Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 68, pp. 1–15.

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

DOI: 10.58997/ejde.2025.??

SINGLE-COMPONENT REGULARITY CRITERION AND INVISCID LIMIT

FOR AXIALLY SYMMETRIC MHD-BOUSSINESQ SYSTEMS

ZHAOJUN XING

Abstract. In this article, we give a critical BKM-type blow-up criterion that only involves the

horizontal swirl component of the velocity for inviscid axially symmetric MHD-Boussinesq sys-
tems. We consider the inviscid limit for viscous MHD-Boussinesq systems, and the convergence

rate as the viscosity coefficient tending to zero.

1. Introduction

The MHD-Boussinesq system models the convection of an incompressible flow, which is driven
by the buoyancy effect of the thermal or density field and the Lorentz force generated by the fluid
magnetic field. In addition, it is closely related to Rayleigh-Bénard convection. This convection
occurs in a horizontal layer of conductive fluid heated from below, with the effect of the magnetic
field. For a more detailed physical background, interested readers are referred to [26, 27, 30, 32]
for further reading. In the following, we present the 3D MHD-Boussinesq system:

∂tu+ u · ∇u− µ∆u+∇p = h · ∇h+ ρe3,

∂th+ u · ∇h− h · ∇u− ν∆h = 0,

∂tρ+ u · ∇ρ− κ∆ρ = 0,

∇ · u = ∇ · h = 0.

(1.1)

Here u ∈ R3 stands for the velocity and h ∈ R3 stands for the magnetic field. p ∈ R denotes the
pressure. µ > 0, ν > 0 and κ > 0 denote the constant kinematic viscosity, magnetic diffusivity,
and thermal diffusivity, respectively.

The MHD-Boussinesq system consists of a coupling between the Boussinesq equation and the
magnetohydrodynamic equations. When the temperature fluctuation can be ignored, the system
(1.1) degenerates into the magnetohydrodynamic system. For the 3D MHD system, Liu [22]
established a regularity criterion for the system, while Jiu-Liu [11] later proved global regularity
for the axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion.
See references therein for more regularity results on the MHD system. On the other hand, when
we ignore the Lorentz force, the system (1.1) reduces to the Boussinesq system. There have been
many studies on the well-posedness of the Boussinesq system. We refer readers to [8, 18, 20]
and references therein for 3D results. For the full MHD-Boussinesq system, there are also some
works concentrated on the global well-posedness of weak and strong solutions. See [1, 2] and
references therein for 2D cases. In the 3D case, Larios-Pei [16] proved the local well-posedness
results in Sobolev space. Liu-Bian-Pu [15] proved the global well-posedness of strong solutions
with nonlinear damping terms in the momentum equations. Li [17] proved the regularity criterion,
which only involves the horizontal swirl component of the vorticity field, to a class of three-
dimensional axisymmetric MHD-Boussinesq system without magnetic impedance and thermal

2020 Mathematics Subject Classification. 35Q35, 76D03, 35B07.

Key words and phrases. MHD-Boussinesq; axially symmetric; regularity criterion; inviscid limit.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 5, 2025. Published July 7, 2025.

1



2 Z. XING EJDE-2025/??

diffusivity. Later, Li-Pan [19] proved the related criterion that only involves the horizontal swirl
component of the velocity field.

In recent years, Bian-Pu [3] proved the global regularity of a family of axially symmetric large
solutions to the MHD-Boussinesq system without magnetic resistivity and thermal diffusivity
under the assumption that the support of the initial thermal fluctuation is away from the z-axis
and its projection on to the z-axis is compact. Later, this result was improved by Pan [31] by
removing the assumption on the data of the thermal fluctuation.

Our first aim is to prove a single-component regularity criterion of the 3D axially symmetric
inviscid MHD-Boussinesq system in Sobolev space Hm(∀m ≥ 3). Setting µ = 0 of (1.1), we obtain

∂tu+ u · ∇u+∇p = h · ∇h+ ρe3,

∂th+ u · ∇h− h · ∇u− ν∆h = 0,

∂tρ+ u · ∇ρ− κ∆ρ = 0,

∇ · u = ∇ · h = 0.

(1.2)

In the cylindrical coordinates (r, θ, z), i.e., for x = (x1, x2, x3) ∈ R3,

r =
√
x2
1 + x2

2, θ = arctan
x2

x1
, z = x3,

a solution of (1.2) is called an axially symmetric solution, if

u = ur(t, r, z)er + uθ(t, r, z)eθ + uz(t, r, z)ez,

h = hr(t, r, z)er + hθ(t, r, z)eθ + hz(t, r, z)ez,

ρ = ρ(t, r, z),

satisfy the system (1.2). Here, the basis vectors er, eθ, ez are

er =
(x1

r
,
x2

r
, 0
)
, eθ =

(
−x2

r
,
x1

r
, 0
)
, ez = (0, 0, 1).

From the local existence and uniqueness results, it is clear that one only needs to assume
h0 · er = h0 · ez ≡ 0, then vanishing of hr and hz holds for all time. In this case, Choosing
ν = κ = 1 for simplicity, (1.2) can be simplified to

∂tur + (ur∂r + uz∂z)ur + ∂rP = − (hθ)
2

r
+

u2
θ

r
,

∂tuθ + (ur∂r + uz∂z)uθ = −uθur

r
,

∂tuz + (ur∂r + uz∂z)uz + ∂zP = ρ,

∂thθ + (ur∂r + uz∂z)hθ −
hθur

r
=

(
∆− 1

r2

)
hθ,

∂tρ+ (ur∂r + uz∂z) ρ−∆ρ = 0,

∇ · u = ∂rur +
ur

r
+ ∂zuz = 0,

(1.3)

where P = p+ 1
2 |hθ|2.

Let

Φk,α(t) ≤ c exp(c exp(· · · exp(ctα)))︸ ︷︷ ︸
k times exponents

, c > 0, k ≥ 1.

Our main result reads as follows.

Theorem 1.1. Given m ∈ N and m ≥ 3. Suppose (u, h, ρ) is the unique strong solution of (1.2)
with initial data (u0, h0, ρ0) ∈ Hm ×Hm ×Hm, rρ0 ∈ L2 and ∇ ·u0 = h0 · er = h0 · ez ≡ 0. Then,
(u, h, ρ)(t, ·) can be smoothly extended before T∗ if and only if a Beale-Kato-Majda-type condition
on the swirl part of the velocity (1.4) holds.∫ T∗

0

∥∇ × (uθeθ)(t, ·)∥L∞dt ≤ C∗ < ∞. (1.4)



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 3

Now, (u, h, ρ)(t, ·) satisfies the temporal asymptotic property

∥(u, h, ρ)(t, ·)∥2Hm ≤ Φ4,3(t), ∀t ≤ T∗.

Remark 1.2. When uθ = 0, the global well-posedness result for the inviscid axisymmetric MHD-
Boussinesq system can be found in [21]. If h = 0, [20] gave a single-component Beale-Kato-Majda-
type regularity criterion for inviscid Boussinesq equations. Our first result can be viewed as a
generalization of the above papers.

