Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)

Lotfi Jlali; Jamel Benameur

doi:10.1515/dema-2024-0013

Open Access Published by De Gruyter Open Access August 6, 2024

Long time decay of incompressible convective Brinkman-Forchheimer in L²(ℝ³)

Lotfi Jlali and Jamel Benameur

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2024-0013

Abstract

In this article, we study the global existence, uniqueness, and continuity for the solution of incompressible convective Brinkman-Forchheimer on the whole space R 3 when 4 μ β ≥ 1 . Additionally, we give an asymptotic type of convergence of the global solution.

Keywords: the convective Brinkman-Forchheimer; global solution; blow-up

MSC 2010: 35Q35; 35A02; 35B44

1 Introduction

The incompressible convective Brinkman-Forchheimer equations on the whole space R 3 are given by

( CBF ) ∂ t u − μ Δ u + u ⋅ ∇ u + α u + β ∣ u ∣ r − 1 u + ∇ p = 0 , in R + × R 3 , div u = 0 , in R + × R 3 , u ( 0 , x ) = u 0 ( x ) , in R 3 ,

where u ( x , t ) = ( u 1 , u 2 , u 3 ) denotes the velocity field, and the scalar function p ( x , t ) represents the pressure. The constant μ denotes the positive Brinkman coefficient (effective viscosity). Besides, α and β , which are two positive constants, denote, respectively, the Darcy (permeability of porous medium) and Forchheimer (proportional to the porosity of the material) coefficients. The exponent r can be greater than or equal to 1.

The equations Convective Brinkman Forchheimer (CBF) describe the motion of incompressible fluid flows in a saturated porous medium. This model has been used in connection with some real-world phenomena, e.g., in the theory of non-Newtonian fluids (see e.g., Shenoy [1]) or in tidal dynamics (see, e.g., Gordeev [2]; Likhtarnikov [3]).

In what follows, we will consider only the critical homogenous CBF equations when r = 3 :

( S ) ∂ t u − μ Δ u + u ⋅ ∇ u + α u + β ∣ u ∣ 2 u + ∇ p = 0 , in R + × R 3 , div u = 0 , in R + × R 3 , u ( 0 , x ) = u 0 ( x ) , in R 3 .

The first equation of system ( S ) has the same scaling as the Navier-Stokes equations ( N S E ) only when the permeability coefficient α is equal to zero. In this case, when ( α = 0 ) is called damped NSE where the damping term is β ∣ u ∣ r − 1 u . The resistance to the motion of the flow, which is caused by physical factors such as porous media flow, drag or friction, or other dissipative mechanisms, is shaped by the damping term β ∣ u ∣ r − 1 u . Several authors studied the damped NSE among them Cai and Jiu [4] proved the global existence of weak solution in

L ∞ ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L r + 1 ( R + , L r + 1 ( R 3 ) ) .

Benameur [5] brought new findings to the field, which is the model damping a ( e b ∣ u ∣ 2 − 1 ) u . He used the Friedrich method and some new tools to prove that there is a global solution u in

L ∞ ( R + , L 2 ( R 3 ) ) ∩ C ( R + , H − 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ ℰ b ,

where ℰ b = { f ∈ L 4 ( R + × R 3 ) : ( e b ∣ f ∣ 2 − 1 ) ∣ f ∣ 2 ∈ L 1 ( R + × R 3 ) } .

Before treating the global existence, we consider the definition of weak solutions.

Definition 1.1

The function pair ( u , p ) is called a weak solution of the problem ( S ) if for any T > 0 , the following conditions are satisfied:

u ∈ L ∞ ( [ 0 , T ] ; L σ 2 ( R 3 ) ) ∩ L 2 ( [ 0 , T ] ; H ˙ 1 ( R 3 ) ) ∩ L 4 ( [ 0 , T ] , L 4 ( R 3 ) ) .
∂ t u − μ Δ u + u ⋅ ∇ u + α u + β ∣ u ∣ 2 u = − ∇ p in D ′ ( [ 0 , T ] × R 3 ) : for any Φ ∈ C 0 ∞ ( [ 0 , T ] × R 3 ) such that div Φ ( t , x ) = 0 for all ( t , x ) ∈ [ 0 , T ] × R 3 and Φ ( T ) = 0 , we have
− ∫ 0 T ( u ; ∂ t Φ ) L 2 + ∫ 0 T ( ∇ u ; ∇ Φ ) L 2 + α ∫ 0 T ( u ; Φ ) L 2 + ∫ 0 T ( ( u ⋅ ∇ ) u ; Φ ) L 2 + β ∫ 0 T ( ∣ u ∣ 2 u ; Φ ) L 2 = ( u 0 ; Φ ( 0 ) ) L 2 .
div u ( x , t ) = 0 for a.e. ( t , x ) ∈ [ 0 , T ] × R 3 . ( ( ; ) L 2 is the inner product in L 2 ( R 3 ) .)

Theorem 1.1

Let u 0 ∈ L 2 ( R 3 ) be a divergence-free vector fields and 4 μ β ≥ 1 , then there is a unique global solution u ∈ C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) of ( S ) . Moreover, we have

(1.1) ‖ u ( t ) ‖ L 2 2 + 2 μ ∫ 0 t ‖ ∇ u ‖ L 2 2 + 2 α ∫ 0 t ‖ u ‖ L 2 2 + 2 β ∫ 0 t ‖ u ‖ L 4 4 d ≤ ‖ u 0 ‖ L 2 2 , ∀ t ≥ 0 ,

(1.2) e 2 α t ‖ u ( t ) ‖ L 2 2 + 2 μ ∫ 0 t e 2 α z ‖ ∇ u ( z ) ‖ L 2 2 d z + 2 β ∫ 0 t e 2 α z ‖ u ( z ) ‖ L 4 4 d z ≤ ‖ u 0 ‖ L 2 2 , ∀ t ≥ 0 .

