Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 23, pp. 1–18.

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

DOI: 10.58997/ejde.2025.23

LOCAL SOLUTIONS FOR A BRINKMAN EQUATION COUPLED

WITH HEAT-CONVECTIVE AND

CONCENTRATION-DIFFUSIVE EQUATIONS AND A

VOLUMETRIC MASS SOURCE

HAKHO HONG

Communicated by Jesus Ildefonso Diaz

Abstract. In this article, we consider a model coupled with the Brinkman

heat-convective and concentration-diffusive equations for a mixed gas flow in a

porous media. The specificity of this model lies in the presence of a volumet-
ric mass source depending on temperature and concentration in mass balance

equation. We will prove the existence and uniqueness of the smooth local solu-
tions for the 3-D Cauchy problem. As a byproduct, we show the convergence

of the approximate solutions based on an iteration scheme.

1. Introduction

The fundamental model we study is based upon the Brinkman equation for the
mixed gas flow and the concentration-diffusive equation for the gas concentrations
in porous media. These make it possible to model the performance of the adsorp-
tion column in a portable oxygen concentrator (see [19]), and the heat-convective
equation for the temperature. Thus, let ρ(x, t),u(x, t), θ(x, t) and wi(x, t) be the
density, velocity, temperature of gas mixture and the concentration of i-gas compo-
nent, respectively, where x is the position, t denotes time and i = 1, 2, . . . , n. Then,
the balance equations of mass, momentum and concentrations are

ρt + div(ρu) = Q0(w, θ),

(ρu)t + div(ρu⊗ u) +∇P + α−1u = 2µdivD(u) + ν∇divu,

wt − df∆w + (u,∇)w +w divu = S(w, θ),

(1.1)

where w = (w1, . . . , wn). The balance equation of energy is

(ρE)t + div(ρuE + Pu) = κ∆θ + div
(
2µD(u)u+ ν divuu

)
+Q1(w, θ), (1.2)

where Q0(w, θ) is the mass source, α > 0 denotes the permeability of the porous
medium, µ > 0 and ν ≥ 0 are the shear and bulk viscosities, respectively, df > 0
is the dispersion coefficient, S(w, θ) = (S1, . . . , Sn), Si is the concentration deposit

2020 Mathematics Subject Classification. 35B40, 35B65, 35L65, 76N05, 76N10, 76T10.
Key words and phrases. Brinkman equation; heat-convective equation; mixed model;

concentration-diffusive equation; existence; uniqueness; iteration scheme.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted September 30, 2024. Published March 3, 2025.

1



2 H. HONG EJDE-2025/23

rate of i-gas component, E = e + 1
2u

2 is the specific total energy, e is the specific
internal energy, κ > 0 is the heat-conducting coefficient, Q1(w, θ) is the heat source,
and D(u) is the deformation tensor given by

D(u) =
1

2

(
∇u+ (∇u)T

)
,

where (∇u)T denotes the transpose of matrix ∇u. The mass source Q0(w, θ),
the heat source Q1(w, θ) and the concentration deposit rate, S(w, θ) are the given
smooth functions for the gas concentration w and the temperature θ. Also, for the
pressure P and the internal energy e, we assume that

P = P (ρ, θ;w), Pρ(ρ, θ;w) > 0, Pwi(ρ, θ;w) > 0, (1.3)

e = e(ρ, θ), eθ(ρ, θ) > 0. (1.4)

The specificity of the system (1.1) lies in the presence of the mass source Q0.
If we set Q0 = 0 and ρ = 1, then the system (1.1)1 and (1.1)2 reduces to the
Brinkman equation

divu = 0,

ut − µ∆u+ (u · ∇)u+∇P + α−1u = 0,
(1.5)

which was first proposed by Brinkman [2] in 1947. The system (1.5) describing the
viscous flow of incompressible fluids in porous media was extensively investigated
in the last several decades (see, e.g., [20, 3, 22, 24, 18, 27, 13, 11, 14, 15, 17, 25, 26]
and references therein). However, the studies in above works depend essentially on
the fact that the velocity is solenoidal, that is, divu = 0.

Recently, the diffusive interface model for tumor growth have been developed
and analyzed, which reduces to the system

divu = S1(φ, χ),

−µ∆u− ν∇ divu+∇P + α−1u = (J + φ)∇χ,

χt + div(χu) = m∆J + S2(φ, χ), J = −∆χ+Ψ′(χ)− χφ,

φt − df∆φ+ div(φu) = S3(φ, χ),

(1.6)

which consists of the Brinkman equation with mass source (for fluid flow), Cahn-
Hilliard (for the tumor density) and reaction-diffusion (for the nutrient or other
chemical factors) equations. The model (1.6) is a description of the evolution of a
two-phase cell mixture, containing tumour cells and healthy host cells, surrounded
by a chemical species acting as nutrients only for the tumour cells, and is trans-
ported by a fluid velocity field. The variable χ denotes the difference in the volume
fractions of the cells, with the region {χ = 1} representing the tumour cells and
{χ = −1} representing the host cells, while φ denotes the concentration of the nu-
trient. The fluid velocity u is taken as the volume averaged velocity, with pressure
p, and J denotes the chemical potential associated to χ. Also, the function Ψ is
potential energy density and Si(i = 1, 2, 3) are generic source terms that can be
specified depending on the application (see [8, 12] for more details in the model
derivation).

An interesting feature of the system (1.6) is that the velocity is not solenoidal,
that is, div u ̸= 0. Recently, considerable progress has been obtained for the theo-
retical and numerical studies of the system (1.6). The existence of weak solutions



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 3

was established in [5] and the analysis of weak and stationary solutions of this sys-
tem with singular potentials was considered in [7]. Numerical investigations can be
found in [9]. A simplified version of this model was investigated in [6], where the
time derivative and the convection term in the nutrient equation are neglected. For
this model, the authors proved strong well-posedness and showed that the solutions
converge to the corresponding Cahn-Hilliard-Darcy model, that is, the system (1.6)
with µ = ν = 0.

On the other hand, there are a few results concerning the Brinkman equation
with non-constant density. If we set Q0 = 0, µ = α = 1 and ν = 0, and assume
that the terms ut and (u · ∇)u vanish, then the system (1.1)1 and (1.1)2 reduces to
the system

ρt + div(ρu) = 0, ∆u− u = ∇p,

where p = p(ρ), which is equivalent to

ρt = div
(
ρ(1−∆)−1∇p(ρ)

)
. (1.7)

For Cauchy problem of the system (1.7), the existence of the strong and smooth
solutions was proved in [1] for 1-D case and in [16] for multi-D case, respectively.
In [23], the existence of weak solutions was established for Cauchy problem of the
following 1-D regularized Brinkman equation

ρt = ∂xxρ+ ∂x
(
ρ(1− ∂xx)

−1∂x(ρ
2)
)
.

