2∥∆̇k+lg∥L2 · 2j
3
q ∥∆̇jh∥Lqdt (by (2.3))

≤ C
∑
|l|≤1

∑
j−k≤3

∫ T

0

2k
3
2q ∥∆̇kf∥L2 · 2(k+l) 3

2q ∥∆̇k+lg∥L2 · 2(2j−2k−l) 3
2q ∥∆̇jh∥Lqdt

≤ C2
3
2q

∑
|l|≤1

∑
m≤3

2m
3
q

∫ T

0

∑
k

2k
3
2q ∥∆̇kf∥L22(k+l) 3

2q ∥∆̇k+lg∥L2

× ∥∆̇k+mh∥Lqdt(j − k = m)

≤ C
∑
|l|≤1

∑
m≤3

2m
3
q

∫ T

0

∥
(
2k

3
2q ∥∆̇kf∥L2

)
k
∥ℓ2∥

(
2(k+l) 3

2q ∥∆̇k+lg∥L2

)
k
∥ℓ2

× ∥
(
∥∆̇k+mh∥Lq

)
k
∥ℓ∞dt

≤ C
∑
|l|≤1

∑
m≤3

2m
3
q

∫ T

0

∥f∥
Ḣ

3
2q
∥g∥

Ḣ
3
2q
∥h∥Ḃ0

q,∞
dt

≤ C

∫ T

0

∥f∥1−
3
2q

L2 ∥∇f∥
3
2q

L2 ∥g∥
1− 3

2q

L2 ∥∇g∥
3
2q

L2 ∥h∥Ḃ0
q,∞

dt

≤ C∥f∥1−
3
2q

L∞
T (L2)∥∇f∥

3
2q

L2
T (L2)

∥g∥1−
3
2q

L∞
T (L2)∥∇g∥

3
2q

L2
T (L2)

∥h∥Lp
T (Ḃ0

q,∞).

(2.17)

Plugging (2.15)-(2.17) into (2.14), we obtain eqrefthm:trilinear:result1. To show (2.13), we deduce
from (2.15)-(2.17) that

I1 ≤ C

∫ T

0

∥f∥2−
3
q

L2 ∥∇f∥
3
q−1

L2 ∥∇g∥L2∥h∥Ḃ0
q,∞

dt

≤ C

∫ T

0

∥h∥Ḃ0
q,∞

∥f∥2−
3
q

L2 ∥(∇f,∇g)∥
3
q

L2dt

≤ C

∫ T

0

∥h∥p
Ḃ0

q,∞
∥f∥2L2dt+

ε

3

∫ T

0

∥(∇f,∇g)∥2L2dt,



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 11

I2 ≤ C

∫ T

0

∥h∥p
Ḃ0

q,∞
∥g∥2L2dt+

ε

3

∫ T

0

∥(∇f,∇g)∥2L2dt,

I3 ≤ C

∫ T

0

∥f∥1−
3
2q

L2 ∥∇f∥
3
2q

L2 · ∥g∥
1− 3

2q

L2 ∥∇g∥
3
2q

L2 · ∥h∥Ḃ0
q,∞

dt

≤
∫ T

0

∥h∥Ḃ0
q,∞

∥(f, g)∥2−
3
q

L2 ∥(∇f,∇g)∥
3
q

L2dt

≤ C

∫ T

0

∥h∥p
Ḃ0

q,∞
∥(f, g)∥2L2dt+

ε

3

∫ T

0

∥(∇f,∇g)∥2L2dt.

Setting f = g , summing these above three inequalities, and putting them into (2.14), we obtain
(2.13) as desired.

If the other case 3 < q < ∞ holds, we resort to the fact that Ḃ0
q,∞(R3) ⊂ Ḃ

− 3
q

∞,∞(R3) and
Theorem 2.7 below to deduce (2.12) and (2.13). □

Theorem 2.7. Let 0 < T ≤ ∞ and ε > 0. Assume that f, g ∈ L∞(0, T ;L2(R3))∩L2(0, T ; Ḣ1(R3))
and h verifies (1.15), that is,

h ∈ L
2

2−r (0, T ; Ḃ−r
∞,∞(R3)), 0 < r < 1, (2.18)

Then we have the estimate∫ T

0

∫
R3

fghdxdt ≤ C∥f∥L∞
T (L2)∥g∥1−r

L∞
T (L2)∥∇g∥rL2

T (L2)∥h∥
L

2
2−r
T (Ḃ−r

∞,∞)

+ ∥f∥1−r
L∞

T (L2)∥∇f∥rL2
T (L2)∥g∥L∞

T (L2)∥h∥
L

2
2−r
T (Ḃ−r

∞,∞)
.

(2.19)

Moreover, if f = g, then∫ T

0

∫
R3

g2hdxdt ≤ C

∫ T

0

∥h∥
2

2−r

Ḃ−r
∞,∞

∥g∥2L2dτ + ε

∫ T

0

∥∇g∥2L2dτ. (2.20)

Proof.∫ T

0

∫
R3

fghdxdt

≤
∫ T

0

∥fg∥Ḃr
1,1

∥h∥Ḃ−r
∞,∞

dt (by Lemma 2.4)

≤ C

∫ T

0

(
∥f∥L2∥g∥Ḃr

2,1
+ ∥f∥Ḃr

2,1
∥g∥L2

)
∥h∥Ḃ−r

∞,∞
dt (by Lemma 2.5)

≤ C

∫ T

0

(
∥f∥L2∥g∥1−r

Ḃ0
2,∞

∥g∥r
Ḃ1

2,∞
+ ∥f∥1−r

Ḃ0
2,∞

∥f∥r
Ḃ1

2,∞
∥g∥L2

)
∥h∥Ḃ−r

∞,∞
dt (by Lemma 2.3)

≤ C

∫ T

0

(
∥f∥L2∥g∥1−r

L2 ∥∇g∥rL2 + ∥f∥1−r
L2 ∥∇f∥rL2∥g∥L2

)
∥h∥Ḃ−r

∞,∞
dt

(
Ḣ1 = Ḃ1

2,2 ⊂ Ḃ1
2,∞

)
≤ C

[
∥f∥L∞

T (L2)∥g∥1−r
L∞

T (L2)∥∇g∥rL2
T (L2) + ∥f∥1−r

L∞
T (L2)∥∇f∥rL2

T (L2)∥g∥L∞
T (L2)

]
∥h∥

L
2

2−r
T (Ḃ−r

∞,∞)
.

