Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 119, pp. 1–42.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.119

NON-STEADY MAGNETO-HYDRODYNAMICS-HEAT SYSTEM WITH

JOULE AND BUOYANCY EFFECTS UNDER MIXED BOUNDARY

CONDITIONS

TUJIN KIM

Communicated by Jesus Ildefonso Diaz

Abstract. This article concerns the non-steady magneto-hydro-dynamics (MHD) heat system
with Joule and buoyancy effects under mixed boundary conditions. The boundary conditions for

fluids can include the stick, total pressure, vorticity, total stress conditions, and for a magnetic

field with non-homogeneous mixed boundary conditions. The boundary condition for temper-
ature may include Dirichlet, Neumann and Robin conditions. We prove that if the parameter

for buoyancy effect is small in accordance with the data of problem, then there exists a solution

with “defect measure” for temperature.

1. Introduction

This article concerns the non-steady magneto-hydro-dynamic (MHD) heat system with Joule
and buoyancy effects

∂v

∂t
− 2∇ ·

(
η(θ)E(v)

)
+ (v · ∇)v +

1

ρ
∇P − µ

ρ
curlH ×H = f0 − α0(θ − θ̄0)g,

div v = 0,

∂H

∂t
+

1

µ
curl

( 1

σ(θ)
curlH

)
+ curl(H × v) = 0,

divH = 0,

∂θ

∂t
−∇ ·

(k(θ)
ρCh

∇θ
)
+ v · ∇θ − 1

ρChσ(θ)
| curlH|2 − η(θ)

ρCh
|E(v)|2 = g0,

v(x, 0) = v0, H(x, 0) = H0, θ(x, 0) = θ0,

(1.1)

under mixed boundary conditions. Here v, P,H, θ are, respectively, velocity, pressure, mag-
netic field, and temperature. The strain tensor E(v) is the one with the components εij(v) =
1
2 (∂xi

vj + ∂xj
vi), f0 - body force, α0 - parameter for buoyancy effect, θ̄0 - constant, g - gravi-

tational acceleration, g0 - heat source. The density ρ, magnetic permeability µ and the specific
heat capacity Ch of fluid are positive constants, and the kinematic viscosity η(θ), electrical con-
ductivity σ(θ) and thermal conductivity k(θ) depend on the temperature. In what follows P

ρ

and k(θ)
ρCh

are, respectively, denoted by p and κ(θ). For two matrices A = {aij} and B = {bij},
A : B =

∑
ij aijbij , and |A|2 =

∑
ij |aij |2. The terms 1

σ(θ) | curlH|2 and η(θ)|E(v)|2 in the fifth

equation mean the Joule heats, respectively, by the electric current and viscous friction in fluid.
The boundary conditions for fluid may include stick on the wall, total pressure, vorticity, total
stress conditions (Navier-slip with friction, the normal total stress, the total stress, outflow con-
dition) together, and the boundary conditions for temperature may include Dirichlet, Neumann

2020 Mathematics Subject Classification. 35Q60, 76D03, 76W05, 85A30.
Key words and phrases. Non-stead MHD-heat system; Joule heating; buoyancy; mixed boundary conditions;

existence of a solution.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted March 18, 2025. Published December 22, 2025.

1



2 T. KIM EJDE-2025/119

and Robin conditions together. For the magnetic field the non-homogeneous mixed boundary
conditions are given.

In [27] (2011) the following density dependent MHD-heat system with Joule effects on a bounded
domain Ω of R3 was studied,

∂tρ+ div(ρv) = 0,

∂tH + curl(H × v) + curl(ν curlH) = 0,

divH = 0,

∂t(ρv) + div(ρv ⊗ v)− div(2ηE(v)) +∇P = curlH ×H,

div v = 0,

Cv(∂t(ρθ) + div(ρvθ))− div(κ∇θ) = 2η|E(v)|2 + ν| curlH|2,

(1.2)

where ν = (µσ)−1 (magnetic diffusivity), η - dynamic viscosity, Cv - specific heat at constant
volume and κ - heat conductivity and these depend on ρ and θ, was studied under the boundary
conditions and the initial conditions

(H · n, curlH × n, v, θ) = (0, 0, 0, 0) on ∂Ω× [0, T ],

(ρ,H, ρv, ρθ) = (ρ0, H0, ρ0v0, ρ0θ0) in Ω.

Under the regular conditions and the compatibility conditions at the initial instance for the initial
data, the existence of a local-in-time unique strong solution to (1.2) was proved. For the other
papers dealt with (1.2), we refer to Introduction of [27] and references therein.

In [28](2015) a density dependent non-Newtonian MHD-heat system with Joule effects on a
bounded domain Ω of R3 was studied,

∂tρ+ div(ρv) = 0,

div v = 0,

∂t(ρv) + div(ρv ⊗ v) +∇P = curlH ×H + divS(ρ, η,D(v)),

∂tH + curl(H × v) + curl(ν curlH) = 0,

divH = 0,

∂t(ρQ(θ))− div(q(ρ, θ,∇θ)) + div(ρQ(θ)v)− S(ρ, θ,D(v)) : ∇v − ν| curlH|2 = 0,

(1.3)

where Q is a function of θ, q - thermal flux vector and D(v) = ∇v +∇vT , under the boundary
conditions

v = 0, H = 0, q · n = 0 on ∂Ω× [0, T ].

However, the assumptions

S(ρ, η,D(v)) ∼ µ(ρ, θ)(ϵ+ |D|2)
r−2
2 D,

S(ρ, η,D(v)) ·D ≥ µ(ϵ+ |D|2)
r−2
2 |D|2,

where 0 < µ ≤ µ(ρ, θ), ((6), (7) of [28]) and the condition

r ≥ 11

5

(the condition in [28, Theorem 2.2]) problem (1.3) excludes problem (1.1) in the case that ρ =
const. For similar results for (1.3), we refer the reader to Introduction of [28]. To the best of our
knowledge, for the density dependent fluid systems on the bounded domains always the boundary
condition for fluid v|∂Ω = 0 or v · n|∂Ω = 0 was used, and so are both [27] and [28].

On the other hand, there are several papers concerned with the Navier-Stokes-heat system with
Joule heating by the viscose friction (the case that H ≡ 0 in (1.1)) ([9]-[19]).

In [24] when α0 = 0 (i.e. without buoyancy effect), the existence of solution to the problem was
studied under homogeneous Dirichlet boundary conditions for velocity and temperature. Also, in
[25] when α0 = 0, the existence of a solution was studied under homogeneous Dirichlet boundary
conditions for velocity and homogeneous Neumann condition for temperature. In [26] the existence
of a solution was studied when |α0| is small enough in accordance with data of the problem, under



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 3

homogeneous Dirichlet boundary conditions for velocity and mixture of homogeneous Dirichlet
condition and homogeneous Neumann condition for temperature. All papers above concerned the
homogeneous Dirichlet boundary condition for velocity. In [19] the existence of a solution was
studied when |α0| is small enough in accordance with data of the problem, under mixed boundary
conditions for fluid and mixture of homogeneous Dirichlet condition and non-homogeneous Neu-
mann and Robin conditions for temperature. The definitions of solutions in [19, 24, 26] include
the “defect measures” in the equation for temperature θ due to bad smoothness of the term for
dissipation of energy. In [10] for Newtonian fluid with α0 = 0 the existence of solution to the
problem was studied under Navier-slip boundary conditions for velocity and homogeneous Dirich-
let condition for temperature. But, the definition of solution for temperature is inequality instead
equality (cf. [10, (11)]), and so in essence the solution holds an equality with a “defect measure”
as [19], [24]-[26].

Dealing with Newtonian fluid with α0 = 0 and g = 0, in [9, 11] instead of the equation
for temperature of (1.1) the equation of total energy was used. Under Navier-slip boundary
conditions for velocity and homogeneous Neumann condition for temperature in [9] and under
periodic boundary conditions for velocity and temperature in [11] the existence of solutions to
the problems is proved, but unlike [19], [24]-[26] the equation (1.1) is satisfied without the defect
measures. For the reason of success in [9], we refer to Introduction of [19].

In this paper we study the existence of a solution to (1.1) under mixed boundary conditions
for fluid, magnetic field and temperature. In our problem the mixed boundary conditions for the
magnetic field, which may include the mixture of the normal components of magnetic field on a
portions of boundary and electric current on the other portions of boundary (see Theorem 1 of
[16]), is somewhat more general than previous ones except [16], [17] (see Remark 3.2). Under the
such non-homogeneous boundary conditions for magnetic field, in the estimation of approximate
solutions of problem, the terms according to Lorentz force in the momentum equation and the
nonlinear term in the magnetic equation are not canceled each other, and so the case of non-
homogeneous magnetic boundary conditions is much more difficult than others (see Introduction
of [16]). In the MHD-Boussinesq system the temperature is estimated by the data of problem
and so the buoyancy term is estimated (see [17]), but in (1.1) we can not do so as. Also the
heat source in the fifth equation of (1.1) includes the terms quadratic with respect to gradients of
fluid velocity and electric current. Thus, dealing (1.1) is difficult than MHD-Boussinesq system
and the Navier-Stoks-heat system with friction Joule effect. Also, as already mentioned, in our
problem the viscosity, magnetic permeability and electrical conductivity of the fluid depend on
the temperature.

This paper consists of 4 sections. In Section 2, the boundary conditions and assumptions are
stated. The boundary conditions for fluid may include the stick, the total pressure, vorticity, total
stress conditions together. We exclude the friction boundary conditions, for the reason of which
we refer to Remark 5.1 of [17].

In Section 3, we obtain the variational formulation for the problem. To this end, first assuming
the existence of a smooth function satisfying non-homogeneous mixed boundary conditions for
magnetic fields, we deform the given problem to problem with the homogeneous mixed boundary
conditions for magnetic fields. Then, for the problem we obtain a variational formulation which
consists of three variational equations (see Problem VE). In the end of Section 3, the main results
of this paper are stated (Theorem 3.4). Theorem 3.4 asserts that if the parameter for buoyancy
effect is small enough in accordance with the data of problem, then there exists a solution with
“defect measure” for temperature.

Following the idea in [19] and correcting some mistakes therein, in Section 4 we prove Theorem
3.4. First in Subsection 4.1 we prove the existence of a solution to an auxiliary problem with
the parameter ε for approximation of the viscosity term and buoyancy term in the momentum
equation and Joule effects in the heat equation (Theorem 4.2). Next, in Subsection 4.2 under
condition |α0| is small in accordance with the data of the problem we obtain estimation of the
solutions independent from ε. Therefore, in Subsection 4.3 passing to the limits as ε goes to zero,
we obtain the existence and estimates of a solution to the problem.



4 T. KIM EJDE-2025/119

Throughout this paper we will use the following notation. Let Ω be a connected bounded open
subset of R3 and

∂Ω = ∪7
i=1Γi = Σν ∪ Στ = ΓD ∪ ΓR,

Στ ,Σν ,ΓD,ΓR be open subsets of ∂Ω such that Στ ∩ Σν = ∅, ΓD ∩ ΓR = ∅. And Γi ∩ Γj = ∅
for i ̸= j, Γi =

⋃
j Γij , where Γij are connected open subsets of ∂Ω and Γij ∈ C2,1 for i = 2, 3, 7

and Γij ∈ C2 for others. When X is a Banach space, X = X3. Let W k,p(Ω) be Sobolev spaces,
H1(Ω) =W 1,2(Ω), and so H1(Ω) = {H1(Ω)}3.

An inner product and norm in the space L2(Ω) or L2(Ω) are, respectively, denoted by (· , ·) and
∥ · ∥; and ⟨· , ·⟩ means the duality pairing between a Banach space X and its dual one. Especially,
(·, ·)Γi

is an inner product in L2(Γi) or L2(Γi); and ⟨·, ·⟩Γi
means the duality pairing between

H
1
2 (Γi) and H− 1

2 (Γi) or between H
1
2 (Γi) and H− 1

2 (Γi). Sometimes ⟨· , ·⟩Γi means the duality

pairing between L2(0, T ;H− 1
2 (Γi)) and L

2(0, T ;H
1
2 (Γi)). The inner product and norms in R3 are,

respectively, denoted by (· , ·)R3 and | · |. Let n(x) and τ(x) be, respectively, outward normal and
tangent unit vectors at x in ∂Ω. For convergence in spaces, → and ⇀ mean, respectively, strong
and weak convergence. Q = Ω× (0, T ), where 0 < T <∞. For the Banach space X its dual space
is denoted by X∗.

2. Boundary conditions and assumptions

The total stress tensors St(θ, v, p) is the ones with components stij = −(p + 1/2|v|2)δij +

2η(θ)εij(v). The total stress vectors on the boundary surface is σt(θ, v, p) = St · n and the values
of normal total stress vectors on the boundary surface is σt

n(θ, v, p) = σt · n.
The boundary conditions for fluid are as follows:

v|Γ1 = 0,

vτ |Γ2
= 0, −

(
p+ 1/2|v|2

)
|Γ2

= ϕ2,

vn|Γ3
= 0, curl v × n|Γ3

= ϕ⃗3/η(θ),

vτ |Γ4
= 0,

(
− p− 1/2|v|2 + 2η(θ)εnn(v)

)
|Γ4

= ϕ4,

vn|Γ5
= 0, 2(η(θ)εnτ (v) + αvτ )|Γ5

= ϕ⃗5, α : a matrix,(
− pn− 1/2|v|2n+ 2η(θ)εn(v)

)
|Γ6 = ϕ⃗6,

vτ |Γ7
= 0,

(
− p− 1/2|v|2 + η(θ)∂v/∂n · n

)
|Γ7

= ϕ7,

(2.1)

where εn(v) = E(v)n, εnn(v) = (E(v)n, n)R3 , εnτ (v) = E(v)n− εnn(v)n, vn = v · n, vτ = v − vnn,

and ϕi, ϕ⃗i, αij (components of matrix α) are given functions or vectors of functions.
For the magnetic field, we consider the mixed boundary conditions

H · n|Στ = q,
( 1

µσ(θ)
curlH +H × v

)
× n

∣∣∣
Στ

= φ⃗τ , H × n|Σν
= ϕ⃗ν , (2.2)

where the compatibility condition

φ⃗τ · n = 0, ϕ⃗ν · n = 0 (2.3)

must be satisfied since n is normal on the boundary.

Remark 2.1. (1) For the physical background of the boundary conditions in (2.1), we refer to
[20, Section 2.2], the Introduction in [21], and references therein.

(2) The boundary condition
(

1
µσ curlH + H × v

)
× n

∣∣
Στ

= φτ of (2.2) corresponds to the

boundary condition 1
µE × n|Στ = φτ in [16, (2.3)] for the steady problem since curlH = σ(E −

µH × v). The boundary condition

H · n|Στ
= q, H × n|Σν

= ϕ⃗ν

implies

H · n|Στ
= q, J · n|Σν

= a(x),



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 5

where J = curlH is the electric current density and a(x) is the surface divergence on Σν of ϕ⃗ν ,

that is, divτ ϕ⃗ν (see [16, Theorem 1]).
(3) In (2.3) the boundary conditionH ·n|Στ

= 0, E×n|Στ
= 0 describes the boundary conditions

on perfectly conducting boundary Στ , and the boundary condition H ×n|Σν
= 0 corresponds to a

perfectly insulating boundary (cf. Introduction of [3] and references therein). Let Ω is a domain
in R3 and Ω′ = R3\Ω. The solution of the Maxwell equations over the hole space satisfies the
following transmission relations across surface ∂Ω,

(H0 −H)× n = −J∂Ω,
(B0 −B) · n = 0,

(E0 − E)× n = 0,

where H = B
µ , J∂Ω the surface density of current and (B,E,H) and (B0, E0, H0) denote the

solutions on Ω and Ω′, respectively (see [12, (4.25), Ch. 1]). Knowing solutions on Ω′, from the
above we obtain non-homogeneous boundary conditions (2.3) (See [14, Section 3.8]).

For temperature we consider the boundary conditions

θ|ΓD
= 0,(

κ(θ)
∂θ

∂n
+ β(x)θ

)∣∣
ΓR

= gR(x), β(x), gR(x)- given functions on ΓR.
(2.4)

Let

V = {u ∈ H1(Ω) : div u = 0, u|Γ1
= 0, uτ |(Γ2∪Γ4∪Γ7) = 0, un|(Γ3∪Γ5) = 0},

HV : the closure of V in L2(Ω),

W 1,r
ΓD

= {y ∈W 1,r(Ω) : y|ΓD
= 0}, 1 < r ≤ ∞.

Since Γ1 ̸= ∅ and ΓD ̸= ∅, by Korn’s inequality and Friedrichs’ inequality we use

(v, u)V = (E(v), E(u)), (y, z)W 1,2
ΓD

(Ω) = (∇y,∇z). (2.5)

Assumption 2.2. (1) Γ1 ̸= ∅, ΓD ̸= ∅ and and

ΓR ⊂ (∪i=1,3,5Γi) . (2.6)

(2) The boundary ∂Ω is the union of a finite number of disjoint closed C2 surfaces; each surface
having finite surface area. Στ ̸= ∅, Σν ̸= ∅ and each has a finite number of disjoint nonempty
open components. The Euclidean distance between disjoint open components of Στ , and of Σν , is
bounded below by a positive number. The boundary of each open component is either empty or
a C1,1 curve.

(3) For the functions of (1.1) assume that

f0 ∈ L2(0, T ;V∗);

g0 ∈ L2
(
0, T ;W 1,2

ΓD
(Ω)∗);

η ∈ C(R), 0 < η0 ≤ η(ξ) ≤ η1 <∞ ∀ξ ∈ R;
σ ∈ C(R), 0 < σ0 ≤ σ(ξ) ≤ σ1 <∞ ∀ξ ∈ R;
κ ∈ C(R), 0 < κ0 ≤ κ(ξ) ≤ κ1 <∞ ∀ξ ∈ R.

(2.7)



6 T. KIM EJDE-2025/119

(4) For the functions of (2.1), (2.2) and (2.4) assume that

ϕi ∈ L2(0, T ;H− 1
2 (Γi)), i = 2, 4, 7;

ϕ⃗i ∈ L2(0, T ;H− 1
2 (Γi)), i = 3, 5, 6;

αij ∈ L∞(Γ5);

φ⃗τ ∈ L2
(
0, T ;H−1/2(Στ )

)
, (φ⃗τ , n)Στ

= 0;

ϕ⃗ν ∈ L∞(
0, T ;H1/2(Σν)

)
, (ϕ⃗ν , n)Σν

= 0;

q ∈ L∞(0, T ;H1/2(Στ );

gR ∈ L2(0, T ;L4/3(ΓR));

β1 ≥ β(x) ≥ 0, β1 - a constant, β(x)− textmeasurable.

(2.8)

Remark 2.3. For (2) in Assumption 2.2, which is necessary for (3.2), we refer to [7, Conditions
B1, B2, B3], or [6, B1, B2].

3. Variational formulation and main result

3.1. Analysis of the boundary conditions for the magnetic fields. For magnetic fields we
use the following spaces.

CΣ0(Ω̄) = {v ∈ C(Ω̄); v × n|Σν = 0, v · n|Στ = 0},
HΣ(Ω) = {v ∈ L2(Ω); curl v = 0, div v = 0, v × n|Σν

= 0, v · n|Στ
= 0},

HDC(Ω) =
{
v ∈ L2(Ω); curl v ∈ L2(Ω), div v ∈ L2(Ω)

}
,

(v, u)HDC(Ω) = (v, u)L2(Ω) + (div v,div u)L2(Ω) + (curl v, curlu)L2(Ω),

HDCΣ(Ω) : the clousue of CΣ0(Ω̄) ∩H1(Ω) in HDC(Ω),

VΣ = {v ∈ HDCΣ(Ω) : div v = 0} ∩ HΣ(Ω)
⊥,

HΣ : the closue of VΣ in L2(Ω),

here and in what follows HΣ(Ω)
⊥ is the orthogonal complement of HΣ(Ω) in L2(Ω). A field v

such that curl v = 0, div v = 0 is called harmonic, and elements of HΣ(Ω) are harmonic.
Note that if v ∈ HDCΣ(Ω), then v × n|Σν = 0 in H−1/2(Σν) and v · n|Στ = 0 in H−1/2(Σν)

since v × n|∂Ω ∈ H−1/2(∂Ω) and v · n|∂Ω ∈ H−1/2(∂Ω) for v ∈ HDC(Ω).
By [6, Theorems 10.2] we have

∥ curl v∥2 + ∥ div v∥2 ≥ c∥v∥2 ∃c > 0, ∀v ∈ HDCΣ(Ω) ∩HΣ(Ω)
⊥,

which implies

∥v∥2HDCΣ(Ω) ≥ ∥ curl v∥2 + ∥div v∥2 ≥ c̃∥v∥2HDCΣ(Ω) ∃c̃ > 0, ∀v ∈ HDCΣ(Ω) ∩HΣ(Ω)
⊥. (3.1)

On the other hand, by [6, Theorem 3.3], the norm in HDCΣ(Ω) is equivalent to the one in
H1(Ω), and for convenience we denote the HDCΣ(Ω)-norm of VΣ by ∥ · ∥VΣ

. Therefore, by (3.1),
the norm of VΣ is equivalent to ∥ curl v∥, and in what follows we use

∥ curl v∥2 = ∥v∥2VΣ(Ω) ∀v ∈ VΣ. (3.2)

Assumption 3.1. (1) For any t ∈ [0, T ] there exists a function H̊(t) ∈ L3(Ω)∩HΣ(Ω)
⊥ such that

curl H̊(t) ∈ L2(Ω), div H̊(t) = 0, H̊(t) · n|Στ
= q(t), H̊(t)× n|Σν

= ϕ⃗ν(t).