The problem of vanishing viscosity limit is one of the most challenging topics in fluid dynamics.
For the inviscid limit problems on bounded domains, we refer readers to [12, 33] and [6]. Since Kato
[12] and Swann [33], many authors have studied the convergence of the solution to the Navier-
Stokes equation as the viscosity approaches zero. In the last decades, Itoh [9] and Itoh-Tani
[10] investigated the inviscid limit of the equation for non-homogeneous incompressible fluids and
demonstrated the convergence in H2 and W 1,p (p > 3), respectively. Dı́az-Lerena [7] investigated
the inviscid and non-resistive limit in the Cauchy problem of an incompressible homogeneous MHD
system by using the C0-semigroup technique developed in [13]. Majda [25] proved that when
µ → 0, the solution uµ of the Navier-Stokes equation converges to the unique solution of the Euler

equation in the L2 norm and the convergence rate is (µt)1/2, assuming u0 ∈ Hs, s > d
2 +2. Later,

Masmoudi [28] improved this result and demonstrated the convergence in the Hs norm under the
weaker assumptions u0 ∈ Hs, s > d

2 + 1. Recently, Liu [23] proved that viscous axisymmetric
swirling flows converge to inviscid swirl-free solutions under a specific condition on initial swirl
velocity, in the exterior of a cylinder with the Navier-slip boundary condition. Maafa-Zerguine
[24] studied the inviscid limit of MHD system in Besov spaces.

Our second goal is to examine the inviscid limit of the 3D MHD-Boussinesq system with κ =
ν = 1 of (1.1)

∂tuµ + uµ · ∇uµ − µ∆uµ +∇pµ = hµ · ∇hµ + ρµe3,

∂thµ + uµ · ∇hµ − hµ · ∇uµ −∆hµ = 0,

∂tρµ + uµ · ∇ρµ −∆ρµ = 0,

∇ · uµ = ∇ · hµ = 0.

(1.5)

Here, the solution is also supposed to be axially symmetric,

uµ = uµ,r(t, r, z)er + uµ,θ(t, r, z)eθ + uµ,z(t, r, z)ez,

hµ = hµ,r(t, r, z)er + hµ,θ(t, r, z)eθ + hµ,z(t, r, z)ez,

ρµ = ρµ(t, r, z).

When the viscosity coefficient µ is vanishing, the MHD-Boussinesq system (1.5) appears to degen-
erate into the system (1.2). Our second main result quantified the rate of this convergence.

Theorem 1.3. Let (u, h, ρ) and (uµ, hµ, ρµ) be strong solutions to system (1.2) and system (1.5)
respectively with the same initial data. Then under the conditions as in Theorem 1.1, the following
asymptotic behavior holds

∥uµ − u∥L∞
t L2 + ∥hµ − h∥L∞

t L2 + ∥ρµ − ρ∥L∞
t L2 ≤ (µt)Φ4,3(t).

Remark 1.4. [24] gave the inviscid limit of MHD system in Besov space, and in Theorem 1.3 we
use a similar method to give the inviscid limit of MHD-Boussinesq system in L2.

The detailed statements of Theorems 1.1 and 1.3 can be found in Sections 3 and 4. We next
introduce the notation, conventions and lemmas to be used in the proof.

2. Preliminaries

In this article, we use Ca,b,c,... to represent a positive constant depending on a, b, c, . . . and it
may be different from line to line. For A ≲ B, it means A ≤ CB. And A ≃ B means both A ≲ B
and B ≲ A. [A,B] = AB − BA denotes the communicator of the operator A and the operator B.
We give the usual Lebesgue space Lp, Sobolev functional space W k,p, and the usual homogeneous
Sobolev space Ẇ k,p. When p = 2, replace W k,p and Ẇ k,p with Hk and Ḣk. f ∈ Lp ∩ Lq with



4 Z. XING EJDE-2025/??

1 ≤ p, q ≤ ∞, we shall denote its Yudovich type norm as ∥f∥Lp∩Lq = max {∥f∥Lp , ∥f∥Lq}. Given
a Banach space X, we say v : [0, T ]× R3 → R belongs to the Bochner-Banach space Lp(0, T ;X),
if ∥v(t, ·)∥X ∈ Lp(0, T ), and we usually use Lp

TX for short notation of Lp(0, T ;X).
In this paper, we do not distinguish functional spaces for scalar or vector-valued functions since

it will be clear from the context.
Now we introduce some essential lemmas. First of all, we give the Gagliardo-Nirenberg inter-

polation inequality, which we will not prove here.

Lemma 2.1 (Gagliardo-Nirenberg). Fix q, r ∈ [1,∞] and j,m ∈ N ∪ {0} with j ≤ m. Suppose

that f ∈ Lq ∩ Ẇm,r(Rd) and there exists a real number α ∈ [j/m, 1] such that

1

p
=

j

3
+ α

(1
r
− m

3

)
+

1− α

q
.

Then f ∈ Ẇ j,p
(
Rd

)
and there exists a constant C > 0 such that

∥∇jf∥Lp ≤ C∥∇mf∥αLr∥f∥1−α
Lq ,

except the following two cases:

(i) j = 0,mr < d and q = ∞; (In this case it is necessary to assume also that either u → 0
at infinity, or u ∈ Ls for some s < ∞).

(ii) 1 < r < ∞ and m− j − 3/r ∈ N. (In this case it is necessary to assume also that α < 1.)

Next, we state the following estimation of the triple product form with commutators, the proof
of which can be found in [20, Lemma 3.8].

Lemma 2.2. Let m ∈ N and m ≥ 2, f, g, k ∈ C∞
0 (R3). The following estimate holds:∣∣ ∫

R3

[∇m, f · ∇]g∇mkdx
∣∣ ≤ C∥∇m(f, g, k)∥2L2∥∇(f, g)∥L∞ .

The following inequality is related to the logarithmic Sobolev inequality.

Lemma 2.3. For any divergence free vector field g such that g : R → R3 and g ∈ H3(R3), it holds

∥∇g∥L∞(R3) ≲ 1 + ∥∇ × g∥L∞ log
(
e+ ∥g∥H3(R3)

)
.

The following lemma can be obtained from the Biot-Savart law and the Lp boundedness of the
Calderon-Zygmund singular integral operator; its proof ican be found in [4, 5].

Lemma 2.4. Let u = urer + uθeθ + uzez be an axially symmetric vector field, ω = ∇ × u =
ωrer + ωθeθ + ωzez and b = urer + uzez. Then we have

∥∇u∥Lp ≤ Cp∥ω∥Lp ,

and
∥∇b∥Lp ≤ Cp∥ωθ∥Lp , (2.1)

for all 1 < p < ∞.

Here is a famous lemma whose proof can be found in [14, (A.5)] and in [29, Prop. 2.5 ].

Lemma 2.5. Define Ω := ωθ

r . For 1 < p < +∞, there exists an absolute constant Cp > 0 such
that

∥∇ur

r
(t, ·)∥Lp ≤ Cp∥Ω(t, ·)∥Lp . (2.2)

Below, we give the one-dimensional Hardy inequality.

Lemma 2.6. If p > 1, σ ̸= 1, f is a nonnegative measurable function and F is defined by

F (x) =

∫ x

0

f(t)dt, (σ > 1), F (x) =

∫ x

0

f(t)dt, (σ < 1).

Then, ∫ ∞

0

x−σF p dx <
( p

|σ − 1|
)
)p

∫ ∞

0

x−σ(xp)p dx,

unless f ≡ 0.