Remark 1.1

Unlike in other blow-up problems (see [6–8]) when the nonlinear capacity methods is used, in this work, the proof of existence and uniqueness is based on the Friedrich method. The continuity and uniqueness are proved by new tools.
The proofs of (1.1) and (1.2) are given by the Friedrich method and Cantor diagonal process, respectively.
The uniqueness and inequality (1.2) imply that ( t ↦ e 2 α t ‖ u ( t ) ‖ L 2 2 ) is decreasing. Indeed, for 0 ≤ t 1 ≤ t 2 , by uniqueness of this solution, we obtain
e 2 α ( t 2 − t 1 ) ‖ u ( t 2 ) ‖ L 2 2 = e 2 α ( t 2 − t 1 ) ‖ u ( t 1 + ( t 2 − t 1 ) ) ‖ L 2 2 ≤ ‖ u ( t 1 ) ‖ L 2 2
and
e 2 α t 2 ‖ u ( t 2 ) ‖ L 2 2 ≤ e 2 α t 1 ‖ u ( t 1 ) ‖ L 2 2 .
The last property does not imply that lim t → ∞ e α t ‖ u ( t ) ‖ L 2 = 0 .
The idea of looking for an asymptotic type of convergence e α t ‖ u ( t ) ‖ L 2 to zero is due to the following: the solution of linear system
( SL ) ∂ t f − μ Δ f + α f = 0 , in R + × R 3 , div f = 0 , in R + × R 3 , f ( 0 , x ) = u 0 ( x ) , in R 3 ,
has the following form:
f ( t , x ) = e − α t e μ t Δ u 0 .
We have
e α t ‖ f ( t ) ‖ L 2 = ‖ e μ t Δ u 0 ‖ L 2 .
The dominated convergence theorem implies that
lim t → ∞ e α t ‖ f ( t ) ‖ L 2 = 0 .

We are now ready to state the second result.

Theorem 1.2

Let u 0 ∈ L 2 ( R 3 ) be a divergence-free vector fields and 4 μ β ≥ 1 . If u ∈ C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) is the unique global solution of (S) given by Theorem 1.1, then

(1.3) ‖ u ( t ) ‖ L 2 = o ( e − α t ) , t → ∞ .

The remainder of this article is organized as follows. In Second 2, we give some notations, definitions, and preliminary results. Section 3 is devoted to prove Theorem 1.1, and this proof is done in two steps. In the first, we give a general result of equicontinuity, while in the second, we apply the Friedritch method to construct a global solution of ( S ) . Moreover, uniqueness and the continuity of solution are also mentioned. In Section 4, we study the large time decay.

2 Notations and preliminary results

2.1 Notations

For a function f : R 3 ⟶ R ¯ and R > 0 , the Friedrich operator J R is defined by
J R ( D ) = ℱ − 1 ( χ B R f ^ ) ,
where B R is the ball of center 0 and radius R .
L σ 2 ( R 3 ) = { f ∈ ( L 2 ( R 3 ) ) 3 ; div f = 0 } ,
The Leray projection P : ( L 2 ( R 3 ) ) 3 ⟶ ( L 2 ( R 3 ) ) 3 is defined by
ℱ − 1 ( P f ) = f ^ ( ξ ) − f ^ ( ξ ) ⋅ ξ ∣ ξ ∣ ξ ∣ ξ ∣ = M ( ξ ) f ^ ( ξ ) ,
where M ( ξ ) is the matrix δ k , ℓ − ξ k ξ ℓ ∣ ξ ∣ 2 1 ≤ k , ℓ ≤ 3 .

2.2 Preliminary results

In this section, we collect some classical results, and we give some lemmas that are well suited to the study of system ( S ) . We start by the following elementary inequality.

Proposition 2.1

[9] Let H be Hilbert space.

If ( x n ) is a bounded sequence of elements in H , then there is a subsequence ( x φ ( n ) ) such that
( x φ ( n ) ∣ y ) → ( x ∣ y ) , ∀ y ∈ H .
If x ∈ H and ( x n ) is a bounded sequence of elements in H such that
( x n ∣ y ) → ( x ∣ y ) , ∀ y ∈ H ,
then ‖ x ‖ ≤ liminf n → ∞ ‖ x n ‖ .
If x ∈ H and ( x n ) is a bounded sequence of elements in H such that
( x n ∣ y ) → ( x ∣ y ) , ∀ y ∈ H
and
limsup n → ∞ ‖ x n ‖ ≤ ‖ x ‖ ,
then lim n → ∞ ‖ x n − x ‖ = 0 .

Lemma 2.1

[10] Let s 1 and s 2 be two real numbers.

If s 1 < 3 ∕ 2 and s 1 + s 2 > 0 , there exists a constant C 1 = C 1 ( s 1 , s 2 ) , such that if f , g ∈ H ˙ s 1 ( R 3 ) ∩ H ˙ s 2 ( R 3 ) , then f . g ∈ H ˙ s 1 + s 2 − 3 2 ( R 3 ) and
‖ f g ‖ H ˙ s 1 + s 2 − 3 2 ≤ C 1 ( ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 + ‖ f ‖ H ˙ s 2 ‖ g ‖ H ˙ s 1 ) .
If s 1 , s 2 < 3 ∕ 2 and s 1 + s 2 > 0 , there exists a constant C 2 = C 2 ( s 1 , s 2 ) such that: if f ∈ H ˙ s 1 ( R 3 ) and g ∈ H ˙ s 2 ( R 3 ) , then f . g ∈ H ˙ s 1 + s 2 − 3 2 ( R 3 ) and
‖ f g ‖ H ˙ s 1 + s 2 − 3 2 ≤ C 2 ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 .

Lemma 2.2

[11] For all x , y ∈ R 3 , we have

( ∣ x ∣ 2 x − ∣ y ∣ 2 ) y ( x − y ) ≥ 1 2 ( ∣ x ∣ 2 + ∣ y ∣ 2 ) ∣ x − y ∣ 2 .

Lemma 2.3

[5] For all s > d 2 : L 1 ( R d ) ↪ H − s ( R d ) . Moreover, we have

‖ f ‖ H − s ( R d ) ≤ σ s , d ‖ f ‖ L 1 ( R d ) , ∀ f ∈ H − s ( R d ) ,

where σ s , d = ∫ R d ( 1 + ∣ ξ ∣ 2 ) − s d ξ 1 ∕ 2 .