Although considerable progress has been obtained for the mathematical studies
concerning the Brinkman equation in different settings, most of these results are
obtained for the case where the fluid density is constant. Moreover, there is little
result for the full system (1.1) except for [19], where they employed the model to
simulate the microporous adsorption of nitrogen in a portable oxygen concentrator
by using COMSOL Multiphysics. Thus, we will prove the existence and uniqueness
of the smooth solutions to the 3-D Cauchy problem for the general system (1.1),
(1.2).

More precisely, we consider the system (1.1), (1.2) in R3 with the initial condi-
tions

(ρ,u,w, θ)(x, 0) = (ρ0,u0,w0, θ0)(x) → (ρ̄,0, w̄, θ̄) as |x| → ∞, (1.8)

where ρ̄, w̄ = (w̄1, . . . , w̄n), θ̄ are given positive constants.

Notation. Lp(R3) and W k
p (R3) denote the usual Lebesgue and Sobolev spaces on

R3, with norms ∥ · ∥Lp and ∥ · ∥Wk
p
, respectively. When p = 2, we denote W k

p (R3)

by Hk(R3) with the norm ∥ · ∥Hk and ∥ · ∥H0 = ∥ · ∥ will be used to denote the usual
L2−norm. The notation ∥(A1, A2, . . . , Al)∥Hk means the summation of ∥Ai∥Hk

from i = 1 to i = l. For an integer m, the symbol ∇m denotes the summation of all
terms Dα with the multi-index α satisfying |α| = m. We omit the spatial domain
R3 in integrals for convenience.

The main result of this paper can be stated as follows.

Theorem 1.1. Assume (1.3), (1.4) and

Q0(w̄, θ̄) = 0, S(w̄, θ̄) = 0, Q1(w̄, θ̄) = 0. (1.9)



4 H. HONG EJDE-2025/23

Suppose that the initial data ρ0,u0, φi0 satisfy

(ρ0 − ρ̄,u0,w0 − w̄, θ0 − θ̄) ∈ HN (R3),

inf
x∈R3

ρ0(x) > 0, inf
x∈R3

w0(x) > 0, inf
x∈R3

θ0(x) > 0
(1.10)

for an integer N ≥ 3. Moreover, there is a constant M0, such that ρ0,u0,w0, θ0
satisfy ∥∥(ρ0 − ρ̄,u0,w0 − w̄, θ0 − θ̄)

∥∥
HN ≤ M0. (1.11)

Then, there exist constants T0 and C depending on M0 such that the unique smooth
solution (ρ,u,w, θ) of the Cauchy problem (1.1), (1.2) , (1.8) exists on the time
interval [0, T0] with the properties:

(ρ− ρ̄,u,w − w̄, θ − θ̄) ∈ C([0, T0];H
N (R3)), ∇ρ ∈ L2(0, T0;H

N (R3)),

u ∈ L2(0, T0;H
N+1(R3)),

(
∇w,∇θ

)
∈ L2(0, T0;H

N (R3))
(1.12)

and ∥∥(ρ− ρ̄,u,w − w̄, θ − θ̄)(t)
∥∥2
HN +

∫ t

0

∥∥∇ρ
∥∥2
HN−1dτ

+

∫ t

0

∥∥u∥∥2
HN+1dτ +

∫ t

0

∥∥(∇w,∇θ
)∥∥2

HNdτ ≤ C

(1.13)

for all t ∈ [0, T0].

Remark 1.2. Assumption (1.9) come from system (1.1) satisfying

Q0(w) = −
n∑

i=1

αiMi(wi − w̄i),

Si = αi(wi − w̄i), i = 1, 2, . . . , n, S(w) = (S1, S2, . . . , Sn),

(1.14)

where αi > 0 is the macroscopic mass transfer rate of i-gas component into ze-
olite particles, and Mi > 0 and w̄i > 0 are the molecular weight and the mean
concentration of i-gas component, respectively (see [19, (4) and (12)]).

Next, we show the convergence of the approximate solutions based on the iter-
ation scheme. To this end, we first reformulate the Cauchy problem (1.1), (1.2) ,

(1.8). By using (1.1)1, E = e(ρ, θ) + u2

2 and

div(ρu⊗ u) = div(ρu)u+ ρ(u,∇)u,

we rewrite (1.1)2 and (1.2) as

ρut + ρ(u,∇)u+Q0(w, θ)u+∇P + α−1u = µ∆u+ (µ+ ν)∇ divu,

ρet + ρu · ∇e+Q0(w, θ)
(
e+

u2

2

)
+ P divu

= Q0(w, θ)u2 + α−1u2 + κ∆θ + 2µD(u) : D(u) + λ(divu)2 +Q1(w, θ).

(1.15)

By using the thermodynamical relation

−ρ2eρ(ρ, θ) = θPθ(ρ, θ;w)− P (ρ, θ;w)



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 5

(see [10, (1.6)]), we rewrite (1.15)2 as

θt + u · ∇θ +
θPθ(ρ, θ;w)

ρeθ(ρ, θ)
divu− κ

ρeθ(ρ, θ)
∆θ

=
2µD(u) : D(u)

ρeθ(ρ, θ)
+

λ (divu)
2

ρeθ(ρ, θ)
+

Q1(w, θ)

ρeθ(ρ, θ)

+
1

ρeθ(ρ, θ)

(1
2
Q0(w, θ)u2 + α−1u2 −Q0(w, θ)e(ρ, θ)

)
.

(1.16)

Setting

φ = ρ− ρ̄, m = w − w̄, ζ = θ − θ̄,

and using assumption (1.9), we rewrite system (1.1)1, (1.15)1, (1.1)3, (1.16) as
follows:

φt + ρ̄divu+ u · ∇φ−∇wQ0(w̄, θ̄) ·m−Q′
0θ(w̄, θ̄)ζ = G1(φ,u,m, ζ),

ut −
µ

ρ̄
∆u− µ+ λ

ρ̄
∇divu+

1

αρ̄
u+

Pρ(ρ̄, θ̄; w̄)

ρ̄
∇φ

+
Pθ(ρ̄, θ̄; w̄)

ρ̄
∇ζ +

n∑
i=1

Pwi(ρ̄, θ̄; w̄)

ρ̄
∇mi

= G2(φ,u,m, ζ),

mt − df∆m+ w̄ divu−DwS(w̄, θ̄)m− S′
θ(w̄, θ̄)ζ = G3(φ,u,m, ζ),

ζt +
θ̄Pθ(ρ̄, θ̄; w̄)