(2) H̊ ∈ L∞(0, T ;H1(Ω)), H̊ ∈W 1,2(0, T ;L3(Ω)).

Remark 3.2. For one case satisfying (1) of Assumption 3.1, we refer to [16, Remark 2].

If we assume the existence of a function H̊ ∈ L3(Ω)) ∩HΣ(Ω)
⊥ such that

curl H̊ = 0, div H̊ = 0, H̊ · n|Στ
= q, H̊ × n|Σν

= ϕ⃗ν



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 7

as in [1, 2, 4] and use (3.3) below, then in the estimation of approximate solutions the terms
corresponding to Lorentz force in the momentum equation and the nonlinear term in the magnetic
equation are canceled each other and the problem is simplified. However, the assumption curl H̊ =
0 is not usual for the magnetic field and excludes the boundary conditions such thatH · n|Στ

= q
J · n|Σν

̸= 0 (where J is electric current density) very important in practice (see 1) of Remark

4 of [16]). Thus, as in [16, 17] we assumed in this paper curl H̊ ∈ L2(Ω), and so the following
calculation is complicate.

We put

H(t) = H̊(t) + H̄(t), (3.3)

where H̊(t) is the function in Assumption 3.1, and H̄(t) ∈ VΣ. Then, for H of (3.3), u, v ∈ H1(Ω)
and C ∈ VΣ we have

−
〈
µ curlH ×H,u

〉
= −

〈
µ curl H̄ × H̄, u

〉
−
〈
µ curl H̄ × H̊, u

〉
−
〈
µ curl H̊ × H̄, u

〉
−
〈
µ curl H̊ × H̊, u

〉
,〈

µH × v, curlC
〉
=

〈
µ curlC × H̄, v

〉
+

〈
µ curlC × H̊, v

〉
.

(3.4)

Also taking into account (2.2), (3.3) and (3.4), for H ∈ H2(Ω), v ∈ H1(Ω) and C ∈ VΣ we have( 1

µ
curl

( 1

σ(θ)
curlH

)
+ curl(H × v), C

)
=

( 1

µσ(θ)
curlH + (H × v), curlC

)
−
( 1

µσ(θ)
curlH +H × v

)
× n,C

)
Στ

=
( 1

µσ(θ)
curl H̄, curlC

)
+
( 1

µσ(θ)
curl H̊, curlC

)
+
(
curlC × H̄, v

)
+
(
curlC × H̊, v

)
− (φ⃗τ , C)Στ

.

(3.5)

3.2. Variational formulation. For v ∈ H2(Ω) ∩V, θ ∈W 1,2(Ω) and u ∈ V, we have

−2⟨∇ · (η(θ)E(v)), u⟩ = 2(η(θ)E(v), E(u))− 2(η(θ)E(v)n, u)∪7
i=2Γi

= 2(η(θ)E(v), E(u)) + 2(η(θ)k(x)v, u)Γ2
− (η(θ) curl v × n, u)Γ3

+ 2(η(θ)Sṽ, ũ)Γ3 − 2(η(θ)εnn(v), un)Γ4 − 2(η(θ)εnτ (v), u)Γ5

− 2(η(θ)εn(v), u)Γ6 −
(
η(θ)

∂v

∂n
, u

)
Γ7

+ (η(θ)k(x)v, u)Γ7 ,

(3.6)

where S is the shape operator of boundary surface, ṽ and ũ are expressions of the vectors v and
u in a local orthogonal curvilinear coordinates on Γ3 and and k(x) = div n(x) (see [18, Theorems
2.1 and 2.2]).

For p ∈ H1(Ω) and u ∈ V we have

(∇p, u) = (p, un)∪7
i=2Γi

= (p, un)Γ2 + (p, un)Γ4∪Γ7 + (pn, u)Γ6 , (3.7)

where un |Γ1∪Γ3∪Γ5
= 0 (see (2.1)) was used.

For θ ∈W 1,2(Ω) and φ ∈W 1,2
ΓD

, by (2.4) we have

⟨−∇ · (κ(θ)∇θ), φ⟩ =
(
κ(θ)∇θ,∇φ

)
−

(
κ(θ)

∂θ

∂n
, φ

)
ΓR

= (κ(θ)∇θ,∇φ) + (βθ − gR, φ)ΓR
. (3.8)

By (2.6) we see that vn = 0 on ΓR, and so for v ∈ V, θ ∈W 1,2(Ω) and φ ∈W 1,2
ΓD

we have

⟨v · ∇θ, φ⟩ = (vnθ, φ)ΓR
− (θv,∇φ) = −(θv,∇φ). (3.9)



8 T. KIM EJDE-2025/119

Taking (v · ∇)v = curl v × v + 1
2grad|v|

2 into account, by (3.3)-(3.9) we can see that smooth
solutions (v, p,H, θ) of problem (1.1), (2.1), (2.2), (2.4) satisfy the following:

(v′(t), u) + 2(η(θ)E(v), E(u)) + ⟨curl v × v, u⟩+ 2(η(θ)k(x)v, u)Γ2

+ 2(η(θ)Sṽ, ũ)Γ3
+ 2(α(x)v, u)Γ5

+ (η(θ)k(x)v, u)Γ7
− 1

ρ
(µ curl H̄ × H̄, u)

− 1

ρ
(µ curl H̄ × H̊, u)− 1

ρ
(µ curl H̊ × H̄, u)

= ⟨(f0 + α0θ̄0g)− α0θg, u⟩+
1

ρ
(µ curl H̊ × H̊, u)

+
∑

i=2,4,7

⟨ϕi, un⟩Γi +
∑

i=3,5,6

⟨ϕ⃗i, u⟩Γi ∀u ∈ V,

(H̄ ′(t), C) +
〈 1

µσ(θ)
curl H̄(t), curlC

〉
+
〈 1

µσ(θ)
curl H̊(t), curlC

〉
+
〈
curlC × H̄(t), v(t)

〉
+
〈
curlC × H̊(t), v(t)

〉
+ (H̊ ′(t), C)

= ⟨φ⃗τ , C⟩Στ ∀C ∈ VΣ,

(θ′(t), φ) + (κ(θ)∇θ(t),∇φ)− (θ(t)v(t),∇φ) + (βθ(t), φ)ΓR

−
〈 1

ρChσ(θ)
| curlH|2, φ

〉
−

〈η(θ)
ρCh

|E(v)|2, φ
〉

= (gR, φ)ΓR
+ ⟨g0(t), φ⟩ ∀φ ∈W 1,∞

ΓD
(Ω).

We define a0⟨θ̃; ·, ·⟩, a1(·, ·, ·) and f1(t) ∈ V∗ by

a0⟨θ̃;w, u⟩ = 2⟨η(θ̃)E(w), E(u)⟩+ 2⟨η(θ̃)k(x)w, u⟩Γ2
+ 2⟨η(θ̃)Sw̃, ũ⟩Γ3

+ 2⟨α(x)w, u⟩Γ5 + ⟨η(θ̃)k(x)w, u⟩Γ7 ∀w, u ∈ V, θ̃ ∈W 1,r
ΓD
,

f(t) = f0(t) + α0θ̄0g +
µ

ρ
curl H̊(t)× H̊(t),

⟨f1(t), u⟩ = ⟨f(t), u⟩+
∑

i=2,4,7

⟨ϕi(t), un⟩Γi +
∑

i=3,5,6

⟨ϕ⃗i(t), u⟩Γi ∀u ∈ V.

(3.10)

We define b0(θ̃; ·, ·) and r1(t) ∈ V∗
Σ by

b0⟨θ̃; H̄, C⟩ =
〈 1

µσ(θ̃)
curl H̄, curlC

〉
∀H̄, C ∈ VΣ, θ̃ ∈W 1,r

ΓD
,

⟨r1(t), C⟩ = ⟨φ⃗τ , C⟩Στ
− ⟨H̊ ′(t), C⟩ ∀C ∈ VΣ.

(3.11)

We define c0⟨θ̃; ·, ·⟩ and g1(t) ∈ (W 1,2
ΓD

(Ω))∗ by

c0⟨θ̃; θ, φ⟩ = ⟨κ(θ̃)∇θ,∇φ⟩+ ⟨β(x)θ, φ⟩ΓR
∀θ̃, θ ∈W 1,r

ΓD
(Ω), φ ∈W 1,r′

ΓD
(Ω),

⟨g1(t), φ⟩ = ⟨gR, φ⟩ΓR
+ ⟨g0(t), φ⟩ ∀φ ∈W 1,2

ΓD
(Ω).

(3.12)

By (2.7), (2.8) and (2) of Assumption 3.1,

f1 ∈ L2(0, T ;V∗), r1 ∈ L2(0, T ;V∗
Σ), g1 ∈ L2(0, T ; (W 1,2

ΓD
)∗). (3.13)

Then, we introduce the following variational formulation for problem (1.1), (2.1), (2.2), (2.4).



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 9

Problem VE. Find (v, H̄, θ) ∈
(
L∞(0, T ;HV) ∩L2(0, T ;V)

)
× (L∞(0, T ;HΣ) ∩L2(0, T ;VΣ)

)
×(

L∞(0, T ;L2(Ω)) ∩ Lr(0, T ;W 1,r
ΓD

)
)
, 1 < r < 5

4 , such that∫ T

0

[
⟨−v(t), u′(t)⟩+ a0⟨θ; v(t), u(t)⟩+ ⟨curl v(t)× v(t), u(t)⟩

+
〈
− µ

ρ
curl H̄(t)× H̄(t)− µ

ρ
curl H̄(t)× H̊(t)− µ

ρ
curl H̊(t)× H̄(t)

− µ

ρ
curl H̊(t)× H̊(t) + α0θ(t)g − f1(t), u(t)

〉]
dt

= ⟨v0, u(x, 0)⟩ ∀u ∈ C1([0, T ];V) with u(·, T ) = 0,∫ T

0

[
⟨−H̄(t), C ′(t)⟩+ b0⟨θ(t); H̄(t), C(t)⟩+

〈 1

µσ(θ)
curl H̊(t), curlC

〉
+
〈
curlC(t)× H̄(t), v(t)

〉
+

〈
curlC(t)× H̊(t), v(t)

〉
− ⟨r1(t), C(t)⟩

]
dt

= ⟨H̄0, C(x, 0)⟩ ∀C ∈ C1([0, T ];VΣ) with C(·, T ) = 0,∫ T

0

[
⟨−θ(t), φ′(t)⟩+ c0⟨θ(t); θ(t), φ(t)⟩ − ⟨θ(t)v(t),∇φ(t)⟩

−
〈 1

ρChσ(θ)
| curlH|2, φ

〉
−

〈η(θ)
ρCh

|E(v)|2, φ
〉
− ⟨g1(t), φ(t)⟩

]
dt

= ⟨θ0(x), φ(x, 0)⟩ ∀φ ∈ C1([0, T ];W 1,∞
ΓD

(Ω))with φ(·, T ) = 0,

(3.14)

where H̄0 = H0 − H̊(0, x).

Remark 3.3. For the case of the variational problem above equivalent to the original PDE
problem (1.1), (2.1), (2.2), (2.4), we refer to [17, Theorem 3.1].

3.3. Main result. We want to obtain a solution satisfying (3.14), but we prove existence of
(v, H̄, θ) with a “defect measure” in the third equation of (3.14).

Theorem 3.4. Let Assumptions 2.2 and 3.1 be satisfied, and let v0 ∈ HV, H̄0 ∈ HΣ, θ0 ∈ L1(Ω).
If |α0| is small enough in accordance with date of the problem (4.101), then there exists (v, H̄, θ) ∈(
L∞(0, T ;HV)∩L2(0, T ;V)

)
×
(
L∞(0, T ;HΣ)∩L2(0, T ;VΣ)

)
×
(
L∞(0, T ;L1(Ω))∩Lr(0, T ;W 1,r

ΓD
)
)

for all r ∈ (1, 5/4) and a Radon measure µ̄ such that∫ T

0

[
− ⟨v(t), u′(t)⟩+ a0⟨θ; v(t), u(t)⟩+ ⟨curl v(t)× v(t), u(t)⟩

+
〈
− µ

ρ
curl H̄(t)× H̄(t)− µ

ρ
curl H̄(t)× H̊(t)− µ

ρ
curl H̊(t)× H̄(t)

− µ

ρ
curl H̊(t)× H̊(t) + α0θ(t)g − f1(t), u(t)

〉]
dt

= ⟨v0, u(x, 0)⟩ ∀u ∈ C1([0, T ];V) with u(·, T ) = 0,∫ T

0

[
⟨−H̄(t), C ′(t)⟩+ b0⟨θ(t); H̄(t), C(t)⟩+

〈 1

µσ(θ)
curl H̊(t), curlC

〉
+
〈
curlC(t)× H̄(t), v(t)

〉
+

〈
curlC(t)× H̊(t), v(t)

〉
− ⟨r1(t), C(t)⟩

]
dt

= ⟨H̄0, C(x, 0)⟩ ∀C ∈ C1([0, T ];VΣ) with C(·, T ) = 0,∫ T

0

[
⟨−θ, ∂φ

∂t
⟩+ c0⟨θ; θ, φ⟩ − ⟨θv,∇φ⟩ −

〈 1

ρChσ(θ)
| curlH|2, φ

〉
−
〈η(θ)
ρCh

|E(u)|2, φ
〉
− ⟨g1, φ⟩

]
dt

= ⟨θ0(x), φ(x, 0)⟩+
∫ T

0

φdµ̄ ∀φ ∈ C1([0, T ];C1
ΓD

(Ω)) with φ(·, T ) = 0,

(3.15)



10 T. KIM EJDE-2025/119

where C1
ΓD

(Ω) = {ϕ ∈ C1(Ω̄) : ϕ|ΓD
= 0}. And (v, H̄, θ) satisfies the estimate

sup
t∈[0,T ]

∥v(t)∥2 + ∥v∥2L2(0,T ;V) + sup
t∈[0,T ]

∥H̄(t)∥2 + ∥H̄∥2L2(0,T ;VΣ)

+ ess sup
t∈[0,T ]

∥θ(t)∥L1(Ω) +

∫
Q

|∇θ|r dx dt+
∫
Q

|∇θ|2

(1 + |θ|)1+δ
dx dt

≤ C1

(
∥v0∥, ∥f1∥L2(0,T ;V∗),

∑
i=2,4,7

∥ϕi∥
L2(0,T ;H− 1

2 (Γi))
,

∑
i=3,5,6

∥ϕ⃗i∥
L2(0,T ;H− 1

2 (Γi))
, ∥H̄0∥, c∗, ∥H̊(t)∥L∞(0,T ;H1(Ω)),

2(1−δ)/δ

1− δ
,

|α0|, ∥θ0∥L1(Ω), ∥gR∥L2(0,T ;L4/3(ΓR)), ∥g0∥L2(0,T ;(W 1,2
ΓD

)∗)

)
∀r

(
1 < r <

5

4

)
, ∀δ

(
0 < δ <

5− 4r

3

)
,

(3.16)

where c∗ is the one in (4.77).

4. Proof of Theorem 3.4

4.1. Existence of a solution to an approximate problem. Let v0 ∈ V, H̄0(≡ H0−H̊(0, x)) ∈
VΣ, θ0 ∈W 1,2

ΓD
(Ω), 0 < ε ≤ 1.

Problem VEA. Find (v, H̄, θ) ∈
(
C([0, T ];HV)∩L6(0, T ;V)

)
×
(
C([0, T ];HΣ)∩L6(0, T ;VΣ)

)
×(

C([0, T ];L2(Ω)) ∩ L2(0, T ;W 1,2
ΓD

)
)
such that

〈∂v
∂t
, u

〉
+ 2

〈
[η(θ) + ε∥v∥4V]E(v), E(u)

〉
+ ⟨curl v × v, u⟩

+ 2(η(θ)k(x)v, u)Γ2
+ 2(η(θ)Sṽ, ũ)Γ3

+ 2(α(x)v, u)Γ5

+ (η(θ)k(x)v, u)Γ7
− ⟨µ

ρ
curl H̄(t)× H̄(t), u⟩ − ⟨µ

ρ
curl H̄(t)× H̊(t), u⟩

− ⟨µ
ρ
curl H̊(t)× H̄(t), u⟩+ ⟨ α0θ

1 + εθ2
g, u⟩

= ⟨f1(t), u(t)⟩ ∀u ∈ L6(0, T ;V),

⟨H̄ ′(t), C⟩+
〈
[

1

µσ(θ)
+ ε∥ curl H̄(t)∥4] curl H̄(t), curlC

〉
+
〈 1

µσ(θ)
curl H̊(t), curlC

〉
+
〈
curlC × H̄(t), v(t)

〉
+
〈
curlC × H̊(t), v(t)

〉
= ⟨r1(t), C⟩ ∀C ∈ L6(0, T ;VΣ),〈∂θ
∂t
, φ

〉
+ (κ(θ)∇θ,∇φ) + (β(x)θ, φ)ΓR

− ⟨vθ,∇φ⟩

−
〈 1

ρChσ(θ)

| curlH|2

1 + ε| curlH|2
, φ

〉
−

〈η(θ)
ρCh

|E(v)|2

1 + ε|E(v)|2
, φ

〉
= ⟨g1, φ⟩ ∀φ ∈ L2(0, T ;W 1,2

ΓD
(Ω)),

v(0) = v0, H̄(0) = H̄0, θ(0) = θ0.

(4.1)

To prove the main result, we use the following result, see [19, Proposition 4.1 and Remark 4.1].

Proposition 4.1. Let X be a real reflexive Banach space and norms of X and X∗ be strictly
convex. Let a linear operator L, where D(L) is a dense linear subspace of X, be maximal monotone.
The operator A : D(L) → X∗ satisfies the following:



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 11

(i) For every sequence such that xn ⇀ x in X, xn, x ∈ D(L), Lxn ⇀ Lx in X∗ and
lim supn→∞ ⟨Axn, xn − x⟩ ≤ 0, there exists a subsequence such that

lim inf
k→∞

⟨Axk, xk − v⟩ ≥ ⟨Ax, x− v⟩ ∀v ∈ X; (4.2)

(ii) There exists a function ψ : R+ → R+, which is bounded on any compact interval, and a
number θ, 0 ≤ θ < 1, such that

∥A(x)∥X∗ ≤ ψ(∥x∥X) + θ∥Lx∥X∗ ∀x ∈ D(L); (4.3)

(iii)

⟨A(x), x⟩
∥x∥X

→ ∞, as ∥x∥X → ∞. (4.4)

Then, for any f ∈ X∗ there exists a solution to Lx+Ax = f .

Let

V := L6(0, T ;V)× L6(0, T ;VΣ)× L2(0, T ;W 1,2
ΓD

),

∥(·, ·)∥V =
(
∥ · ∥2L6(0,T ;V) + ∥ · ∥2L6(0,T ;VΣ) + ∥ · ∥2

L2(0,T ;W 1,2
ΓD

)

)1/2
.

Then

V ∗ := L6/5(0, T ;V∗)× L6/5
(
0, T ;V∗

Σ)× L2
(
0, T ; (W 1,2

ΓD
)∗
)
,

∥(·, ·)∥V ∗ =
(
∥ · ∥2L6/5(0,T ;V∗) + ∥ · ∥2

L6/5
(
0,T ;V∗

Σ)
+ ∥ · ∥2

L2
(
0,T ;(W 1,2

ΓD
)∗
))1/2.

Theorem 4.2. There exists a solution (vε, H̄ε, θε) ∈ V to Problem VEA.