Lemma 2.4

[5] Let p ∈ ( 1 , ∞ ) and Ω ≠ ∅ be an open subset of R 4 . If ( f n ) is a bounded sequence in L 2 ( Ω ) ∩ L p ( Ω ) and f ∈ L 2 ( Ω ) such that

lim n ⟶ ∞ ( f n ∣ g ) L 2 = ( f ∣ g ) L 2 , ∀ g ∈ L 2 ( Ω ) .

Then, f ∈ L p ( Ω ) and

‖ f ‖ L 2 ≤ liminf n ⟶ ∞ ‖ f n ‖ L 2 , ‖ f ‖ L p ≤ liminf n ⟶ ∞ ‖ f n ‖ L p .

3 Proof of Theorem 1.1

This proof is done in two steps.

3.1 Step 1

In this step, we prove a general result for bounded sequence in energy space of system ( S ) .

Proposition 3.1

Let ν 1 , ν 2 , ν 3 ∈ [ 0 , ∞ ) , r 1 , r 2 , r 3 ∈ ( 0 , ∞ ) , and f 0 ∈ L σ 2 ( R 3 ) . For n ∈ N , let F n : R + × R 3 → R 3 be a measurable function in C 1 ( R + , L 2 ( R 3 ) ) such that

A n ( D ) F n = F n , F n ( 0 , x ) = A n ( D ) f 0 ( x )

and

( E 1 ) ∂ t F n + ∑ k = 1 3 ν k ∣ D k ∣ 2 r k F n + A n ( D ) div ( F n ⊗ F n ) + α F n + β A n ( D ) [ ∣ F n ∣ 2 F n ] = 0 , ( E 2 ) ‖ F n ( t ) ‖ L 2 2 + 2 ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ ∣ D k ∣ r k F n ‖ L 2 2 + 2 α ∫ 0 t ‖ F n ‖ L 2 2 + 2 β ∫ 0 t ‖ F n ‖ L 4 4 ≤ ‖ f 0 ‖ L 2 2 .

Then, for every ε > 0 , there is δ = δ ( ε , α , β , ‖ f 0 ‖ L 2 ) > 0 such that for all t 1 , t 2 ∈ R + , we have

(3.1) ( ∣ t 2 − t 1 ∣ < δ ⇒ ‖ F n ( t 2 ) − F n ( t 1 ) ‖ H − s 0 < ε ) , ∀ n ∈ N ,

with s 0 = max { 3 , 2 max 1 ≤ i ≤ 3 r i } .

Proof

This proof is inspired from [5]-Proposition 3.1. Integrating ( E 1 ) on the interval [ t 1 , t 2 ] ⊂ R + and taking the inner product in H − s 0 , we obtain

‖ F n ( t 2 ) − F n ( t 1 ) ‖ H − s 0 ≤ I 1 , n ( t 1 , t 2 ) + I 2 , n ( t 1 , t 2 ) + I 3 , n ( t 1 , t 2 ) + I 4 , n ( t 1 , t 2 ) ,

with

I 1 , n ( t 1 , t 2 ) = ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ ∣ D k ∣ 2 r k F n ‖ H − s 0 , I 2 , n ( t 1 , t 2 ) = ∫ t 1 t 2 ‖ A n ( D ) div ( F n ⊗ F n ) ‖ H − s 0 , I 3 , n ( t 1 , t 2 ) = α ∫ t 1 t 2 ‖ F n ‖ H − s 0 , I 4 , n ( t 1 , t 2 ) = β ∫ t 1 t 2 ‖ A n ( D ) [ ∣ F n ∣ 2 F n ] ‖ H − s 0 .

Let ε > 0 be a positive real, let us find a positive real δ > 0 such that if ∣ t 2 − t 1 ∣ < δ , we obtain

I k , n < ε 4 , ∀ k = { 1 , 2 , 3 , 4 } .

• To estimate I 1 , n ( t 1 , t 2 ) , we write

I 1 , n ( t 1 , t 2 ) = ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ ∣ D k ∣ 2 r k F n ‖ H − s 0 ≤ ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ F n ‖ H 2 r k − s 0 ≤ ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ F n ( τ ) ‖ H 0 d τ ≤ ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 2 d τ ≤ ∑ k = 1 3 ν k ‖ f 0 ‖ L 2 ( t 2 − t 1 ) .

Then, if ∣ t 2 − t 1 ∣ < δ 1 = ε 4 ∑ k = 1 3 ν k ‖ f 0 ‖ L 2 + 4 , we obtain

I 1 , n ( t 1 , t 2 ) < ε 4 .

• To estimate I 2 , n ( t 1 , t 2 ) , from Lemma (2.3), we have

I 2 , n ( t 1 , t 2 ) = ∫ t 1 t 2 ‖ A n ( D ) div ( F n ⊗ F n ) ‖ H − s 0 ≤ ∫ t 1 t 2 ‖ div ( F n ⊗ F n ) ( τ ) ‖ H − 3 d τ ≤ ∫ t 1 t 2 ‖ ( F n ⊗ F n ) ( τ ) ‖ H − 2 d τ ≤ σ 2 , 3 ∫ t 1 t 2 ‖ ( F n ⊗ F n ) ( τ ) ‖ L 1 d τ ≤ σ 2 , 3 ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 2 2 d τ ≤ σ 2 , 3 ‖ f 0 ‖ L 2 2 ( t 2 − t 1 ) .

Then, if ∣ t 2 − t 1 ∣ < δ 2 = ε 4 σ 2 , 3 ‖ f 0 ‖ L 2 2 + 4 , we obtain

I 2 , n ( t 1 , t 2 ) < ε 4 .

• To estimate I 3 , n ( t 1 , t 2 ) , we write

I 3 , n ( t 1 , t 2 ) = α ∫ t 1 t 2 ‖ F n ‖ H − s 0 ≤ α ∫ t 1 t 2 ‖ F n ( τ ) ‖ H 0 d τ ≤ α ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 2 d τ ≤ α ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 2 d τ ≤ α ‖ f 0 ‖ L 2 ( t 2 − t 1 ) .

Then, if ∣ t 2 − t 1 ∣ < δ 3 = ε 4 α ‖ f 0 ‖ L 2 + 4 , we obtain

I 3 , n ( t 1 , t 2 ) < ε 4 .