ρ̄eθ(ρ̄, θ̄)
divu

=
κ

ρ̄eθ(ρ̄, θ̄)
∆ζ +

∇wQ1(w̄, θ̄) ·m+Q′
1θ(w̄, θ̄)ζ

ρ̄eθ(ρ̄, θ̄)

+
e(ρ̄, θ̄)

ρ̄eθ(ρ̄, θ̄)

(
∇wQ0(w̄, θ̄) ·m+Q′

0θ(w̄, θ̄)ζ
)
+G4(φ,u,m, ζ),

(1.17)

where DwS(w̄, θ̄) =
(

∂Sj(w̄,θ̄)
∂wi

)n

i,j=1
,

G1(φ,u,m, ζ)

= −φdivu+
(
Q0(w, θ)−Q0(w̄, θ̄)−∇wQ0(w̄, θ̄) ·m−Q′

0θ(w̄, θ̄)ζ
)
,

(1.18)

G2(φ,u,m, ζ)

= −(u,∇)u− (Q0(w, θ)−Q0(w̄, θ̄))u+ h1(φ, ζ,m)∇φ+ h2(φ, ζ,m)∇ζ

+

n∑
i=1

h3i(φ, ζ,m)∇mi − h4(φ)
(
µ∆u+ (µ+ ν)∇ divu

)
+ α−1h4(φ)u,

h1(φ, ζ,m) =
Pρ(ρ̄, θ̄; w̄)

ρ̄
− Pρ(ρ, θ;w)

ρ
,

h2(φ, ζ,m) =
Pθ(ρ̄, θ̄; w̄)

ρ̄
− Pθ(ρ, θ;w)

ρ
,

h3i(φ, ζ,m) =
Pwi

(ρ̄, θ̄; w̄)

ρ̄
− Pwi

(ρ, θ;w)

ρ
, h4(φ) =

1

ρ̄
− 1

ρ

(1.19)



6 H. HONG EJDE-2025/23

G3(φ,u,m, ζ)

= −(u,∇)m−m divu+
(
S(w, θ)− S(w̄, θ̄)−DwS(w̄, θ̄)m− S′

θ(w̄, θ̄)ζ
)
,

(1.20)
and

G4(φ,u,m, ζ)

= −u · ∇ζ − κh5(φ, ζ)∆ζ − h6(φ, ζ,m) divu

+
2µD (u) : D (u)

ρeθ(ρ, θ)
+

λ (divu)
2

ρeθ(ρ, θ)
+

1

ρeθ(ρ, θ)

(
1

2
Q0(w, θ) + α−1

)
u2

+ h5(φ, ζ)
(
Q1(w, θ)−Q1(w̄, θ̄)

)
− h7(φ, ζ)

(
Q0(w, θ)−Q0(w̄, θ̄)

)
+

1

ρ̄eθ(ρ̄, θ̄)

(
Q1(w, θ)−Q1(w̄, θ̄)−∇wQ1(w̄, θ̄) ·m−Q′

1θ(w̄, θ̄)ζ
)

+
e(ρ̄, θ̄)

ρ̄eθ(ρ̄, θ̄)

(
Q0(w, θ)−Q0(w̄, θ̄)−∇wQ0(w̄, θ̄) ·m−Q′

0θ(w̄, θ̄)ζ
)
,

h5(φ, ζ) =
1

ρ̄eθ(ρ̄, θ̄)
− 1

ρeθ(ρ, θ)
,

h6(φ, ζ,m) =
θ̄Pθ(ρ̄, θ̄; w̄)

ρ̄eθ(ρ̄, θ̄)
− θPθ(ρ, θ;w)

ρeθ(ρ, θ)
,

h7(φ, ζ) =
e(ρ̄, θ̄)

ρ̄eθ(ρ̄, θ̄)
− e(ρ, θ)

ρeθ(ρ, θ)
.

(1.21)

Also, the initial condition (1.8) is reformulated as

(φ,u,m, ζ)(x, 0) = (φ0,u0,m0, ζ0)(x)

≡
(
ρ0(x)− ρ̄,u0,w0 − w̄, θ0(x)− θ̄

)
.

(1.22)

Based on the above reformulation, we construct the following iteration scheme:

φt + ρ̄divu+ uj−1 · ∇φ−∇wQ0(w̄, θ̄) ·m−Q′
0θ(w̄, θ̄)ζ

= G1(φ
j−1,uj−1,mj−1, ζj−1),

ut −
µ

ρ̄
∆u− µ+ λ

ρ̄
∇divu+

1

αρ̄
u+

Pρ(ρ̄, θ̄; w̄)

ρ̄
∇φ

+
Pθ(ρ̄, θ̄; w̄)

ρ̄
∇ζ +

n∑
i=1

Pwi
(ρ̄, θ̄; w̄)

ρ̄
∇mi

= G2(φ
j−1,uj−1,mj−1, ζj−1),

mt − df∆m+ w̄ divu−DwS(w̄, θ̄)m− S′
θ(w̄, θ̄)ζ

= G3(φ
j−1,uj−1,mj−1, ζj−1),

ζt +
θ̄Pθ(ρ̄, θ̄; w̄)

ρ̄eθ(ρ̄, θ̄)
divu

=
κ

ρ̄eθ(ρ̄, θ̄)
∆ζ +

∇wQ1(w̄, θ̄) ·m+Q′
1θ(w̄, θ̄)ζ

ρ̄eθ(ρ̄, θ̄)

+
e(ρ̄, θ̄)

ρ̄eθ(ρ̄, θ̄)

(
∇wQ0(w̄, θ̄) ·m+Q′

0θ(w̄, θ̄)ζ
)
+G4(φ

j−1,uj−1,mj−1, ζj−1),



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 7

(φ,u,m, ζ)(x, 0) = (φ0,u0,m0, ζ0)(x), x ∈ Ω, (1.23)

for each fixed (φj−1,uj−1,mj−1, ζj−1) ∈ XN (0, t0), j = 1, 2, . . . , where

(φ0,u0,m0, ζ0) = (φ0,u0,m0, ζ0)

and

XN (0, t0) =
{
(φ,u,m, ζ) ∈ C([0, t0];H

N (R3)) : ∇φ ∈ L2(0, t0;H
N−1(R3)),

u ∈ L2(0, t0;H
N+1(R3)), (∇m,∇ζ) ∈ L2(0, t0;H

N (R3))
}
.

Theorem 1.3. Let {(φj ,uj ,mj , ζj)}∞i=1 be the sequence of the approximate sulu-
tions given by the iteration scheme (1.23). Then, under the assumptions of Theorem
1.1, it holds that

(φj ,uj ,mj , ζj) → (φ,u,m, ζ) *-weak in L∞(0, T0;H
N (R3)),

uj → u weak in L2(0, T0;H
N+1(R3)),(

∇mj ,∇ζj
)
→

(
∇m,∇ζ

)
weak in L2(0, T0;H

N (R3)),

(φj ,uj ,mj , ζj) → (φ,u,m, ζ) strong in L2(0, T0;H
N−1
loc (R3)),

where HN−1
loc (R3) denotes the space HN−1(Ω) for any bounded domain Ω of R3,

and (φ,u,m, ζ) is the smooth unique solution to the Cauchy problem (1.17), (1.22)
on [0, T0] satisfying (1.12).