Proof. In (4.1) let us make changes of the unknown functions

(v, H̄, θ) −→ (w, z, θ)

by v = ek1tw, H̄ = ek1tz where k1 is a constant to be determined later((4.34)). Owing to the
changes, differently from [16], where the steady problems were studied, we need not assume that
Γ2j ,Γ3j ,Γ7j are convex and the matrix α is positive, and the datum corresponding to the non-
homogeneous magnetic boundary condition is small. Then putting

ŵ = w − v0, ẑ = z − H̄0, θ̂ = θ − θ0

and multiplying the first and second equations of (4.1), respectively, by e−k1t and µ
ρ e

−k1t, we have

a modified evolution problem with zero initial conditions equivalent to Problem VEA. □

Problem VEA’ Find (ŵ, Ĥ, θ̂) ∈
(
C([0, T ];HV)∩L6(0, T ;V)

)
×
(
C([0, T ];HΣ)∩L6(0, T ;VΣ)

)
×(

C([0, T ];L2(Ω)) ∩ L2(0, T ;W 1,2
ΓD

)
)
such that

〈∂ŵ
∂t
, u

〉
+ 2

〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(u)

〉
+ ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩+ k1⟨ŵ + v0, u⟩+ 2⟨η(θ)k(x)(ŵ + v0), u⟩Γ2

+ 2⟨η(θ)S( ˜̂w + v0), ũ⟩Γ3 + 2⟨α(x)(ŵ + v0), u⟩Γ5 + ⟨η(θ)k(x)(ŵ + v0), u⟩Γ7

− µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0) + curl(ẑ(t) + H̄0)× H̊(t)

+ curl H̊(t)× (ẑ(t) + H̄0), u
〉
+ e−k1t

〈 α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
g, u

〉
= e−k1t⟨f1(t), u⟩ ∀u ∈ L6(0, T ;V),



12 T. KIM EJDE-2025/119

µ

ρ

[
⟨∂ẑ(t)
∂t

, C⟩+
〈
[

1

µσ(θ)
+ εe4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curlC

〉
+
〈
e−k1t

1

µσ(θ)
curl H̊(t), curlC

〉
+ k1⟨(ẑ(t) + H̄0), C⟩

+
〈
curlC × (ẑ(t) + H̄0), e

k1tw(t)
〉
+

〈
curlC × H̊(t), w(t)

〉]
=
µ

ρ
e−k1t⟨ r1(t), C⟩ ∀C ∈ L6(0, T ;VΣ),

〈∂θ̂
∂t
, φ

〉
+ ⟨κ(θ)(∇θ̂ +∇θ0),∇ϕ⟩+ ⟨β(x)(θ̂ + θ0), ϕ⟩ΓR

− ek1t⟨(ŵ + v0)(θ̂ + θ0),∇ϕ⟩ −
〈 1

ρChσ(θ)

| curl(H̄ + H̊)|2

1 + ε| curl(H̄ + H̊)|2
, φ

〉
−
〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, φ

〉
= ⟨g1(t), φ⟩ ∀φ ∈ L2(0, T ;W 1,2

ΓD
(Ω)),

ŵ(0) = 0, Ĥ(0) = 0, θ̂(0) = 0. (4.5)

We define an operator L : D(L ) → V ∗ by

D(L ) :=
{
(ŵ, ẑ, θ̂) ∈ V : (ŵ′, ẑ′, θ̂′) ∈ V ∗, ŵ(0) = 0, ẑ(0) = 0, θ̂(0) = 0

}
,

∥(ŵ, ẑ, θ̂)∥D(L ) := ∥(ŵ, ẑ, θ̂)∥V + ∥(ŵ′,
µ

ρ
ẑ′, θ̂′)∥V ∗ ,

L :=
( d
dt
,
µ

ρ

d

dt
,
d

dt

)
.

Then, L is a linear, maximal monotone operator; see [23, Section 2.1, Ch. 3 ].
We define an operator Aε : D(L ) → V ∗ by

〈
Aε(ŵ, ẑ, θ̂), (u,C, ϕ)

〉
=

∫ T

0

[
2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(u)

〉
+ ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩+ k1(ŵ + v0, u) + 2⟨η(θ)k(x)(ŵ + v0), u⟩Γ2

+ 2⟨η(θ)S( ˜̂w + ṽ0), ũ⟩Γ3
+ 2⟨α(x)(ŵ + v0), u⟩Γ5

+ ⟨η(θ)k(x)(ŵ + v0), u⟩Γ7

− µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0) + curl(ẑ(t) + H̄0)× H̊(t)

+ curl H̊(t)× (ẑ(t) + H̄0), u
〉
+ e−k1t

〈 α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
g, u

〉
+
〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curlC

〉
+
〈
e−k1t

1

ρσ(θ)
curl H̊(t), curlC

〉
+ k1⟨(ẑ + H̄0), C⟩

+
〈µ
ρ
ek1t curlC × (ẑ(t) + H̄0), (ŵ(t) + v0)

〉
+
〈µ
ρ
curlC × H̊(t), (ŵ(t) + v0)

〉
+ ⟨κ(θ)(∇θ̂ +∇θ0),∇ϕ⟩+ ⟨β(x)(θ̂ + θ0), ϕ⟩ΓR

− ek1t⟨(ŵ + v0)(θ̂ + θ0),∇ϕ⟩

−
〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, ϕ

〉
−
〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, ϕ

〉]
dt

(4.6)

for (ŵ, ẑ, θ̂) ∈ D(L ) and all (u,C, ϕ) ∈ V , where and in what follows H̄(t) = ek1tz(t) ≡ ek1t(ẑ +

H̄0) and θ = θ̂ + θ0. Estimate (4.43) below shows that this operator is well-defined.



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 13

Then, the existence of a solution to Problem VEA’ is equivalent to the existence of a solution
to

L (ŵ, ẑ, θ̂) + Aε(ŵ, ẑ, θ̂) = F , F (t) =

 e−k1tf1(t)
µ
ρ e

−k1tr1(t)

g1(t)

 .

Remark 4.3. By definition ofD(L ), from the fact that (ŵ, ẑ, θ̂) ∈ D(L ) it follows that (ŵ, ẑ, θ̂) ∈(
C([0, T ];HV)× C([0, T ];HΣ)× C([0, T ];L2(Ω))

)
. (See [20, Theorem 1.37].)

Relying on Proposition 4.1, we will prove the existence of a solution to the equation above.

(i) Let us prove property (iii) of Proposition 4.1 for Aε(ŵ, ẑ, θ̂):

1

∥(ŵ, ẑ, θ̂)∥V

〈
Aε(ŵ, ẑ, θ̂), (ŵ, ẑ, θ̂)

〉
→ ∞ as ∥(ŵ, ẑ, θ̂)∥V → ∞. (4.7)

〈
Aε(ŵ, ẑ, θ̂), (ŵ, ẑ, θ̂)

〉
=

∫ T

0

[
2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(ŵ)

〉
+ ek1t⟨curl(ŵ + v0)× (ŵ + v0), ŵ⟩+ k1(ŵ + v0, ŵ) + 2⟨η(θ)k(x)(ŵ + v0), ŵ⟩Γ2

+ 2⟨η(θ)S( ˜̂w + ṽ0), ˜̂w⟩Γ3
+ 2⟨α(x)(ŵ + v0), ŵ⟩Γ5

+ ⟨η(θ)k(x)(ŵ + v0), ŵ⟩Γ7

− µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0) + curl(ẑ(t) + H̄0)× H̊(t)

+ curl H̊(t)× (ẑ(t) + H̄0), ŵ
〉
+ e−k1t⟨ α0θ

1 + εθ2
g, ŵ⟩

+
〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curl ẑ

〉
+
〈
e−k1t

1

ρσ(θ)
curl H̊(t), curl ẑ

〉
+ k1⟨(ẑ(t) + H̄0), ẑ(t)⟩

+
〈µ
ρ
ek1t curl ẑ(t)× H̊(t), (ŵ(t) + v0)

〉
+

〈µ
ρ
curl ẑ(t)× H̄0, (ŵ(t) + v0)

〉
+ ⟨κ(θ)(∇θ̂ +∇θ0),∇θ̂⟩+ ⟨β(x)(θ̂ + θ0), θ̂⟩ΓR

− ek1t⟨(ŵ + v0)(θ̂ + θ0),∇θ̂⟩

−
〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, θ̂
〉
−

〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, θ̂
〉]
dt

(4.8)

for all (ŵ, ẑ, θ̂) ∈ D(L ).

Using ⟨rotw × v, v⟩ = 0 and ⟨wθ̂,∇θ̂⟩ = 0 (cf. (3.9)) and arranging, we obtain〈
Aε(ŵ, ẑ, θ̂), (ŵ, ẑ, θ̂)

〉
=

∫ T

0

[
2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(ŵ)

〉
+ ek1t⟨curl ŵ × v0, ŵ⟩+ ek1t⟨curl v0 × v0, ŵ⟩+ k1(ŵ + v0, ŵ)

+ 2⟨η(θ)k(x)(ŵ + v0), ŵ⟩Γ2
+ 2⟨η(θ)S( ˜̂w + ṽ0), ˜̂w⟩Γ3

+ 2⟨α(x)(ŵ + v0), ŵ⟩Γ5
+ ⟨η(θ)k(x)(ŵ + v0), ŵ⟩Γ7

− µ

ρ

〈
ek1t curl H̄0 × (ẑ(t) + H̄0) + curl H̄0 × H̊(t) + curl H̊(t)× (ẑ(t) + H̄0), ŵ

〉
+ e−k1t⟨ α0θ

1 + εθ2
g, ŵ⟩

+
〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curl ẑ

〉
+
〈
e−k1t

1

ρσ(θ)
curl H̊(t), curl ẑ

〉
+ k1⟨(ẑ(t) + H̄0), ẑ(t)⟩



14 T. KIM EJDE-2025/119

+
〈µ
ρ
ek1t curl ẑ × (ẑ(t) + H̄0), v0

〉
+
〈µ
ρ
curl ẑ × H̊(t), v0

〉
+ ⟨κ(θ)(∇θ̂ +∇θ0),∇θ̂⟩+ ⟨β(x)(θ̂ + θ0), θ̂⟩ΓR

− ek1t⟨(ŵ + v0)θ0,∇θ̂⟩

−
〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, θ̂
〉
−
〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, θ̂
〉]
dt (4.9)

for all (ŵ, ẑ, θ̂) ∈ D(L ).
Let us estimate the terms on the right-hand side the above. Taking into account (2.5), by

Hölder’s and Young’s inequalities, we have∫ T

0

[
2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(ŵ)

〉
≥ 2η0∥ŵ∥2L2(0,T ;V) +

∫ T

0

2
(
η(θ)E(v0), E(ŵ)

)
dt

+ ε

∫ T

0

2
〈
e4k1t∥ŵ + v0∥4VE(ŵ + v0), E(ŵ + v0)

〉
dt

− ε

∫ T

0

2
〈
e4k1t∥ŵ + v0∥4VE(ŵ + v0), E(v0)

〉
dt

≥ η0∥ŵ∥2L2(0,T ;V) + 2ε∥ŵ + v0∥6L6(0,T ;V)

− 2cε∥ŵ + v0∥5L6(0,T ;V)∥v0∥L6(0,T ;V) − kT∥v0∥2V
≥ η0∥ŵ∥2L2(0,T ;V) + ε∥ŵ∥6L6(0,T ;V) −K1,

(4.10)

whereK1 depends on ∥v0∥V, ε, T . By Hölder’s and Young’s inequalities, an interpolation inequality

∥ŵ∥L3(Ω) ≤ ∥ŵ∥1/2W1,2(Ω)∥ŵ∥
1/2, we have

ek1T ⟨curl ŵ × v0, ŵ⟩ ≤ ek1T ∥ŵ∥2V∥v0∥L3 ≤ ε

3
∥ŵ∥6V +

2

3ε

(
ek1T ∥v0∥L3

)3/2
,

ek1T ⟨curl v0 × v0, ŵ⟩ ≤ ek1T ∥v0∥2V + c∥v0∥2V∥ŵ∥2L3

≤ ek1T ∥v0∥2V + c∥v0∥2V∥ŵ∥V∥ŵ∥

≤ η0
14

∥ŵ∥2V + ek1T ∥v0∥2V + c1∥v0∥4V∥ŵ∥2,

where c1 depends on η0, but is independent from k1. Then∣∣∣ ∫ T

0

ek1t
[
⟨curl ŵ × v0, ŵ⟩+ ⟨curl v0 × v0, ŵ⟩

]
dt
∣∣∣

≤ η0
14

∥ŵ∥2L2(0,T ;V) +
ε

3
∥ŵ∥6L6(0,T ;V) +K2 + k11∥ŵ∥2L2(0,T ;L2),

(4.11)

where K2 = 2T
3ε

(
ek1T ∥v0∥L3

)3/2
+ Tek1T ∥v0∥2V and k11 = c1∥v0∥4V. Also,∫ T

0

k1(ŵ + v0, ŵ) ≥ k1∥ŵ∥2L2(0,T ;L2) − L1∥ŵ∥L2(0,T ;L2), (4.12)

where L1 depends on ∥v0∥, k1.
Because Γ2j ,Γ3j ,Γ7j are in C2.1(Γij), there exists a constant M such that

∥S(x)∥∞, ∥k(x)∥∞, ∥α∥L∞(Γ5) ≤M.

Thus,∣∣∣ ∫ T

0

(
2⟨η(θ)k(x)ŵ, ŵ⟩Γ2

+ 2⟨η(θ)S ˜̂w, ˜̂w⟩Γ3
+ 2⟨α(x)ŵ, ŵ⟩Γ5

+ ⟨η(θ)k(x)ŵ, ŵ⟩Γ7

)
dt
∣∣∣

≤ η0
14

∥ŵ∥2L2(0,T ;V) + k12∥ŵ∥2L2(0,T ;L2)

(4.13)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 15

(cf. [15, Theorem 1.5.1.10 ]), and∣∣∣ ∫ T

0

(
2⟨η(θ)k(x)v0, ŵ⟩Γ2

+ 2⟨η(θ)Sṽ0, ˜̂w⟩Γ3
+ 2⟨α(x)v0, ŵ⟩Γ5

+ ⟨η(θ)k(x)v0, ŵ⟩Γ7

)
dt
∣∣∣

≤ η0
14

∥ŵ∥2L2(0,T ;V) +K3,

(4.14)

where K3 depends on ∥v0∥V, η0, T .
Let us estimate∫ T

0

µ

ρ
ek1t

〈
curl H̄0 × (ẑ(t) + H̄0), ŵ

〉
dt

=

∫ T

0

µ

ρ
ek1t

〈
curl H̄0 × ẑ(t), ŵ

〉
dt+

∫ T

0

µ

ρ
ek1t

〈
curl H̄0 × H̄0, ŵ

〉
dt

By Hölder’s and Young’s inequalities and the interpolation inequality

∥u∥L3(Ω) ≤ ∥u∥1/2W1,2(Ω)∥u∥
1/2
L2(Ω),

we obtain∫ T

0

µ

ρ
ek1t|

〈
curl H̄0 × ẑ(t), ŵ

〉
| dt

≤
∫ T

0

µ

ρ
ek1t∥ curl H̄0∥∥ẑ(t)∥VΣ∥ŵ∥L3 dt

≤ T
µ2

ρ2
e2k1T ∥H̄0∥2H1(Ω) +

∫ T

0

∥ẑ∥2VΣ
∥ŵ∥V∥ŵ∥ dt

≤ T
µ2

ρ2
e2k1T ∥H̄0∥2H1(Ω) +

µε

4ρ
∥ẑ∥6L6(0,T ;VΣ) +

2

3

√
4ρ

3µε

∫ T

0

∥ŵ∥3/2V ∥ŵ∥3/2 dt

≤ µε

4ρ
∥ẑ∥6L6(0,T ;VΣ) +

ε

6
∥ŵ∥6L6(0,T ;V) + k13∥ŵ∥2L2(0,T ;L2) +K41,

(4.15)

where k13 depends on 1
ε2 , ρ, µ, and K41 = T µ2

ρ2 e
k1T ∥H̄0∥2H1(Ω).∫ T

0

µ

ρ
ek1t| curl H̄0 × H̄0, ŵ| dt ≤

∫ T

0

µ

ρ
ek1t∥H̄0∥2H1∥ŵ∥V dt ≤ η0

14
∥ŵ∥2L2(0,T ;V) +K42, (4.16)

where K42 depends on ∥H0∥4H1(Ω), η0, ρ, µ and ek1T . By (4.15) and (4.16), we have∣∣∣ ∫ T

0

µ

ρ
ek1t

〈
curl H̄0 × (ẑ(t) + H̄0), ŵ

〉
dt
∣∣∣

≤ µε

4ρ
∥ẑ∥6L6(0,T ;VΣ) +

η0
14

∥ŵ∥2L2(0,T ;V) +
ε

3
∥ŵ∥6L6(0,T ;V) + k13∥ŵ∥2L2(0,T ;L2) +K4,

(4.17)

where K4 = K41 +K42.
Also, by Hölder’s and Young’s inequalities we have∣∣∣ ∫ T

0

µ

ρ

〈
curl H̄0 × H̊(t), ŵ

〉
dt
∣∣∣ ≤ η0

14
∥ŵ∥2L2(0,T ;V) +K5, (4.18)

where K5 depends on η0 and (∥H̄0∥2H1(Ω)∥H̊∥2L∞(0,T ;H1(Ω))).

Let us estimate∫ T

0

µ

ρ

〈
curl H̊(t)× (ẑ(t) + H̄0), ŵ

〉
dt

=

∫ T

0

µ

ρ

〈
curl H̊(t)× ẑ(t), ŵ

〉
dt+

∫ T

0

µ

ρ

〈
curl H̊(t)× H̄0, ŵ

〉
dt.



16 T. KIM EJDE-2025/119

Similarly to (4.15) we have∣∣∣ ∫ T

0

µ

ρ

〈
curl H̊(t)× ẑ(t), ŵ

〉
dt
∣∣∣

≤
∣∣∣µ
ρ

∫ T

0

∥ curl H̊(t)∥∥ẑ(t)∥VΣ∥ŵ∥L3 dt
∣∣∣

≤ 1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) + c∥H̊∥2L∞(0,T ;H1(Ω))

∫ T

0

∥ŵ∥V∥ŵ∥ dt

≤ 1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) +

η0
28

∥ŵ∥2L2(0,T ;V) + k14∥ŵ∥2L2(0,T ;L2),

(4.19)

where k14 depends on ∥H̊∥2L∞(0,T ;H1(Ω)), ρ, σ1, η0.∣∣∣ ∫ T

0

µ

ρ

〈
curl H̊(t)× H̄0, ŵ

〉
dt
∣∣∣ ≤ ∣∣∣µ

ρ

∫ T

0

∥ curl H̊(t)∥∥H̄0∥VΣ
∥ŵ∥L3 dt

∣∣∣
≤ c∥H̊∥2L∞(0,T ;H1(Ω))∥H̄0∥2VΣ

+

∫ T

0

∥ŵ∥V∥ŵ∥ dt

≤ η0
28

∥ŵ∥2L2(0,T ;V) + k15∥ŵ∥2L2(0,T ;L2) +K6,

(4.20)

where k15 depends on η0 and K6 depends on ∥H̊∥2L∞(0,T ;H1(Ω)), ∥H̄0∥2VΣ
, ρ, µ.

From (4.19) and (4.20), we obtain∣∣∣ ∫ T

0

µ

ρ

〈
curl H̊(t)× (ẑ(t) + H̄0), ŵ

〉
dt
∣∣∣

≤ 1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) +

η0
14

∥ŵ∥2L2(0,T ;V) + (k14 + k15)∥ŵ∥2L2(0,T ;L2) +K6.

(4.21)

Since
ξ

1 + εξ2
≤ 1

2
√
ε

∀ξ ∈ [0,∞), (4.22)

we have ∣∣∣ ∫ T

0

e−k1t⟨ α0θ

1 + εθ2
g, ŵ⟩ dt

∣∣∣ ≤ c√
ε
∥ŵ∥L2(0,T ;L2). (4.23)

Taking into account (3.2), in the same way as (4.10), we have∫ T

0

〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curl ẑ

〉
dt

≥ 1

ρσ1
∥ẑ∥2L2(0,T ;VΣ) +

µε

ρ
∥ẑ∥6L6(0,T ;VΣ) −K7,

(4.24)

where K7 depends on ∥H̄0∥VΣ
, T, ε, ρ, µ. Also we obtain∫ T

0

〈
e−k1t

1

ρσ(θ)
curl H̊(t), curl ẑ

〉
dt ≤ L2∥ẑ(t)∥L2(0,T ;VΣ), (4.25)

where L2 depends on ∥H̄0∥L∞(0,T ;H1), ρ, σ0.∫ T

0

k1⟨(ẑ(t) + H̄0), ẑ(t)⟩ dt ≥ k1∥ẑ∥2L2(0,T ;L2) − L3∥ẑ∥L2(0,T ;L2), (4.26)

where L3 depends on k1, ∥H̄0∥.
Let us estimate∫ T

0

〈µ
ρ
ek1t curl ẑ × (ẑ(t) + H̄0), v0

〉
dt

=

∫ T

0

〈µ
ρ
ek1t curl ẑ × ẑ(t), v0

〉
dt+

∫ T

0

〈µ
ρ
ek1t curl ẑ × H̄0, v0

〉
dt.