To show (2.20), it suffices to let f = g in the above inequality, and modify the last line as∫ T

0

∫
R3

g2hdx ≤ C

∫ T

0

(
∥g∥L2∥g∥1−r

L2 ∥∇g∥rL2 + ∥g∥1−r
L2 ∥∇g∥rL2∥g∥L2

)
∥h∥Ḃ−r

∞,∞
dt

≤ C

∫ T

0

∥h∥Ḃ−r
∞,∞

∥g∥2−r
L2 ∥∇g∥2L2dt

≤ C

∫ T

0

∥h∥
2

2−r

Ḃ−r
∞,∞

∥g∥2L2dτ + ε

∫ T

0

∥∇g∥2L2dτ. □



12 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

3. Energy-type equality

A Leray weak solution satisfies the energy inequality. As we will show, under condition (1.12)
or (1.15), the energy inequality becomes energy equality.

Theorem 3.1. Let 0 < T ≤ ∞. Assume that u and v are Leray weak solutions to (1.1) and (1.3)
with u0,w0 ∈ L2(R3) and f ,g ∈ L2(0, T ;L2(R3)) respectively. If u satisfies (1.12) or (1.15), then
we have the following energy-type equality∫

R3

v(t) · u(t)dx+

∫ t

0

∫
R3

{2∇v : ∇u+ [(u · ∇)u] · v + [(v · ∇)v] · u}dxdτ

=

∫
R3

v0 · u0dx+

∫ t

0

∫
R3

f · vdxdτ +

∫ t

0

∫
R3

(f + g) · udxdτ, 0 ≤ t ≤ T.

(3.1)

Proof. Let η(t) ≥ 0 be a smooth radial function on R supported in the unit ball, and satisfies

1 =

∫ 1

−1

η(t)dt ⇒
∫ 1

0

η(t)dt =
1

2
.

Set

ηn(t) = n · η(nt) (n ≥ 1), un(τ) =

∫ t

0

ηn(|τ − σ|)u(σ)dσ,

vn(τ) =

∫ t

0

ηn(|τ − σ|)v(σ)dσ (0 ≤ τ ≤ t).

then un,vn ∈ C1((0, t); Ḣ1(R3)), and we may test (1.1)1 and (1.3)1 by vn and un respectively,
and obtain ∫

R3

u(t) · vn(t)dx+

∫ t

0

∫
R3

{∇u : ∇vn + [(u · ∇)u] · vn} dxdτ

=

∫ t

0

∫
R3

u · ∂τvndxdτ +

∫
R3

u0 · vn(0)dx+

∫ t

0

∫
R3

f · vndxdτ,

as well as ∫
R3

v(t) · un(t)dx+

∫ t

0

∫
R3

{∇v : ∇un + [(v · ∇)v] · un} dxdτ

=

∫ t

0

∫
R3

v · ∂τundxdτ +

∫
R3

v0 · un(0)dx+

∫ t

0

∫
R3

(f + g) · undxdτ.

Summing these above two equalities, and noticing the following cancellation property∫ t

0

∫
R3

(v · ∂τun + u · ∂τvn)dxdτ

=

∫ t

0

∫
R3

[
v(τ) ·

∫ t

0

∂τηn(|τ − σ|)u(σ)dσ + u(τ) ·
∫ t

0

∂τηn(|τ − σ|)v(σ)dσ
]
dxdτ

=

∫ t

0

∫
R3

∫ t

0

[∂τηn(|τ − σ|)v(τ) · u(σ)− ∂σηn(|τ − σ|)u(τ) · v(σ)] dσdxdτ

= 0 (by interchanging τ and σ),

we deduce ∫
R3

[u(t) · vn(t) + v(t) · un(t)]dx+

∫ t

0

∫
R3

(∇u · ∇vn +∇v : ∇un) dxdτ

+

∫ t

0

∫
R3

{[(u · ∇)u] · vn + [(v · ∇)v] · un} dxdτ

=

∫
R3

[v0 · un(0) + u0 · vn(0)]dx+

∫ t

0

∫
R3

[f · vn + (f + g) · un]dxdτ.

(3.2)



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 13

We now pass to limit n → ∞ in (3.2). For the first integral in the left-hand side of (3.2), we
compute directly as∣∣∣ ∫

R3

[u(t) · vn(t) + v(t) · un(t)]dx−
∫
R3

u(t) · v(t)dx
∣∣∣

=
∣∣∣ ∫

R3

∫ t

0

ηn(|σ|)[u(t) · v(t− σ) + v(t) · u(t− σ)]dσdx

− 2

∫
R3

∫ t

0

ηn(|σ|)u(t) · v(t)dσdx
∣∣∣ (if n ≥ 1

t
)

=
∣∣∣ ∫ t

0

ηn(|σ|)
{∫

R3

[u(t) · v(t− σ) + v(t) · u(t− σ)− 2u(t) · v(t)]dx
}
dσ

∣∣∣
≤ 1

2
sup

0<σ< 1
n

∣∣∣ ∫
R3

[u(t) · v(t− σ) + v(t) · u(t− σ)− 2u(t) · v(t)]dx
∣∣∣ → 0 (n → ∞).

(3.3)

Similarly,

lim
n→∞

∫
R3

[v0 · un(0) + u0 · vn(0)]dx =

∫
R3

u0 · v0dx. (3.4)

Using the regularities of weak solutions u,v and that of f ,g, we have

lim
n→∞

∫ t

0

∫
R3

(∇u : ∇vn +∇v : ∇un)dxdt = 2

∫ t

0

∫
R3

∇u : ∇vdxdτ, (3.5)

as well as

lim
n→∞

∫ t

0

∫
R3

[f · vn + (f + g) · un]dxdτ =

∫ t

0

∫
R3

f · vdxdτ +

∫ t

0

∫
R3

(f + g) · udxdτ. (3.6)

For the convection terms, we employ Theorem 2.6 or Theorem 2.7 to deduce that∣∣∣ ∫
R3

[(u · ∇)u] · vndx−
∫
R3

[(u · ∇)u] · vdx
∣∣∣ = ∣∣∣ ∫

R3

[(u · ∇)u] · (vn − v)dx
∣∣∣

can be dominated by

C∥u∥2−
3
q

L∞
T (L2)∥∇u∥

3
q−1

L2
T (L2)