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 5

By Lemma 2.6, we can get the following result whose poof can be found in [20].

Lemma 2.7.

∥uθ

r
(t, ·)∥L∞ ≤ 1

2
∥ωz(t, ·)∥L∞ ,

for any t > 0.

3. Proof of Theorem 1.1

Firstly, we derive the fundamental estimates for system (1.2) from the basic energy estimate.
Secondly, we define four special quantities to establish a reformulated system and derive a self-
closed estimate. Then, we derive a one-component BKM-type criterion for the MHD-Boussinesq
system. Finally, we conclude the proof of Theorem 1.1.

First of all, we give the fundamental energy estimates of the system (1.2).

Proposition 3.1. We define H := hθ

r . Let (u, h, ρ) be a smooth solution of (1.2), we have

(i) for p ∈ [2,∞) and t ∈ R+,

∥H(t, ·)∥pLp +

∫ t

0

∫
R3

|∇H(s, x)|2|H(s, x)|p−2 dx ds ≤ ∥H0∥pLp ,

∥H(t, ·)∥L∞ ≤ ∥H0∥L∞ ,

∥ρ(t, ·)∥pLp +

∫ t

0

∫
R3

|∇ρ(s, x)|2|ρ(s, x)|p−2 dx ds ≤ ∥ρ0∥pLp ,

∥ρ(t, ·)∥L∞ ≤ ∥ρ0∥L∞ .

(3.1)

(ii) for (u0, h0, ρ0) ∈ L2 and t ∈ R+,

∥(u, h)(t, ·)∥2L2 +

∫ t

0

∥∇h (s, ·) ∥2L2 ds ≤ C0(1 + t)2, (3.2)

where C0 depends only on ∥(u0, h0, ρ0)∥L2 .

Proof. From (1.3)4, we derive that

∂tH+ (ur∂r + uz∂z)H =
(
∆+

2

r
∂r

)
H.

Multiplying the above equation by p|H|p−2H, integrating over R3, and using ∇u = 0, we obtain

1

p

d

dt
∥H(t, ·)∥pLp +

∫
R3

|∇H(t, ·)|2|H(t, ·)|p−2dx ≤ 0,

and integrating over (0, t) to obtain (3.1)1,2.
Similarly for equation of ρ in (1.3)5, one derives (3.1)3,4. The estimation in (3.2), can be

obtained by applying the standard L2 inner product estimation for system (1.2) and using (3.1)1.
Also, refer to [31, Proposition 2.1]. □

Next, we establish a reformulated system.
The vorticity of the axially symmetric velocity field u is given by

ω(t, r, z) = ∇× u = ωr(t, r, z)er + ωθ(t, r, z)eθ + ωz(t, r, z)ez,

where
ωr = −∂zuθ, ωθ = ∂zur − ∂ruz, ωz = ∂ruθ +

uθ

r
.

By taking special derivatives of (1.3)1,2,3, one concludes that (ωr, ωθ, ωz) satisfies

∂tωr + (ur∂r + uz∂z)ωr = (ωr∂r + ωz∂z)ur,

∂tωθ + (ur∂r + uz∂z)ωθ =
ur

r
ωθ +

1

r
∂z

(
u2
θ

)
− 1

r
∂z

(
h2
θ

)
− ∂rρ,

∂tωz + (ur∂r + uz∂z)ωz = (ωr∂r + ωz∂z)uz.

(3.3)

We denote

Ω :=
ωθ

r
, J :=

ωr

r
, N :=

∂rρ

r
, ∇H := ∇hθ

r
.



6 Z. XING EJDE-2025/??

From (3.3) combined with (1.3)4,5, we obtain the following reformulated system for (Ω, J,N ,∇H):

∂tΩ+ u · ∇Ω = −2
uθωr

r2
− ∂zH2 −N ,

∂tJ + u · ∇J = (ωr∂r + ωz∂z)
ur

r
,

∂tN + u · ∇N −
(
∆+

2

r
∂r

)
N = ∂zuzN − ∂ruz

∂zρ

r
,

∂t∇H+ u · ∇∇H+∇b · ∇H −
(
∆+

2

r
∂r

)
∇H−∇

(2
r

)
∂rH = 0.

(3.4)

Proposition 3.2. Let (Ω, J,N ,∇H) be defined as above, which solves (3.4) with initial data

(Ω0, J0,N0,H0) ∈ (L2 ∩ L6)× (L2 ∩ L6)× L2 × (L∞ ∩H1).

Then, the following space-time estimate holds for any t ∈ (0, T∗]:

∥(Ω, J)(t, ·)∥2L2∩L6 + ∥(N ,∇H)(t, ·)∥2L2 +

∫ t

0

∥(∇N ,∇2H)(s, ·)∥2L2ds ≤ Φ1,3(t). (3.5)

Proof. For the proof of (3.5), we first derive an estimate about ∇H from Ω, and then an estimate
about N . Then, the estimate of (Ω, J) is obtained from the first two parts. Finally, we combine
these estimates to get (3.5). The following is specific to the process:

At the beginning, we derive the estimate of ∇H. By performing L2 inner product of ∇H, (3.4)4
follows that:

1

2

d

dt
∥∇H(t, ·)∥2L2 + ∥∇2H(t, ·)∥2L2 −

∫
R3

2

r
∂r∇N · ∇N dx−

2∑
i=1

∫
R3

∂i(
2

r
)∂rH∂iH dx

= −
3∑

i,j=1

∫
R3

∂iuj∂jH∂iH dx.

Here, one can follow the same method as in [21, Proposition 3.2] to carry out estimates. This ends
up with

∥∇H(t, ·)∥2L2 +

∫ t

0

∥∇2H(s, ·)∥2L2 ds ≲ ∥∇H0∥2L2 +

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇h(s, ·)∥2L2 ds. (3.6)

Then, we obtain the estimate of N . By taking the L2 inner product with N for (3.4)3, then
integrating on R3, one has

1

2

d

dt
∥N (t, ·)∥2L2 + ∥∇N (t, ·)∥2L2 + 2π

∫ ∞

0

|N (t, 0, z)|2dz

=

∫
R3

∂zuzN 2 dx−
∫
R3

∂ruz
∂zρ

r
N dx.

(3.7)

Using the method in the proof in [19, Proposition 3.2], (3.7) can be written as

∥N (t, ·)∥2L2 +

∫ t

0

∥∇N (s, ·)∥2L2ds ≲ ∥N0∥2L2 +

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇ρ∥2L2ds. (3.8)

Next is the estimate of (Ω, J). we perform Lp (2 ≤ p ≤ 6) energy estimates of (3.4)1 and (3.4)2
respectively to obtain

d

dt
∥Ω(t, ·)∥Lp ≲ ∥uθ

r
(t, ·)∥L∞∥J(t, ·)∥Lp + ∥∂zH2(t, ·)∥Lp + ∥N (t, ·)∥Lp ,

d

dt
∥J(t, ·)∥Lp ≲ ∥(ωr, ωz)(t, ·)∥L∞∥∇ur

r
(t, ·)∥Lp .