Before finishing this section, we recall the following useful Lemmas which we will
use extensively.

Lemma 1.4 ([4]). Let Ω = Rd, s1 ≥ s and s2 ≥ s be such that either

s1 + s2 − s ≥ d
( 1

q1
+

1

q2
− 1

q

)
≥ 0, sj − s > d

( 1

qj
− 1

q

)
, j = 1, 2

or

s1 + s2 − s > d
( 1

q1
+

1

q2
− 1

q

)
≥ 0, sj − s ≥ d

( 1

qj
− 1

q

)
, j = 1, 2,

then (u, v) 7→ u ·v is a continuous bilinear map from W s1
q1 (Ω)×W s1

q1 (Ω) into W s
q (Ω).

Lemma 1.5 ([21, Lemma 2.5]). Let f(φ) and f(φ,w) be smooth functions of φ
and (φ,w), respectively, with bounded derivatives of any order, and ∥φ∥L∞(R3) +
∥w∥L∞(R3) ≤ C. Then for any integer m ≥ 1, we have

∥∇mf(φ)∥Lp ≤ C∥∇mφ∥Lp ,

∥∇mf(φ,w)∥Lp ≤ C∥∇m(φ,w)∥Lp ,
(1.24)

for any 1 ≤ p ≤ ∞, where C may depend on f and m.

Lemma 1.6 ([21, Lemma 2.6]). Let α be any multi-index with |α| = k and 1 <
p < ∞. Then there exists a constant C > 0 such that

∥Dα(fg)∥Lp ≤ C(∥f∥Lp1∥∇kg∥Lp2 + ∥∇kf∥Lp3∥g∥Lp4 ),

∥[Dα, f ]g∥Lp ≤ C(∥∇f∥Lp1∥∇k−1g∥Lp2 + ∥∇kf∥Lp3∥g∥Lp4 ),
(1.25)

where f, g ∈ S is the Schwartz class, 1 ≤ p2, p3 ≤ ∞ such that 1
p = 1

p1
+ 1

p2
= 1

p3
+ 1

p4
,

and [Dα, f ]g = Dα(fg)− fDαg.



8 H. HONG EJDE-2025/23

2. Linearized Problem

In this section, we study the linearized system

φt + ρ̄divu+A · ∇φ− ā0 ·m− b̄0ζ = G1,

ut −
µ

ρ̄
∆u− µ+ λ

ρ̄
∇divu+

1

αρ̄
u+

P̄ρ

ρ̄
∇φ+

P̄θ

ρ̄
∇ζ +

n∑
i=1

P̄wi

ρ̄
∇mi = G2,

mt − df∆m+ w̄ divu− M̄m− B̄ζ = G3,

ζt +
θ̄P̄θ

ρ̄ēθ
divu =

κ

ρ̄ēθ
∆ζ +

ā1 ·m+ b̄1ζ

ρ̄ēθ
+

ē

ρ̄ēθ

(
ā0 ·m+ b̄0ζ

)
+G4,

(2.1)

where A, G1,G2,G3 and G4 are the given functions, and we used the following
notations: ā0 = ∇wQ0(w̄, θ̄), b̄0 = Q′

0θ(w̄, θ̄), P̄ρ = Pρ(ρ̄, θ̄; w̄), P̄θ = Pθ(ρ̄, θ̄; w̄),
P̄wi

= Pwi
(ρ̄, θ̄; w̄), M̄ = DwS(w̄, θ̄), B̄ = S′

θ(w̄, θ̄), ēθ = eθ(ρ̄, θ̄), ē = e(ρ̄, θ̄),
ā1 = ∇wQ1(w̄, θ̄) and b̄1 = Q′

1θ(w̄, θ̄).
We first consider an estimate for the linearized system (2.1).

Theorem 2.1. Let 0 < T < ∞. Assume that (φ0,u0,m0, ζ0) ∈ HN (R3) for an
integer N ≥ 3. Also, suppose that

A ∈ L∞(0, T ;HN (R3)), G1 ∈ L2(0, T ;HN (R3))

G2,G3, G4 ∈ L2(0, T ;HN−1(R3)).
(2.2)

Let

(φ,u,m, ζ) ∈ C([0, T ];HN (R3)), ∇φ ∈ L2(0, T ;HN−1(R3)),

u ∈ L2(0, T ;HN+1(R3)), (∇m,∇ζ) ∈ L2(0, T ;HN (R3))
(2.3)

and (φ,u,m, ζ) be a solution of (2.1). Then there exist positive constants C0 and
c0, independent of t, such that

∥(φ,u,m, ζ)(t)∥2HN +

∫ t

0

∥∇φ∥2HN−1dτ +

∫ t

0

∥u∥2HN+1dτ +

∫ t

0

∥(∇m,∇ζ)∥2HNdτ

≤ C0

[
∥(φ,u,m, ζ)(0)∥2HN +

∫ T

0

(
∥G1∥2HN + ∥

(
G2,G3, G4

)
∥2HN−1

)
dτ

]
×
[
1 + ec0

∫ T
0

(1+∥A∥2
HN )dτ

∫ T

0

(
1 + ∥A∥2HN

)
dτ

]
(2.4)

for each t ∈ [0, T ].



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 9

Proof. Applying ∇k to (2.1) yields

∇kφt + ρ̄div∇ku+∇k
(
A · ∇φ

)
− ā0 · ∇km− b̄0∇kζ = ∇kG1,

∇kut −
µ

ρ̄
∆∇ku− µ+ λ

ρ̄
∇div∇ku+

1

αρ̄
∇ku

+
P̄ρ

ρ̄
∇k+1φ+

P̄θ

ρ̄
∇k+1ζ +

n∑
i=1

P̄wi

ρ̄
∇k+1mi = ∇kG2,

∇kmt − df∆∇km+ w̄ div∇ku− M̄∇km− B̄∇kζ = ∇kG3,

∇kζt +
θ̄P̄θ

ρ̄ēθ
div∇ku− κ

ρ̄ēθ
∆∇kζ − ā1 · ∇km+ b̄1∇kζ

ρ̄ēθ

− ē

ρ̄ēθ

(
ā0 · ∇km+ b̄0∇kζ

)
= ∇kG4,

(2.5)