∥∇(vn − v)∥L2
T (L2)∥∇u∥Lp

T (Ḃ0
q,∞)

+ C∥vn − v∥2−
3
q

L∞
T (L2)∥∇(vn − v)∥

3
q−1

L2
T (L2)

∥∇u∥L2
T (L2)∥∇u∥Lp

T (Ḃ0
q,∞)

+ C∥u∥1−
3
2q

L∞
T (L2)∥∇u∥

3
2q

L2
T (L2)

∥vn − v∥1−
3
2q

L∞
T (L2)∥∇(vn − v)∥

3
2q

L2
T (L2)

∥∇u∥Lp
T (Ḃ0

q,∞),

if (1.12) with 3
2 < q ≤ 3 holds, by

C∥u∥L∞
T (L2)∥vn − v∥1−

3
q

L∞
T (L2)∥∇(vn − v)∥

3
q

L2
T (L2)

∥∇u∥Lp
T (Ḃ0

q,∞)

+ C∥u∥1−
3
q

L∞
T (L2)∥∇u∥

3
q

L2
T (L2)

∥vn − v∥L∞
T (L2)∥∇u∥Lp

T (Ḃ0
q,∞),

if (1.12) with 3 < q < ∞ holds, and by

C∥u∥L∞
T (L2)∥vn − v∥1−r

L∞
T (L2)∥∇(vn − v)∥rL2

T (L2)∥∇u∥
L

2
2−r
T (Ḃ−r

∞,∞)

+ ∥u∥1−r
L∞

T (L2)∥∇u∥rL2
T (L2)∥vn − v∥L∞

T (L2)∥∇u∥
L

2
2−r
T (Ḃ−r

∞,∞)
,

if (1.15) holds.
Notice that each term in the above three formulas has a factor ∥vn − v∥L∞

T (L2) or ∥∇(vn −
v)∥L2

T (L2), whose powers have at least one greater than 0; and hence we deduce that

lim
n→∞

∣∣∣ ∫
R3

[(u · ∇)u] · vndx−
∫
R3

[(u · ∇)u] · vdx
∣∣∣ = 0. (3.7)

Similarly,

lim
n→∞

∫
R3

[(v · ∇)v] · undxdτ =

∫
R3

[(v · ∇)v] · udxdτ. (3.8)



14 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

Collecting (3.3)-(3.8), we find that (3.2) becomes (3.1) as n → ∞. □

Taking v = u and w0 = g = 0 in (3.1), we have the following result.

Theorem 3.2 (Energy equality). . Let T > 0, and assume that u is a Leray weak solution to
(1.1) with u0 ∈ L2(R3) and f ∈ L2(0, T ;L2(R3)). If u satisfies (2.10) or (2.18), then we have the
energy equality∫

R3

|u(t)|2dx+ 2

∫ t

0

∫
R3

|∇u(τ)|2dxdτ =

∫
R3

|u0|2dx+ 2

∫ t

0

∫
R3

f · udxdτ. (3.9)

4. Uniform stability under small initial perturbation

In this section, we study the uniform stability of Leray weak solutions u to the Navier-Stokes
system (1.1) under small perturbation of initial data and external force, and provide the proof of
Theorem 1.2 and Theorem 1.3.

By Theorem 3.2,∫
R3

|u(t)|2dx+ 2

∫ t

0

∫
R3

|∇u(τ)|2dxdτ =

∫
R3

|u0|2dx+ 2

∫ t

0

∫
R3

f · udxdτ. (4.1)

From the definition of weak solution v of (1.3), we have∫
R3

|v(t)|2dx+ 2

∫ t

0

∫
R3

|∇v(τ)|2dxdτ ≤
∫
R3

|u0 +w0|2dx+ 2

∫ t

0

∫
R3

(f + g) · vdxdτ. (4.2)

It then follows from (4.1) + (4.2)− 2× (3.1) that∫
R3

|v(t)− u(t)|2dx+ 2

∫ t

0

∫
R3

|∇(v − u)|2dxdτ

≤ 2

∫ t

0

∫
R3

{[(u · ∇)u] · v + [(v · ∇)v] · u} dxdτ +

∫
R3

|w0|2dx+ 2

∫ t

0

∫
R3

g · (v − u)dxdτ.

(4.3)

Denoting w(x, t) = v(x, t)− u(x, t), and noticing the fact that∫
R3

{[(u · ∇)u] · v + [(v · ∇)v] · u}dx =

∫
R3

{−[(u · ∇)v] · u+ [(v · ∇)v] · u} dx

=

∫
R3

[((v − u) · ∇)v] · udx

=

∫
R3

[((v − u) · ∇)(v − u)] · udx

= −
∫
R3

[((v − u) · ∇)u] · (v − u)dx,

we derive from (4.3) that∫
R3

|w(t)|2dx+ 2

∫ t

0

∫
R3

|∇w|2dxdτ

≤ 2

∫ t

0

∫
R3

[(w · ∇)u] ·wdxdτ +

∫
R3

|w0|2dx+ 2

∫ t

0

∥g∥L2∥w∥L2dτ.

(4.4)

By (2.13),

∥w(t)∥2L2 + 2

∫ t

0

∥∇w∥2L2dτ ≤ C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
∥w∥2L2dτ +

∫ t

0

∥∇w∥2L2dτ

+ ∥w0∥2L2 +

∫ t

0

∥g∥L2dτ + C

∫ t

0

∥g∥L2∥w∥2L2dτ,



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 15

which is equivalent to

∥w(t)∥2L2 +

∫ t

0

∥∇w∥2L2dτ

≤
[
∥w0∥2L2 +

∫ t

0

∥g∥L2dτ
]
+ C

∫ t

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
∥w∥2L2dτ.

(4.5)

Applying the Gronwall inequality yields

∥w(t)∥2L2 +

∫ t

0

∥∇w∥2L2dτ ≤
(
∥w0∥2L2 +

∫ t

0

∥g∥L2dτ
){

1 + C

∫ t

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ

× exp
[
C

∫ T

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ

]}
.

This shows (1.13), and (1.14) follows immediately. The proof of Theorem 1.2 is complete.
For the proof of Theorem 1.3, it suffices to estimate the first integral on the right-hand side of

(4.4) by (2.20), and proceed as above.