By Lemma 2.7 and using the identity ∇× (uθeθ) = ωrer + ωzez, together with (2.2) of Lemma
2.5, we obtain

d

dt
∥Ω(t, ·)∥Lp ≲ ∥ωz(t, ·)∥L∞∥J(t, ·)∥Lp + ∥∂zH2(t, ·)∥Lp + ∥N (t, ·)∥Lp

≲ ∥∇ × (uθeθ)(t, ·)∥L∞∥J(t, ·)∥Lp + ∥∂zH2(t, ·)∥Lp + ∥N (t, ·)∥Lp ,



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 7

d

dt
∥J(t, ·)∥Lp ≲ ∥∇ × (uθeθ)(t, ·)∥L∞∥∇ur

r
(t, ·)∥Lp

≲ ∥∇ × (uθeθ)(t, ·)∥L∞∥Ω(t, ·)∥Lp .

Integrating with time and using (3.1)2, one derives that

∥Ω(t, ·)∥Lp ≲ ∥Ω0∥Lp +

∫ t

0

∥∇ × (uθeθ)(s, ·)∥L∞∥J(s, ·)∥Lpds

+ ∥H0∥L∞

∫ t

0

∥∂zH(s, ·)∥Lpds+

∫ t

0

∥N (s, ·)∥Lpds,

(3.9)

∥J(t, ·)∥Lp ≲ ∥J0∥Lp +

∫ t

0

∥∇ × (uθeθ)(s, ·)∥L∞∥Ω(s, ·)∥Lpds. (3.10)

Combining (3.9) and (3.10), we have

∥(Ω, J)(t, ·)∥Lp ≲ ∥(Ω0, J0)∥Lp +

∫ t

0

∥∇ × (uθeθ)(s, ·)∥L∞∥(Ω, J)(s, ·)∥Lpds

+ ∥H0∥L∞

∫ t

0

∥∂zH(s, ·)∥Lpds+

∫ t

0

∥N (s, ·)∥Lpds.

Using Grönwall’s inequality, we have

∥(Ω, J)(t, ·)∥Lp ≲
(
∥(Ω0, J0)∥Lp + ∥H0∥L∞

∫ t

0

∥∂zH(s, ·)∥Lpds

+

∫ t

0

∥N (s, ·)∥Lpds
)
exp

(∫ t

0

∥∇ × (uθeθ)(s, ·)∥L∞ds
)
.

By (1.4), for any t ≤ T∗, we conclude for 2 ≤ p ≤ 6 that

∥(Ω, J)(t, ·)∥Lp

≤ CC∗

(
∥(Ω0, J0)∥Lp + ∥H0∥L∞

∫ t

0

∥∂zH(s, ·)∥Lpds+

∫ t

0

∥N (s, ·)∥Lpds
)
.

(3.11)

Finally, choosing p = 2 and p = 6 in (3.11), one derives

∥Ω(t, ·)∥2L2∩L6 ≲ ∥Ω0∥2L2∩L6 + ∥H0∥2L∞

(∫ t

0

∥∂zH(s, ·)∥L2 ds+

∫ t

0

∥∂zH(s, ·)∥L6ds
)2

+
(∫ t

0

∥N (s, ·)∥L2 ds+

∫ t

0

∥N (s, ·)∥L6 ds
)2

.

Using the Sobolev inequality and the Hölder’s inequality, one deduces

∥Ω(t, ·)∥2L2∩L6 ≲ ∥Ω0∥2L2∩L6 + t∥H0∥2L∞

(∫ t

0

∥∇H(s, ·)∥2L2 ds+

∫ t

0

∥∇2H(s, ·)∥2L2 ds
)

+
(
t2 sup

s∈(0,t)

∥N (s, ·)∥2L2 + t

∫ t

0

∥∇N (s, ·)∥2L2 ds
)
.

Substituting (3.6) and (3.8) in the right-hand side, we have

∥Ω(t, ·)∥2L2∩L6

≲ ∥Ω0∥2L2∩L6 + t∥H0∥L∞

(
∥H0∥2L2 + ∥∇H0∥2L2 +

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇h(s, ·)∥2L2 ds
)

+ (1 + t2)
(
∥N0∥2L2 +

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇ρ(s, ·)∥2L2 ds
)
.

This indicates, for any t ≤ T ∗,

∥Ω(t, ·)∥2L2∩L6 ≲ ∥Ω0∥2L2∩L6 + t∥H0∥2L∞∥H0∥2H1 + t∥H0∥2L∞

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇h(s, ·)∥2L2 ds



8 Z. XING EJDE-2025/??

+ (1 + t2)∥N0∥2L2 + (1 + t2)

∫ t

0

∥Ω(s, ·)∥2L2∩L6∥∇ρ(s, ·)∥2L2 ds.

Thus by the Grönwall’s inequality:

∥Ω(t, ·)∥2L2∩L6

≲
(
∥Ω0∥2L2∩L6 + t∥H0∥2L∞∥H0∥2H1 + (1 + t2)∥N0∥2L2

)
× exp

(
t∥H0∥2L∞

∫ t

0

∥∇h(s, ·)∥2L2 ds+ (1 + t2)

∫ t

0

∥∇ρ(s, ·)∥2L2 ds

)
.

Using the fundamental energy estimates (3.1)1 and (3.1)3, one has

∥Ω(t, ·)∥2L2∩L6 ≤ C0,C∗(1 + t2) exp
(
C0

(
1 + t3

))
≤ Φ1,3(t), ∀t ∈ (0, T∗] . (3.12)

Substituting (3.12) in (3.6) and (3.8) respectively, using (3.1)1 and (3.1)3, one concludes

∥N (t, ·)∥2L2 +

∫ t

0

∥∇N (s, ·)∥2L2 ds+ ∥∇H(t, ·)∥2L2 +

∫ t

0

∥∇2H(s, ·)∥2L2 ds

≤ Φ1,3(t)

∫ t

0

(
∥∇ρ(s, ·)∥2L2 + ∥∇h(s, ·)∥2L2

)
ds

≤ Φ1,3(t), ∀t ∈ (0, T∗].

(3.13)

Thus the proposition is proved by combining (3.12) and (3.13). □

The purpose of this next part is to derive a one-component BKM-type criterion for MHD-
Boussinesq system through the following steps: we first derive the ∥rρ∥L∞

t L2∩L2
t Ḣ

1 and then

get the ∥∇ρ∥L∞
t L2∩L2

t Ḣ
1 . Next, we derive the ∥∇u∥L∞

t (L2∩L6). It then follows that deduce the

∥(∇∂zH,∇h,∇2h,∇2ρ)∥L∞
t L2∩L2

t Ḣ
1 . Finally, we deduce the ∥(ωθ,∇h,∇ρ)∥L1

tL
∞ .

We first give L∞
t L2

t ∩ L2
t Ḣ

1 estimate of rρ and ∇ρ.

Proposition 3.3. Under the conditions of Theorem 1.1, rρ satisfies the space-time estimate

∥rρ(t, ·)∥2L2 +

∫ t

0

∥∇(rρ)(s, ·)∥2L2 ds ≤ C0(1 + t)3, (3.14)

where C0 > 0 is a constant depending only on the initial data u0, h0, and ρ0.

Next, we will use the weighted estimate of rρ of (3.14) in Proposition 3.3 to establish the

L∞
t L2 ∩L2

t Ḣ
1 estimate of ∇ρ. This proposition can be found in [20, Proposition 3.4], so we omit

proof here.