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 17

Taking into account ∥ẑ∥L3(Ω) ≤ ∥ẑ∥1/2VΣ
∥ẑ∥1/2L2(Ω), we obtain∣∣∣ ∫ T

0

〈µ
ρ
ek1t curl ẑ × ẑ(t), v0

〉
dt
∣∣∣

≤ c

∫ T

0

µ

ρ
ek1t∥ẑ(t)∥VΣ

∥ẑ(t)∥L3∥v0∥L6 dt

≤ c′
Tµ2

ρ2
e2k1T ∥v0∥2V +

∫ T

0

∥ẑ(t)∥3VΣ
∥ẑ(t)∥ dt

≤ c′
Tµ2

ρ2
e2k1T ∥v0∥2V +

µε

4ρ
∥ẑ∥6L6(0,T ;VΣ) + k16∥ẑ∥2L2(0,T ;L2),

(4.27)

where k16 depends on ρ, µ, ε.∣∣∣ ∫ T

0

〈µ
ρ
ek1t curl ẑ × H̄0, v0

〉
dt
∣∣∣

≤ c

∫ T

0

µ

ρ
ek1t∥ẑ(t)∥VΣ

∥H̄0∥L3∥v0∥L6 dt

≤ 1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) + c′′

Tµ2

ρ2
e2k1T ∥H̄0∥2L3∥v0∥2L6 ,

(4.28)

where c′′ depends on ρ, σ1.
By (4.27) and (4.28), we have∫ T

0

〈µ
ρ
ek1t curl ẑ × (ẑ(t) + H̄0), v0

〉
dt

≤ µε

4ρ
∥ẑ∥6L6(0,T ;V) +

1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) + k16∥ẑ∥2L2(0,T ;L2) +K8,

(4.29)

where K8 = c′ Tµ2

ρ2 e
2k1T ∥v0∥2V + c′′ Tµ2

ρ2 e
2k1T ∥H̄0∥2L3∥v0∥2L6 . Also we obtain∣∣∣ ∫ T

0

〈µ
ρ
curl ẑ × H̊(t), v0

〉
dt
∣∣∣ ≤ 1

6ρσ1
∥ẑ∥2L2(0,T ;VΣ) +K9, (4.30)

where K9 depends on ∥H̊∥2L∞(0,T ;L3), ∥v0∥
2
L6 , ρ, µ, σ1. Also, we have∫ T

0

⟨κ(θ)(∇θ̂ +∇θ0),∇θ̂⟩ dt ≥ κ0∥θ̂∥2L2(0,T ;W 1,2
ΓD

)
− κ0

6
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
−K10,∫ T

0

(β(x)θ̂, θ̂)ΓR
dt ≥ 0,∣∣∣ ∫ T

0

(β(x)θ0, θ̂)ΓR
dt
∣∣∣ ≤ κ0

6
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+K11,

(4.31)

where K10 = T∥θ0∥2W 1,2
ΓD

and K11 depends on ∥θ0∥W 1,2(Ω) and T .

Let us estimate∣∣∣ ∫ T

0

ek1t⟨(ŵ + v0)θ0,∇θ̂⟩ dt
∣∣∣ = ∣∣∣ ∫ T

0

[
ek1t⟨ŵθ0,∇θ̂⟩+ ek1t⟨v0θ0,∇θ̂⟩

]
dt
∣∣∣.

First, we have∣∣∣ ∫ T

0

ek1t⟨ŵθ0,∇θ̂⟩ dt
∣∣∣ ≤ ∫ T

0

ek1t∥ŵ∥L3∥θ0∥L6∥∇θ̂∥ dt

≤ κ0
12

∥θ̂∥2
L2(0,T ;W 1,2

ΓD
)
+
c′Te4k1T

ε
∥θ0∥4W 1,2 +

ε

6
∥ŵ∥6L6(0,T ;V).

Also ∣∣∣ ∫ T

0

ek1t⟨v0θ0,∇θ̂⟩ dt
∣∣∣ ≤ κ0

12
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+ c′′Te2k1T ∥v0∥2V∥θ0∥2W 1,2 ,



18 T. KIM EJDE-2025/119

where c′′ depends on κ0. Combining above two inequalities yields∣∣∣ ∫ T

0

ek1t⟨(ŵ + v0)θ0,∇θ̂⟩ dt
∣∣∣ ≤ κ0

6
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+
ε

6
∥ŵ∥6L6(0,T ;V) +K12, (4.32)

where K12 = c′Te4k1T

ε ∥θ0∥4W 1,2 + c′′Te2k1T ∥v0∥2V∥θ0∥2W 1,2 .
In the same way as for (4.22), we obtain∣∣∣ ∫ T

0

〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, θ̂
〉
dt
∣∣∣ ≤ c

ε
∥θ̂∥L2(0,T ;L2(Ω)),∣∣∣ ∫ T

0

〈η(θ)
ρCh

|E(w)|2

e−2k1t + ε|E(w)|2
, θ̂
〉
dt
∣∣∣ ≤ c

ε
∥θ̂∥L2(0,T ;L2(Ω)).

(4.33)

Taking

k1 = max
{ 5∑

i=1

k1i, k16

}
, (4.34)

by (4.10)-(4.14), (4.17)-(4.18), (4.21)-(4.26), (4.29)-(4.33), we have from (4.9) that〈
Aε(ŵ, ẑ, θ̂), (ŵ, ẑ, θ̂)

〉
≥ min

{η0
2
,
ε

3
,

1

2ρσ1
,
µε

2ρ
,
κ0
2

}(
∥ŵ∥2L2(0,T ;V) + ∥ŵ∥6L6(0,T ;V) + ∥ẑ∥2L2(0,T ;VΣ)

+ ∥ẑ∥6L6(0,T ;VΣ) + ∥θ̂∥2
L2(0,T ;W 1,2

ΓD
)

)
−

12∑
i=1

Ki − (L1 +
c√
ε
)∥ŵ∥L2(0,T ;L2)

− L2∥ẑ∥L2(0,T ;VΣ) − L3∥ẑ∥L2(0,T ;L2) −
2c

ε
∥θ̂∥L2(0,T ;L2) ∀(ŵ, ẑ, θ̂) ∈ D(L ),

(4.35)

which implies (4.7).

(ii) Let us prove (ii) of Proposition 4.1 for Aε(ŵ, ẑ, θ̂). Let us estimate

I1 ≡ 1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

[
2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(u)

〉
+ ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩+ k1(ŵ + v0, u) + 2⟨η(θ)k(x)(ŵ + v0), u⟩Γ2

+ 2⟨η(θ)S( ˜̂w + ṽ0), ũ⟩Γ3
+ 2⟨α(x)(ŵ + v0), u⟩Γ5

+ ⟨η(θ)k(x)(ŵ + v0), u⟩Γ7

]
dt
∣∣∣.

Applying Hölder’s and Young’s inequalities and |a+ b|p ≤ 2p(|a|p + |b|p), p ∈ (1,∞), we have

I1 ≤ c
(
∥ŵ∥5L6(0,T ;V) +K

)
, (4.36)

where K depends on v0 and T .

We see that for (ŵ, ẑ, θ̂) ∈ D(L )

(ŵ, ẑ, θ̂)) ∈ C([0, T ];HV)× C([0, T ];L2(Ω))× C([0, T ];L2(Ω)),

∥ŵ∥C([0,T ];HV) ≤ c∥ŵ′∥1/2
L6/5(0,T ;V∗)

∥ŵ∥1/2L6(0,T ;V),

∥ẑ∥C([0,T ];L2) ≤ c∥ẑ′∥1/2
L6/5(0,T ;(VΣ)∗)

∥ẑ∥1/2L6(0,T ;VΣ),

∥θ̂∥C([0,T ];L2) ≤ c∥θ̂′∥1/2
L2
(
0,T ;(W 1,2

ΓD
)∗
)∥θ̂∥1/2

L2(0,T ;W 1,2
ΓD

)

(4.37)

(cf. [20, Theorem 1.37 and (1.27)].) and

∥w∥L3(Ω) ≤ c∥w∥1/2L2(Ω)∥w∥
1/2
V ∀w ∈ V,

∥z∥L3(Ω) ≤ c∥z∥1/2L2(Ω)∥z∥
1/2
VΣ

∀z ∈ VΣ,

∥θ∥L3(Ω) ≤ c∥θ∥1/2L2(Ω)∥θ∥
1/2

W 1,2
ΓD

(Ω)
∀θ ∈W 1,2

ΓD
(Ω).

(4.38)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 19

Taking into account (4.37), (4.38) and applying the inequality |a + b|p ≤ 2p(|a|p + |b|p), p ∈
(1,∞), we have

I2 ≡ 1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

[
ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩

]
dt
∣∣∣

≤ c

∥u∥L6(0,T ;V)

∫ T

0

∥ curl(ŵ + v0)∥L2∥(ŵ + v0)∥1/2L2 ∥(ŵ + v0)∥1/2V ∥u∥L6 dt

≤ ∥ŵ + v0∥1/2C(0,T ;L2)

( c

∥u∥L6(0,T ;V)
∥ŵ + v0∥3/2L9/5(0,T ;V)

∥u∥L6(0,T ;V)

)
≤ ∥ŵ + v0∥C([0,T ];HV) + c∥(ŵ + v0)∥3L9/5(0.T ;V)

≤ c∥ŵ′∥1/2
L6/5(0,T ;V∗)

∥ŵ∥1/2L6(0,T ;V) + ∥v0∥+ c∥ŵ + v0∥3L6(0,T ;V)

≤ 1

2
∥ŵ′∥L6/5(0,T ;V∗) + c

(
∥ŵ∥3L6(0,T ;V) +K

)
.

(4.39)

where K depends on v0 and T .
Let us estimate

I3 ≡ 1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

[
− µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0)

+ curl(ẑ(t) + H̄0)× H̊(t) + curl H̊(t)× (ẑ(t) + H̄0), u
〉

+ e−k1t
〈 α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
g, u

〉]
dt.

Taking into account (4.37), (4.38), we obtain

1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0), u

〉
dt
∣∣∣

≤ c

∥u∥L6(0,T ;V)

∫ T

0

∥ẑ(t) + H̄0∥H1∥ẑ(t) + H̄0∥1/2L2 ∥ẑ(t) + H̄0∥1/2H1 ∥u(t)∥V dt

≤ ∥ẑ + H̄0∥1/2C(0,T ;L2)

( c

∥u∥L6(0,T ;V)
∥ẑ(t) + H̄0∥3/2L9/5(0,T ;V)

∥u∥L6(0,T ;V)

)
≤ ∥ẑ + H̄0∥C(0,T ;L2) + c∥ẑ(t) + H̄0∥3L2(0,T ;V)

≤ ∥ẑ∥C(0,T ;L2) + ∥H̄0∥+ c
(
∥ẑ∥3L2(0,T ;V) + ∥ẑ∥2L2(0,T ;V) + ∥ẑ∥L2(0,T ;V) + 1

)
≤ 1

4
∥ẑ′∥L6/5(0,T ;(VΣ)∗) + c∥ẑ∥L6(0,T ;VΣ) + c

(
∥ẑ∥3L2(0,T ;VΣ) +K

)
.

Taking into account Assumption 3.1, we obtain

1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

µ

ρ

〈
ek1t curl(ẑ(t) + H̄0)× H̊(t), u

〉
dt
∣∣∣

≤ c

∥u∥L6(0,T ;V)

∫ T

0

∥ẑ(t) + H̄0∥H1∥H̊(t)∥L3∥u(t)∥V dt

≤ ∥H̊∥L∞(0,T ;L3)

( c

∥u∥L6(0,T ;V)
∥ẑ(t) + H̄0∥L6/5(0,T ;VΣ)∥u∥L6(0,T ;V)

)
≤ c

(
∥ẑ(t)∥L6(0,T ;VΣ) +K

)
,

where c depends on ∥H̊∥L∞(0,T ;L3), and K depends on H̄0. Taking into account (4.37), (4.38), we
obtain

1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

µ

ρ

〈
ek1t curl H̊(t)× (ẑ(t) + H̄0), u

〉
dt
∣∣∣



20 T. KIM EJDE-2025/119

≤ c

∥u∥L6(0,T ;V)

∫ T

0

∥H̊∥H1∥ẑ(t) + H̄0∥1/2L2 ∥ẑ(t) + H̄0∥1/2H1 ∥u(t)∥V dt

≤ c

∥u∥L6(0,T ;V)

∫ T

0

∥H̄0∥H1∥ẑ(t) + H̄0∥1/2L2 ∥ẑ(t) + H̄0∥1/2H1 ∥u(t)∥V dt

≤ ∥ẑ + H̄0∥1/2C(0,T ;L2)

(c∥H̊∥L∞(0,T ;H1)

∥u∥L6(0,T ;V)
∥ẑ(t) + H̄0∥1/2L1(0,T ;VΣ)∥u∥L2(0,T ;V)

)
≤ 1

4
∥ẑ′∥L6/5(0,T ;(VΣ)∗) + c

(
∥ẑ∥L6(0,T ;VΣ) + ∥ẑ∥L2(0,T ;VΣ) +K

)
,

where c depends on ∥H̊∥L∞(0,T ;H1) and K depends on H̄0. Also taking into account (4.22), we
obtain

1

∥u∥L6(0,T ;V)

∣∣∣ ∫ T

0

e−k1t
〈 α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
g, u

〉
dt ≤ c

2
√
ε
.

The four estimations above and Young’s inequality yield

I3 ≤ 1

4
∥ẑ′∥L6/5(0,T ;(VΣ)∗) + c

(
∥ẑ∥L6(0,T ;VΣ) + ∥ẑ∥3L2(0,T ;VΣ) +K +

1√
ε

)
. (4.40)

Let us estimate

I4 ≡ 1

∥C∥L6(0,T ;VΣ)

∣∣∣ ∫ T

0

[〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curlC

〉
+
〈
e−k1t

1

ρσ(θ)
curl H̊(t), curlC

〉
+ k1⟨(z̄ + H̄0), C⟩

+
〈µ
ρ
ek1t curlC × (ẑ(t) + H̄0), (ŵ(t) + v0)

〉
+
〈µ
ρ
curlC × H̊(t), (ŵ(t) + v0)

〉]
dt
∣∣∣.

First, we obtain

1

∥C∥L6(0,T ;VΣ)

∣∣∣ ∫ T

0

[〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curlC

〉
+
〈
e−k1t

1

ρσ(θ)
curl H̊(t), curlC

〉
+ k1⟨(z̄ + H̄0), C⟩

]
dt

≤ c
(
∥ẑ∥5L6(0,T ;VΣ) + ∥H̊∥L2(0,T ;VΣ) +K

)
,

where K depends on H̄0, ρ, σ1, µ, k1. In much the same way as (4.39), we obtain

1

∥C∥L6(0,T ;VΣ)

∣∣∣ ∫ T

0

〈µ
ρ
ek1t curlC × (ẑ(t) + H̄0), (ŵ(t) + v0)

〉
dt
∣∣∣

≤ c

∥C∥L6(0,T ;VΣ)

∫ T

0

∥C(t)∥VΣ
∥ẑ(t) + H̄0∥1/2L2 ∥ẑ(t) + H̄0∥1/2H1 ∥ŵ(t) + v0∥V dt

≤ ∥ẑ(t) + H̄0∥1/2C(0,T ;L2)

(
∥ẑ(t) + H̄0∥1/2L3/2(0,T ;H1)

∥ŵ + v0∥L2(0,T ;V)

)
≤ 1

2
∥ẑ′∥L6/5(0,T ;(VΣ)∗) + c

(
∥ẑ∥L6(0,T ;VΣ) + ∥ẑ(t)∥2L2(0,T ;VΣ) + ∥ŵ∥4L6(0,T ;V) +K

)
,

where K depends on H̄0, v0. Also, we obtain

1

∥C∥L6(0,T ;VΣ)

∣∣∣ ∫ T

0

〈µ
ρ
curlC × H̊(t), (ŵ(t) + v0)

〉
dt

≤ c

∥C∥L6(0,T ;VΣ)

∫ T

0

∥C(t)∥VΣ
∥H̊(t)∥L3∥ŵ(t) + v0∥V dt ≤ c(∥ŵ∥2L2(0,T ;V) +K),

where K depends on ∥H̊∥L∞(0,T ;L3) and v0.



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 21

The three estimations above yield

I4 ≤ 1

4
∥ẑ′∥L6/5(0,T ;(VΣ)∗) + c

(
∥ẑ∥5L6(0,T ;VΣ) + ∥ŵ∥4L6(0,T ;V) +K1

)
, (4.41)

where K1 depends on H̄0, v0 and ∥H̊∥2L∞(0,T ;L3). Let us estimate

I5 ≡ 1

∥ϕ∥L2(0,T ;W 1,2
ΓD

)

∣∣∣ ∫ T

0

[
⟨κ(θ)(∇θ̂ +∇θ0),∇ϕ⟩

+ ⟨β(x)(θ̂ + θ0), ϕ⟩ΓR
− ek1t⟨(ŵ + v0)(θ̂ + θ0),∇ϕ⟩

−
〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, ϕ

〉
−

〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, ϕ

〉]
dt
∣∣∣.

It is easy to obtain

1

∥ϕ∥L2(0,T ;W 1,2
ΓD

)

∣∣∣ ∫ T

0

[
⟨κ(θ)(∇θ̂ +∇θ0),∇ϕ⟩+ ⟨β(x)(θ̂ + θ0), ϕ⟩ΓR

]
dt
∣∣∣

≤ c
(
∥θ̂∥L2(0,T ;W 1,2

ΓD
) +K

)
,

where K depends on θ0 and κ1. In much the same way as (4.39), we obtain

c

∥ϕ∥L2(0,T ;W 1,2
ΓD

)

∫ T

0

∥ŵ + v0∥L6∥θ̂ + θ0∥1/2L2 ∥θ̂ + θ0∥1/2W 1,2
ΓD

∥∇ϕ∥L2 dt

≤ ∥θ̂ + θ0∥1/2C([0,T ];L2)

1

∥ϕ∥L2(0,T ;W 1,2
ΓD

)

∫ T

0

∥ŵ + v0∥L6∥θ̂ + θ0∥1/2W 1,2
ΓD

∥∇ϕ∥L2 dt

≤ c∥θ̂ + θ0∥C([0,T ];L2) + ∥ŵ + v0∥2L6(0,T ;V)∥θ̂ + θ0∥L3/2(0,T ;W 1,2
ΓD

)

≤ c∥θ̂′∥1/2
L2(0,T ;(W 1,2

ΓD
)∗)

∥θ̂∥1/2
L2(0,T ;W 1,2

ΓD
)
+ ∥θ0∥

+ ∥ŵ + v0∥2L6(0,T ;V)∥θ̂ + θ0∥L3/2(0,T ;W 1,2
ΓD

)

≤ 1

2
∥θ̂′∥L2(0,T ;(W 1,2

ΓD
)∗) + c∥θ̂∥L2(0,T ;W 1,2

ΓD
) + ∥θ0∥

+ ∥ŵ + v0∥4L6(0,T ;V) + ∥θ̂ + θ0∥2L3/2(0,T ;W 1,2
ΓD

)

≤ 1

2
∥θ̂′∥L2(0,T ;(W 1,2

ΓD
)∗) + c

(
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+ ∥ŵ∥4L6(0,T ;V) +K

)
,

where K depends on v0, θ0. It is easy to obtain

1

∥ϕ∥L2(0,T ;W 1,2
ΓD

)

∫ T

0

∣∣∣〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2

+
η(θ)

ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, ϕ

〉∣∣∣ dt ≤ c

ε
.