where k = 0, 1, . . . , N .
Multiplying (2.5)2 by ∇ku and using (2.5)1, we have

1

2

d

dt

(
∥∇ku∥2 + P̄ρ

ρ̄2
∥∇kφ∥2

)
+

1

ρ̄

∫ (
µ|∇k+1u|2 + (λ+ µ)|div∇ku|2

)
dx

+
1

αρ̄2
∥∇ku∥2 − P̄θ

ρ̄

∫
∇kζ div∇kudx−

n∑
i=1

P̄wi

ρ̄

∫
∇kmi div∇kudx

− P̄ρ

ρ̄2

∫
∇kφ

(
ā0 · ∇km+ b̄0∇kζ

)
dx

=

∫
∇kG2 · ∇ku dx+

P̄ρ

ρ̄2

∫
∇kφ∇kG1dx− P̄ρ

ρ̄2

∫
divA|∇kφ|2 dx

− P̄ρ

ρ̄2

∫ [
∇k,A

]
· ∇φ∇kφdx,

(2.6)

where we used

∇k (A · ∇φ) = A · ∇∇kφ−
[
∇k,A

]
· ∇φ.

Multiplying (2.5)3 by
( P̄w1

ρ̄w̄1
∇km1, . . . ,

P̄wn

ρ̄w̄n
∇kmn

)
, we have

n∑
i=1

P̄wi

ρ̄w̄i

( d

2dt
∥∇kmi∥2 + df∥∇k+1mi∥2

)
+

n∑
i=1

P̄wi

ρ̄

∫
div∇ku∇kmi dx

−
∫ (

M̄∇km+ B̄∇kζ
)
·
( P̄w1

ρ̄w̄1
∇km1, . . . ,

P̄wn

ρ̄w̄n
∇kmn

)
dx

=

∫
∇kG3 ·

( P̄w1

ρ̄w̄1
∇km1, . . . ,

P̄wn

ρ̄w̄n
∇kmn

)
dx.

(2.7)

Multiplying (2.5)4 by ēθ
θ̄
∇kζ, we have

ēθ
2θ̄

d

dt
∥∇kζ∥2 + κ

ρ̄θ̄
∥∇k+1ζ∥2 + P̄θ

ρ̄

∫
div∇ku∇kζ dx

− 1

ρ̄θ̄

∫ (
ā1 · ∇km+ b̄1∇kζ

)
∇kζ dx+

ē

ρ̄θ̄

∫ (
ā0 · ∇km+ b̄0∇kζ

)
∇kζ dx

=
ēθ
θ̄

∫
∇kG2∇kζdx.

(2.8)



10 H. HONG EJDE-2025/23

Adding (2.6)-(2.8) yields

1

2

d

dt

(
∥∇ku∥2 + P̄ρ

ρ̄2
∥∇kφ∥2 +

n∑
i=1

P̄wi

ρ̄w̄i
∥∇kmi∥2 +

ēθ
θ̄
∥∇kζ∥2

)
+

1

ρ̄

∫ (
µ|∇k+1u|2 + (λ+ µ)|div∇ku|2

)
dx+

1

αρ̄2
∥∇ku∥2

+

n∑
i=1

df P̄wi

ρ̄w̄i
∥∇k+1mi∥2 +

κ

ρ̄θ̄
∥∇k+1ζ∥2

= J1
k + J2

k + J3
k ,

(2.9)

where

J1
k = −

∫ (
M̄∇km+ B̄∇kζ

)
·
( P̄w1

ρ̄w̄1
∇km1, . . . ,

P̄wn

ρ̄w̄n
∇kmn

)
dx

− P̄ρ

ρ̄2

∫
∇kφ

(
ā0 · ∇km+ b̄0∇kζ

)
dx− 1

ρ̄θ̄

∫ (
ā1 · ∇km+ b̄1∇kζ

)
∇kζ dx

+
ē

ρ̄θ̄

∫ (
ā0 · ∇km+ b̄0∇kζ

)
∇kζ dx,

J2
k = − P̄ρ

ρ̄2

∫
divA|∇kφ|2dx− P̄ρ

ρ̄2

∫ [
∇k,A

]
· ∇φ∇kφdx,

J3
k =

∫
∇kG2 · ∇ku dx+

P̄ρ

ρ̄2

∫
∇kφ∇kG1dx+

ēθ
θ̄

∫
∇kG4∇kζ dx

+

∫
∇kG3 ·

( P̄w1

ρ̄w̄1
∇km1, . . . ,

P̄wn

ρ̄w̄n
∇kmn

)
dx.

(2.10)
Summing (2.9) for k = 0, 1, . . . , N we obtain

1

2

d

dt

(
∥u∥2HN +

P̄ρ

ρ̄2
∥φ∥2HN +

n∑
i=1

P̄wi

ρ̄w̄i
∥mi∥2HN +

ēθ
θ̄
∥ζ∥2HN

)
+

1

ρ̄
∥∇u∥2HN +

1

αρ̄2
∥u∥2HN +

df P̄min

ρ̄w̄min
∥∇m∥2HN +

κ

ρ̄θ̄
∥∇ζ∥2HN

≤
N∑

k=0

J1
k +

N∑
k=0

J2
k +

N∑
k=0

J3
k ,

(2.11)

where w̄min = min{w̄1, . . . , w̄n} > 0 and P̄min = min{P̄1, . . . , P̄n} > 0 because of
(1.3).

Using Hölder inequality and (1.25), we obtain from (2.10) that

N∑
k=0

J1
k ≤ C

N∑
k=0

(
∥∇km∥+ ∥∇kζ∥

)(
∥∇km∥+ ∥∇kφ∥+ ∥∇kζ∥

)
≤ C

(
∥m∥2HN + ∥φ∥2HN + ∥ζ∥2HN

)
,

(2.12)

N∑
k=0

J2
k ≤ C

N∑
k=0

(
∥∇A∥L∞∥∇kφ∥+ ∥∇φ∥L∞∥∇kA∥

)
∥∇kφ∥

≤ C∥A∥HN ∥φ∥2HN

(2.13)



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 11

and

N∑
k=0

J3
k ≤ C

N−1∑
k=0

(
∥∇kG2∥∥∇ku∥+ ∥∇kG3∥∥∇km∥+ ∥∇kG4∥∥∇kζ∥

)
+ C

N∑
k=0

∥∇kG1∥∥∇kφ∥+ C
∣∣ ∫ ∇N−1G2 · ∇N+1udx

∣∣
+ C

∣∣ ∫ ∇N−1G4∇N+1ζdx
∣∣

+ C
∣∣∣ ∫ ∇N−1G3 ·

( P̄w1

ρ̄w̄1
∇N+1m1, . . . ,

P̄wn

ρ̄w̄n
∇N+1mn

)
dx

∣∣∣
≤ C

(
∥G1∥HN ∥φ∥HN + ∥(G2,G3, G4)∥HN−1∥∇(u,m, ζ)∥HN

)
.