Proposition 3.4. Under the conditions of Theorem 1.1, ∇ρ satisfies the space-time estimate

∥∇ρ(t, ·)∥2L2 +

∫ t

0

∥∇2ρ(s, ·)∥2L2 ds ≤ Φ1,3(t). (3.15)

Before listing the estimate of the critical proof of the velocity field, we give a proposition to be
used.

Proposition 3.5. Under the conditions of Theorem 1.1, the Lp estimate of hθ satisfies

∥hθ(t, ·)∥Lp ≤ Φ2,3(t), (3.16)

where for p ∈ [2,∞) is uniform.

Proof. For any p ≥ 2, taking Lp inner product of hθ on (1.3)4, one derives

1

p

d

dt
∥hθ(t, ·)∥pLp ≤ ∥ur

r
(t, ·)∥L∞∥hθ(t, ·)∥pLp −

∫
R3

|hθ|p

r2
dx− (p− 1)

∫
R3

|∇hθ|2 |hθ|p−2
dx

≤ ∥ur

r
(t, ·)∥L∞∥hθ(t, ·)∥pLp .



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 9

Here using Lemma 2.1, Lemma 2.5, and (3.12), one derives that:

∥ur

r
(t, ·)∥L∞ ≲ ∥ur

r
(t, ·)∥1/2L6 ∥∇ur

r
(t, ·)∥1/2L6 ≲ ∥∇ur

r
(t, ·)∥1/2L2 ∥∇ur

r
(t, ·)∥1/2L6

≲ ∥Ω(t, ·)∥1/2L2 ∥Ω(t, ·)∥1/2L6 ≤ Φ1,3(t),
(3.17)

then using the Grönwall’s inequality, one finds that

∥hθ(t, ·)∥Lp ≤ ∥h0 · eθ∥Lp exp
(∫ t

0

∥ur

r
(s, ·)∥L∞ ds

)
≤ Φ2,3(t), uniformly for p ∈ [2,∞).

□

Based on Propositions 3.2, 3.4, and 3.5, we can now obtain the estimate of the velocity field.

Proposition 3.6. Under the conditions of Theorem 1.1, the L2 ∩ L6 estimate of ∇u,

∥∇u(t, ·)∥L2∩L6 ≤ Φ2,3(t),

holds uniformly for 0 ≤ t ≤ T∗.

Proof. From (1.3)2,
uθ

r satisfies

∂t
uθ

r
+ (u · ∇)

uθ

r
+ 2

ur

r
· uθ

r
= 0.

Multiplying |uθ

r |p−2 uθ

r and integrating over R3, one derives

d

dt
∥uθ

r
(t, ·)∥Lp ≤ 2∥ur

r
(t, ·)∥L∞∥uθ

r
(t, ·)∥Lp .

Using Grönwall’s inequality and (3.17), we have

∥uθ

r
(t, ·)∥Lp ≤ ∥u0

r
· eθ∥Lp exp

(
2

∫ t

0

∥ur

r
(s, ·)∥L∞ ds

)
≤ Φ2,3(t), for any p ∈ [2,∞).

Thus, we can obtain the Lp estimate of (3.3)2:

∥ωθ(t, ·)∥Lp ≲ ∥ω0 · eθ∥Lp + ∥ωr∥L1
tL

∞∥uθ

r
∥L∞

t Lp + ∥hθ∥L∞
t L∞∥∂zH∥L1

tL
p + ∥∇ρ∥L1

tL
p

+

∫ t

0

∥ωθ(s, ·)∥Lp∥ur

r
(s, ·)∥L∞ds.

By the Grönwall’s inequality, Sobolev inequality, Proposition 3.1, (3.13), (3.15) and (3.16), it
follows that

∥ωθ(t, ·)∥L2 ≤
(
∥ω0 · eθ∥L2 + ∥ωr∥L1

tL
∞∥uθ

r
∥L∞

t L2 + ∥hθ∥L∞
t L∞∥∇H∥L1

tL
2 + ∥∇ρ∥L1

tL
2

)
× exp

(∫ t

0

∥ur

r
(s, ·)∥L∞ds

)
≤ [1 + Φ2,3(t) +

√
tΦ2,3(t) +

√
t]Φ2,3(t) ≤ Φ2,3(t),

∥ωθ(t, ·)∥L6 ≤
(
∥ω0 · eθ∥L6 + ∥ωr∥L1

tL
∞∥uθ

r
∥L∞

t L6 + ∥hθ∥L∞
t L∞∥∇H∥L1

tL
6 + ∥∇ρ∥L1

tL
6

)
× exp

(∫ t

0

∥ur

r
(s, ·)∥L∞ ds

)
≤ [1 + Φ2,3(t) + Φ1,3(t)Φ2,3(t) + Φ1,3(t)]Φ2,3(t)

≤ Φ2,3(t),

for all t ≤ T∗. Then from (2.1) in Lemma 2.4, we have

∥∇b(t, ·)∥L2∩L6 ≤ Φ2,3(t).

Next, we also estimate ∇ (uθeθ). From ∇× (uθeθ) = ωrer + ωzez and div (uθeθ) = 0 and
using the argument of the Calderon-Zygmund singular integral operator, it only needs to prove



10 Z. XING EJDE-2025/??

the same estimate for (ωr, ωz). Therefore, performing the Lp estimates for (3.3)1 and (3.3)3, one
derives

max
{ d

dt
∥ωr(t, ·)∥pLp ,

d

dt
∥ωz(t, ·)∥pLp

}
≲

∫
R3

(|ωr|p + |ωz|p) |∇b| dx

≲ ∥∇ × (uθeθ) (t, ·)∥L∞

∫
R3

(
|ωr|p−1 + |ωz|p−1

)
|∇b| dx.

Using Hölder’s inequality, it follows that

max
{ d

dt
∥ωr(t, ·)∥pLp ,

d

dt
∥ωz(t, ·)∥pLp

}
≲ ∥∇ × (uθeθ) (t, ·)∥L∞

(
∥ωr(t, ·)∥p−1

Lp + ∥ωz(t, ·)∥p−1
Lp

)
∥∇b(t, ·)∥Lp .

By dividing
(
∥ωr(t, ·)∥p−1

Lp + ∥ωz(t, ·)∥p−1
Lp

)
on both sides, it indicates that

max
{ d

dt
∥ωr(t, ·)∥Lp ,

d

dt
∥ωz(t, ·)∥Lp

}
≲ ∥∇ × (uθeθ) (t, ·)∥L∞∥∇b(t, ·)∥Lp .

Integrating with t, for any p ∈ [2, 6], one concludes that:

∥ (ωr, ωz) (t, ·)∥Lp ≲ ∥ (ω0 · er, ω0 · ez) ∥Lp + ∥∇b∥L∞
t Lp

∫ t

0

∥∇ × (uθeθ) (s, ·)∥L∞ ds ≤ Φ2,3(t).

So, the proof of Proposition 3.6 is complete. □

Next, we derive the L1
tL

∞ estimate for the vector field (∇ × u,∇ × h,∇ρ), which is the key
to obtaining a higher-order estimation of the solution. And before we do that, we need to get an
estimate of ∇∂zH,∇h,∇2h and ∇2ρ.

Proposition 3.7. Under the conditions of Theorem 1.1, the following space-time estimate of
∇∂zH,∇h,∇2h and ∇2ρ holds:

∥(∇∂zH,∇h,∇2h,∇2ρ)(t, ·)∥2L2 +

∫ t

0

∥∇(∇∂zH,∇h,∇2h,∇2ρ)(s, ·)∥2L2 ≤ Φ2,3(t).