• To estimate I 4 , n ( t 1 , t 2 ) , from Lemma (2.3), we have

I 4 , n ( t 1 , t 2 ) ≤ β ∫ t 1 t 2 ‖ A n ( D ) [ ∣ F n ∣ 2 F n ] ‖ H − 3 ≤ σ 3 , 3 β ∫ t 1 t 2 ‖ ( ∣ F n ∣ 2 F n ) ( τ ) ‖ L 1 d τ .

Now, for ( n , R , t ) ∈ N × ( 0 , + ∞ ) × [ 0 , + ∞ ) , we consider the following subset of R 3 :

X n ( R , t ) = { x ∈ R 3 ∕ ∣ F n ( t , x ) ∣ ≤ R } ,

I 4 , n ( t 1 , t 2 ) ≤ σ 3 , 3 β ∫ t 1 t 2 ∫ X n ( R , t ) ∣ F n ( τ , x ) ∣ 3 d x d τ + σ 3 , 3 β ∫ t 1 t 2 ∫ X n ( R , t ) c ∣ F n ( τ , x ) ∣ 3 d x d τ ≤ σ 3 , 3 β ∫ t 1 t 2 ∫ X n ( R , t ) R ∣ F n ( τ , x ) ∣ 2 d x d τ + σ 3 , 3 β ∫ t 1 t 2 ∫ X n ( R , t ) c ∣ F n ( τ , x ) ∣ 4 R d x d τ ≤ σ 3 , 3 β R ∫ t 1 t 2 ∫ X n ( R , t ) ∣ F n ( τ , x ) ∣ 2 d x d τ + σ 3 , 3 β R ∫ t 1 t 2 ∫ X n ( R , t ) c ∣ F n ( τ , x ) ∣ 4 d x d τ ≤ σ 3 , 3 β R ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 2 2 d τ + σ 3 , 3 β R ∫ t 1 t 2 ‖ F n ( τ ) ‖ L 4 4 d τ .

Using ( E 2 ) , we obtain

I 4 , n ( t 1 , t 2 ) ≤ σ 3 , 3 β R ‖ f 0 ‖ L 2 2 ( t 2 − t 1 ) + σ 3 , 3 β 2 α R ‖ f 0 ‖ L 2 2 .

Hence, with the choices R = R ε = 4 σ 3 , 3 ‖ f 0 ‖ L 2 2 + 4 ε and δ 4 = ε 8 α σ 3 , 3 β ‖ ∣ f 0 ∣ 2 ‖ L 2 2 + 8 , we obtain

∣ t 2 − t 1 ∣ < δ 4 ⇒ I 4 , n ( t 1 , t 2 ) < ε 4 .

To conclude, it suffices to take δ = min { δ 1 , δ 2 , δ 3 , δ 4 } .□

3.2 Step 2

In this step, we construct a global solution of ( S ) , where we use a method inspired by [12]. For this, consider the approximate system with the parameter n ∈ N :

( S n ) ∂ t u − μ Δ J n u + J n ( J n u ⋅ ∇ J n u ) + α J n u + β J n [ ∣ J n u ∣ 2 J n u ] = − ∇ P n , in R + × R 3 P n = ( − Δ ) − 1 ( div J n ( J n u ⋅ ∇ J n u + β div J n [ ∣ J n u ∣ 2 J n u ] ) ) div u = 0 , in R + × R 3 , u ( 0 , x ) = J n u 0 ( x ) , in R 3 .

By the Cauchy-Lipschitz theorem, we obtain a unique solution u n ∈ C 1 ( R + , L 2 ( R 3 ) ) of ( S n ) , with the following properties:
div u n = 0 , J n u n = u n .
Moreover, u n satisfies
∂ t u n − μ Δ u n + J n ( u n . ∇ u n ) + α u n + β J n [ ∣ u n ∣ 2 u n ] = − ∇ p n , in R + × R 3 , p n = ( − Δ ) − 1 ( div J n ( u n . ∇ u n + β div J n [ ∣ u n ∣ 2 u n ] ) ) , div u n = 0 , in R + × R 3 , u n ( 0 , x ) = J n u 0 ( x ) , in R 3 .
and the following energy estimate
(3.2) ‖ u n ( t ) ‖ L 2 2 + 2 μ ∫ 0 t ‖ ∇ u n ‖ L 2 2 + 2 α ∫ 0 t ‖ u n ‖ L 2 2 + 2 β ∫ 0 t ‖ u n ‖ L 4 4 ≤ ‖ u 0 ‖ L 2 2 , ∀ t ≥ 0 ,

(3.3) e 2 α t ‖ u n ( t ) ‖ L 2 2 + 2 μ ∫ 0 t e 2 α z ‖ ∇ u n ( z ) ‖ L 2 2 d z + 2 β ∫ 0 t e 2 α z ‖ u n ( z ) ‖ L 4 4 d z ≤ ‖ u 0 ‖ L 2 2 , ∀ t ≥ 0 .
By Inequality (3.2), we obtain ( u n ) that is bounded in
L ∞ ( R + , L 2 ( R 3 ) ) ∩ L l o c 2 ( R + , H 1 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) .
Using Proposition 3.1, we deduce that
(3.4) the sequence ( u n ) is equicontinuous in C b ( R + , H − 1 ( R 3 ) ) .
Let ( T q ) q ∈ ( 0 , ∞ ) N such that T q < T q + 1 and T q → ∞ as q → ∞ . Let ( θ q ) q ∈ N be a sequence in C 0 ∞ ( R 3 ) such that: for all q ∈ N ,
θ q ( x ) = 1 , ∀ x ∈ B 0 , q + 1 + 1 4 , θ q ( x ) = 0 , ∀ x ∈ B ( 0 , q + 2 ) c , 0 ≤ θ q ≤ 1 .
Using (3.2)–(3.4) and classical argument by combining Ascoli’s theorem and the Cantor diagonal process, we obtain a nondecreasing φ : N → N and
u ∈ L ∞ ( R + , L 2 ( R 3 ) ) ∩ C ( R + , H − 3 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) ,
such that for all q ∈ N , we have
(3.5) lim n → ∞ ‖ θ q ( u φ ( n ) − u ) ‖ L ∞ ( [ 0 , T q ] , H − 4 ) = 0 .
Combining the aforementioned inequalities, we obtain (1.1) and (1.2).
It remains to show that u is a solution of system ( S ) . Using the same idea in [5], we prove that u satisfies
∂ t u − μ Δ u + u ⋅ ∇ u + α u + β ∣ u ∣ 2 u = − ∇ p , in D ′ ( R + × R 3 ) .
Then, u is a solution of our system ( S ) .