5. Asymptotic stability under large perturbation of initial data and external
force

In this section, we prove Theorem 1.6, namely, we show that under large perturbation of initial
data and external force, the Leray weak solution v of (1.3) converges asymptotically to the solution
u of (1.1), under the assumption that (1.12). For this purpose, we need the following time-space
decay estimate.

Lemma 5.1. Under the assumptions of Theorem 1.6, we have

lim
t→∞

∫ t

t−1

∥w(τ)∥2L2dτ = 0. (5.1)

Proof. It suffices to consider the case where (1.19) holds. Indeed, if (1.20) holds, we could follow

the arguments below, replacing
( ∫ t

s
∥∇u∥p

Ḃ0
q,∞

dτ
)1/p

by
( ∫ t

s
∥∇u∥

2
2−r

Ḃ−r
∞,∞

dτ
) 2−r

2 .

Subtracting (1.1) from (1.3) shows that w = v − u satisfies

∂tw −∆w + (u · ∇)w + (w · ∇)u+ (w · ∇)w +∇Πw = g, w|t=0 = w0 (5.2)

in the weak sense.
Motivated by [6, 15], we choose the test function

ϕn(τ) =

∫ t

s

ηn(|τ − σ|)(1−∆)−γe(t−τ)∆e(t−σ)∆w(σ)dσ, 0 < s < t < ∞,

where ηn is defined in Section 3 and γ is an arbitrary constant satisfying 1
4 ≤ γ < 1.

Testing (5.2) by ϕn gives∫
R3

w(t) · ϕn(t)dx−
∫
R3

w(s) · ϕn(s)ds+

∫ t

s

∫
R3

(−w · ∂τϕn +∇w : ∇ϕn) dxdτ

= −
∫ t

s

∫
R3

[(u · ∇)w + (w · ∇)u+ (w · ∇)w] · ϕndx+

∫ t

s

∫
R3

g · ϕndxdτ.

(5.3)

We now study the convergence of each term in (5.3) as n → ∞. For the first integral in the
left-hand side of (5.3), we invoke the Plancherel theorem and the Lebesgue dominated convergence



16 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

theorem to conclude that∫
R3

w(t) · ϕn(t)dx =

∫
R3

∫ t

s

ηn(|t− σ|)(1−∆)−γe(t−σ)∆w(σ) ·w(t)dσdx

=

∫
R3

∫ t

s

ηn(|t− σ|)(1 + |ξ|2)−γe−(t−σ)|ξ|2ŵ(σ) · ŵ(t)dσdξ

=

∫
R3

∫ t−s

0

ηn(|τ |)(1 + |ξ|2)−γe−τ |ξ|2ŵ(t− τ) · ŵ(t)dτdξ (t− σ = τ)

→ 1

2

∫
R3

(1 + |ξ|2)−γŵ(t) · ŵ(t)dx (
1

t− s
≤ n → ∞)

=
1

2

∫
R3

|(1−∆)−γ/2w(t)|2dx.

(5.4)

Similarly,∫
R3

w(s) · ϕn(s)dx

=

∫
R3

∫ t

s

ηn(|s− σ|)(1−∆)−γe(t−s)∆e(t−σ)∆w(σ) ·w(s)dσdx

=

∫
R3

∫ t

s

ηn(|s− σ|)(1 + |ξ|2)−γe−(t−s)|ξ|2e−(t−σ)|ξ|2ŵ(σ) · ŵ(s)dσdξ

=

∫
R3

∫ 0

s−t

ηn(|τ |)(1 + |ξ|2)−γe−(t−s)|ξ|2e−(t−s+τ)|ξ|2ŵ(s− τ) · ŵ(s)dσdξ (s− σ = τ)

→ 1

2

∫
R3

(1 + |ξ|2)−γe−2(t−s)|ξ|2ŵ(s) ·w(s)dξ (
1

t− s
≤ n → ∞)

=
1

2

∫
R3

|e(t−s)∆(1−∆)−γ/2w(s)|2dx.

(5.5)

For the third integral in the left-hand side of (5.3), we just need to integrate by parts,∫ t

s

∫
R3

(−w · ∂τϕn +∇w : ∇ϕn)dxdτ

=
[ ∫ t

s

∫
R3

∫ t

s

−∂τηn(|τ − σ|)(1−∆)−γe(t−τ)∆e(t−σ)∆w(σ) ·w(τ)dσdxdτ

−
∫ t

s

∫
R3

∫ t

s

ηn(|τ − σ|)(1−∆)−γe(t−τ)∆(−∆)e(t−σ)∆w(σ) ·w(τ)sdσdxdτ
]

+

∫ t

s

∫
R3

∫ t

s

ηn(|τ − σ|)(1−∆)−γe(t−τ)∆e(t−σ)∆∇w(σ) : ∇w(τ)dσdxdτ

=

∫ t

s

∫
R3

∫ t

s

∂σηn(|τ − σ|)(1−∆)−γe(t−τ)∆e(t−σ)∆w(σ) ·w(τ)dσdxdτ = 0,

(5.6)

interchanging τ and σ in the last integral J gives J = −J
For the convergence of remaining terms in (5.3), we need some estimate of ϕn(τ). By the

Plancherel theorem and (1.13)/(1.16),

sup
s<τ<t

∥ϕn(τ)∥L2 = sup
s<τ<t

∥
∫ t

s

ηn(|τ − σ|)(1 + |ξ|2)−γe−(t−τ)|ξ|2e−(t−σ)|ξ|2ŵ(σ)dσ∥L2

≤ sup
s<τ<t

∥
∫ t

s

ηn(|τ − σ|)|ŵ(σ)|dσ∥L2

= sup
s<τ<t

∫ τ−s

τ−t

ηn(|ν|)∥ŵ(τ − ν)∥L2dν (τ − σ = ν)

≤
∫
R
ηn(|ν|)dν · sup

s<σ<t
∥ŵ(σ)∥L2 ≤ sup

s<σ<t
∥w(σ)∥L2 ≤ E0,



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 17

where E2
0 is the right-hand side of (1.13) with T = ∞, that is,

E2
0 =

(
∥w0∥2L2 +

∫ ∞

0

∥g∥L2dτ
)

×
{
1 + C

∫ ∞

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ exp

[
C

∫ ∞

0

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ

]}
.