Proof. Firstly, we can apply ∂z on (3.4)4 and perform the L2 inner product to handle the ∇∂zH.
Then we obtain

∥∇∂zH(t, ·)∥2L2 +

∫ t

0

∥∇2∂zH(s, ·)∥2L2 ds ≤ Φ2,3(t). (3.18)

Secondly, we deal with ∇h and ∇2h. Taking ∇ and ∇2 on (1.2)2 and performing the L2 inner
product respectively, we can conclude that

∥∇h(t, ·)∥2L2 +

∫ t

0

∥∇2h(s, ·)∥2L2 ds ≤ Φ2,3(t),

∥∇2h(t, ·)∥2L2 +

∫ t

0

∥∇3h(s, ·)∥2L2 ds ≤ Φ2,3(t).

(3.19)

Finally, applying ∇2 on (1.3)5 and performing the L2 inner product, we derive the estimate for ρ:

∥∇2ρ(t, ·)∥2L2 +

∫ t

0

∥∇3ρ(s, ·)∥2L2 ds ≤ Φ2,3(t). (3.20)

The proofs of the above estimates can be found in [21]. By combining (3.18), (3.19) and (3.20),
we complete the proof of Proposition 3.7. □

Now we give the L1
tL

∞ estimate of ∇× u, ∇× h, and ∇ρ.

Proposition 3.8. Under the conditions of Theorem 1.1, the following L1
tL

∞ estimates of ∇ ×
u,∇× h and ∇ρ follows ∫ t

0

∥(ωθ,∇h,∇ρ)(s, ·)∥L∞ ds ≤ Φ2,3(t).



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 11

Proof. From the definition of wθ,

∂tωθ + (ur∂r + uz∂z)ωθ =
ur

r
ωθ +

1

r
∂z (uθ)

2 − 1

r
∂z (hθ)

2 − ∂rρ.

Integrating this along the particle trajectory start at x ∈ R3, one knows that

ωθ(t,X(t, x)) = (ω0 · eθ) (x) +
∫ t

0

(ur

r
ωθ +

1

r
∂z (uθ)

2 − 1

r
∂z (hθ)

2 − ∂rρ
)
(s,X(s, x)) ds.

Taking the L∞ norm with x ∈ R3, one derives from the previous estimates:

∥ωθ(t, ·)∥L∞

≲ ∥ω0 · eθ∥L∞ +

∫ t

0

∥ur

r
(s, ·)∥L∞∥ωθ(s, ·)∥L∞ ds+ ∥ωr∥L1(0,t,L∞)∥

uθ

r
∥L∞(0,t,L∞)

+ ∥∂zH∥L1(0,t,L∞)∥hθ∥L∞(0,t,L∞) + ∥∇ρ∥L1(0,t,H2)

≤ Φ2,3(t) +

∫ t

0

∥ur

r
(s, ·)∥L∞∥ωθ(s, ·)∥L∞ ds.

Using Grönwall’s inequality, it indicates that

∥ωθ(t, ·)∥L∞ ≤ Φ2,3(t) exp
(∫ t

0

∥ur

r
(s, ·)∥L∞ ds

)
≤ Φ2,3(t). (3.21)

By Lemma 2.1 and Hölder’s inequality, together with (3.19), one derives∫ t

0

∥∇h(s, ·)∥L∞ ds ≲
∫ t

0

∥∇h(s, ·)∥
1
4

L2∥∇3h(s, ·)∥
3
4

L2 ds

≲ ∥∇h∥L∞(0,t,L2)

(∫ t

0

∥∇3h(s, ·)∥2L2 ds
)3/8

t5/8

≤ Φ2,3(t) (Φ2,3(t))
3/8

t5/8 ≤ Φ2,3(t).

(3.22)

Similarly, using the estimates (3.15), we obtain∫ t

0

∥∇ρ(s, ·)∥L∞ ds ≤ Φ2,3(t). (3.23)

Combining (3.21), (3.22) and (3.23), we complete the proof. □

3.1. Completion of the proof of Theorem 1.1. In this part, we will use the estimates obtained
above to reach the conclusion of Theorem 1.1. Applying ∇m(m ∈ N,m ≥ 3) to (1.2)1,2,3 and

performing the L2 energy estimate, we can obtain

1

2

d

dt
∥∇m(u, h, ρ)(t, ·)∥2L2 + ∥∇m+1h(t, ·)∥2L2 + ∥∇m+1ρ(t, ·)∥2L2

= −
∫
R3

[∇m, u · ∇]u∇mu dx︸ ︷︷ ︸
I1

+

∫
R3

[∇m, h · ∇]h∇mu dx︸ ︷︷ ︸
I2

−
∫
R3

[∇m, u · ∇]h∇mh dx︸ ︷︷ ︸
I3

+

∫
R3

[∇m, h · ∇]u∇mh dx︸ ︷︷ ︸
I4

−
∫
R3

[∇m, u · ∇] ρ∇mρ dx︸ ︷︷ ︸
I5

+

∫
R3

∇mρ∇mu dx︸ ︷︷ ︸
I6

,

(3.24)

where we have used ∫
R3

h · ∇∇mh · ∇mu dx+

∫
R3

h · ∇∇mu · ∇mh dx = 0.

By Lemma 2.2, we have that I1, . . . , I5 satisfy

Ij ≲ ∥∇m(u, h, ρ)(t, ·)∥2L2∥∇(u, h, ρ)(t, ·)∥L∞ , ∀j = 1, 2, 3, 4, 5, (3.25)

and I6 satisfies

I6 ≤ ∥∇mρ(t, ·)∥L2∥∇mu(t, ·)∥L2 ≤ ∥∇m(u, h, ρ)(t, ·)∥2L2 . (3.26)



12 Z. XING EJDE-2025/??

Substituting (3.25) and (3.26) into (3.24), we obtain

d

dt
∥(u, h, ρ)(t, ·)∥2Hm ≤ C (1 + ∥∇(u, h, ρ)(t, ·)∥L∞) ∥(u, h, ρ)(t, ·)∥2Hm . (3.27)

Let

Em(t) := ∥(u, h, ρ)(t, ·)∥2Hm , ∀t ≤ T∗.

Then by Lemma 2.3, (3.27) can be written as

E′
m(t) ≲ (1 + ∥(∇× u,∇× h,∇ρ)(t, ·)∥L∞ log(e+ Em(t))) (e+ Em(t)).

Using Grönwall’s inequality twice, one arrives at

e+ Em(t) ≤ (e+ Em(0))exp(C
∫ t
0
(1+∥(∇×u,∇h,∇ρ)(s,·)∥L∞ds), ∀t ≤ T∗. (3.28)

Recalling Proposition 3.8, one has∫ t

0

(1 + ∥(∇× u,∇h,∇ρ)(s, ·)∥L∞) ds ≤ Φ2,3(t).

Substituting in (3.28), one concludes that

sup
0≤s≤t

∥(u, h, ρ)(s, ·)∥2Hm ≤ Φ4,3(t),

for all m ∈ N. This completes the proof of Theorem 1.1.