3.3 Continuity of the solution in L 2 ( R 3 )

By Inequality (1.1), we have
limsup t ⟶ 0 ‖ u ( t ) ‖ L 2 ≤ ‖ u 0 ‖ L 2 .
Then, Proposition 2.1 implies that
limsup t ⟶ 0 ‖ u ( t ) − u 0 ‖ L 2 = 0 .
This ensures the continuity of the solution u at 0.
To prove the continuity on ( 0 , ∞ ) , consider the function
v n , ε ( t ) = u φ ( n ) ( t + ε ) , p n , ε ( t ) = p φ ( n ) ( t + ε ) , for n ∈ N and ε > 0 .
We have
∂ t u φ ( n ) − μ Δ u φ ( n ) + J φ ( n ) ( u φ ( n ) ⋅ ∇ u φ ( n ) ) + α u φ ( n ) + β J φ ( n ) ( ∣ u φ ( n ) ∣ 2 u φ ( n ) ) + ∇ p φ ( n ) = 0 , ∂ t v n , ε − μ Δ v n , ε + J φ ( n ) ( v n , ε ⋅ ∇ v n , ε ) + α v n , ε + β J φ ( n ) ( ∣ v n , ε ∣ 2 v n , ε ) + ∇ p n , ε = 0 .
The function w n , ε = u φ ( n ) − v n , ε fulfills the following:
∂ t w n , ε − μ Δ w n , ε + α w n , ε + β J φ ( n ) ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) + ∇ ( p φ ( n ) − p n , ε ) = J φ ( n ) ( w n , ε ⋅ ∇ w n , ε ) − J φ ( n ) ( w n , ε ⋅ ∇ u φ ( n ) ) − J φ ( n ) ( u φ ( n ) ⋅ ∇ w n , ε ) .
Taking the scalar product with w n , ε in L 2 ( R 3 ) and using the fact that ⟨ w n , ε ⋅ ∇ w n , ε ; w n , ε ⟩ = 0 and div w n , ε = 0 , we obtain
(3.6) 1 2 ‖ w n , ε ( t ) ‖ L 2 2 + μ ‖ ∇ w n , ε ( t ) ‖ L 2 2 + α ‖ w n , ε ( t ) ‖ L 2 2 + β ⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) ; w n , ε ⟩ L 2 = − ⟨ J φ ( n ) ( w n , ε ⋅ ∇ u φ ( n ) ) ; w n , ε ⟩ L 2 .
From Lemma 2.2, we have
⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) ; w n , ε ⟩ L 2 = ⟨ ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) ; J φ ( n ) w n , ε ⟩ L 2 = ⟨ ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) ; w n , ε ⟩ L 2 ≥ 1 2 ∫ R 3 ( ∣ u φ ( n ) ∣ 2 + ∣ v n , ε ∣ 2 ) ∣ w n , ε ∣ 2 ,
which implies
(3.7) β ⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ 2 u φ ( n ) − ∣ v n , ε ∣ 2 v n , ε ) ; w n , ε ⟩ L 2 ≥ β 2 ∫ R 3 ( ∣ u φ ( n ) ∣ 2 + ∣ v n , ε ∣ 2 ) ∣ w n , ε ∣ 2 .
Taking into account the equality
I n , ε = ⟨ J φ ( n ) ( w n , ε ⋅ ∇ u φ ( n ) ) ; w n , ε ⟩ L 2 = ⟨ J φ ( n ) ( w n , ε ⋅ ∇ v n , ε ) ; w n , ε ⟩ L 2
and, combining the Cauchy-Schwarz and Young’s inequalities, we obtain, for f = v n , ε or f = u φ ( n ) ,
(3.8) ∣ ⟨ J φ ( n ) ( w n , ε ⋅ ∇ f ) ; w n , ε ⟩ L 2 ∣ ≤ ∫ R 3 ∣ w n , ε ∣ ∣ f ∣ ∣ ∇ w n , ε ∣ = ∫ R 3 ∣ w n , ε ∣ 2 ∣ f ∣ 2 1 2 ∫ R 3 ∣ ∇ w n , ε ∣ 2 1 2 ≤ β ∫ R 3 ∣ w n , ε ∣ 2 ∣ f ∣ 2 + 1 4 β ‖ ∇ w n , ε ‖ L 2 2 .
This inequality is true for f = v n , ε and f = u φ ( n ) , then using the elementary inequality
min ( a , b ) ≤ 1 2 ( a + b ) ,
with a = ∫ R 3 ∣ w n , ε ∣ 2 ∣ v n , ε ∣ 2 and b = ∫ R 3 ∣ w n , ε ∣ 2 ∣ u φ ( n ) ∣ 2 , we obtain
∣ I n , ε ∣ ≤ β 2 ∫ R 3 ∣ w n , ε ∣ 2 ( ∣ v n , ε ∣ 2 + ∣ u φ ( n ) ∣ 2 ) + 1 4 β ‖ ∇ w n , ε ‖ L 2 2 .
Combining Inequalities (3.7) and (3.8), we deduce that
(3.9) 1 2 d d t ‖ w n , ε ( t ) ‖ L 2 2 + μ − 1 4 β ‖ ∇ w n , ε ( t ) ‖ L 2 2 + α ‖ w n , ε ( t ) ‖ L 2 2 ≤ 0 .
We have
1 2 d d t ‖ w n , ε ( t ) ‖ L 2 2 + μ − 1 4 β ‖ ∇ w n , ε ( t ) ‖ L 2 2 ≤ 0 .
Hence, we obtain
‖ w n , ε ( t ) ‖ L 2 2 ≤ ‖ w n , ε ( 0 ) ‖ L 2 2
and
‖ u φ ( n ) ( t + ε ) − u φ ( n ) ( t ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 .
For t 0 > 0 and ε ∈ ( 0 , t 0 ∕ 2 ) , we have
‖ u φ ( n ) ( t 0 + ε ) − u φ ( n ) ( t 0 ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 , ‖ u φ ( n ) ( t 0 − ε ) − u φ ( n ) ( t 0 ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 .
Thus,
‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 = ‖ J φ ( n ) u φ ( n ) ( ε ) − J φ ( n ) u φ ( n ) ( 0 ) ‖ L 2 2 = ‖ J φ ( n ) ( u φ ( n ) ( ε ) − u 0 ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) − u 0 ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) ‖ L 2 2 + ‖ u 0 ‖ L 2 2 − 2 Re ⟨ u φ ( n ) ( ε ) ; u 0 ⟩ L 2 ≤ 2 ‖ u 0 ‖ L 2 2 − 2 Re ⟨ u φ ( n ) ( ε ) ; u 0 ⟩ .
Since liminf n ⟶ ∞ ⟨ u φ ( n ) ( ε ) ; u 0 ⟩ L 2 = ⟨ u ( ε ) ; u 0 ⟩ L 2 ,
liminf n ⟶ ∞ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 ≤ 2 ‖ u 0 ‖ L 2 2 − 2 Re ⟨ u ( ε ) ; u 0 ⟩ L 2 .
Moreover, for all q , N ∈ N ,
‖ J N ( θ q ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ) ‖ L 2 2 ≤ ‖ θ q ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ‖ L 2 2 .
For q big enough using (3.5), we obtain
‖ J N ( θ q ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ) ‖ L 2 2 ≤ liminf n ⟶ ∞ ‖ u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 .
Then,
‖ J N ( θ q ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ) ‖ L 2 2 ≤ 2 ( ‖ u 0 ‖ L 2 2 − Re ⟨ u ( ε ) ; u 0 ⟩ L 2 ) .
By applying the Monotone convergence theorem in the order N and q , we obtain
‖ u ( t 0 ± ε ) − u ( t 0 ) ‖ L 2 2 ≤ 2 ( ‖ u 0 ‖ L 2 2 − Re ⟨ u ( ε ) ; u 0 ⟩ L 2 ) .
Using the continuity at 0 and make ε ⟶ 0 , we obtain the continuity at t 0 .