The three estimations above yield

I5 ≤ 1

2
∥θ̂′∥L2(0,T ;(W 1,2

ΓD
)∗) + c

(
∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+ ∥ŵ∥4L6(0,T ;V) +K +

c1
ε

)
. (4.42)



22 T. KIM EJDE-2025/119

Owing to (4.36), (4.39)-(4.42), from (4.9) we have

∥Aε(ŵ, ẑ, θ̂)∥V ∗

= sup
∥(u,C,ϕ)∥V =1

〈
Aε(ŵ, ẑ, θ̂), (u,C, ϕ)

〉
≤ c

(
∥ŵ∥5L6(0,T ;V) + ∥ẑ∥5L6(0,T ;VΣ) + ∥θ̂∥2

L2(0,T ;W 1,2
ΓD

)
+K +

1

ε
+

1√
ε

)
+

1

2
∥L (ŵ, ẑ, θ̂)∥V ∗ ,

(4.43)

which shows the property (4.3) for Aε(ŵ, ẑ, θ̂).

(iii) Let us prove property (i) of Proposition 4.1 for Aε(ŵ, ẑ, θ̂). Let {(ŵk, ẑk, θ̂k)} ⊂ D(L ) be
a sequence such that

(ŵk, ẑk, θ̂k)⇀ (ŵ, ẑ, θ̂) ∈ D(L ) in V ,

L (ŵk, ẑk, θ̂k)⇀ L (ŵ, ẑ, θ̂) in V ∗,

lim sup
k→∞

〈
Aε(ŵk, ẑk, θ̂k), (ŵk, ẑk, θ̂k)− (ŵ, ẑ, θ̂)

〉
≤ 0.

(4.44)

In what follows wk = ŵk + w0, zk = ẑk + H̄0 and θk = θ̂k + θ0. Then, by (4.9),

cmin{η0,
1

ρσ1
, κ0}

∫ T

0

(
∥ŵk(t)− ŵ(t)∥2V + ∥ẑk(t)− ẑ(t)∥2VΣ

+ ∥θ̂k(t)− θ̂(t)∥2
W 1,2

ΓD

)
dt

= cmin{η0,
1

ρσ1
, κ0}

∫ T

0

(
∥wk(t)− w(t)∥2V + ∥zk(t)− z(t)∥2VΣ

+ ∥θk(t)− θ(t)∥2
W 1,2

ΓD

)
dt

≤
∫ T

0

2
(
[η(θk) + e4k1tε∥wk∥4V]E(wk), E(wk − w)

)
dt−

∫ T

0

2
(
η(θk)E(w), E(wk − w)

)
dt

−
∫ T

0

2e4k1tε∥wk∥4V
〈
E(wk), E(wk − w)

〉
dt

+

∫ T

0

〈
[

1

ρσ(θk)
+
µε

ρ
e4k1t∥ curl zk∥4] curl zk, curl(zk − z)

〉
dt

+

∫ T

0

〈µε
ρ
e4k1t∥ curl zk∥4 curl zk, curl(z − zk)

〉
dt−

∫ T

0

〈 1

ρσ(θk)
curl z, curl(zk − z)

〉
dt

+

∫ T

0

(
κ(θk)∇θk,∇θk −∇θ

)
dt−

∫ T

0

(
κ(θk)∇θ,∇θk −∇θ

)
dt

≤
〈
Aε(ŵk, ẑk, θ̂k), (wk − w, zk − z, θk − θ)

〉
−
∫ T

0

2
(
η(θk)E(w), E(wk − w)

)
dt

+

∫ T

0

2e4k1tε
〈
∥wk∥4VE(wk), E(w − wk)

〉
dt−

∫ T

0

ek1t⟨curlwk × wk, wk − w⟩ dt

−
∫ T

0

[
⟨k1(wk, wk − w⟩+ 2⟨η(θk)k(x)wk, wk − w⟩Γ2 + 2⟨(η(θk)S(w̃k, w̃k − w̃⟩Γ3

+ 2⟨α(x)wk, wk − w⟩Γ5
+ ⟨η(θk)k(x)wk, wk − w⟩Γ7

]
dt

+

∫ T

0

[µ
ρ

〈
ek1t curl zk(t)× zk(t) + curl zk(t)× H̊(t) + curl H̊(t)× zk(t)

− e−k1tα0θk

1 + εθk
2 g, wk − w

〉]
dt

+

∫ T

0

〈µε
ρ
e4k1t∥ curl zk∥4 curl zk, curl(z − zk)

〉
dt



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 23

−
∫ T

0

〈 1

ρσ(θk)
curl zk, curl(zk − z)

〉
dt

−
∫ T

0

[〈 e−k1t

ρσ(θk)
curl H̊(t), curl(zk − z)

〉
+ k1⟨zk, zk − z⟩

]
dt

−
∫ T

0

[〈µ
ρ
ek1t curl(zk − z)× zk(t), wk(t)

〉
+

〈µ
ρ
curl(zk − z)× H̊(t), wk(t)

〉]
dt

−
∫ T

0

[〈
κ(θk)∇θ,∇θk −∇θ

〉
+

〈
β(x)θk, θk − θ

〉
ΓR

− ek1t⟨wkθk,∇(θk − θ)⟩

−
〈 1

ρChσ(θk)

| curl
(
H̄k(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄k(t) + H̊(t)

)
|2

+
η(θk)

ρCh

|E(wk)|2

e−2k1t + ε|E(wk)|2
, θk − θ

〉]
dt. (4.45)

Let us estimate the right-hand side of the above expression By (4.44) (see [20, Corollary 1.1]), we
see that ∫ T

0

2
(
η(θk)E(w), E(wk − w)

)
dt→ 0. (4.46)

We have

lim
k→∞

sup

∫ T

0

2
(
e4k1tε∥wk∥4VE(wk), E(w − wk)

)
dt

≤
∫ T

0

2e4k1tε lim
k→∞

sup ∥wk∥4V · lim
k→∞

sup
(
E(wk), E(w − wk)

)
dt

≤
∫ T

0

2e4k1tε lim
k→∞

sup ∥wk∥4V ·
(
∥E(w)∥2 − lim

k→∞
inf ∥E(wk)∥2

)
dt

≤
∫ T

0

2e4k1tε lim
k→∞

sup ∥wk∥4V ·
(
∥E(w)∥2 − ∥E(w)∥2

)
dt = 0.

(4.47)

By taking a subsequence and denoting with same subindex if necessary, from (4.44) we obtain

wk → w in L6(0, T ;Wα,2(Ω)) (9/10 < α < 1),

zk → z in L6(0, T ;Wα,2(Ω)) (9/10 < α < 1),

θk → θ in L2(0, T ;Wα,2(Ω)) (9/10 < α < 1).

(4.48)

Then, we have ∫ T

0

ek1t⟨curlwk × wk, wk − w⟩ dt

≤ c∥wk∥L4(Q)∥∇wk∥L2(Q)∥wk − w∥L4(Q) → 0 as k → ∞.

(4.49)

Also, by trace theorem and Remark 3.4 of [20] we have

−
∫ T

0

[
(k1(ŵk, wk − w) + 2(µ(θ̂k)k(x)ŵk, wk − w)Γ2

+ 2(µ(θ̂k)S ˜̂wk, w̃k − w̃)Γ3

+ 2(α(x)ŵk, wk − w)Γ5
+ (µ(θ̂k)k(x)ŵk, wk − w)Γ7

+ (β(x)θ̂k, θk − θ)ΓR

]
dt→ 0

(4.50)

as k → ∞. Let us prove that

I6 ≡
∫ T

0

µ

ρ

〈
ek1t curl zk(t)× zk(t) + curl zk(t)× H̊(t) + curl H̊(t)× zk(t)

+ e−k1t
α0θk

1 + εθk
2g, wk − w

〉
dt→ 0 as k → ∞.

(4.51)

Since {zk} is bounded in L6(0, T ;L6),

I6 ≤
[
∥ curl zk∥L2(Q)∥zk∥L6(Q) + ∥ curl zk∥L2(Q)∥H̊∥L6(Q)

+ ∥ curl H̊∥L2(Q)∥zk∥L6(Q) +
g√
ε

]
∥wk − w∥L3(Q) → 0,



24 T. KIM EJDE-2025/119

which implies (4.51). In much the same way as for (4.47), we obtain

lim
k→∞

sup

∫ T

0

〈µε
ρ
e4k1t∥ curl zk∥4 curl zk, curl(zk − z)

〉
dt ≤ 0. (4.52)

Let us estimate

I7 ≡ −
∫ T

0

[〈 1

ρσ(θk)
curl(zk(t)), curl(zk − z)

〉
+
〈
e−k1t

1

ρσ(θk)
curl H̊(t), curl(zk − z)

〉
+ k1⟨zk, zk − z⟩

]
dt.

(4.53)

By [20, Corollaries 1.1 and 1.2], we obtain

− lim inf
〈 1

ρσ(θk)
curl(zk(t)), curl(zk(t))

〉
≤ −

〈 1

ρσ(θ)
curl(z(t)), curl(z(t))

〉
,
〈 1

ρσ(θk)
curl(zk(t)), curl(z(t))

〉
→

〈 1

ρσ(θ)
curl(z(t)), curl(z(t))

〉
.

(4.54)

Taking into account (4.54), from (4.53) we have

lim inf
k→∞

I7 ≤ 0. (4.55)

It is easy to prove

−
∫ T

0

[〈
e−k1t

1

ρσ(θk)
curl H̊(t), curl(zk − z)

〉
+ k1⟨zk, zk − z⟩

]
dt→ 0 as k → ∞. (4.56)

Let us prove

I8 ≡ −
∫ T

0

[〈µ
ρ
ek1t curl(zk − z)× zk(t), wk(t)

〉
+
〈µ
ρ
curl(zk − z)× H̊(t), wk(t)

〉]
dt→ 0 (4.57)

as k → ∞. Taking into account (4.48), the fact that ziwj ∈ L3(Q) and curl(zk − z) ⇀ 0 in
L2(0, T ;L2), we have ∫ T

0

〈µ
ρ
ek1t curl(zk − z)× zk(t), wk(t)

〉
dt

≤ c∥ curl(zk − z)∥L2(Q)∥zk∥L6(Q)∥wk(t)− w(t)∥L3(Q)

+ c∥ curl(zk − z)∥L2(Q)∥(zk − z)∥L3(Q)∥w(t)∥L6(Q)

+ c

∫ T

0

∣∣∣〈 curl(zk − z)× z(t), w(t)
〉∣∣∣ dt ⇀ 0.

In a similar way we have ∫ T

0

∣∣∣〈µ
ρ
curl(zk − z)× H̊(t), wk(t)

〉∣∣∣ dt ⇀ 0.

The 2 formulas above give (4.57).
Let us prove

I9 ≡ −
∫ T

0

[〈
κ(θk)∇θ,∇θk −∇θ

〉
+

〈
β(x)θk, θk − θ

〉
ΓR

− ek1t⟨wkθk,∇(θk − θ)⟩

−
〈 1

ρChσ(θ)

| curl
(
H̄k(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄k(t) + H̊(t)

)
|2
, θk − θ

〉
−
〈η(θk)
ρCh

|E(wk)|2

e−2k1t + ε|E(wk)|2
, θk − θ

〉]
dt → 0.

(4.58)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 25

It is easy to verify that∫ T

0

[〈
κ(θk)∇θ,∇θk −∇θ

〉
+

〈
β(x)θk, θk − θ

〉
ΓR

]
dt→ 0 as k → ∞. (4.59)

(See [20, Corollary 1.1] and (4.48).) Let us verify that∫ T

0

ek1t⟨wkθk,∇(θk − θ)⟩ dt→ 0. (4.60)

Since {θk} is bounded in L2(0, T ;W 1,2(Ω)) ∩ L∞(0, T ;L2(Ω)), by [20, Theorem 1.33 and (1.22)]
it is bounded in L3(0, T ;W 2/3,2(Ω)), and {θk} is relatively compact in L3(Q). Then, by taking
a subsequence and denoting with same subindex if necessary, we have that θk → θ in L3(Q). By
taking into account (3.9), we have∫ T

0

∣∣∣ek1t⟨wkθk,∇(θk − θ)⟩
∣∣∣ dt ≤ c∥wk∥L6(Q)∥θk − θ∥L3(Q)∥∇θk∥L2(Q) → 0,

which yields (4.60). Taking into account θk → θ in L3(Q) for the subsequence, we can see easily
the convergence of the last two terms of I9 to zero. Thus, (4.59)-(4.60) implies (4.58).

Taking into account (4.46)-(4.47), (4.49)-(4.52), (4.55)-(4.58) and the last formula of (4.44),
from (4.45) we have

cmin{η0,
1

ρσ0
, κ0} lim

k→∞
sup

∫ T

0

(
∥ŵk(t)− ŵ(t)∥2V + ∥ẑk(t)− ẑ(t)∥2VΣ

+ ∥θ̂k(t)− θ̂(t)∥2
W 1,2

ΓD

)
dt

≤ lim
k→∞

sup
〈
Aε(ŵk, ẑk, θ̂k), (wk − w, zk − z, θk − θ)

〉
≤ 0,

which implies that

ŵk → ŵ in L2(0, T ;V),

ẑk → ẑ in L2(0, T ;VΣ),

θ̂k → θ̂ in L2(0, T ;W 1,2
ΓD

),

∇ŵk → ∇ŵ, ∇ẑk → ∇ẑ, ∇θ̂k → ∇θ̂ a.e. in Q.

(4.61)

When (u,C, ϕ) ∈ V , by definition of Aε, we have〈
Aε(ŵk, ẑk, θ̂k), (ŵk − u, ẑk − C, θ̂k − ϕ)

〉
=

∫ T

0

[
2
〈
[η(θk) + e4k1tε∥ŵk + v0∥4V]E(ŵk + v0), E(ŵk − u)

〉
+ ek1t⟨curl(ŵk + v0)× (ŵk + v0), ŵk − u⟩+ k1(ŵk + v0, ŵk − u)

+ 2⟨η(θk)k(x)(ŵk + v0), ŵk − u⟩Γ2
+ 2⟨η(θk)S( ˜̂wk + ṽ0), ˜̂wk − ũ⟩Γ3

+ 2⟨α(x)(ŵk + v0), ŵk − u⟩Γ5 + ⟨η(θk)k(x)(ŵk + v0), ŵk − u⟩Γ7

− µ

ρ

〈
ek1t curl(ẑk(t) + H̄0)× (ẑk(t) + H̄0) + curl(ẑk(t) + H̄0)× H̊(t)

+ curl H̊(t)× (ẑk(t) + H̄0), ŵk − u
〉
+ e−k1t

〈 α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
g, ŵk − u

〉
+
〈
[

1

ρσ(θk)
+
µε

ρ
e4k1t∥ curl(ẑk(t) + H̄0)∥4] curl(ẑk(t) + H̄0), curl(ẑk − C)

〉
+
〈
e−k1t

1

ρσ(θk)
curl H̊(t), curl(ẑk − C)

〉
+ k1⟨(z̄k + H̄0), (ẑk − C)⟩

+
〈µ
ρ
ek1t curl(ẑk − C)× (ẑk(t) + H̄0), (ŵk(t) + v0)

〉
+
〈µ
ρ
curl(ẑk − C)× H̊(t), (ŵk(t) + v0)

〉
+ ⟨κ(θk)(∇θ̂k +∇θ0),∇(θ̂k − ϕ)⟩

+ ⟨β(x)(θ̂k + θ0), (θ̂k − ϕ)⟩ΓR
− ek1t⟨(ŵk + v0)(θ̂k + θ0),∇(θ̂k − ϕ)⟩



26 T. KIM EJDE-2025/119

−
〈 1

ρChσ(θk)

| curl
(
H̄k(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄k(t) + H̊(t)

)
|2
, θ̂k − ϕ

〉
−
〈η(θk)
ρCh

|E(ŵk + v0)|2

e−2k1t + ε|E(ŵk + v0)|2
, θ̂k − ϕ

〉]
dt. (4.62)

By [20, Corollary 1.1] and (4.61),

lim
k→∞

∫ T

0

2
(
η(θk)E(ŵk + v0), E(ŵk − u) dt =

∫ T

0

2
(
η(θ)E(ŵ + v0), E(ŵ − u) dt. (4.63)

The sequence {e4k1t∥ŵk+v0∥4Vεij(ŵk+v0)} is bounded in L6/5(0, T ;L2(Ω)) and εij(u) ∈ L6(0, T ;L2(Ω)),
and by taking a subsequence and denoting with same subindex, we know that

lim
k→∞

∫ T

0

2e4k1t
〈
∥ŵk + v0∥4VE(ŵk + v0), E(u)

〉
dt

→
∫ T

0

2e4k1t
〈
∥ŵ + v0∥4VE(ŵ + v0), E(u)

〉
dt.

(4.64)

Since wk ⇀ w in L6(0, T ;V) and (4.61), there exist subsequences, respectively, (which is expressed
as before with sub-indexes k) such that wk(t) ⇀ w(t) and ⟨E(ŵk + v0)(t), E(ŵk)(t)⟩ → ⟨E(ŵ +
v0)(t), E(ŵ)(t)⟩ for a.e. t ∈ [0, T ] (cf. [22, Sec. 2, Ch. VII]). Thus

lim inf
k→∞

∫ T

0

∥ŵk + v0∥4V
(
E(ŵk + v0)(t), E(ŵk)(t)

)
dt

≥
∫ T

0

∥ŵ + v0∥4V
(
E(ŵ + v0), E(ŵ)

)
dt.

(4.65)

(See [5, Exercises 2, (c) in pp. 173].)
By (4.63)-(4.65), we have

lim inf
k→∞

∫ T

0

2
〈
[η(θk) + e4k1tε∥ŵk + v0∥4VE(ŵk + v0), E(ŵk − u)

〉
dt

≥
∫ T

0

2
〈
[η(θ) + e4k1tε∥ŵ + v0∥4V]E(ŵ + v0), E(ŵ − u)

〉
dt.

(4.66)

Since ek1t⟨curl(ŵk + v0) × (ŵk + v0), ŵk − u⟩ = ek1t⟨curl(ŵk + v0) × (ŵk + v0),−v0 − u⟩, taking
into account the fact that curl(ŵk + v0) → curl(ŵ + v0) in L2(Q) ((4.61)), ŵk + v0 → ŵ + v0 in
L3(Q) ((4.48)) and −v0 − u ∈ L6(Q), we can see that∫ T

0

ek1t⟨curl(ŵk + v0)× (ŵk + v0), ŵk − u⟩ dt

→
∫ T

0

ek1t⟨curl(ŵ + v0)× (ŵ + v0), ŵ − u⟩ dt.
(4.67)

Let us consider ∫ T

0

−µ
ρ

〈
ek1t curl(ẑk(t) + H̄0)× (ẑk(t) + H̄0), ŵk − u

〉
dt

=

∫ T

0

−µ
ρ

〈
ek1t curl(ẑk(t) + H̄0)× (ẑk(t) + H̄0), ŵk − ŵ

〉
dt

+

∫ T

0

−µ
ρ

〈
ek1t curl(ẑk(t) + H̄0)× (ẑk(t) + H̄0), ŵ − u

〉
dt.

Since curl(ẑk+H̄0) and (ẑk(t)+H̄0) are bounded in L2(Q) and L6(Q), respectively, and ŵk → ŵ
in L3(Q) ((4.48)), the first term of the right hand side of the above equality converges to zero.



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 27

Also, since curl(ẑk+H̄0) → curl(ẑ+H̄0) in L2(Q) ((4.61)), (ẑk+H̄0) → (ẑ+H̄0) in L3(Q) ((4.48))
and ŵ − u ∈ L6(Q), we have for a subsequence (which is denoted with the same subindex k)∫ T

0

−µ
ρ

〈
ek1t curl(ẑk(t) + H̄0)× (ẑk(t) + H̄0), ŵk − u

〉
dt

→
∫ T

0

−µ
ρ

〈
ek1t curl(ẑ(t) + H̄0)× (ẑ(t) + H̄0), ŵ − u

〉
dt.

(4.68)

In much the same way as above, we obtain∫ T

0

µ

ρ

〈
curl(ẑk(t) + H̄0)× H̊(t) + curl H̊(t)× (ẑk(t) + H̄0), ŵk − u

〉
→

∫ T

0

µ

ρ

〈
curl(ẑ(t) + H̄0)× H̊(t) + curl H̊(t)× (ẑ(t) + H̄0), ŵ − u

〉
.

(4.69)

Since
α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
→ α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
a.e. in Q

and
α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
is bounded in L2(0, T ;L2),

we have
α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
⇀

α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
in L2(0, T ;L2)

(cf. [23, Lemma 1.3, Ch. 1]). Owing to (4.61) and the above, we have∫ T

0

e−k1t⟨ α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
g, wk − u⟩ dt

=

∫ T

0

e−k1t⟨ α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
g, wk − w⟩ dt+

∫ T

0

e−k1t⟨ α0(θ̂k + θ0)

1 + ε|θ̂k + θ0|2
g, w − u⟩ dt

→
∫ T

0

e−k1t⟨ α0(θ̂ + θ0)

1 + ε|θ̂ + θ0|2
g, w − u⟩ dt.