4. Proof of Theorem 1.3

This section is devoted to study the inviscid limit of the viscous system (1.5). Let (uµ, hµ, ρµ)
and (u, h, ρ) be a solution of (1.5) and (1.2), respectively. We denote

ūµ = uµ − u, h̄µ = hµ − h, p̄µ = pµ − p, ρ̄µ = ρµ − ρ .

Direct calculations show that (ūµ, h̄µ, ρ̄µ) satisfies

∂tūµ + (ūµ + u) · ∇ūµ − µ∆(ūµ + u) = −ūµ · ∇u+ h̄µ · ∇h+ hµ · ∇h̄µ −∇p̄µ + ρ̄µe3,

∂th̄µ + uµ · ∇h̄µ −∆h̄µ = −ūµ · ∇h+ h̄µ · ∇u+ hµ · ∇ūµ,

∂tρ̄µ + uµ · ∇ρ̄µ −∆ρ̄µ = −ūµ · ∇ρ,

∇ · ūµ = ∇ · h̄µ = 0.

(4.1)

By a standard L2-energy method, combined with ∇ · ūµ = ∇ · h̄µ = 0, we deduce from (4.1)1 that

1

2

d

dt
∥ūµ(t)∥2L2 + µ

∫
R3

|∇ūµ(t, x)|2dx

= µ

∫
R3

∆u · ūµ dx−
∫
R3

(ūµ · ∇u) · ūµ dx+

∫
R3

(
h̄µ · ∇h

)
· ūµ dx

−
∫
R3

(hµ · ∇ūµ) · h̄µ dx+

∫
R3

(ρ̄µ · ūµ) dx.

Using the same method for h̄µ and ρ̄µ, we obtain

1

2

d

dt
∥h̄µ(t)∥2L2 +

∫
R3

∣∣∇h̄µ(t, x)
∣∣2 dx

= −
∫
R3

(ūµ · ∇h) · h̄µ dx+

∫
R3

(
h̄µ · ∇u

)
· h̄µ dx+

∫
R3

(hµ · ∇ūµ) · h̄µ dx,

and
1

2

d

dt
∥ρ̄µ(t)∥2L2 +

∫
R3

|∇ρ̄µ(t, x)|2 dx = −
∫
R3

(ūµ · ∇ρ) · ρ̄µ dx.

Combining with the above estimates, and

µ

∫
R3

|∇ūµ(t, x)|2dx ≥ 0,

∫
R3

|∇h̄µ(t, x)|2dx ≥ 0,

∫
R3

|∇ρ̄µ(t, x)|2dx ≥ 0,



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 13

we derive

1

2

d

dt

(
∥ūµ(t)∥2L2 + ∥h̄µ(t)∥2L2 + ∥ρ̄µ(t)∥2L2

)
≤ µ

∫
R3

∆u · ūµ dx−
∫
R3

(ūµ · ∇u) · ūµ dx+

∫
R3

(
h̄µ · ∇h

)
· ūµ dx−

∫
R3

(ūµ · ∇h) · h̄µ dx

+

∫
R3

(
h̄µ · ∇u

)
· h̄µ dx−

∫
R3

(ūµ · ∇ρ) · ρ̄µ dx+

∫
R3

(ρ̄µ · ūµ) dx.

Using the Cauchy-Schwartz inequality, it follows that

1

2

d

dt

(
∥ūµ(t)∥2L2 + ∥h̄µ(t)∥2L2 + ∥ρ̄µ(t)∥2L2

)
≤ ∥ūµ∥L2 (µ∥∆u∥L2) + ∥ūµ∥L2∥ūµ · ∇u∥L2 + ∥ūµ∥L2∥h̄µ · ∇h∥L2 + ∥h̄µ∥L2∥ūµ · ∇h∥L2

+ ∥h̄µ∥L2∥h̄µ · ∇u∥L2 + ∥ρ̄µ∥L2∥ūµ · ∇ρ∥L2 + ∥ρ̄µ∥L2∥ūµ∥L2 .

Now, we integrate the above inequality over time. Noticing that (ūµ, h̄µ, ρ̄µ) has zero initial data,
we derive

∥ūµ∥2L∞
t L2 + ∥h̄µ∥2L∞

t L2 + ∥ρ̄µ∥2L∞
t L2

≲ ∥ūµ∥L∞
t L2∥ūµ · ∇u∥L1

tL
2 + ∥ūµ∥L∞

t L2∥h̄µ · ∇h∥L1
tL

2 + ∥h̄µ∥L∞
t L2∥ūµ · ∇h∥L1

tL
2

+ ∥h̄µ∥L∞
t L2∥h̄µ · ∇u∥L1

tL
2 + ∥ρ̄µ∥L∞

t L2∥ūµ · ∇ρ∥L1
tL

2 + ∥ρ̄µ∥L∞
t L2∥ūµ∥L1

tL
2

+ ∥ūµ∥L∞
t L2

(
µ∥∆u∥L1

tL
2

)
.

Young’s inequality yields

∥ūµ∥2L∞
t L2 + ∥h̄µ∥2L∞

t L2 + ∥ρ̄µ∥2L∞
t L2

≲ ∥ūµ · ∇u∥2L1
tL

2 + ∥h̄µ · ∇h∥2L1
tL

2 + ∥ūµ · ∇h∥2L1
tL

2

+ ∥h̄µ · ∇u∥2L1
tL

2 + ∥ūµ · ∇ρ∥2L1
tL

2 + ∥ūµ∥2L1
tL

2 +
(
µ∥∆u∥L1

tL
2

)2
.

(4.2)

From Young’s inequality it follows that

2
(
∥ūµ∥L∞

t L2∥h̄µ∥L∞
t L2 + ∥ūµ∥L∞

t L2∥ρ̄µ∥L∞
t L2 + ∥h̄µ∥L∞

t L2∥ρ̄µ∥L∞
t L2

)
≤ 2

(
∥ūµ∥2L∞

t L2 + ∥h̄µ∥2L∞
t L2 + ∥ρ̄µ∥2L∞

t L2

)
.

(4.3)

Combining with (4.2), (4.3) and employing that (a+ b+ c)2 = a2 + b2 + c2 + 2ab+ 2ac+ 2bc, we
obtain (

∥ūµ∥L∞
t L2 + ∥h̄µ∥L∞

t L2 + ∥ρ̄µ∥L∞
t L2

)2
≲ ∥ūµ · ∇u∥2L1

tL
2 + ∥h̄µ · ∇h∥2L1

tL
2 + ∥ūµ · ∇h∥2L1

tL
2

+ ∥h̄µ · ∇u∥2L1
tL

2 + ∥ūµ · ∇ρ∥2L1
tL

2 + ∥ūµ∥2L1
tL

2 +
(
µ∥∆u∥L1

tL
2

)2
.

This indicates that

∥ūµ∥L∞
t L2 + ∥h̄µ∥L∞

t L2 + ∥ρ̄µ∥L∞
t L2

≲ ∥ūµ · ∇u∥L1
tL

2 + ∥h̄µ · ∇h∥L1
tL

2 + ∥ūµ · ∇h∥L1
tL

2

+ ∥h̄µ · ∇u∥L1
tL

2 + ∥ūµ · ∇ρ∥L1
tL

2 + ∥ūµ∥L1
tL

2 + µ∥∆u∥L1
tL

2 .