3.4 Uniqueness of u in L 2 ( R 3 )

Let u and v be two solutions of ( S ) in the space

C ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) .

Set w = u − v . Then, the function w satisfies the following:

∂ t w − μ Δ w + α w + β ( ∣ u ∣ 2 u − ∣ v ∣ 2 v ) + ∇ ( p u − p v ) = w ⋅ ∇ w − w ⋅ ∇ u − u ⋅ ∇ w .

Taking the scalar product in L 2 with w , we obtain

1 2 d d t ‖ w ‖ L 2 2 + μ ‖ ∇ w ‖ L 2 2 + α ‖ w ‖ L 2 2 + β ⟨ ( ∣ u ∣ 2 u − ∣ v ∣ 2 v ) ; w ⟩ L 2 = − ⟨ w ⋅ ∇ u ; w ⟩ L 2 .

By adapting the same method for the proof of the continuity of such solution in L 2 ( R 3 ) , with u , v , and w instead of u φ ( n ) , v n , ε , and w n , ε in order, we find

β ⟨ ( ∣ u ∣ 2 u − ∣ v ∣ 2 v ) ; w ⟩ L 2 ≥ β 2 ∫ R 3 ( ∣ u ∣ 2 + ∣ v ∣ 2 ) ∣ w ∣ 2

and

∣ ⟨ w ⋅ ∇ u ; w ⟩ L 2 ∣ = ∣ ⟨ w ⋅ ∇ v ; w ⟩ L 2 ∣ ≤ β ∫ R 3 ∣ w ∣ 2 ∣ u ∣ 2 + 1 4 β ‖ ∇ w ‖ L 2 2 β ∫ R 3 ∣ w ∣ 2 ∣ v ∣ 2 + 1 4 β ‖ ∇ w ‖ L 2 2 ≤ β 2 min ∫ R 3 ∣ w ∣ 2 ∣ u ∣ 2 , ∫ R 3 ∣ w ∣ 2 ∣ v ∣ 2 + 1 4 β ‖ ∇ w ‖ L 2 2 ≤ β 2 ∫ R 3 ∣ w ∣ 2 ( ∣ u ∣ 2 + ∣ v ∣ 2 ) + 1 4 β ‖ ∇ w ‖ L 2 2 .

Combining the aforementioned inequalities, we find the following energy estimate:

1 2 d d t ‖ w ‖ L 2 2 + μ − 1 4 β ‖ ∇ w ‖ L 2 2 + α ‖ w ‖ L 2 2 ≤ 0 .

Using the fact μ − 1 4 β ≥ 0 and α > 0 , we obtain

‖ w ( t ) ‖ L 2 2 ≤ ‖ w 0 ‖ L 2 2 .

As w 0 = 0 , then w = 0 and u = v , which implies the uniqueness. And the proof of Theorem 1.1 is completed.

4 Proof of Theorem (1.2)

Let χ ∈ C 0 ∞ ( R 3 ) be a regular function, such that

0 ≤ χ ≤ 1 , χ = 1 , on B ( 0 , 1 ) , χ = 0 , on B ( 0 , 2 ) c .

Also, for δ > 0 , put the following functions χ δ and ψ δ defined by

χ δ ( ξ ) = χ ( δ ξ ) , ψ δ = ℱ − 1 ( χ δ ) .

We have

‖ ψ δ ‖ L 1 = ‖ ψ 1 ‖ L 1 , ∀ δ > 0 .

In addition, we define the operator S δ by

S δ ( f ) = χ δ ( D ) f = ψ δ ⁎ f .

For δ > 0 , which will be fixed later, put the following functions:

v δ = ψ δ ⁎ u = ℱ − 1 ( χ δ ( ξ ) u ^ ) , w δ = u − v δ = ℱ − 1 ( ( 1 − χ δ ( ξ ) ) u ^ ) .

Let ε > 0 be a fixed real.