(5.7)

By the Sobolev inequality and the Plancherel theorem, we obtain

sup
s<τ<t

∥ϕn(τ)∥L3 ≤ C sup
s<τ<t

∥ϕn(τ)∥Ḣ1/2

≤ C sup
s<τ<t

∥
∫ t

s

ηn(|τ − σ|)|ξ|1/2(1 + |ξ|2)−γe−(t−τ)|ξ|2e−(t−σ)|ξ|2ŵ(σ)dσ∥L2

≤ C sup
s<τ<t

∥
∫ t

s

ηn(|τ − σ|)|ŵ(σ)|dσ∥L2 (since γ ≥ 1

4
)

≤ C sup
s<τ<t

∥ŵ(τ)∥L2 ≤ C sup
s<τ<t

∥w(τ)∥L2 ≤ CE0,

and ∫ t

s

∥∇ϕn(τ)∥2L2dτ =

∫ t

s

∥∇
∫ t

s

ηn(|τ − σ|)(1−∆)−γe(t−τ)∆e(t−σ)∆w(σ)dσ∥2L2dτ

=

∫ t

s

∥
∫ t

s

ηn(|τ − σ|)(1 + |ξ|2)−γe−(t−τ)|ξ|2e−(t−σ)|ξ|2∇̂w(σ)dσ∥2L2dτ

≤
∫ t

s

∥
∫ t

s

ηn(|τ − σ|)|∇̂w(ξ)|dσ∥2L2dτ

≤
∫ t

s

∥∇̂w(τ)∥2L2dτ =

∫ t

s

∥∇w(τ)∥2L2dτ ≤ E2
0 .

With the above estimates, we are ready to bound the right-hand side of (5.3),

−
∫ t

s

∫
R3

[(u · ∇)w] · ϕndxdτ ≤
∫ t

s

∥u∥L6∥∇w∥L2∥ϕn∥L3dτ

≤ C
(∫ t

s

∥∇u∥2L2dτ
)1/2(∫ t

s

∥∇w∥2L2dτ
)1/2

sup
s<τ<t

∥ϕn∥L3

≤ CE2
0

(∫ t

s

∥∇u∥2L2dτ
)1/2

,

(5.8)

−
∫ t

s

∫
R3

[(w · ∇u] · ϕndxdτ

≤ CE2
0

(∫ t

s

∥∇u∥p
Ḃ0

q,∞
dτ

)1/p

(by Theorems 2.6 and 1.2),

(5.9)

−
∫ t

s

∫
R3

[(w · ∇)w] · ϕndx ≤
∫ t

s

∥w∥L6∥∇w∥L2∥ϕn∥L2dτ ≤ E0

∫ t

s

∥∇w∥2L2dτ, (5.10)

−
∫ t

s

∫
R3

g · ϕndxdτ ≤
∫ t

s

∥g∥L2∥ϕn∥L2dτ ≤ CE0

∫ t

s

∥g∥L2dτ. (5.11)

It then follows from (5.4)-(5.11) that (5.3) converges, as n → ∞, to

1

2

∫
R3

|(1−∆)−γ/2w(t)|2dx

≤ 1

2

∫
R3

|e(t−s)∆(1−∆)−γ/2w(s)|2dx+ CE2
0

(∫ t

s

∥∇u∥2L2dτ
)1/2

+ CE2
0

(∫ t

s

∥∇u∥p
Ḃ0

q,∞
dτ

)1/p

+ E0

∫ t

s

∥∇w∥2L2dτ + CE0

∫ t

s

∥g∥L2dτ.

(5.12)



18 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

Employing the Plancherel theorem, the Lebesgue dominated convergence theorem, the first integral
in the right-hand side of (5.12) converges to zero as t → ∞,

lim
t→∞

∫
R3

|e(t−s)∆(1−∆)−γ/2w(s)|2dx = lim
t→∞

∫
R3

|e−(t−s)|ξ|2(1 + |ξ|2)−γ/2ŵ(s)|2dξ = 0.

Consequently, taking limit t → ∞ in (5.12) gives

lim sup
t→∞

∫
R3

|(1−∆)−γ/2w(t)|2dx ≤ CE2
0

(∫ ∞

s

∥∇u∥2L2dτ
)1/2

+ CE2
0

(∫ ∞

s

∥∇u∥p
Ḃ0

q,∞
dτ

)1/p

+ E0

∫ ∞

s

∥∇w∥2L2dτ + CE0

∫ ∞

s

∥g∥L2dτ.

Passing to limit s → ∞, we find that

lim sup
t→∞

∫
R3

|(1−∆)−γ/2w(t)|2dx ≤ 0 ⇒ lim
t→∞

∫
R3

|(1−∆)−γ/2w(t)|2dx = 0. (5.13)

From

∥w∥2L2 =

∫
R3

|ŵ|2dξ

=

∫
R3

[
(1 + |ξ|2)−γ |ŵ|2

] 1
1+γ

[
(1 + |ξ|2)|ŵ|2

] γ
1+γ dξ (since γ < 1)

≤
[ ∫

R3

(1 + |ξ|2)−γ |ŵ|2dξ
] 1

1+γ
[ ∫

R3

(1 + |ξ|2)|ŵ|2dξ
] γ

1+γ

=
[ ∫

R3

|(1−∆)−γ/2w|2dx
] 1

1+γ
[ ∫

R3

(|w|2 + |∇w|2)dx
] γ

1+γ

,

we obtain

lim
t→∞

∫ t

t−1

∥w(τ)∥2L2dτ

≤ lim
t→∞

∫ t

t−1

{[∫
R3

|(1−∆)−γ/2w(τ)|2dx
] 1

1+γ
[ ∫

R3

(|w|2 + |∇w(τ)|2)dx
] γ

1+γ
}
dτ

≤ lim
t→∞

[ ∫ t

t−1

∫
R3

|(1−∆)−γ/2w(τ)|2dxdτ
] 1

1+γ
[ ∫ t

t−1

∫
R3

(|w|2 + |∇w(τ)|2)dxdτ
] γ

1+γ

≤ E
2γ

1+γ

0 lim
t→∞

[
sup

t−1<τ<t

∫
R3

|(1−∆)−γ/2w(τ)|2dx
] 1

1+γ

(by (1.13) and (5.7))

= 0 (by (5.13)).