(4.4)

We denote

L (t) := ∥ūµ∥L∞
t L2 + ∥h̄µ∥L∞

t L2 + ∥ρ̄µ∥L∞
t L2 .

Clearly, L (0) = 0. Then applying Hölder’s inequality and Young’s inequality for the right-hand
side of (4.4), we obtain

L (t) ≲ µ

∫ t

0

∥∆u(τ)∥L2dτ +

∫ t

0

(1 + ∥∇u(τ)∥L∞ + ∥∇h(τ)∥L∞ + ∥∇ρ(τ)∥L∞)L (τ)dτ .



14 Z. XING EJDE-2025/??

Grönwall’s inequality leads to

L (t) ≲ e
∫ t
0
(1+∥∇u(τ)∥L∞+∥∇h(τ)∥L∞+∥∇ρ(τ)∥L∞ ) dτ

(
µ

∫ t

0

∥∆u(τ)∥L2dτ
)
. (4.5)

Using Sobolev interpolation and Hölder’s inequality, if m ≥ 3, from (4.5) it follows that

∥ūµ∥L∞
t L2 + ∥h̄µ∥L∞

t L2 + ∥ρ̄µ∥L∞
t L2 ≤ (µt)∥u∥L∞

t Hm .

Recalling Theorem 1.1, we arrive at

∥uµ − u∥L∞
t L2 + ∥hµ − h∥L∞

t L2 + ∥ρµ − ρ∥L∞
t L2 ≤ (µt)Φ4,3(t).

This completes the proof of Theorem 1.3.

References

[1] Bian, D.; Gui, G.; On 2-D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ.,

261 (3) (2016), 1669–1711.
[2] Bian, D.; Liu, J.; Initial-boundary value problem to 2D Boussinesq equations for MHD convection with strat-

ification effects. J. Differ. Equ,. 263 (12) (2017), 8074–8101.

[3] Bian, D.; Pu, X.; Global smooth axisymmetic solutions of the Boussinesq equations for magnetohydrodynamics
convection, J. Math. Fluid Mech., 22 (2020) (2), (12).

[4] Chae, D.; Lee, J.; On the regularity of the axisymmetric solutions of the Navier-Stokes equations. Math. Z.,
239 (4) (2002), 645–671.

[5] Chen, Q.; Zhang, Z.; Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations.

J. Math. Anal. Appl., 331 (2) (2007), 1384–1395.
[6] Constantin, P.; Kukavica, I.; and Vicol, V.; On the inviscid limit of the Navier-Stokes equations. Proc. Amer.

Math. Soc., 143 (2015), 3075–3090.

[7] Dı́az, J.; Lerena, M.; On the inviscid and non-resistive limit for the equations of incompressible magnetohy-
drodynamics. Math. Models Methods Appl. Sci., 12 (10) (2002), 1401–1419.

[8] Hmidi, T.; Rousset, F.; Global well-posedness for the Euler-Boussinesq system with axisymmetric data. J.

Funct. Anal. 260 (3) (2011), 745–796.
[9] Itoh, S.; On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incom-

pressible fluids. Glasg Math J., 36 (1994), 123–129.

[10] Itoh, S.; Tani, A.; Solvability of nonstationary problems for nonhomogeneous incompressible fluids and con-
vergence with vanishing viscosity. Tokyo J. Math. 22 (1999), 17–42.

[11] Jiu, Q.; Liu, J.; Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and
vertical magnetic diffusion. Discrete Contin. Dyn. Syst. 35 (1) (2015), 301–322.

[12] Kato, T.; Nonstationary flows of viscous and ideal fluids in R3. J. Funct. Anal., 9 (1972), 296–305.

[13] Kato, T.; Quasi-linear equations of evolution, with applications to partial differential equations. Lect Notes in
Math. 48 (1975), 25–70.

[14] Lei, Z.; On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259

(7) (2015), 3202-3215.
[15] Liu, H.; Bian, D.; Pu, X.; Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion. Z.

Angew. Math. Phys. 70 (2019), 81.

[16] Larios, A.; Pei, Y.; On the local well-posedness and a Prodi-Serrin-type regularity criterion of the three-
dimensional MHD-Boussinesq system without thermal diffusion. J. Differential Equations, 263 (2) (2017),

1419–1450.
[17] Li, Z.; Critical conditions on ωθ imply the regularity of axially symmetric MHD-Boussinesq systems. J. Math.

Anal. Appl., 505 (1) (2022), 125451.
[18] Li, Z.; A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal

diffusivity. J. Differ. Equ., 374 (2023), 737–760.

[19] Li, Z.; Pan, X.; One component regularity criteria for the axially symmetric MHD-Boussinesq system. Discrete

Contin. Dyn. Syst. 42 (5) (2022), 2333–2353.
[20] Li, Z.; Pan, X.; BKM-type blow-up criterion of the inviscid axially symmetric Boussinesq system involving a

single component of velocity. Z. Angew. Math. Phys., 74 (2023), 9.
[21] Li, Z.; Xing, Z.; Yang, M.; On the Large Data Global Well-Posedness of Inviscid Axially Symmetric MHD-

Boussinesq System. Acta Appl. Math., 187 (2023), 15.

[22] Liu, J.; On regularity criterion to the 3D axisymmetric incompressible MHD equations. Math. Methods Appl.

Sci., 39 (15) (2016), 4535–4544.
[23] Liu, J.; Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder. Nonlin-

earity. 37 (5) (2024), 21.
[24] Maafa, Y.; Zerguine, M.; Inviscid limit for the full viscous MHD system with critical axisymmetric initial

data. Applicable Analysis, 102 (2022), 343–383.

[25] Majda, A.; Vorticity and the mathematical theory of an incompressible fluid flow. Commun. Pure Appl. Math.,
39 (1986), 187–220.



EJDE-2025/?? SYMMETRIC MHD-BOUSSINESQ SYSTEMS 15

[26] Majda, A.; Introduction to PDEs and Waves for the Atmosphere and Ocean. AMS/CIMS, Providence, 2003.

[27] Marchioro, C.; Pulvirenti, M.; Mathematical Theory of Incompressible Nonviscous Fluids. Springer, New York,

1994.
[28] Masmoudi, N.; Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys., 270

(2007), 777–788.

[29] Miao, C., Zheng, X.; On the global well-posedness for the Boussinesq system with horizontal dissipation.
Commun. Math. Phys., 321 (1) (2013), 33–67.

[30] Mulone, G.; Rionero, S.; Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard

problem. Arch. Ration. Mech. Anal., 166 (3) (2003), 197–218.
[31] Pan, X.; Global regularity for the 3D non-diffusive MHD-Boussinesq system with axisymmetric data. Acta

Appl. Math. 180 (6) (2022), 18.

[32] Pedlosky, J.; Geophysical Fluid Dynamics. Springer, New York, 1987.
[33] Swann, H.; The Convergence with Vanishing Viscosity of Nonstationary Navier-Stokes Flow to Ideal Flow in

R3. Trans. Amer. Math. Soc. 157 (1971), 373–397.

Zhaojun Xing
School of Mathematics and Statistics, Nanjing University of Information Science and Technology,

Nanjing 210044, China

Email address: zhaojunxing@nuist.edu.cn


	1. Introduction
	2. Preliminaries
	3. Proof of Theorem 1.1
	3.1. Completion of the proof of Theorem 1.1

	4. Proof of Theorem 1.3
	References