(4.70)

In much the same way as (4.66), we obtain

lim inf
k→∞

∫ T

0

〈
[

1

ρσ(θk)
+
µε

ρ
e4k1t∥ curl(ẑk(t) + H̄0)∥4] curl(ẑk(t) + H̄0), curl(ẑk − C)

〉
dt

≥
∫ T

0

〈
[

1

ρσ(θ)
+
µε

ρ
e4k1t∥ curl(ẑ(t) + H̄0)∥4] curl(ẑ(t) + H̄0), curl(ẑ − C)

〉
dt.

(4.71)

This is rewritten as∫ T

0

〈µ
ρ
ek1t curl(ẑk − C)× (ẑk(t) + H̄0), (ŵk(t) + v0)

〉
dt

=

∫ T

0

〈µ
ρ
ek1t curl(ẑk − ẑ)× (ẑk(t) + H̄0), (ŵk(t) + v0)

〉
dt

+

∫ T

0

〈µ
ρ
ek1t curl(ẑ − C)× (ẑk(t) + H̄0), (ŵk(t) + v0)

〉
dt.

By the same argument as (4.68), for a subsequence (which is denoted with the same subindex k),
we have ∫ T

0

〈µ
ρ
ek1t curl(ẑk − C)× (ẑk(t) + H̄0), (ŵk(t) + v0)

〉
dt

→
∫ T

0

〈µ
ρ
ek1t curl(ẑ − C)× (ẑ(t) + H̄0), (ŵ(t) + v0)

〉
dt.

(4.72)



28 T. KIM EJDE-2025/119

Similarly, taking a subsequence, we have∫ T

0

〈µ
ρ
curl(ẑk − C)× H̊(t), (ŵk(t) + v0)

〉
dt→

∫ T

0

〈µ
ρ
curl(ẑ − C)× H̊(t), (ŵ(t) + v0)

〉
dt,∫ T

0

ek1t⟨(ŵk + v0)(θ̂k + θ0),∇(θ̂k − ϕ)⟩ dt→
∫ T

0

ek1t⟨(ŵ + v0)(θ̂ + θ0),∇(θ̂ − ϕ)⟩ dt.

(4.73)
By [20, Corollary 1.1] and (4.61), we obtain∫ T

0

⟨κ(θk)∇θk,∇(θ̂k − ϕ)⟩ dt =
∫ T

0

⟨κ(θk)∇θk,∇(θ̂k − θ̂)⟩ dt−
∫ T

0

⟨κ(θk)∇θk,∇(θ̂ − ϕ)⟩ dt

→
∫ T

0

(κ(θ)∇θ,∇(θ̂ − ϕ)) dt.

(4.74)
In much the same arguments as for (4.70) and (4.71), we have∫ T

0

〈 1

ρChσ(θk)

| curl
(
H̄k(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄k(t) + H̊(t)

)
|2
, θ̂k − ϕ

〉
dt

→
∫ T

0

〈 1

ρChσ(θ)

| curl
(
H̄(t) + H̊(t)

)
|2

1 + ε| curl
(
H̄(t) + H̊(t)

)
|2
, θ̂ − ϕ

〉
dt,∫ T

0

〈η(θk)
ρCh

|E(ŵk + v0)|2

e−2k1t + ε|E(ŵk + v0)|2
, θ̂k − ϕ

〉
dt

→
∫ T

0

〈η(θ)
ρCh

|E(ŵ + v0)|2

e−2k1t + ε|E(ŵ + v0)|2
, θ̂ − ϕ

〉
dt.

(4.75)

It is easy to prove convergence of other terms in the right hand side of (4.62). Thus, by (4.62)
and (4.66)-(4.75), we have the existence of a subsequence {(ŵk, ẑk, η̂k)} such that

lim inf
k→∞

〈
Aε(ŵk, ẑk, θ̂k), (ŵk − u, ẑk − C, θ̂k − ϕ)

〉
≥

〈
Aε(ŵ, ẑ, θ̂), (ŵ − u, ẑ − C, θ̂ − ϕ)

〉
,

by which property (i) of Proposition 4.1 for Aε(w, θ) is proved.
Therefore, by Proposition 4.1 there exists a solution to Problem VEA’, which gives us the

conclusion of Theorem 4.2. □

4.2. Estimates of solutions to the approximate problem. Let 0 < ε < 1. For v0 ∈ HV,
H̄0 ∈ HΣ, θ0 ∈ L1(Ω), there exist v0ε ∈ V, H̄0ε ∈ VΣ, θ0ε ∈W 1,2

ΓD
, such that

∥v0 − v0ε∥HV
≤ ε, ∥H̄0 − H̄0ε∥HΣ

≤ ε, ∥θ0 − θ0ε∥L1(Ω) ≤ ε. (4.76)

Let (vε(t), H̄ε(t), θε(t)) be a solution to Problem VEA with the initial function (v0ε, H̄0ε, θ0ε) by
Theorem 4.2.

We define Aε(θ) : V → V∗ by

⟨Aε(θ)v, w⟩ = a0(θ; v, w) + 2ε∥v∥4V
〈
E(v), E(w)

〉
+ ⟨curl v × v, w⟩ ∀v, w ∈ V,

where a0(θ; v, w) is as in (3.10). Them there exists c∗ ≥ 0 such that

|ν(k(x)z, z)Γ2
+ 2ν(Sz̃, z̃)Γ3

+ (α(x)z, z)Γ5
+ ν(k(x)z, z)Γ7

| ≤ η0∥z∥2V + c∗∥z∥2 ∀z ∈ V (4.77)

(cf. [15, Theorem 1.5.1.10]).

Remark 4.4. Note that c∗ depends on shape of Γi, i = 2, 3, 7, and the norm of matrix α on Γ5,
and so for the fixed domain Ω it depends only on |α|.

By (2.5), (3.10), (4.77), we have

⟨Aε(θε)z, z⟩ ≥ η0∥z∥2V + 2ε∥z∥6V − c∗∥z∥2, ∀z ∈ V,

| ⟨Aε(θε)z, w⟩ | ≤ c2
(
∥z∥V + ε∥z∥5V

)
∥w∥V ∀z, w ∈ V.

(4.78)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 29

Putting u = vε in the first equation of (4.1) and taking (4.78), we have

1

2

d

dt
∥vε(t)∥2 + η0∥vε(t)∥2V + 2ε∥vε(t)∥6V − c∗∥vε(t)∥2 −

[
⟨µ
ρ
curl H̄ε(t)× H̄ε(t), vε(t)⟩

+ ⟨µ
ρ
curl H̄ε(t)× H̊(t), vε(t)⟩+ ⟨µ

ρ
curl H̊(t)× H̄ε(t), vε(t)⟩

]
+ ⟨ α0θε

1 + εθ2ε
g, vε(t)⟩

≤ ⟨f1(t), vε(t)⟩.

(4.79)

Putting C = H̄ε and multiplying µ
ρ , from the second equation of (4.1), we obtain

µ

2ρ

d

dt
∥H̄ε(t)∥2 +

〈
[

1

ρσ(θε)
+ ε∥ curl H̄ε(t)∥4] curl H̄ε(t), curl H̄ε(t)

〉
+

1

µσ(θε)

〈
curl H̊(t), curl H̄ε

〉
+

〈µ
ρ
curl H̄ε(t)× H̄ε(t), vε(t)

〉
+
〈µ
ρ
curl H̄ε(t)× H̊(t), vε(t)

〉
= ⟨µ

ρ
r1(t), H̄ε(t)⟩.

(4.80)

Adding (4.79) and (4.80), and integrating on [0, t] yield

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 + 2

∫ t

0

(
η0∥vε∥2V + 2ε∥vε∥6V

)
ds

+ 2

∫ t

0

( 1

ρσ(θε)
∥H̄ε(t)∥2VΣ

+ ε∥H̄ε(t)∥6VΣ

)
ds− 2

∫ t

0

⟨µ
ρ
curl H̊(t)× H̄ε(t), vε(t)⟩ ds

+ 2

∫ t

0

⟨ α0θε
1 + εθ2ε

g, vε(t)⟩ ds+ 2

∫ t

0

1

µσ(θε)

〈
curl H̊(t), curl H̄ε(t)

〉
ds

≤ ∥v0ε∥2 +
µ

ρ
∥H̄0ε∥2 + 2

∫ t

0

c∗∥vε(s)∥2 ds+ 2

∫ t

0

(
⟨f1(s), vε(t)⟩+ ⟨µ

ρ
r1(s), H̄ε(t)⟩

)
ds.

(4.81)

Taking into account the first one of (4.38), by Hölder’s inequality, we have

2
∣∣∣µ
ρ

〈
curl H̊ × H̄ε, vε

〉∣∣∣ ≤ c∥H̄ε∥VΣ
∥ curl H̊∥∥vε∥1/2V ∥vε∥1/2

≤ 1

4ρσ1
∥H̄ε∥2VΣ

+
η0
4
∥vε∥2V + cρση1∥ curl H̊∥4∥vε∥2,

2|⟨ α0θε
1 + εθ2ε

g, vε(t)⟩| ≤ c|α0g|∥θε∥
L

6
5 (Ω)

∥vε∥V,

2
∣∣∣ 1

ρσ(θε)

〈
curl H̊, curl H̄ε

〉∣∣∣ ≤ c∥ curl H̊∥∥H̄ε∥VΣ

≤ 1

4ρσ1
∥H̄ε∥2VΣ

+ cρσ1∥ curl H̊∥2,

2|⟨f1(t), vε⟩| ≤
η0
4
∥vε∥2V + cη∥f1(t)∥2V∗ (see (3.13)),

2⟨µ
ρ
r1(t), H̄ε(t)⟩ ≤

1

4ρσ1
∥H̄ε∥2VΣ

+ cρσ2∥r1(t)∥2V∗
Σ
.

(4.82)



30 T. KIM EJDE-2025/119

Taking into account (4.78) and (4.82), from (4.81), we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

∫ t

0

(3
2
η0∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 5

4ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds

≤ ∥v0ε∥2 +
µ

ρ
∥H̄0ε∥2 +

∫ t

0

(
2c∗ + cρση1∥ curl H̊∥4

)
∥vε∥2 ds

+ c|α0g|
∫ t

0

∥θε∥
L

6
5 (Ω)

∥vε∥V ds+ cρσ1

∫ t

0

∥ curl H̊∥2 ds

+ cη

∫ t

0

∥f1(t)∥2V∗ ds+ cρσ2

∫ t

0

∥r1(t)∥2V∗
Σ
ds.

(4.83)

For a fixed 0 < δ < 1, we define

Φ(ξ) :=
(
1− 1

(1 + |ξ|)δ
)
sign ξ, ξ ∈ R,

Ψ(ξ) :=

{∫ ξ

0
Φ(τ) dτ for ξ ≥ 0

−
∫ 0

ξ
Φ(τ) dτ for ξ < 0.

(cf. [24]). Then

Φ′(ξ) =
δ

(1 + |ξ|)1+δ
∀ξ ∈ R,

|Φ(ξ)| ≤ 1, 0 ≤ Φ′(ξ) ≤ δ ∀ξ ∈ R,
Ψ′(ξ) = Φ(ξ),

Ψ(ξ) = |ξ|+ 1

1− δ

(
1− (1 + |ξ|)1−δ

)
∀ξ ∈ R,

|ξ|
2

− 2(1−δ)/δ

1− δ
≤ Ψ(ξ) ≤ |ξ| ∀ξ ∈ R.

(4.84)

Since θε ∈ L2(0, T ;W 1,2
ΓD

(Ω)) ∩ C([0, T ];L2(Ω)), we see that

Φ(θε(t)) ∈W 1,2
ΓD

(Ω), Ψ(θε(t)) ∈W 1,2
ΓD

(Ω) for a.a. t ∈ [0, T ],

⟨vεθε,∇Φ(θε(t))⟩ = −⟨(vε · ∇θε),Φ(θε(t))⟩ = −⟨vε,∇Ψ(θε(t))⟩ = ⟨div vε,Ψ(θε(t))⟩ = 0,
(4.85)

where v · n|ΓR
= 0(cf. (2.3)) and Φ(θε) = 0 on ΓD were used. Also∫ t

0

〈∂θε
∂t

,Φ(θε)
〉
ds =

∫
Ω

Ψ(θε(t)) dx−
∫
Ω

Ψ(θ0ε) dx ∀t ∈ [0, T ],

(κ(θε)∇θε,∇Φ(θε)) =

∫
Ω

κ(θε)|∇θε|2Φ′(θε) dx,

(β(x)θε,Φ(θε))ΓR
≥ 0.

(4.86)

Putting φ(t) = Φ(θε(t)) in the third equation of (4.1) and using (4.85), (4.86), we have∫
Ω

Ψ(θε(t)) dx+

∫ t

0

∫
Ω

κ(θε)|∇θε|2Φ′(θε) dx+

∫ t

0

(β(x)θε,Φ(θε))ΓR
ds

=

∫
Ω

Ψ(θ0ε) dx+

∫ t

0

〈 1

ρChσ(θε)

| curl(H̄ε + H̊)|2

1 + ε| curl(H̄ε + H̊)|2
,Φ(θε)

〉
dt

+

∫ t

0

〈η(θε)
ρCh

|E(vε)|2

1 + ε|E(vε)|2
,Φ(θε)

〉
ds+

∫ t

0

〈
g1(t),Φ(θε)

〉
ds.

(4.87)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 31

By (4.84), (4.86), from (4.87), we have

1

2
∥θε(t)∥L1(Ω) −

2(1−δ)/δ

1− δ
measΩ + δ

∫ t

0

∫
Ω

κ0
|∇θε|2

(1 + |θε|)1+δ
dxds

≤ ∥θ0ε∥L1(Ω) +
2

ρChσ0

∫ t

0

∥H̄∥2VΣ
dt+

2

ρChσ0

∫ t

0

∥H̊∥2H1 dt

+
η1
ρCh

∫ t

0

∥vε∥2V ds+

∫ t

0

∥g1(t)∥(W 1,2
ΓD

)∗ ds.

(4.88)

Putting

β = min
{ρCh

η1
· η0
2
,

ρChσ0
2

· 1

4ρσ1
, 1

}
(4.89)

and summing (4.83) and (4.88) multiplied by β, we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

β

2
∥θε(t)∥L1(Ω) +

∫ t

0

(
η0∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 1

ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds+ δβ

∫ t

0

∫
Ω

κ0
|∇θε|2

(1 + |θε|)1+δ
dxds

≤ Λδε +

∫ t

0

(
2c∗ + cρση1∥ curl H̊∥4

)
∥vε∥2 ds+ c|α0g|

∫ t

0

∥θε∥
L

6
5 (Ω)

∥vε∥V ds,

(4.90)

where

Λδε = c
(2(1−δ)/δ

1− δ
measΩ + ∥v0ε∥2 + ∥H̄0ε∥2 + ∥θ0ε∥L1(Ω) +

∫ T

0

∥H̊∥2H1 dt

+

∫ T

0

∥ curl H̊∥2 dt+
∫ T

0

∥f1(t)∥2V∗ dt+

∫ T

0

∥r1(t)∥2V∗
Σ
dt+

∫ T

0

∥g1(t)∥(W 1,2
ΓD

)∗ dt
)
.

Taking into account (4.76) and putting

Λδ = c
(2(1−δ)/δ

1− δ
measΩ + 1 + ∥v0∥2 + ∥H̄0∥2 + ∥θ0∥L1(Ω) +

∫ T

0

∥H̊∥2H1 dt

+

∫ T

0

∥ curl H̊∥2 dt+
∫ T

0

∥f1(t)∥2V∗ dt+

∫ T

0

∥r1(t)∥2V∗
Σ
dt+

∫ T

0

∥g1(t)∥(W 1,2
ΓD

)∗ dt
)
,

(4.91)

we obtain Λδε < Λδ. By Hölder’s inequality and Young’s inequality,

c|α0g|
∫ t

0

∥θε∥
L

6
5 (Ω)

∥vε∥V ds ≤ c|α0g|
(∫ t

0

∥θε∥2L6/5 ds
)1/2(∫ t

0

∥vε∥2V ds
)1/2

≤ |α0|cη0

∫ t

0

∥θε∥2L6/5 ds+
η0
2

∫ t

0

∥vε∥2V ds.

(4.92)

By complex interpolation with exponent 13
18 between L1 and L5/2(cf. [20, Theorem 1.12]),

∥θε∥L6/5(Ω) ≤ ∥θε∥13/18L1(Ω)∥θε∥
5/18

L5/2(Ω)
. (4.93)

Taking into account (4.91)-(4.93), from (4.90), we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

β

2
∥θε(t)∥L1(Ω) +

∫ t

0

(η0
2
∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 1

ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds+ δβκ0

∫ t

0

∫
Ω

|∇θε|2

(1 + |θε|)1+δ
dxds

≤ Λδ + c∗
∫ t

0

∥vε∥2 ds+ |α0|cη0

∫ t

0

∥θε∥13/9L1 ∥θε∥5/9L5/2 ds,

(4.94)

where c∗ = 2c∗ + cρση1∥H̊∥4L∞(0,T ;H1(Ω)). For 0 < ζ < 1 we define

ηε :=
|θε|

(1 + |θε|)(1+ζ)/2
a.e. in Q.



32 T. KIM EJDE-2025/119

Then, by elementary calculation we see that

(1 + |θε|)(1−ζ)/2 ≤ 1 + ηε, |∇ηε| ≤
|∇θε|

(1 + |θε|)(1+ζ)/2
a.e. in Q. (4.95)

In fact, adding |θε| and dividing by (1 + |θε|)(1+ζ)/2, from

1 ≤ (1 + |θε|)(1+ζ)/2

we obtain the first inequality of (4.95). From

∇ηε =
signθε(1 + |θε|)(1+ζ)/2 − |θε| · 1+ζ

2 (1 + |θε|)(−1+ζ)/2signθε

(1 + |θε|)(1+ζ)
∇θε,

we obtain

|∇ηε| ≤ |∇θε|
1− |θε| · 1+ζ

2 (1 + |θε|)−1

(1 + |θε|)(1+ζ)/2
≤ |∇θε|

(1 + |θε|)(1+ζ)/2
,

which is the second part of (4.95).

Taking ζ = 1
6 , from the first inequality of (4.95) we obtain |θε|

5
2 ·

1
6 ≤ 1 + ηε, which yields

∥θε∥5/2L5/2(Ω)
≤ 26

(
measΩ + ∥ηε∥6L6(Ω)

)
.

Taking 3
√

in two sides of the above and using |a|+ |b| ≤
(
|a| 13 + |b| 13

)3
, we obtain

∥θε∥5/6L5/2(Ω)
≤ 22

(
(measΩ)

1
3 + ∥ηε∥2L6(Ω)

)
,

which yields

∥θε(s)∥5/6L5/2(Ω)
≤ c6(1 + ∥ηε(s)∥2L6(Ω).

Thus, from (4.94), we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

β

2
∥θε(t)∥L1(Ω) +

∫ t

0

(η0
2
∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 1

ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds+ δβκ0

∫ t

0

∫
Ω

|∇θε|2

(1 + |θε|)1+δ
dxds

≤ Λδ + c∗
∫ t

0

∥vε(s)∥2 ds+ |α0|cη0

∫ t

0

∥θε(s)∥13/9L1(Ω)(1 + ∥η∥2L6(Ω))
2/3 ds.

(4.96)

By Gronwoll’s inequality, from (4.96), we have

∥vε(t)∥2 ≤
(
Λδ + |α0|cη0

∫ t

0

∥θε(s)∥13/9L1(Ω)

(
1 + ∥η∥2L6(Ω)

)2/3
ds
)
ec

∗t,

which yields

c∗
∫ t

0

∥vε(s)∥2 ds ≤ c∗ec
∗TT

(
Λδ + |α0|cη0

∫ t

0

∥θε(s)∥13/9L1(Ω)

(
1 + ∥η∥2L6(Ω)

)2/3
ds
)
.

Taking into account the above, from (4.96), we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

β

2
∥θε(t)∥L1(Ω) +

∫ t

0

(η0
2
∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 1

ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds+ δβκ0

∫ t

0

∫
Ω

|∇θε|2

(1 + |θε|)1+δ
dxds

≤ Λδ(1 + c∗ec
∗TT ) + (1 + c∗ec

∗TT )|α0|cη0

∫ t

0

∥θε(s)∥13/9L1(Ω)(1 + ∥η(s)∥2L6(Ω))
2/3 ds.

(4.97)

We put

ϕ(t) := ∥θε∥C([0,t];L1(Ω)) + ∥η∥2L2(0,t;L6(Ω)).