This completes the proof of Lemma 5.1. □

Now, we return to show Theorem 1.6. Just as the proof of Lemma 5.1, it suffices to consider
only the case where (1.19) holds. Checking the derivation of (4.5), we see that (4.5) still holds if
we replace the time interval (0, t) to be (s, t),∫

R3

|w(t)|2dx+

∫ t

s

∫
R3

|∇w(τ)|2dτ

≤
[
∥w(s)∥2L2 +

∫ t

s

∥g∥L2dτ
]
+ C

∫ t

s

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
∥w∥2L2dτ

≤
[
∥w(s)∥2L2 +

∫ t

s

∥g∥L2dτ
]
+ C

∫ t

s

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
E2

0dτ

≤ ∥w(s)∥2L2 + C

∫ t

s

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ.



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 19

Integrating the above inequality with respect to s over (t− 1, t) yields∫
R3

|w(t)|2dx ≤
∫ t

t−1

∥w(s)∥2L2ds+ C

∫ t

t−1

∫ t

s

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτds

≤
∫ t

t−1

∥w(s)∥2L2ds+ C

∫ t

t−1

∫ t

t−1

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτds

≤
∫ t

t−1

∥w(s)∥2L2ds+ C

∫ t

t−1

(
∥g∥L2 + ∥∇u∥p

Ḃ0
q,∞

)
dτ → 0

as t → ∞, by Lemma 5.1 and the assumptions of Theorem 1.6. This completes the proof of
Theorem 1.6.

6. Upper bounds estimates

In this and latter sections, we shall prove Theorem 1.9. First, let us first treat the upper bound
in (1.25). Without loss of generality, we may assume (1.19). The case where (1.20) can be treated
in a similar way.

As in (5.2), the difference w of the solutions of the perturbed Navier-Stokes system (1.3) and
the original Navier-Stokes system (1.1) satisfies

∂tw −∆w + (v · ∇)w + (w · ∇)u+∇Πw = g, w|t=0 = w0. (6.1)

To investigate the optimal convergence rates, we need the pointwise bound of ŵ.

Lemma 6.1. Under the assumptions of Theorem 1.9, we have

|ŵ(ξ, t)| ≤ |ŵ0(ξ)|e−t|ξ|2 + C|ξ|
∫ t

0

∥w(s)∥L2ds. (6.2)

Proof. It suffices to show (6.2) formally, since the rigorous derivation can be obtained as [11,
Lemma 3.1]. Applying the Fourier transform to both sides of (6.1) gives

∂tŵ + |ξ|2ŵ = −F [(v · ∇)w + (w · ∇)u]−F [∇Πw] = J1(ξ, t) + J2(ξ, t).

Solving this ordinary differential equation yields

ŵ(t) = ŵ0(ξ)e
−t|ξ|2 +

∫ t

0

e−(t−s)|ξ|2(J1 + J2)ds.

Consequently,

|ŵ(t)| ≤ |ŵ0(ξ)|e−t|ξ|2 +

∫ t

0

(|J1|+ |J2|)ds. (6.3)

For J1, it follows from the divergence-free condition, the Hölder inequality, and the definition of
weak solutions that

|J1(ξ, s)| =
∣∣ 1

(2π)3/2

∫
R3

e−iξ·x[(v · ∇)w + (w · ∇)u]dx
∣∣

=
∣∣ 1

(2π)3/2

3∑
i=1

∫
R3

e−iξ·x[∂i(viw) + ∂i(wiu)]dx
∣∣

≤ C

3∑
i=1

|ξ|
∫
R3

|viw + wiu|dx

≤ C|ξ|(∥v∥L2 + ∥u∥L2)∥w∥L2

≤ C|ξ|∥w∥L2 .

(6.4)

To estimate J2, we need a representation of Πw in terms of u, v, and w. Taking the divergence
of (6.1) gives

−∆Πw =

3∑
i,j=1

∂i∂j(viwj + wivj),



20 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

and hence

∇Πw = ∇
3∑

i,j=1

(−∆)−1∂i∂j(viwj + wivj).

Consequently,

|J2(ξ, s)| ≤ |ξ|
3∑

i,j=1

∫
R3

|viwj + wiuj |dx

≤ C|ξ| (∥v∥L2 + ∥u∥L2) ∥w∥L2

≤ C|ξ|∥w∥L2 .

(6.5)

Putting (6.4)-(6.5) into (6.3), we obtain (6.2) as desired. □

We are now ready to show the upper bound in (1.25). Taking the inner product of (6.1) with
w in L2(R3), and invoking Theorem 2.6, we obtain

1

2

d

dt
∥w∥2L2 + ∥∇w∥2L2 = −

∫
R3

[(w · ∇)u] ·wdx

≤ C∥∇u∥p
Ḃ0

q,∞
∥w∥2L2 +

1

2
∥∇w∥2L2

≤ C∥∇u∥p
Ḃ0

q,∞
∥w∥2L2 +

1

2
∥∇w∥2L2 ,

or equivalently,
d

dt
∥w∥2L2 + ∥∇w∥2L2 ≤ C∥∇u∥p

Ḃ0
q,∞

∥w∥2L2 . (6.6)

Employing the Plancherel theorem, it follows that

d

dt

∫
R3

|w(t)|2dξ +
∫
R3

|ξ|2|ŵ(t)|2dξ ≤ C∥∇u∥p
Ḃ0

q,∞
∥w∥L2 . (6.7)

Multiplying both sides of (6.7) by (1+t)a with a > 0 sufficiently large to be determined, we obtain

d

dt

[
(1 + t)a

∫
R3

|w(t)|2dξ
]
+ (1 + t)a

∫
R3

|ξ|2|ŵ(t)|2dξ

≤ a(1 + t)a−1

∫
R3

|ŵ(t)|2dξ + C∥∇u∥p
Ḃ0

q,∞
(1 + t)a∥w∥L2 .

(6.8)

Let

B(t) =
{
ξ ∈ R3; |ξ|2 ≤ a

1 + t

}
. (6.9)

Then we split R3 into B(t) and its complement Bc(t). Consequently,

(1 + t)a
∫
R3

|ξ|2|ŵ(t)|2dξ ≥ (1 + t)a
∫
Bc(t)

|ξ|2|ŵ(t)|2dξ ≥ a(1 + t)a−1

∫
Bc(t)

|ŵ(t)|2dξ

= a(1 + t)a−1

∫
R3

|ŵ(t)|2dξ − a(1 + t)a−1

∫
B(t)

|ŵ(t)|2dξ.