(2.14)

Substituting (2.12)-(2.14) into (2.11), we obtain

d

dt

(
∥u∥2HN +

P̄ρ

ρ̄2
∥φ∥2HN +

n∑
i=1

P̄wi

ρ̄w̄i
∥mi∥2HN +

ēθ
θ̄
∥ζ∥2HN

)
+

1

ρ̄
∥∇u∥2HN +

2

αρ̄2
∥u∥2HN +

df P̄min

ρ̄w̄min
∥∇m∥2HN +

κ

ρ̄θ̄
∥∇ζ∥2HN

≤ C
(
∥m∥2HN + ∥φ∥2HN + ∥ζ∥2HN

)
+ C∥A∥HN ∥φ∥2HN

+ C
(
∥G1∥2HN + ∥(G2,G3, G4)∥2HN−1

)
.

(2.15)

On the other hand, noticing that∫
∇kut · ∇k+1φdx

=
d

dt

∫
∇ku · ∇k+1φdx− ρ̄∥∇k divu∥2 −

∫
∇k divu∇k

(
A · ∇φ

)
dx

+

∫
∇k divu

(
ā0 · ∇km+ b̄0∇kζ

)
dx+

∫
∇k divu∇kG1dx

because of (2.5)1, and multiplying (2.5)2 by ∇k+1φ we have

d

dt

∫
∇ku · ∇k+1φdx+

P̄ρ

ρ̄
∥∇k+1φ∥2 = J4

k + J5
k + ρ̄∥∇k divm∥2, (2.16)

where

J4
k =

∫
∇kG2 · ∇k+1φdx− P̄θ

ρ̄

∫
∇k+1ζ · ∇k+1φdx

− 1

αρ̄

∫
∇ku · ∇k+1φdx

+
µ

ρ̄

∫
∆∇ku · ∇k+1φdx+

µ+ λ

ρ̄

∫
∇ div∇ku · ∇k+1φdx

−
∫

∇k divu∇kG1dx−
n∑

i=1

P̄wi

ρ̄

∫
∇k+1mi · ∇k+1φdx

−
∫

∇k divu
(
ā0 · ∇km+ b̄0∇kζ

)
dx

(2.17)



12 H. HONG EJDE-2025/23

and

J5
k =

∫
∇k divu∇k

(
A · ∇φ

)
dx. (2.18)

Using Hölder inequality and (1.25), we obtain from (2.17) and (2.18) that

N−1∑
k=0

J4
k ≤ C

(
∥G2∥HN−1 + ∥(ζ,u)∥HN + ∥∇u∥HN

)
∥φ∥HN

+ C∥u∥HN ∥G1∥HN−1 + C∥m∥HN ∥φ∥HN

+ C∥u∥HN (∥m∥HN−1 + ∥ζ∥HN−1)

≤ C
(
∥G1∥2HN + ∥G1∥2HN−1 + ∥(φ,u,m, ζ)∥2HN

)
+ C∥∇u∥HN ∥φ∥HN

(2.19)

and
N−1∑
k=0

J3
k ≤ C

N−1∑
k=0

∥∇k+1u∥
(
∥A∥L∞∥∇k+1φ∥+ ∥∇φ∥L∞∥∇kA∥

)
≤ C∥A∥HN ∥∇u∥HN−1∥φ∥HN ,

(2.20)

respectively.
Summing (2.16) for k = 0, 1, . . . , N − 1, and using (2.19) and (2.20), we obtain

d

dt

∫
∇ku · ∇k+1φdx+

P̄ρ

ρ̄
∥∇k+1φ∥2

≤ ∥∇u∥2HN + C
(
1 + ∥A∥2HN

)
∥φ∥2HN

+ C
(
∥G1∥2HN + ∥G1∥2HN−1 + ∥(φ,u,m, ζ)∥2HN

)
.

(2.21)

We can assume 0 < ϵ ≤ 1 without loss of generality. And we choose β ∈ (0, 1] to
be suitably small. Then, adding (2.15) and β × (2.21), we obtain

d

dt
E(t) +

βP̄ρ

ρ̄
∥∇φ∥2HN−1 + (ρ̄−1 − β)∥∇m∥2HN

+
2

αρ̄2
∥u∥2HN +

df P̄min

ρ̄w̄min
∥∇m∥2HN +

κ

ρ̄θ̄
∥∇ζ∥2HN

≤ C
(
∥G1∥2HN + ∥(G2,G3, G4)∥2HN−1

)
+ C

(
1 + ∥A∥2HN

)
∥
(
φ,u,m, ζ

)
∥2HN ,

(2.22)

where

E(t) := ∥u∥2HN + β

N−1∑
k=0

∫
∇ku · ∇k+1φdx+

P̄ρ

ρ̄2
∥ϱ∥2HN

+

n∑
i=1

P̄wi

ρ̄w̄i
∥mi∥2HN +

ēθ
θ̄
∥ζ∥2HN .

(2.23)

By (2.23) and (1.3), we can choose a small β ∈ (0, 1
2ρ̄ ] independent ϵ ∈ (0, 1]

such that

E(t) ⋍ ∥(φ,u,m, ζ)(t)∥2HN (2.24)

uniformly for all t ∈ [0, T ].



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 13

Integrating (2.22) over t ∈ [0, T ], and using (2.24), (1.3) and the smallest of β,
we have

∥(φ,u,m, ζ)(t)∥2HN +

∫ t

0

∥∇φ∥2HN−1dτ +

∫ t

0

∥u∥2HN+1dτ +

∫ t

0

∥
(
∇m,∇ζ

)
∥2HNdτ

≤ ∥(φ,u,m, ζ)(0)∥2HN + C

∫ t

0

(
1 + ∥A∥2HN

)
∥
(
φ,u,m, ζ

)
∥2HNdτ

+ C

∫ t

0

(
∥G1∥2HN + ∥(G2,G3, G4)∥2HN−1

)
dτ.

(2.25)
Applying Gronwall inequality to (2.25) yields

∥(φ,u,m, ζ)(t)∥2HN ≤ CΛ1(T )e
CΛ2(T ), (2.26)

where

Λ1(T ) = ∥(φ,u,m, ζ)(0)∥2HN +

∫ T

0

(
∥G1∥2HN + ∥(G2,G3, G4)∥2HN−1

)
dτ,

Λ2(T ) =

∫ T

0

(
1 + ∥A∥2HN

)
dτ.

By (2.25) and (2.26), we obtain (2.4). The proof of Theorem 2.1 is complete. □

Next, applying Theorem 2.1, it is easy to prove the existence and uniqueness of
the smooth solution for the linearized system (2.1) by the standard methods. We
will omit the proof for brevity.