Taking into account the second inequality of (4.95) and

∥ηε(s)∥L6(Ω) ≤ c̄∥∇ηε(s)∥L2(Ω),



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 33

from (4.97), we have

ϕ(t) ≤ ∥θε∥C([0,t];L1(Ω)) + c̄

∫ t

0

∫
Ω

|∇θε|2

(1 + |θε|)1+δ
dxds

≤ c3Λδ(1 + c∗ec
∗TT ) + (1 + c∗ec

∗TT )|α0|c4
∫ t

0

∥θε(s)∥13/9L1(Ω)(1 + ∥η(s)∥2L6(Ω))
2
3 ds,

(4.98)

where c3 = max{ 2
β ,

c̄
δβκ0

} and c4 = c3cη0
. Let us estimate the second term of the right hand side

of (4.98). Using Hölder’s inequality with exponents 3/2, 3, we have

(1 + c∗ec
∗TT )|α0|c4

∫ t

0

∥θε(s)∥13/9L1(Ω)(1 + ∥η(s)∥2L6(Ω))
2/3 ds

≤ (1 + c∗ec
∗TT )|α0|c4∥θε(s)∥13/9C([0,t];L1(Ω))

(∫ t

0

(1 + ∥η(s)∥2L6(Ω)) ds
)2/3

t1/3

≤ (1 + c∗ec
∗TT )|α0|c4T 1/3∥θε(s)∥13/9C([0,t];L1(Ω))

(
t+ ∥η(s)∥2L2(0,t;L6(Ω))

)2/3
≤ (1 + c∗ec

∗TT )|α0|c4T 1/3
(
t+ ∥ϕ(t)∥

)19/9
.

(4.99)

Therefore, from (4.98), we have

t+ ϕ(t) ≤ K0 + (1 + c∗ec
∗TT )|α0|c4T 1/3

(
t+ ϕ(t)

)19/9
, (4.100)

where

K0 = c3Λδ(1 + c∗ec
∗TT ) + T.

Now we use the following lema.

Lemma 4.5 ([26, Lemma 1 in Appendix]). Let Ψ : [0, T ] → [0,+∞) be a continuous non-
decreasing function such that

Ψ(0) ≤ K0,

Ψ(t) ≤ K0 + βΨ(t)γ ∀t ∈ [0, T ],

where K0 = const > 0, γ = const > 1 and β = 1
3 (2K0)

1−γ . Then

Ψ(t) ≤ 2K0 ∀t ∈ [0, T ].

By (4.98) we see that ϕ(0) ≤ K0. According to assumption of the main theorem that the
buoyancy effect (|α0|) is small enough than the data of problem and K0 depends to the data of
problem, we can assume

(1 + c∗ec
∗TT )α0c4T

1/3 ≤ 1

3
(2K0)

(1−19/9). (4.101)

Then, putting Ψ(t) = t+ ϕ(t) and applying Lemma 4.5 to (4.100), we have

t+ ϕ(t) ≤ 2K0 ∀t ∈ [0, T ]. (4.102)

Taking into account (4.99), (4.101) and (4.102) in the second term of the right hand side of (4.97),
we have

∥vε(t)∥2 +
µ

ρ
∥H̄ε(t)∥2 +

β

2
∥θε(t)∥L1(Ω) +

∫ t

0

(η0
2
∥vε∥2V + 4ε∥vε∥6V

)
ds

+

∫ t

0

( 1

ρσ1
∥H̄ε(t)∥2VΣ

+ 2ε∥H̄ε(t)∥6VΣ

)
ds+ δβκ0

∫ t

0

∫
Ω

|∇θε|2

(1 + |θε|)1+δ
dxds

≤ C(Λδ, c
∗),

(4.103)

where C(Λδ, c
∗) is independent from ε and depends on δ ∈ (0, 1) and c∗.



34 T. KIM EJDE-2025/119

We will obtain an estimate for θε in Lr(0, T ;W 1,r) (1 < r < 5/4). To this end, we need to
obtain an estimate on ∇θε. By (4.103), Hölder’s inequality with exponents 2

r ,
2

2−r , inequalities

|a+ b|p ≤ 2p(|a|p + |b|p), |a|+ |b| ≤ (|a|
1
p + |b|

1
p )p, p ∈ (1,∞), we have∫

Q

|∇θε|r dx dt ≤
(∫

Q

|∇θε|2

(1 + |θε|)1+δ
dx dt

)r/2(∫
Q

(
1 + |θε|)r(1+δ)/(2−r) dx dt

)(2−r)/2

≤ c2r(1+δ)/2C(Λδ, c
∗)r/2

[
(measQ)(2−r)/2 +

(∫
Q

|θε|r(1+δ)/(2−r) dx dt
)(2−r)/2]

,

(4.104)
where C(Λδ, c

∗) is the one in (4.103).
To use the property W 1,r(Ω) ⊂ Lq(Ω), let us take q such that 1

r − 1
3 = 1

q , i.e. q =
3r
3−r . Let us

take 1 > δ0 > 0 such that

1 < r(1 + δ0)/(2− r) < q,

which holds if

0 < δ0 < (3− 2r)/(3− r). (4.105)

Set s = r(1 + δ0)/(2− r) and take λ such that

1

s
=
λ

1
+

1− λ

q
.

Then, λ = q−s
s(q−1) , 0 < λ < 1 and

∥θε∥Ls(Ω) ≤ c∥θε∥λL1(Ω)∥θε∥
1−λ
Lq(Ω) ≤ c∥θε∥λL1(Ω)∥∇θε∥

1−λ
Lr(Ω) ∀θε ∈W 1.r

ΓD
(Ω) (4.106)

(see [20, Theorem 1.12]). Therefore, by (4.103) and (4.106) we have∫
Q

|θε|r(1+δ0)/(2−r) dx dt ≤ cC(Λδ0 , c
∗)λs

∫ T

0

(∫
Ω

|∇θε|r dx
)(1−λ)s/r

dt.

We will fix r and δ0 so that

1 < r <
5

4
, 0 < δ0 <

5− 4r

3
,

which ensures that (4.105). Taking into account λ = q−s
s(q−1) , q =

3r
3−r and s = r(1+δ0)

2−r , we have

(1− λ)s/r =
1

r

q(s− 1)

q − 1
=

3(2r − 2 + rδ0)

(4r − 3)(2− r)
<

3
(
2r − 2 + r 5−4r

3

)
(4r − 3)(2− r)

= 1.

Then, by Hölder’s inequality, we have∫
Q

|θε|r(1+δ0)/(2−r) dx dt ≤ cC(Λδ0 , c
∗)λs

(∫ T

0

∫
Ω

|∇θε|r dx dt
)(1−λ)s/r

,

and (∫
Q

|θε|r(1+δ0)/(2−r) dx dt
)(2−r)/2

≤ cC(Λδ0 , c
∗)λs(2−r)/2

(∫
Q

|∇θε|r dx dt
)(1−λ)s(2−r)/2r

,

(4.107)

where (1−λ)s(2−r)
2r = 1

2 (1− λ)(1 + δ0) < 1. Then, by Young’s inequality and (4.107), we obtain

2r(1+δ0)/2C(Λδ0 , c
∗)r/2

(∫
Q

|θε|r(1+δ0)/(2−r) dx dt
)(2−r)/2

≤ 1

2

∫
Q

|∇θε|r dx dt+R(C(Λδ0 , c
∗)),

(4.108)
where R(·) is a nonnegative continuous function on [0,∞). Substituting (4.108) into (4.104) with
δ = δ0, we have∫

Q

|∇θε|r dx dt ≤ cC(Λδ0 , c
∗)r/2 + 2R(C(Λδ0 , c

∗)), 1 < r <
5

4
, 0 < δ0 <

5− 4r

3
, (4.109)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 35

which yields

θε ∈ Lr(0, T :W 1,r
ΓD

), 1 < r <
5

4
,

∥θε∥Lr(0,T :W 1,r
ΓD

) ≤
(
cC(Λδ0 , c

∗)r/2 + 2R(C(Λδ0 , c
∗))

)1/r

, 0 < δ0 <
5− 4r

3
.

(4.110)

Taking r = 6/5, from (4.110), we have

∥θε∥
6
5

L6/5(0,T ;L2)
≤ cC(Λδ0 , c

∗)3/5 + 2R(C(Λδ0 , c
∗)). (4.111)

On the other hand, from the first equation of (4.1), we obtain〈∂vε
∂t

(t), u
〉
+
〈
Aε(θε)vε(t), u

〉
+ ⟨curl vε(t)× vε(t), u⟩ − ⟨µ

ρ
curl H̄ε(t)× H̄ε(t), u⟩

− ⟨µ
ρ
curl H̄ε(t)× H̊(t), u⟩ − ⟨µ

ρ
curl H̊(t)× H̄ε(t), u⟩+ ⟨ α0θε(t)

1 + εθε(t)2
g, u⟩

= ⟨f1(t), u(t)⟩ ∀u ∈ V.

Taking (4.111) implies

∥v′ε(t)∥V∗ ≤ ∥Aε(θε)vε∥V∗ + ∥ curl vε∥L2∥vε∥L3 +
µ

ρ
∥ curl H̄ε(t)∥L2∥H̄ε(t)∥L3

+
µ

ρ
∥ curl H̄ε(t)∥L2∥H̊(t)∥L3 +

µ

ρ
∥ curl H̊(t)∥L2∥H̄ε(t)∥L3 + |α0g|∥θε∥+ ∥f1(t)∥V∗ .

Thus, by (4.103) and (4.111) the right hand side of the next inequality is bounded.

∥v′ε∥L1(0,T ;V∗) ≤ c
(
∥vε∥L2(0,T ;V) + ε∥vε∥5L5(0,T ;V) + ∥vε∥2L2(0,T ;V) + ∥H̄ε∥2L2(0,T ;VΣ)

+ ∥H̄ε∥L2(0,T ;VΣ)∥H̊∥L∞(0,T ;L3) + ∥ curl H̊∥L∞(0,T ;L2)∥H̄ε∥L2(0,T ;VΣ)

+ ∥θε∥L6/5(0,T ;L2) + ∥f1∥L2(0,T ;V∗)

)
.

(4.112)

Therefore, by (4.103), there exists v and a subsequence {vεk} such that

vεk
∗
⇀ v in L∞(0, T ;H),

vεk → v in L2(0, T ;W
9
10 ,2(Ω)),

vεk ⇀ v in L2(0, T ;V)

(4.113)

as εk → 0.
From the second equation of (4.1), we obtain

⟨H̄ ′
ε(t), C⟩+

〈
[

1

µσ(θ)
+ ε∥ curl H̄(t)∥4] curl H̄(t), curlC

〉
+
〈 1

µσ(θε)
curl H̊(t), curlC

〉
+
〈
curlC × H̄ε(t), vε(t)

〉
+
〈
curlC × H̊(t), vε(t)

〉
= ⟨r1(t), C⟩ ∀C ∈ VΣ,

which yields

∥H̄ ′(t)∥V∗
Σ
≤ c

(
∥H̄(t)∥VΣ

+ ε∥ curl H̄(t)∥5VΣ
+ ∥ curl H̊(t)∥L2

+ ∥H̄(t)∥L6∥v(t)∥L3 + ∥H̊(t)∥L6∥v(t)∥L3 + ∥r1(t)∥V∗
Σ

)
.

Thus, by (4.103), (2) of Assumption 3.1 and (3.13) we have

∥H̄ ′
ε∥L1(0,T ;V∗

Σ) ≤ c
(
∥H̄ε∥L2(0,T ;VΣ) + ε∥ curl H̄(t)∥5L6(0,T ;VΣ) + ∥H̊∥L∞(0,T ;L2)

+ ∥H̄ε∥L2(0,T ;VΣ)∥vε(t)∥L2(0,T ;V) + ∥H̊∥L∞(0,T ;H1)∥vε∥L2(0,T ;V)

+ ∥r1(t)∥L2(0,T ;V∗
Σ)

)
.

(4.114)

Therefore, by (4.103), there exists H̄ and a subsequence {H̄εk} such that

H̄εk
∗
⇀ H̄ in L∞(0, T ;L2(Ω)),

H̄εk → H̄ in L2(0, T ;W
9
10 ,2(Ω)),

H̄εk ⇀ H̄ in L2(0, T ;VΣ)

(4.115)



36 T. KIM EJDE-2025/119

as εk → 0.
We will obtain an estimate on θ′ε. As above, let 1 < r < 5

4 , and r
′ := r/(r − 1) > 5.

|⟨κ(θε)∇θε,∇φ⟩| ≤ κ1∥∇θε∥Lr∥∇φ∥Lr′ ≤ c∥∇θε∥Lr∥φ∥
W 1,r′

ΓD
(Ω)
,

|(β(x)θε, φ)ΓR
| ≤ β1∥θε∥L1(ΓR)∥φ∥L∞(ΓR) ≤ c∥θε∥W 1,r

ΓD
(Ω)∥φ∥W 1,r′

ΓD
(Ω)
.

(4.116)

In the last inequality a trace theorem (cf. [20, Theorem 1.25]) was used. As [24] let us estimate
⟨vεθε,∇φ⟩ (cf. [24, pp. 1901 ]). Let r satisfy 21

20 < r < 5
4 . Then there exists 4

3 < σ < q such that

q(σ − 1)

σ(q − 1)
=

5r

8
,

where q = 3r
3−r . We define

ϱ :=
4σ

3σ − 4
, λ1 :=

q − σ

σ(q − 1)
.

Then
1

4
+

1

σ
+

1

ϱ
= 1,

1

σ
=
λ1
1

+
1− λ1
q

, 1− λ1 =
5r

8
.

Taking into account

∥θε∥Lσ ≤ c∥θε∥λ1

L1∥θε∥1−λ1

Lq ≤ c∥θε∥λ1

L1∥∇θε∥1−λ1

Lr ,

∥vε∥L4(Ω) ≤ c∥vε∥1/4L2(Ω)∥∇vε∥
3/4
L2(Ω),

we have

|⟨vεθε,∇φ⟩| ≤ c∥vε∥L4∥θε∥Lσ∥∇φ∥Lϱ

≤ c∥vε∥1/4L∞(0,T ;H)∥θε∥
λ1

L∞(0,T ;L1)∥∇vε∥
3/4
L2(Ω)∥∇θε∥

5r
8

Lr∥φ∥W 1,ϱ
ΓD

,
(4.117)

where 1− λ1 = 5r
8 was used. Also,∣∣∣〈 1

ρChσ(θε)

| curlHε|2

1 + ε| curlHε|2
, φ

〉∣∣∣ ≤ c
2

σ0

(
∥H̄ε∥2VΣ

+ ∥H̊∥2H1

)
max
x∈Ω

|φ|

≤ c
(
∥H̄ε∥2VΣ

+ ∥H̊∥2H1

)
∥φ∥

W 1,r′
ΓD

∀φ ∈W 1,r′

ΓD
(Ω),

(4.118)

∣∣∣∣〈η(θε)ρCh

|E(vε)|2

1 + ε|E(vε)|2
, φ

〉∣∣∣∣ ≤ cη1∥vε∥2V max
x∈Ω

|φ| ≤ c∥vε∥2V∥φ∥
W 1,r′

ΓD

∀φ ∈W 1,r′

ΓD
(Ω). (4.119)

Let τ := max{r′, ϱ}. Then, by (4.116)-(4.119), from the third equation of (4.1), we have∣∣∣〈∂θε
∂t

, φ
〉∣∣∣ ≤ c

(
∥∇θε∥Lr + ∥vε∥1/4L∞(0,T ;H)∥θε∥

λ
L∞(0,T ;L1)∥∇vε∥

3/4
L2(Ω)∥∇θε∥

5r
8

Lr

+
(
∥H̄ε∥2VΣ

+ ∥H̊∥2H1

)
+ ∥vε∥2V + ∥g1∥(W 1,2

ΓD
)∗

)
∥φ∥W 1,τ

ΓD

∀φ ∈W 1,τ
ΓD

(Ω).

Hence, taking into account (4.103), the second one of (4.110), 2) of Assumption 3.1 and (3.13),

we see that θ′ε ∈ L1
(
0, T ; (W 1,τ

ΓD
)∗
)
and

∥θ′ε∥L1
(
0,T ;(W 1,τ

ΓD
)∗
)

≤ c
(
∥θε∥

L1
(
0,T ;W 1,r

ΓD

) + ∥vε∥1/4L∞(0,T ;H)∥θε∥
1− 5r

8

L∞(0,T ;L1)∥vε∥
3/4
L2(0,T ;V)∥θε∥

5r
8

Lr(0,T ;W 1,r
ΓD

)

+ ∥H̄ε∥2L2(0,T ;VΣ) + ∥H̊∥2L∞(0,T ;H1) + ∥vε∥2L2(0,T ;V) + ∥g1∥L2(0,T ;(W 1,2
ΓD

)∗)

)
,

(4.120)

where 21
20 < r < 5

4 . To obtain the second term of the right hand side of the above it was used the
fact that by Hölder’s inequality with exponents 8/3 and 8/5∫ T

0

∥∇vε(t)∥3/4L2(Ω)∥∇θε(t)∥
5r
8

Lr(Ω) dt ≤ ∥vε∥3/4L2(0,T ;V)∥θε∥
5r
8

Lr(0,T ;W 1,r
ΓD

)
.



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 37

Thus, by (4.110) and (4.120), there exists θ and a subsequence {θεk} such that

θεk
∗
⇀ θ in L∞(0, T ;L1(Ω)),

θεk ⇀ θ in Lr(0, T ;W 1,r
ΓD

(Ω)) ∀r (1 < r < 5/4),

θεk → θ in Lr(0, T ;Wα,r
ΓD

(Ω)) ∀r (1 < r < 5/4) ∀α (
1

r
< α < 1) and a.e. in Q.

(4.121)

The last formula the above is obtained as follows. First, by W 1,τ
ΓD

(Ω) ↪→ L6(Ω), we obtain

L6/5(Ω) ↪→ (W 1,τ
ΓD

(Ω))∗ [13, Remark 5.14, Ch. 1], which combing with the fact that Wα,r
ΓD

(Ω) ⊂
L3/2(Ω) yields Wα,r

ΓD
(Ω) ⊂ (W 1,τ

ΓD
)∗. Then, since W 1,r

ΓD
(Ω) ↪→ Wα,r

ΓD
(Ω) is compact, we obtain the

fact that {θε} is relatively compact in Lr(0, T ;Wα,r
ΓD

(Ω)) (cf. [20, Theorem 1.39]).

4.3. Passing to the limit as ε → 0. By passing to limit of solutions in Theorem 4.2 we will
prove Theorem 3.4. Owing to (4.113), (4.115) and (4.121), we can extract subsequences, which is
denoted as before, such that

vεk ⇀ v in L2(0, T ;V),

vεk
∗
⇀ v in L∞(0, T ;H),

vεk → v in L2(0, T ;W
9
10 ,2(Ω)),

H̄εk
∗
⇀ H̄ in L∞(0, T ;L2(Ω)),

H̄εk → H̄ in L2(0, T ;W
9
10 ,2(Ω)),

H̄εk ⇀ H̄ in L2(0, T ;VΣ),

θεk ⇀ θ in Lr(0, T ;W 1,r
ΓD

(Ω)) ∀r (1 < r < 5/4),

θεk → θ in Lr(0, T ;Wα,r
ΓD

(Ω)(Ω)) ∀r (1 < r < 5/4), ∀α
(1
r
< α < 1

)
and a.e. in Q

(4.122)

as εk → 0.
We will obtain the first equation of (3.15). Let u ∈ C1([0, T ];V) with u(T ) = 0. By (4.76) and

(4.122), we have ∫ T

0

〈∂vεk
∂t

, u
〉
dt→

∫ T

0

〈
− v(t),

∂u

∂t

〉
dt− ⟨v0, u(x, 0)⟩ (4.123)

Taking into account the above, by [20, Corollary 1.1] we obtain∫ T

0

⟨µ(θεk)E(vεk), E(u)
〉
dt→

∫ T

0

⟨µ(θ)E(v), E(u)
〉
dt. (4.124)

By (4.103) we have∣∣∣ ∫ T

0

εk∥vεk∥4V
(
E(vεk), E(u)

)
dt
∣∣∣

≤ cεk
1/6

∫ T

0

εk
5/6∥vεk∥5V∥u(t)∥V dt

≤ cεk
1/6

(∫ T

0

(
εk

5/6∥vεk∥5V
)6/5

dt
)5/6(∫ T

0

∥u(t)∥6V dt
)1/6

≤ cεk
1/6C(Λδ, c

∗)5/6∥u∥L6(0,T ;V) → 0 as εk → 0.