Plugging the above inequality into (6.8), we find that

d

dt

[
(1 + t)a∥w(t)∥2L2

]
≤ a(1 + t)a−1

∫
B(t)

|ŵ(t)|2dξ + C∥∇u∥p
Ḃ0

q,∞
(1 + t)a∥w∥2L2 + (1 + t)a∥g∥L2 .

(6.10)

We apply an iterative process to derive the optimal upper bound estimate in (1.25). Suppose now

∥w(t)∥L2 ≤ C(1 + t)−bn , ∀t ≥ 0, (6.11)

with b0 = 0 by the definition of weak solutions.



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 21

Integrating (6.10) with respect to time over [0, t] gives

(1 + t)a∥w(t)∥2L2 ≤ ∥w(0)∥2L2 + a

∫ t

0

(1 + s)a−1

∫
B(s)

|ŵ(s)|2dξds

+ C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥w(s)∥2L2ds.

(6.12)

We denote by K the first integral in the right-hand side of (6.12). We shall estimate K by Lemma
6.1,

K = a

∫ t

0

(1 + s)a−1

∫
B(s)

|ŵ(s)|2dξds

≤ a

∫ t

0

(1 + s)a−1

∫
B(s)

[
|ŵ0|e−s|ξ|2 + C|ξ|

∫ s

0

∥w(τ)∥L2dτ

]2
dξds

≤ C

∫ t

0

(1 + s)a−1∥F(es∆w0)∥2L2ds

+ C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2
[ ∫ s

0

∥w(τ)∥L2dτ
]2
dξds

≡ K1 +K2.

(6.13)

By the Plancherel theorem and Lemma 1.8,

K1 = C

∫ t

0

(1+s)a−1∥es∆w0∥2L2ds ≤ C

∫ t

0

(1+s)a−1(1+s)−γds ≤ C(1+t)a−γ (if a > γ ) . (6.14)

For K2, we resort to (6.11),

K2 ≤ C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2
[∫ s

0

(1 + τ)−bndτ

]2
dξds

≤ C

∫ t

0

(1 + s)a−1+2(1−bn)

∫
B(s)

|ξ|2dξds (if bn < 1 )

≤ C

∫ t

0

(1 + s)a−1+2(1−bn)
1

(1 + t)
5
2

ds

≤ C(1 + t)a−2bn− 1
2 (if a > 2bn +

1

2
).

(6.15)

Collecting (6.14)-(6.15) into (6.13), inequality (6.12) becomes

(1 + t)a∥w(t)∥2L2 ≤ C(1 + t)a−γ + C(1 + t)a−2bn− 1
2 + C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥w(s)∥2L2ds.

Invoking the Gronwall inquality gives

(1 + t)a∥w(t)∥2L2

≤ C
[
(1 + t)a−γ + (1 + t)a−2bn− 1

2

]{
1 + C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
dτ exp

[
C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
dτ

]}
≤ C(1 + t)max{a−γ,a−2bn− 1

2}

= C(1 + t)a−min{γ,2bn+ 1
2};

or equivalently,

∥w(t)∥L2 ≤ C(1 + t)−min{ γ
2 ,bn+

1
2}, ∀t ≥ 0.

This implies that bn+1 = min{γ
2 , bn + 1

2},

b0 = 0, b1 =
1

4
, b2 =

1

2
, b3 =

3

4
, b4 = 1.



22 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

After four iterations, we should estimate K2 in a different manner,

K2 ≤ C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2
[ ∫ s

0

(1 + τ)−b4dτ
]2
dξds

≤ C

∫ t

0

(1 + s)a−1 ln2(1 + s) · 1

(1 + s)
5
2

ds (b4 = 1)

≤ C

∫ t

0

(1 + s)
a−( 5

2 )− · 1

(1 + s)
7
2−(

5
2 )−

ln2(1 + s)ds

≤ C(1 + t)a−( 5
2 )− (if a ≥ 5

2
),

(6.16)

where ( 52 )− represents any positive real number less than 5
2 . Gathering (6.14), (6.16) into (6.13),

we find (6.12) reduces to

(1 + t)a∥w(t)∥2L2 ≤ C(1 + t)a−γ + C(1 + t)
a−( 5

2 )− +

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥w(s)∥2L2ds.

Employing the Gronwall inequality and noticing that γ < 5
2 , we deduce

∥w(t)∥L2 ≤ C(1 + t)
−min

{
γ
2 ,(

5
4 )−

}
= C(1 + t)−γ/2, ∀t ≥ 0. (6.17)

With this estimate, we could not iterate further as before to derive finer decay. Indeed, (6.17)
implies that ∫ s

0

∥w(τ)∥L2dτ ≤ C

∫ s

0

(1 + τ)−γ/2dτ =
2
[
1− (1 + t)1−

γ
2

]
γ − 2

≤ 2

γ − 2
,

since γ > 2. This completes the proof of the upper bound estimate in (1.25).
To end this section, let us show (1.26) under the assumption γ ≥ 5

2 . In this circumstance, we
could still iterate as before by using (1.25),

K2 = C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2
[ ∫ s

0

∥w(τ)∥L2dτ
]2
dξds

≤ C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2dξds

≤ C

∫ t

0

(1 + s)a−1 · 1

(1 + s)
5
2

ds

≤ C(1 + t)a−
5
2 (if a >

5

2
).

This and (6.14) imply

(1 + t)a∥w(t)∥2L2 ≤ C(1 + t)a−γ + C(1 + t)a−
5
2 + C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥w(s)∥2L2ds.

Arguing as before, we obtain

∥w(t)∥L2 ≤ C(1 + t)−min{ γ
2 ,

5
4} ≤ C(1 + t)−5/4, ∀t ≥ 0,

as desired. Finally, we can choose a > max{γ, 5
2} .

7. Lower bounds estimates

Recall that the solution difference w = v − u satisfies (6.1), and W(x, t) = et∆w0(x) be the
solution of the heat equation (1.23). We denote V = w −W. Then

∂tV −∆V + (v · ∇)w + (w · ∇)u+∇Πw = 0,

∇ ·V = 0,

V|t=0 = 0.