Estimate of v δ : We have
(4.1) e 2 α t ‖ v δ ( t ) ‖ L 2 2 + 2 μ ∫ 0 t e 2 α z ‖ ∇ v δ ( z ) ‖ L 2 2 d z ≤ A δ ( t ) + B δ ( t ) ,
where
A δ ( t ) = 2 ∫ 0 t e 2 α z ∣ ⟨ S δ ( u ⋅ ∇ u ) ( z ) ; v δ ( z ) ⟩ L 2 ∣ d z , B δ ( t ) = 2 β ∫ 0 t e 2 α z ∣ ⟨ S δ ( ∣ u ∣ 2 u ) ( z ) ; v δ ( z ) ⟩ L 2 ∣ d z .
To estimate A δ , we write
A δ ( t ) = ∫ 0 t e 2 α z ∣ ⟨ S δ ( u ⋅ ∇ u ) ( z ) ; v δ ( z ) ⟩ L 2 ∣ d z ≤ ∫ 0 t e 2 α z ‖ S δ ( u ⋅ ∇ u ) ( z ) ‖ L 2 ‖ v δ ( z ) ‖ L 2 d z ≤ c 0 ∫ 0 t e 2 α z ∫ ∣ ξ ∣ < 2 δ ∣ ξ ∣ 2 ∣ ℱ ( u ⊗ u ) ( z , ξ ) ∣ 2 d ξ 1 ∕ 2 ‖ v δ ( z ) ‖ L 2 d z ≤ 2 c 0 δ 1 ∕ 2 ∫ 0 t e 2 α z ∫ ∣ ξ ∣ < δ ∣ ξ ∣ ∣ ℱ ( u ⊗ u ) ( z , ξ ) ∣ 2 d ξ 1 ∕ 2 ‖ v δ ( z ) ‖ L 2 d z ≤ 2 c 0 δ 1 ∕ 2 ∫ 0 t e 2 α z ‖ ( u ⊗ u ) ( z ) ‖ H ˙ 1 ∕ 2 ‖ v δ ( z ) ‖ L 2 d z ≤ C δ 1 ∕ 2 ∫ 0 t e 2 α z ‖ ∇ u ( z ) ‖ L 2 2 ‖ v δ ( z ) ‖ L 2 d z .
Then,
(4.2) A δ ( t ) ≤ C δ 1 ∕ 2 ‖ u 0 ‖ L 2 3 ∀ t ≥ 0 .
To estimate B δ , by the Hölder inequality, we obtain
B δ ( t ) = ∫ 0 t e 2 α z ∣ ⟨ S δ ( ∣ u ∣ 2 u ) ( z ) ; v δ ( z ) ⟩ L 2 ∣ d z ≤ ∫ 0 t e 2 α z ‖ S δ ( ∣ u ∣ 2 u ) ( z ) ‖ L 4 ∕ 3 ‖ v δ ( z ) ‖ L 4 d z ≤ ∫ 0 t e 2 α z ‖ ψ δ ⁎ ( ∣ u ∣ 2 u ) ( z ) ‖ L 4 ∕ 3 ‖ v δ ( z ) ‖ L 4 d z ≤ ∫ 0 t e 2 α z ‖ ψ δ ‖ L 1 ‖ ( ∣ u ∣ 2 u ) ( z ) ‖ L 4 ∕ 3 ‖ v δ ( z ) ‖ L 4 d z ≤ ‖ ψ 1 ‖ L 1 ∫ 0 t e 2 α z ‖ ( ∣ u ∣ 2 u ) ( z ) ‖ L 4 ∕ 3 ‖ v δ ( z ) ‖ L 4 d z ≤ ‖ ψ 1 ‖ L 1 ∫ 0 ∞ e 2 α z ‖ u ( z ) ‖ L 4 3 ‖ v δ ( z ) ‖ L 4 d z ≤ ‖ ψ 1 ‖ L 1 ∫ 0 ∞ e 2 α z ‖ u ( z ) ‖ L 4 4 d z 3 ∕ 4 ∫ 0 ∞ e 2 α z ‖ v δ ( z ) ‖ L 4 4 d z 1 ∕ 4 .
Using (1.2), we obtain
B δ ( t ) ≤ ‖ ψ 1 ‖ L 1 ‖ u 0 ‖ L 2 2 2 β 3 ∕ 4 ∫ 0 ∞ e 2 α z ‖ v δ ( z ) ‖ L 4 4 d z 1 ∕ 4 .
By interpolation, we obtain
‖ v δ ( z ) ‖ L 4 4 ≤ C ‖ v δ ( z ) ‖ L 2 ‖ ∇ v δ ( z ) ‖ L 2 3 ≤ 2 C δ ‖ v δ ( z ) ‖ L 2 2 ‖ ∇ v δ ( z ) ‖ L 2 2 ≤ 2 C δ ‖ u 0 ‖ L 2 2 ‖ ∇ v δ ( z ) ‖ L 2 2 ≤ 2 C δ ‖ u 0 ‖ L 2 2 ‖ ∇ u ( z ) ‖ L 2 2 .
Then,
∫ 0 ∞ e 2 α z ‖ v δ ( z ) ‖ L 4 4 d z ≤ 2 C δ ‖ u 0 ‖ L 2 2 ∫ 0 ∞ e 2 α z ‖ ∇ u ( z ) ‖ L 2 2 d z ≤ C μ − 1 ‖ u 0 ‖ L 2 4 δ
and
(4.3) B δ ( t ) ≤ ‖ ψ 1 ‖ L 1 ‖ u 0 ‖ L 2 2 2 β 3 ∕ 4 ( C μ − 1 ) 1 ∕ 4 ‖ u 0 ‖ L 2 δ 1 ∕ 4 .
Combining Inequalities (4.1)–(4.3), we obtain
e 2 α t ‖ v δ ( t ) ‖ L 2 2 ≤ C ( u 0 ) δ 1 ∕ 4 ( 1 + δ 1 ∕ 4 ) ,
which implies the existence of δ 0 > 0 , such that
(4.4) sup t ≥ 0 e 2 α t ‖ v δ ( t ) ‖ L 2 2 < ( ε ∕ 2 ) 2 , ∀ δ ∈ ( 0 , δ 0 ] .
Prove that w δ 0 ∈ L 2 ( R + , L 2 ( R 3 ) ) . We have
∫ 0 ∞ e 2 α t ‖ w δ 0 ( t ) ‖ L 2 2 d t = c 0 ∫ 0 ∞ e 2 α z ‖ w ^ δ 0 ( t ) ‖ L 2 2 d t = c 0 ∫ 0 ∞ ∫ ∣ ξ ∣ > δ 0 e 2 α z ∣ u ^ ( t , ξ ) ∣ 2 d ξ d t ≤ c 0 δ 0 − 2 ∫ 0 ∞ e 2 α z ∫ ∣ ξ ∣ > δ 0 ∣ ξ ∣ 2 ∣ u ^ ( t , ξ ) ∣ 2 d ξ d t ≤ c 0 δ 0 − 2 ∫ 0 ∞ e 2 α z ‖ ∇ u ^ ( t ) ‖ L 2 2 d t ≤ δ 0 − 2 ∫ 0 ∞ e 2 α z ‖ ∇ u ( t ) ‖ L 2 2 d t ≤ 2 μ − 1 δ 0 − 2 ‖ u 0 ‖ L 2 2 .
Then, ( t ↦ e 2 α t ‖ w δ 0 ( t ) ‖ L 2 2 ) ∈ C ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , L 2 ( R 3 ) ) .