(4.125)

In a rather routine way we can prove that∫ T

0

⟨curl vεk(t)× vεk(t), u(t)⟩ dt→
∫ T

0

⟨curl v × v, u⟩ dt as εk → 0. (4.126)

(See [20, (6.46)-(6.49)].) Owing to (4.122),

vεk → v in L2(0, T ;L2(∂Ω)),



38 T. KIM EJDE-2025/119

and taking a subsequence if necessary, by [20, Lemma 1.3]

2(µ(θεk)k(x)vεk , u)Γ2 + 2(µ(θεk)Sṽεk , ũ)Γ3 + 2(α(x)vεk , u)Γ5 + (µ(θεk)k(x)vεk , u)Γ7

→ 2(µ(θ)k(x)v, u)Γ2 + 2(µ(θ)Sṽ, ũ)Γ3 + 2(α(x)v, u)Γ5 + (µ(θ)k(x)v, u)Γ7 .
(4.127)

Let us prove ∫ T

0

〈µ
ρ
curl H̄εk(t)× H̄εk(t), u(t)

〉
dt→

∫ T

0

〈µ
ρ
curl H̄ × H̄, u(t)

〉
dt. (4.128)

Writing as

µ

ρ

∫ T

0

∣∣∣⟨curl H̄εk(t)× H̄εk(t), u(t)⟩ − ⟨curl H̄ × H̄, u(t)⟩
∣∣∣ dt

≤ µ

ρ

∫ T

0

|⟨curl H̄εk(t)× (H̄εk(t)− H̄), u(t)⟩| dt+ µ

ρ

∫ T

0

|⟨curl H̄εk(t)− curl H̄, H̄ × u(t)⟩| dt

≤ µ

ρ
∥ curl H̄εk(t)∥L2(0,T ;L2)∥(H̄εk(t)− H̄)∥L2(0,T ;L3)∥u∥L∞(0,T ;L6)

+
µ

ρ

∫ T

0

|⟨curl H̄εk(t)− curl H̄, H̄ × u(t)⟩| dt

and taking into account (4.122) and that H̄ × u ∈ L2(0, T ;L2), we obtain (4.128).

Similarly, taking into account H̊(t)×u(t) ∈ L2(0, T ;L2(Ω)), u(t)×curl H̊(t) ∈ L∞(0, T ;L
6
5 (Ω))

and (4.122), we obtain

µ

ρ

∫ T

0

〈
curl H̄εk × H̊, u(t)

〉
dt =

µ

ρ

∫ T

0

〈
H̊ × u(t), curl H̄εk

〉
dt→

∫ T

0

〈µ
ρ
curl H̄ × H̊, u(t)

〉
dt,

µ

ρ

∫ T

0

〈
curl H̊ × H̄εk , u(t)

〉
dt =

µ

ρ

∫ T

0

〈
u(t)× curl H̊(t), H̄εk

〉
dt

→
∫ T

0

〈µ
ρ
curl H̊ × H̄, u(t)

〉
dt

(4.129)
as εk → 0. By (4.122) and [20, Lemma 1.5], we have∫ T

0

〈( α0θεk
1 + εkθ2εk

− α0θ
)
g, u(t)

〉
dt

=

∫ T

0

〈α0(θεk − θ)

1 + εkθ2εk
g, u(t)

〉
dt+

∫ T

0

〈( 1

1 + εkθ2εk
− 1

)
α0θg, u(t)

〉
dt→ 0,

which shows that∫ T

0

〈 α0θεk
1 + εkθ2εk

g, u(t)
〉
dt→

∫ T

0

⟨α0θg, u(t)⟩ dt→ 0 as εk → 0. (4.130)

Therefore, by (4.123)-(4.130) and (4.76), from the first equation of (4.1) we have the first equation
of (3.16).

We will obtain the second equation of (3.16). Let C ∈ C1([0, T ];VΣ) with C(·, T ) = 0. As in
(4.123) we obtain∫ T

0

⟨H̄ ′
εk
(t), C(t)⟩ dt→

∫ T

0

⟨−H̄ ′(t), C(t)⟩ dt+ ⟨H̄0, C(x, 0)⟩ as εk → 0. (4.131)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 39

At the same way as (4.124) and (4.125), we see that∫ T

0

b0(θεk ; H̄εk , C(t)) dt→
∫ T

0

b0(θ; H̄, C(t)) dt,∫ T

0

〈
εk∥ curl H̄εk(t)∥4 curl H̄εk(t), curlC

〉
dt→ 0,∫ T

0

〈 1

µσ(θεk)
curl H̊, curlC(t)

〉
dt→

∫ T

0

〈 1

µσ(θ)
curl H̊, curlC(t)

〉
dt

(4.132)

as εk → 0. Since∫ T

0

|
〈
curlC(t)× H̄εk , vεk

〉
−
〈
curlC(t)× H̄, v

〉
| dt

≤ ∥ curlC∥L∞(0,T ;L2)∥H̄εk − H̄∥L2(0,T ;L3)∥vεk∥L2(0,T ;L6)

+ ∥ curlC∥L∞(0,T ;L2)∥H̄∥L2(0,T ;L6)∥vεk − v∥L2(0,T ;L3),

we have ∫ T

0

〈
curlC(t)× H̄εk , vεk

〉
→

∫ T

0

〈
curlC(t)× H̄, v

〉
dt. (4.133)

Also, by (2) of Assumption 3.1,

curlC(t)× H̊(t) ∈ L2(0, T ;L
6
5 (Ω)),

and taking into account (4.122), we have∫ T

0

〈
curlC(t)× H̊(t), vεk

〉
→

∫ T

0

〈
curlC(t)× H̊(t), v

〉
dt as εk → 0. (4.134)

Using (4.131)-(4.134) and (4.76), from the second equation of (4.1) we obtain the second one of
(3.16).

We will obtain the third equation of (3.16). Let φ ∈ C1([0, T ];C1
ΓD

(Ω)). We have from the
third equation of (4.1)∫ t

0

〈∂θεk
∂t

, φ
〉
ds+

∫ t

0

(κ(θεk)∇θεk ,∇φ) ds+
∫ t

0

(β(x)θεk , φ)ΓR
ds

−
∫ t

0

⟨vεkθεk ,∇φ⟩ ds−
∫ t

0

〈 1

ρChσ(θεk)

∣∣ curl (H̄εk(s) + H̊(s)
)∣∣2

1 + ε
∣∣ curl (H̄εk(s) + H̊(s)

)∣∣2 , φ〉 ds
−
∫ t

0

〈η(θεk)
ρCh

|E(vεk)|2

1 + ε|E(vεk)|2
, φ

〉
ds

=

∫ t

0

〈
g1, φ

〉
ds ∀t ∈ [0, T ].

As (4.123), we obtain∫ T

0

〈∂θεk
∂t

, φ
〉
dt→

∫ T

0

〈
− θ(t),

∂φ

∂t

〉
dt− ⟨θ0, φ(x, 0)⟩. (4.135)

By [20, Corollary 1.1] we have∫ t

0

(κ(θεk)∇θεk ,∇φ) ds→
∫ t

0

(κ(θ)∇θ,∇φ) ds as εk → 0. (4.136)

It is easy to prove that∫ t

0

(β(x)θεk , φ)ΓR
ds→

∫ t

0

(β(x)θ, φ)ΓR
ds as εk → 0. (4.137)

Since vεk → v in L2(0, T ;W
9
10 ,2(Ω)) and W

9
10 ,2(Ω) ⊂ L5(Ω) (cf. [20, Theorem 1.20]), we have

vεk → v in L2(0, T ;L5(Ω)). (4.138)



40 T. KIM EJDE-2025/119

Since {θεk} ⊂
(
L6/5(0, T ;W 1,6/5(Ω)) ∩ L∞(0, T ;L1(Ω))

)
and W 1,6/5(Ω) ⊂ L2(Ω), by complex

interpolation with exponent 2
5 we have

{θεk} ⊂ L2(0, T ;L
10
7 (Ω)) ⊂ L2(0, T ;L

5
4 (Ω))

(cf. [20, Theorems 1.34 and 1.12]). Taking into account (4.138) and the above, we have∫ t

0

⟨vεkθεk ,∇φ⟩ ds→
∫ t

0

⟨vθ,∇φ⟩ ds as εk → 0. (4.139)

Since when φ ≥ 0,∫ t

0

〈 1

ρChσ(θεk)

∣∣ curl (H̄εk(s) + H̊(s)
)∣∣2

1 + ε
∣∣ curl (H̄εk(s) + H̊(s)

)∣∣2 , φ〉 ds
=

∫ t

0

〈 1

ρChσ(θεk)

∣∣ curl (H̄εk(s) + H̊(s)
)∣∣√φ√

1 + ε
∣∣ curl (H̄εk(s) + H̊(s)

)∣∣2 ,
∣∣ curl (H̄εk(s) + H̊(s)

)∣∣√φ√
1 + ε

∣∣ curl (H̄εk(s) + H̊(s)
)∣∣2

〉
ds,

∫ t

0

〈η(θεk)
ρCh

|E(vεk)|2

1 + εk|E(vεk)|2
, φ

〉
ds =

∫ t

0

〈η(θεk)
ρCh

|E(vεk)|
√
φ√

1 + εk|E(vεk)|2
,

|E(vεk)|
√
φ√

1 + εk|E(vεk)|2
〉
ds,

by [20, Lemma 1.4 and Corollary 1.2] we have

lim inf
εk→0

∫ t

0

〈 1

ρChσ(θεk)

∣∣ curl (H̄εk(s) + H̊(s)
)∣∣2

1 + ε
∣∣ curl (H̄εk(s) + H̊(s)

)∣∣2 , φ〉 ds
≥

∫ t

0

〈 1

ρChσ(θ)

∣∣ curl (H̄(s) + H̊(s)
)∣∣2, φ〉 ds,

lim inf
εk→0

∫
Ω

〈η(θεk)
ρCh

|E(vεk)|2

1 + εk|E(vεk)|2
, φ

〉
ds ≥

∫ t

0

〈η(θ)
ρCh

|E(v)|2, φ
〉
ds.

(4.140)

Therefore, taking into account (4.135)-(4.137), (4.139), (4.140) and (4.76), we have

〈
θ(t), φ(t)

〉
+

∫ t

0

[
⟨−θ, ∂φ

∂s

〉
+ ⟨κ(θ)∇θ,∇φ⟩+ ⟨β(x)θ, φ⟩ΓR

− ⟨vθ,∇φ⟩
]
ds

≥
∫ t

0

[〈 1

ρChσ(θ)

∣∣ curl (H̄(s) + H̊(s)
)∣∣2, φ〉+

〈η(θ)
ρCh

|E(v)|2, φ
〉
+ ⟨g1(s), φ⟩

]
ds

+
〈
θ0, φ(x, 0)

〉
for a.a. t ∈ [0, T ] ∀φ ∈ C1([0, T ];C1

ΓD
(Ω)), φ ≥ 0.

(4.141)

Now, following [24] we can prove that θ satisfies the third formula of (3.16) with “defect
measure”. For convenience of readers we give here proof.

By (4.103),
η(θεk )

ρCh
|E(vεk)|2 ∈ L1(Q), and these give continuous linear functionals on C(Q).

Since Q is compact, every
η(θεk )

ρCh
|E(vεk)|2 is identified with a Radon measure(cf. [8, Theorem

7.10.4, pp.1]). Since
{η(θεk )

ρCh
|E(vεk)|2

}
is bounded in L1(Q) and C(Q) is separable, there exists a

weak∗ convergent subsequence (cf. Corollary, 1. Polar Sets, Appendix to Chap. V of [29]) and a
positive Radon measure µ̄0 ∈ M (Q̄) such that∫ T

0

〈η(θεk)
ρCh

|E(vεk)|2, φ
〉
dt→

∫
Q̄

φdµ̄0 ∀φ ∈ C(Q). (4.142)

By the same argument, there exists a weak∗ convergent subsequence and a positive Radon measure
µ̄1 ∈ M (Q̄) such that∫ T

0

[〈 1

ρChσ(θεk)

∣∣ curl (H̄εk(t) + H̊(s)
)∣∣2, φ〉 dt→ ∫

Q̄

φdµ̄1 ∀φ ∈ C(Q). (4.143)



EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 41

Therefore, taking into account (4.135)-(4.137), (4.139), (4.142) and (4.143), we have∫ T

0

[
⟨−θ, ∂φ

∂t

〉
+ ⟨κ(θ)∇θ,∇φ⟩+ ⟨β(x)θ, φ⟩ΓR

− ⟨vθ,∇φ⟩
]
dt

=

∫
Q̄

φd(µ̄1 + µ̄0) +
〈
θ0, φ(x, 0)

〉
+

∫ T

0

〈
g1, φ

〉
dt

(4.144)

for all φ ∈ C1([0, T ];C1
ΓD

(Ω)), φ(·, T ) = 0. From (4.142) and (4.144) we have∫
Q̄

φd(µ̄1 + µ̄0) ≥
∫ T

0

[〈 1

ρChσ(θ)

∣∣ curl (H̄(t) + H̊(t)
)∣∣2 + η(θ)

ρCh
|E(v)|2, φ

〉]
dt

for all φ ∈ C1([0, T ];C1
ΓD

(Ω)), φ ≥ 0, and φ(·, T ) = 0. We define

µ̄ = (µ̄1 + µ̄0)−
( 1

ρChσ(θ)

∣∣ curl (H̄ + H̊
)∣∣2 + η(θ)

ρCh
|E(v)|2

)
λ4|B(Q),

where λ4 is the Lebesgue measure in R4 and B(Q) is the σ-algebra of Borel subsets of Q. Then,
the third formula of (3.16) is valid with a “defect measure” µ̄.

To get estimate (??), we use (4.103) and (4.109). Let us prove that∫
Q

|∇θ|2

(1 + |θ|)1+δ
dx dt ≤ lim inf

εk→0

∫
Q

|∇θεk |2

(1 + |θεk |)1+δ
dx dt 0 < δ < 1. (4.145)

We put

zεk,i :=
∂θεk/∂xi

(1 + |θεk |)(1+δ)/2
a.e. in Q (i = 1, 2, 3).

By (4.122), {zεk} is bounded in Lr(Q), and by passing to a subsequence if necessary, we have

zεk,i ⇀ zi in Lr(Q). (4.146)

To obtain (4.145), it is sufficient to prove that

zi =
∂θ/∂xi

(1 + |θ|)(1+δ)/2
. (4.147)

By (4.122),

∂θεk/∂xi ⇀ ∂θ/∂xi in Lr(Q), 1 < r <
5

4
,

and by Lebesgue Theorem,

1

(1 + |θεk |)(1+δ)/2
→ 1

(1 + |θ|)(1+δ)/2
in Lr′(Q).

Then
∂θεk/∂xi

(1 + |θεk |)(1+δ)/2
→ ∂θ/∂xi

(1 + |θ|)(1+δ)/2
in L1(Q),

from which we obtain

∂θεk/∂xi
(1 + |θεk |)(1+δ)/2

→ ∂θ/∂xi
(1 + |θ|)(1+δ)/2

a.e. in Q (4.148)

for a subsequence if necessary. By (4.146) and (4.148), we obtain (4.147) (cf. [23, Lemma 1.3,
Ch.1]), which yields (4.145). By (4.109), we obtain∫

Q

|∇θ|r dx dt ≤ lim inf
εk→0

∫
Q

|∇θεk |r dx dt ≤ cC(Λδ0 , c
∗)r/2 + 2R(C(Λδ0 , c

∗)),

1 < r <
5

4
, 0 < δ0 <

5− 4r

3
.

(4.149)

Taking into account (4.91), (4.145), and (4.149), from (4.103), we obtain (??). □



42 T. KIM EJDE-2025/119

References

[1] G. Alekseev; Mixed boundary value problems for stationary magnetohydrodynamic equations of a viscous heat-
conducting fluid, J. Math. Fluid Mech., 18 (2016), 591-607.

[2] G. V. Alekseev; Mixed boundary value problems for steady-state magnetohydrodynamic equations of viscous

incompressible fluid, Comput. Math. Math. Phys., 56:8 (2016), 1426-1439.
[3] G. V. Alekseev, R. V. Brizitskii; Solvability of the boundary value problem for stationary magnetohydrodynamic

equations under mixed boundary conditions for the magnetic field, Appl. Math. Lett., 32 (2014), 13-19.

[4] G. Alekseev, R. V. Brizitskii, V. V. Pukhnachev; Solvability of the inhomogeneous mixed boundary value
problem for stationary magnetohydrodynamic equations, Dokl. Phys., 59 (2014), 467-471.

[5] H. Amann, J. Escher; Analysis I, Birkhäuser Verlag, 2005.
[6] G. Auchmuty; The main inequality of 3D vector analysis, Math. Models Methods Appl. Sci., 14(1) (2004),

79-103.

[7] G. Auchmuty, J. C. Alexander; Finite-energy solutions of mixed 3D div-curl systems, Quart. Appl. Math.,
64(2) (2006), 335-357.

[8] V. I. Bogachev; Measure Theory II, Springer (2007).

[9] M. Buĺıček, E. Feireisl, J. Málek; A Navier-Stokes-Fourier system for incompressible fluids with temperature
dependent material coefficients, Nonlinear Anal. Real World Appl., 10 (2009), 992-1015.

[10] L. Consiglieri; Existence of proper weak solutions to the Navier-Stokes-Fourier system, Funkcial. Ekvac., 55

(2012), 347-365.
[11] E. Feireisl, J. Málek; On the Navier-Stokes equations with temperature-dependent transport coefficients, Differ.

Equ. Nonlinear Mech., (2006) 14 pp. (digital) Art. ID 90616.

[12] R. Dautray, J. L. Lions; Mathematical analysis and numerical methods for science and technology I,
Springer,1990.

[13] H. Gajewski, K. Gröger, K. Zacharias; Nichtlinear operatorgleichungen und operatordifferentialgleichungen,
Academie-Verlag (1974) (Russian, 1978).

[14] J. F. Gerbeau, C. L. Bris, T. Lelièvre; Mathematical methods for the magnetohydrodynamics of liquid metals,

Oxford University Press, 2006.
[15] P. Grisvard; emphElliptic problems in nonsmooth domains, Pitman Advanced Publishing Program, Boston,

London, Melbourne,1985.

[16] T. Kim; Existence of a solution to the steady magnetohydrodynamics-Boussinesq system with mixed boundary
conditions, Math. Methods Appl. Sci., 2022;1-42. doi: 10.1002/mma.8297.

[17] T. Kim; Existence of a solution to the non-steady magnetohydrodynamics-Boussinesq system with mixed bound-

ary conditions, J. Math. Anal. Appl., 525 (2023), 127183.
[18] T. Kim, D. Cao; Some properties on the surfaces of vector fields and its application to the Stokes and Navier-

Stokes problems with mixed boundary conditions, Nonlinear Anal., 113 (2015), 94-114.

[19] T. Kim, D. Cao; A non-steady system with friction boundary conditions for flow of heat-conducting incom-
pressible viscous fluids, J. Math. Anal. Appl., 484 (2020), 123676

[20] T. Kim, D. Cao; Equations of motion for incompressible viscous fluids, Birkhäuser, 2021.

[21] T. Kim, D. Cao; The steady Navier-Stokes and Stokes systems with mixed boundary conditions including
one-sided leaks and pressure, Methods Appl. Anal. 23(4) (2016), 329-364.

[22] A. N. Kolmogorov, S. V. Fomin; Element of theory of functions and functional analysis, Moscow “Science”
(Russian) (1981).

[23] J. L. Lions; Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars,
Paris (1969) (Russian, 1972).

[24] J. Naumann; On the existence of weak solutions to the equations of non-stationary motion of heat-conducting
incompressible viscous fluids, Math. Methods Appl. Sci., 29 (2006), 1883-1906.

[25] J. Naumann; An existence theorem for weak solutions to the equations of non-stationary motion of heat-
conducting incompressible viscous fluids, J. Nonlinear Convex Anal., 7(3) (2006), 483-497.

[26] J. Naumann, J. Wolf; Existence of weak solutions to the equations of natural convection with dissipative heating,
In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, 367-384, Springer-Verlag
Berlin Heidelberg, 2010.

[27] H. Wu; Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math.

Appl., 61 (2011), 2742-2753.
[28] W. Yan; Existence of weak solutions to the three-dimensional density-dependent generalized incompressible

magnetohydrodynamic flow, Discrete Contin. Dyn. Syst., 35(3) (2015), 1359-1385.
[29] K. Yosida; Functional analysis, Springer, 1995.

Tujin Kim

Institute of Mathematics, State Academy of Sciences, Pyongyang, DPR Korea

Email address: math.tujin@star-co.net.kp