Theorem 2.2. Let 0 < T < ∞. Assume that (φ0,u0,m0, ζ0) ∈ HN (R3) for an
integer N ≥ 3. Also, suppose that (2.2) holds. Then, there exists a unique solution
(φ,m,u, ζ) of the linearized system (2.1) satisfying (2.3) and (2.4).

3. Proof of main results

We first prove the existence of the smooth local solutions to the reformulated
system (1.17), (1.22) using Theorem 2.1. To this end, we define a set

ZM (0, t0) =
{
(φ,u,m, ζ) ∈ XN (0, t0) : ∥(φ,u,m, ζ)∥2XN (0,t0)

≤ M,

0 < m−1
0 ≤ ρ̄+ φ(x, t), w̄ +m(x, t), θ̄ + ζ(x, t) ≤ m0

}
for a positive constant m0 > 1 and an integer N ≥ 3, where

M = 4C0∥(φ0,u0,m0, ζ0)∥2HN , C0 is the constant determined in (2.4), (3.1)

and

∥(φ,u,m, ζ)∥2XN (0,t0)
= sup

0≤t≤t0

∥(φ,u,m, ζ)(t)∥2HN +

∫ t0

0

∥∇φ∥2HN−1dτ

+

∫ t0

0

∥u∥2HN+1dτ +

∫ t

0

∥
(
∇m,∇ζ

)
∥2HNdτ.

(3.2)

Noticing that

Q0(w, θ)−Q0(w̄, θ̄)−∇wQ0(w̄, θ̄) ·m−Q′
0θ(w̄, θ̄)ζ = O(m2 + ζ2),



14 H. HONG EJDE-2025/23

and using Lemma 1.4 and (1.24), we have∫ t0

0

∥φj−1 divuj−1∥2HNdτ ≤ C sup
0≤t≤t0

∥φj−1∥2HN

∫ t0

0

∥∇uj−1∥2HNdτ ≤ CM2,

and∫ t0

0

∥Q0(w
j−1, θj−1)−Q0(w̄, θ̄)−∇wQ0(w̄, θ̄) ·mj−1 −Q′

0θ(w̄, θ̄)ζj−1∥2HNdτ

≤ C sup
0≤t≤t0

∥(mj−1, ζj−1)∥2HN

∫ t0

0

∥(mj−1, ζj−1)∥2HNdτ ≤ CM2t0

for (φj−1,uj−1,mj−1, ζj−1) ∈ ZN (0, t0), j = 1, 2, . . . . Therefore, from (1.18) we
obtain that

G1(φ
j−1,uj−1,mj−1, ζj−1) ∈ L2(0, t0;H

N (R3)),∫ t0

0

∥G1(φ
j−1,uj−1,mj−1, ζj−1)∥2HNdτ ≤ CM2(1 + t0).

(3.3)

By using Lemma 1.4 and (1.24), we have∫ t0

0

∥(uj−1,∇)uj−1∥2HN−1dτ ≤ C sup
0≤t≤t0

∥uj−1∥2HN

∫ t0

0

∥∇uj−1∥2HNdτ ≤ CM2,∫ t0

0

∥(Q0(w, θ)−Q0(w̄, θ̄))u∥2HN−1dτ

≤ C sup
0≤t≤t0

∥(mj−1, ζj−1)∥2HN sup
0≤t≤t0

∥uj−1∥2HN−1t0 ≤ CM2t0,∫ t0

0

∥h1(φ
j−1, ζj−1,mj−1)∇φj−1∥2HN−1dτ

≤ C sup
0≤t≤t0

∥(φj−1, ζj−1,mj−1)∥2HN

∫ t0

0

∥∇φj−1∥2HNdτ ≤ CM2,∫ t0

0

∥h2(φ
j−1, ζj−1,mj−1)∇ζj−1∥2HN−1dτ

≤ C sup
0≤t≤t0

∥(φj−1, ζj−1,mj−1)∥2HN

∫ t0

0

∥∇ζj−1∥2HNdτ ≤ CM2,

n∑
i=1

∫ t0

0

∥h3i(φ
j−1, ζj−1,mj−1)∇mj−1

i ∥2HN−1dτ

≤ C sup
0≤t≤t0

∥(φj−1, ζj−1,mj−1)∥2HN

∫ t0

0

∥∇mj−1∥2HNdτ ≤ CM2,∫ t0

0

∥h4(φ
j−1)

(
µ∆uj−1 + (µ+ ν)∇divuj−1

)
∥2HN−1dτ

≤ C sup
0≤t≤t0

∥φj−1∥2HN

∫ t0

0

∥∇uj−1∥2HNdτ ≤ CM2,∫ t0

0

∥h4(φ
j−1)uj−1∥2HN−1dτ ≤ C sup

0≤t≤t0

∥φj−1∥2HN sup
0≤t≤t0

∥uj−1∥2HN−1t0 ≤ CM2t0



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 15

for (φj−1,uj−1,mj−1, ζj−1) ∈ ZN (0, t0), j = 1, 2, . . . . Therefore, from (1.19) we
obtain

G2(φ
j−1,uj−1,mj−1, ζj−1) ∈ L2(0, t0;H

N−1(R3)),∫ t0

0

∥G2(φ
j−1,uj−1,mj−1, ζj−1)∥2HN−1dτ ≤ CM2(1 + t0).

(3.4)

By similar arguments, we obtain from (1.20) and (1.21), respectively, that

G3(φ
j−1,uj−1,mj−1, ζj−1),∫ t0

0

∥G3(φ
j−1,uj−1,mj−1, ζj−1)∥2HN−1dτ ≤ CM2(1 + t0)

(3.5)

and
G4(φ

j−1,uj−1,mj−1, ζj−1),∫ t0

0

∥G4(φ
j−1,uj−1,mj−1, ζj−1)∥2HN−1dτ ≤ CM3(1 + t0).