Put the following subset of R + :
F ε = { t ≥ 0 : e 2 α t ‖ w δ 0 ( t ) ‖ L 2 2 ≥ ( ε ∕ 2 ) 2 } .
As ( t ↦ e 2 α t ‖ w δ 0 ( t ) ‖ L 2 2 ) ∈ C ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , L 2 ( R 3 ) ) , then there exists t ε ∈ R + \ F ε such that
(4.5) e 2 α t ε ‖ w δ 0 ( t ε ) ‖ L 2 2 < ( ε ∕ 2 ) 2 .
Combining (4.4)–(4.5), we obtain
(4.6) e 2 α t ε ‖ u ( t ε ) ‖ L 2 < ε .
Now, put the following system:
( S ) ∂ t f − μ Δ f + f ⋅ ∇ f + α f + β ∣ f ∣ 2 f + ∇ q = 0 in R + × R 3 div f = 0 in R + × R 3 f ( 0 , x ) = u ( t ε , x ) in R 3 .
By the first step, there is a unique solution f ε ∈ C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H 1 ( R 3 ) ) ∩ L 4 ( R + , L 4 ( R 3 ) ) of (2) satisfying for all t ≥ 0 :
e 2 α t ‖ f ε ( t ) ‖ L 2 2 + 2 μ ∫ 0 t e 2 α z ‖ ∇ f ε ( z ) ‖ L 2 2 d z + 2 β ∫ 0 t e 2 α z ‖ f ε ( z ) ‖ L 4 4 d z ≤ ‖ u ( t ε ) ‖ L 2 2 .
The uniqueness of the solution implies that f ( t ) = u ( t ε + t ) for all t ≥ 0 and
e 2 α t ‖ u ( t ε + t ) ‖ L 2 2 ≤ ‖ u ( t ε ) ‖ L 2 2 ,
which implies that
e 2 α t ‖ u ( t ) ‖ L 2 2 ≤ e 2 α t ε ‖ u ( t ε ) ‖ L 2 2 < ε 2 , ∀ t ≥ t ε .
Then, the proof of Theorem 1.2 is completed.

5 Conclusion

This work deals with a detailed study of the global existence, uniqueness, and continuity for the solution of incompressible convective Brinkman-Forchheimer on the whole space R 3 when 4 μ β ≥ 1 . An asymptotic type of convergence of the global solution associated with the aforementioned problem is also established.

Acknowledgments

The authors appreciate the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for supporting and supervising this project.

Funding information: The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research through the Project Number IFP-IMSIU-2023018.
Author contributions: All authors contributed equally in this work. All authors have accepted responsibility for entire content of the manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to the article as no datasets were generated or analyzed during this study.

References

[1] A. V. Shenoy, Non-Newtonian fluid heat transfer in porous media, Adv. Heat Transfer 24 (1994), 101–190. 10.1016/S0065-2717(08)70233-8Search in Google Scholar

[2] R. G. Gordeev, The existence of a periodic solution in a certain problem of tidal dynamics. In: Problems of Mathematical Analysis, No. 4: Integral and Differential Operators. Differential equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1973. Search in Google Scholar

[3] A. L. Likhtarnikov, Existence and stability of bounded and periodic solutions in a nonlinear problem of tidal dynamics. In The Direct Method in the Theory of Stability and its Applications (Irkutsk, 1979), “Nauka” Sibirsk. Otdel., Novosibirsk, vol. 276, 1981, pp. 83–91. Search in Google Scholar

[4] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl. 343 (2008), no. 2, 799–809. 10.1016/j.jmaa.2008.01.041Search in Google Scholar

[5] J. Benameur, Global weak solution of 3D-NSE with exponential damping, Open Mathematics 20 (2022), 590–607. 10.1515/math-2022-0050Search in Google Scholar

[6] M. Jleli, B. Samet, and D. Ye, Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains, J. Differential Equations 268 (2020), no. 6, 3035–3056. 10.1016/j.jde.2019.09.051Search in Google Scholar

[7] M. Jleli and B. Samet, New blow-up results for nonlinear boundary value problems in exterior domains, Nonlinear Analysis. 178 (2019), 348–365. 10.1016/j.na.2018.09.003Search in Google Scholar

[8] M. Jleli, M. A. Ragusa, and B. Samet, Nonlinear Liouville-type theorems for generalized Baouendi-Grushin operator on Riemannian manifolds, Adv. Differential Equations 28 (2023), no. 1/2, 143–168. 10.57262/ade028-0102-143Search in Google Scholar

[9] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1996. Search in Google Scholar

[10] H. Bahouri, J. Y Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Verlag, Berlin, vol. 343, 2011. 10.1007/978-3-642-16830-7Search in Google Scholar

[11] M. Blel and J. Benameur, Long time decay of Leray solution of 3D-NSE with exponential damping, Fractals 30 (2022), 10. 10.1142/S0218348X22402368Search in Google Scholar

[12] J.-Y. Chemin, About Navier-Stokes Equations, Publications of Jaques-Louis Lions Laboratoiry, Paris VI University, 1996. Search in Google Scholar

Received: 2023-11-03

Revised: 2024-02-02

Accepted: 2024-03-19

Published Online: 2024-08-06

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