(7.1)



EJDE-2025/79 STABILITY OF SOLUTIONS TO NAVIER-STOKES EQUATIONS 23

We shall first examine the decay rates of the solution of (7.1). Taking the inner product of (7.1)
with V in L2(R3), we obtain

1

2

d

dt
∥V∥2L2 + ∥∇V∥2L2

= −
∫
R3

[(v · ∇)w] ·Vdx−
∫
R3

[(w · ∇)u] ·Vdx

= −
∫
R3

[(v · ∇)(V +W)] ·Vdx−
∫
R3

[((V +W) · ∇)u] ·Vdx

= −
∫
R3

(v · ∇)W] ·Vdx−
∫
R3

[(V · ∇)u] ·Vdx−
∫
R3

[(W · ∇)u] ·Vdx

≡ L1 + L2 + L3.

(7.2)

It follows from Hölder’s inequality, see [5],

divF = 0, curlG = 0 ⇒ ∥F ·G∥H1 ≤ C∥F∥L2∥G∥L2 ,

and the Sobolev inequality (see [1, Theorem 1.48]) that

L1 = −
∫
R3

[(v · ∇)W] ·Vdx =

∫
R3

[(v · ∇)V] ·Wdx

= −
3∑

i=1

∫
R3

(v · ∇)ViWidx ≤ C

3∑
i=1

∥(v · ∇)Vi∥H1∥Wi∥BMO

≤ C∥v∥L2∥∇V∥L2∥W∥Ḣ3/2 ≤ 1

6
∥∇V∥2L2 + C∥W∥2

Ḣ3/2

≤ 1

6
∥∇V∥2L2 + C(1 + t)−(γ+

3
2 ) (by Lemma 1.8).

(7.3)

For L2, we utilize Lemma 2.6 as

L2 = −
∫
R3

[(V · ∇)u] ·Vdx ≤ 1

6
∥∇V∥2L2 + C∥∇u∥p

Ḃ0
q,∞

∥V∥2L2 . (7.4)

The third term can be bounded as

L3 = −
∫
R3

[(W · ∇)u] ·Vdx =

∫
R3

[(W · ∇)V] · udx

≤ ∥W∥L6∥∇V∥L2∥u∥L3 ≤ C∥∇W∥Ḣ1∥∇V∥L2 · ∥u∥1/2L2 ∥∇u∥1/2L2

≤ 1

6
∥∇V∥2L2 + C∥W∥2

Ḣ1∥∇u∥L2

≤ 1

6
∥∇V∥2L2 + C(1 + t)−(γ+1)∥∇u∥L2 (by Lemma 1.8).

(7.5)

Collecting (7.3)-(7.5) into (7.2), we deduce that

d

dt
∥V∥2L2 + ∥∇V∥2L2 ≤ C(1 + t)−(γ+

3
2 ) + C∥∇u∥p

Ḃ0
q,∞

∥V∥2L2 + C(1 + t)−(γ+1)∥∇u∥L2 . (7.6)

We then apply the developed Fourier splitting methods as in the previous section. Recalling (6.9),
we derive the following analogy of (6.12),

(1 + t)a∥V(t)∥2L2 ≤ a

∫ t

0

(1 + s)a−1

∫
B(s)

|V̂|2dξds+ C

∫ t

0

(1 + s)a−(γ+
3
2 )ds

+ C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥V(s)∥2L2ds

+ C

∫ t

0

(1 + s)a−(γ+1)∥u(s)∥2L2ds.

(7.7)



24 Z. ZHANG, W. YUAN, Z. YAO EJDE-2025/79

To estimate the first integral in the right-hand side of (7.7), it suffices to establish a bound of |V̂|
similar to Lemma 6.1,

|V̂(ξ, t)| ≤ C|ξ|
∫ t

0

∥w(s)∥L2ds ≤ C|ξ|
∫ t

0

(1 + s)−γ/2ds (by (6.17))

≤ C|ξ|.
(7.8)

With (7.8) in hand, we obtain

a

∫ t

0

(1 + s)a−1

∫
B(s)

|V̂|2dξds = a

∫ t

0

(1 + s)a−1

∫
B(s)

|V̂|2dξds

≤ C

∫ t

0

(1 + s)a−1

∫
B(s)

|ξ|2dξds

≤ C

∫ 1

0

(1 + s)a−1 1

(1 + s)
5
2

ds

≤ C(1 + t)a−
5
2 .

(7.9)

Furthermore, we can bound the last integral in the right-hand side of (7.7) as

C

∫ t

0

(1 + s)a−(γ+1)∥u(s)∥2L2ds ≤ C
{∫ t

0

(1 + s)2[a−(γ+1)]ds
}1/2[ ∫ t

0

∥∇u(s)∥2L2ds
]1/2

≤ C(1 + t)
2[a−(γ+1)]+1

2 ≤ C(1 + t)a−γ− 1
2 .

(7.10)

Putting (7.9) and (7.10) into (7.7) yields

(1 + t)a∥V(t)∥2L2 ≤ C(1 + t)a−
5
2 + C(1 + t)a−(γ+

3
2 )+1

+ C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥V(s)∥2L2ds+ C(1 + t)a−γ− 1

2

≤ C(1 + t)a−
5
2 + C

∫ t

0

∥∇u∥p
Ḃ0

q,∞
(1 + s)a∥V(s)∥2L2ds.

Applying the Gronwall inequality, we obtain

∥V(t)∥L2 ≤ C(1 + t)−5/4. (7.11)

From Lemma 1.8, we have

∥W(t)∥L2 ≥ C1(1 + t)−γ/2.

This and (7.11) imply

∥w(t)∥L2 = ∥V(t)−W(t)∥L2 ≥ ∥W(t)∥L2 − ∥V(t)∥L2 ≥ C1(1 + t)−γ/2, (7.12)

which completes the proof of the lower bound estimate in (1.25). The proof of Theorem 1.9 is
complete.

Acknowledgments. This work is supported by the National Key Research and Development
Program of China (2020YFA0712500).

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Zujin Zhang (corresponding author)

School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi,
China

Email address: zhangzujin361@163.com

Weijun Yuan
School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi,

China.

No. 3 Middle School of Yichun City, Jiangxi Province, Yi chun 336000 Jiangxi, China
Email address: 1453540745@qq.com

Zhengan Yao
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Email address: mcsyao@mail.sysu.edu.cn