(3.6)

Also, we have

uj−1 ∈ L∞(0, t0;H
N (R3)), sup

0≤t≤t0

∥uj−1∥HN ≤ M. (3.7)

From (3.3)-(3.7) and Theorem 2.2, there exists a unique solution (φj ,uj ,mj , ζj) ∈
XN (0, t0) to the linearized system (1.23) satisfying

∥(φj ,uj ,mj , ζj)(t)∥2HN +

∫ t

0

∥∇φj∥2HN−1dτ +

∫ t

0

∥∇uj∥2HN+1dτ

+

∫ t

0

∥
(
∇mj ,∇ζj

)
∥2HNdτ

≤ C0

[
∥(φ0,u0,m0, ζ0)∥2HN +

∫ t0

0

(
∥Gj−1

1 ∥2HN + ∥(Gj−1
2 ,Gj−1

3 , Gj−1
4 )∥2HN−1

)
dτ

]
×
[
1 + t0

(
1 + sup

0≤t≤t0

∥uj−1∥2HN

)
e
c0t0

(
1+sup0≤t≤t0

∥uj−1∥2
HN

)]
(3.8)

for any t ∈ [0, t0], where

Gj−1
1 = G1(φ

j−1,uj−1,mj−1, ζj−1), Gj−1
2 = G2(φ

j−1,uj−1,mj−1,

ζj−1), Gj−1
3 = G3(φ

j−1,uj−1,mj−1, ζj−1),

Gj−1
4 = G4(φ

j−1,uj−1,mj−1, ζj−1),

and we used that∫ t0

0

(
1 + ∥uj−1∥2HN

)
dτ ≤ t0

(
1 + sup

0≤t≤t0

∥uj−1∥2HN

)
.

Therefore, choosing t0 small such that

t0

(
1 + sup

0≤t≤t0

∥uj−1∥2HN

)
e
c0t0

(
1+sup0≤t≤t0

∥uj−1∥2
HN

)
≤ 1

and∫ t0

0

(
∥Gj−1

1 ∥2HN + ∥(Gj−1
2 ,Gj−1

3 , Gj−1
4 )∥2HN−1

)
dτ ≤ ∥(φ0,u0,m0, ζ0)∥2HN ,



16 H. HONG EJDE-2025/23

which is possible because of (3.3)-(3.7), from (3.8) we obtain

∥(φj ,uj ,mj , ζj)(t)∥2HN +

∫ t

0

∥∇φj∥2HN−1dτ +

∫ t

0

∥∇uj∥2HN+1dτ

+

∫ t

0

∥
(
∇mj ,∇ζj

)
∥2HNdτ

≤ 4C0∥(φ0,u0,m0, ζ0)∥2HN

(3.9)

for any t ∈ [0, t0]. By using (1.10), ρ̄ + φ(x, t) = ρ0(x) + (φ(x, t) − φ(x, 0)),
w̄+m(x, t) = w0(x)+(m(x, t)−m(x, 0)), and ζ̄+φ(x, t) = θ0(x)+(ζ(x, t)−ζ(x, 0)),
we can choose the small t0 > 0 such that

m−1
0 ≤ ρ̄+ φ(x, t), w̄ +m(x, t), θ̄ + ζ(x, t) ≤ m0. (3.10)

By (3.9), (3.10), (3.1) and (3.2), we obtain

(φj ,uj ,mj , ζj) ∈ ZN (0, t0)

for j = 1, 2, . . . . Moreover, by using (3.9) and (1.11), we have

{(φj ,uj ,mj , ζj)}∞j=1 is bounded in L∞(0, t0;H
N (R3)),

{∇φj}∞j=1 is bounded in L2(0, t0;H
N−1(R3)),

{uj}∞j=1 is bounded in L2(0, t0;H
N+1(R3)),

{
(
∇mj ,∇ζj

)
}∞j=1 is bounded in L2(0, t0;H

N (R3)).

(3.11)

Also, by using (3.11) and (3.3)-(3.7), from (1.23) we obtain

{∂t(φj ,uj ,mj , ζj)}∞j=1 is bounded in L2(0, t0;H
N−1(R3)). (3.12)

By using (3.11), (3.12) and the compactness result, there exist the subsequences of
{(φj ,uj ,mj , ζj)} (denoting them as {(φj ,uj ,mj , ζj)} still) such that when j → ∞,
it holds that

(φj ,uj ,mj , ζj) → (φ,u,m, ζ) *-weak in L∞(0, t0;H
N (R3)),

∇φj → ∇φ weak in L2(0, t0;H
N−1(R3)),

uj → u weak in L2(0, t0;H
N+1(R3)),(

∇mj ,∇ζj
)
→

(
∇m,∇ζ

)
weak in L2(0, t0;H

N (R3)),

(φj ,uj ,mj , ζj) → (φ,u,m, ζ) strong in L2(0, t0;H
N−1
loc (R3)),

(3.13)

and

(φ,u,m, ζ) ∈ L∞(0, t0;H
N (R3)), ∇φ ∈ L2(0, t0;H

N−1(R3)),

u ∈ L2(0, t0;H
N+1(R3)),

(
∇m,∇ζ

)
∈ L2(0, t0;H

N (R3)).
(3.14)



EJDE-2025/23 LOCAL SOLUTIONS FOR A BRINKMAN SYSTEM 17

By using (3.13) and (1.19)-(1.21), we have∫ t0

0

∫
uj−1 · ∇φjzη(t) dx dt →

∫ t0

0

∫
u · ∇φzη(t) dx dt,∫ t0

0

∫
G1(φ

j−1,uj−1,mj−1, ζj−1)zη(t) dx dt →
∫ t0

0

∫
G1(φ,u,m, ζ)zη(t) dx dt,∫ t0

0

∫
G2(φ

j−1,uj−1,mj−1, ζj−1) · zη(t) dx dt →
∫ t0

0

∫
G2(φ,u,m, ζ) · zη(t)dxdt,∫ t0

0

∫
G3(φ

j−1,uj−1,mj−1, ζj−1) · zη(t) dx dt →
∫ t0

0

∫
G3(φ,u,m, ζ) · zη(t)dxdt,∫ t0

0

∫
G4(φ

j−1,uj−1,mj−1, ζj−1)zη(t) dx dt →
∫ t0

0

∫
G4(φ,u,m, ζ)zη(t) dx dt

(3.15)
for all z ∈ {C∞

0 (R3)}3, z ∈ C∞
0 (R3) and η ∈ C∞

0 (0, t0). Moreover, by using (3.13)
and (3.15), it is easy to check that (φ,u,m, ζ) is a solution to the system (1.17),
(1.22). Then, setting

ρ = ρ̄+ φ, w = w̄ +m, θ = θ̄ + ζ,

and by using (1.17), (1.22), (1.18)-(1.21), and (3.14), we know that (ρ,u,w, θ) is
a solution to the Cauchy problem (1.1), (1.2) , (1.8) satisfying (1.12). Also, the
estimate (1.13) follows from (3.9) and (3.13). The proof for the uniqueness of the
local solution is standard, so we will omit it for brevity. The proof of Theorem 1.1
is complete.

Theorem 1.3 follows form (3.13) and the uniqueness of the solutions to the
Cauchy problem (1.1), (1.2) , (1.8).

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Hakho Hong

Institute of Mathematics, State Academy of Sciences, Pyongyang, Democratic People’s

Republic of Korea
Email address: hhhong@star-co.net.kp


	1. Introduction
	Notation

	2. Linearized Problem
	3. Proof of main results
	References