Home Mathematics Global weak solution of 3D-NSE with exponential damping
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Global weak solution of 3D-NSE with exponential damping

  • Jamel Benameur EMAIL logo
Published/Copyright: August 12, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this paper, we prove the global existence of incompressible Navier-Stokes equations with damping α ( e β ∣ u ∣ 2 − 1 ) u , where we use the Friedrich method and some new tools. The delicate problem in the construction of a global solution is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between L ∞ ( R + , L 2 ( R 3 ) ) and the space of functions f such that ( e β ∣ f ∣ 2 − 1 ) ∣ f ∣ 2 ∈ L 1 ( R + × R 3 ) . Fourier analysis and standard techniques are used.

Keywords: Navier-Stokes equations; Friedrich method; weak solution
MSC 2010: 35-XX; 35Q30; 76D05; 76N10

1 Introduction

In this paper, we study the global existence of weak solution to the modified incompressible Navier-Stokes equations (NSE) in R 3 :

( S ) ∂ t u − ν Δ u + u . ∇ u + α ( e β ∣ u ∣ 2 − 1 ) u = − ∇ p in R + × R 3 , div u = 0 in R + × R 3 , u ( 0 , x ) = u 0 ( x ) in R 3 ,

where u = u ( t , x ) = ( u 1 , u 2 , u 3 ) and p = p ( t , x ) denote, respectively, the unknown velocity and the unknown pressure of the fluid at the point ( t , x ) ∈ R + × R 3 , and the viscosity of fluid ν > 0 . Also α , β > 0 denote the parameters of damping term. The fact that div u = 0 , allows to write the term ( u . ∇ u ) ≔ u 1 ∂ 1 u + u 2 ∂ 2 u + u 3 ∂ 3 u in the following form:

div ( u ⊗ u ) ≔ ( div ( u 1 u ) , div ( u 2 u ) , div ( u 3 u ) ) .

While u 0 = ( u 1 0 ( x ) , u 2 0 ( x ) , u 3 0 ( x ) ) is an initial given velocity. If u 0 is quite regular, the divergence-free condition determines the pressure p . We recall in our case it was assumed that the viscosity is unitary ( ν = 1 ) to simplify the calculations and the proofs of our results. The global existence of the weak solution of the initial value problem of classical incompressible Navier-Stokes was proved by Leray and Hopf (see [1,2, 3,4]) long before. Uniqueness remains an open problem for the dimensions d ≥ 3 . The damping is from the resistance to the motion of the flow. It describes various physical situations such as porous media flow, drag or friction effects, and some dissipative mechanisms (see [2,3,5,6] and references therein). The polynomial damping α ∣ u ∣ β − 1 u is studied in [7] by Cai and Jiu, where they proved the global existence of weak solution in

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

The purpose of this paper is to study the well posedness of the incompressible NSEs with damping α ( e β ∣ u ∣ 2 − 1 ) u . We will show that the Cauchy problem ( S ) has global weak solutions for any α , β ∈ ( 0 , ∞ ) . We apply the Friedrich method to construct the approximate solutions and make more delicate estimates to proceed to compactness arguments. In particular, we obtain new more a priori estimates:

‖ u ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u ( z ) ‖ L 2 2 d z + 2 α ∫ 0 t ‖ ( e β ∣ u ( z ) ∣ 2 − 1 ) ∣ u ( z ) ∣ 2 ‖ L 1 d z ≤ ‖ u 0 ‖ L 2 2 ,

comparing with the NSEs, to guarantee that solution u belongs to

L ∞ ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ ℰ β ,

where

ℰ β = { f : R + × R 3 → R measurable ; ( e β ∣ f ∣ 2 − 1 ) ∣ f ∣ 2 ∈ L 1 ( R + × R 3 ) } .

Before treating the global existence, we give the definition of solution of ( S ) .

Definition 1.1

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

  1. u ∈ L ∞ ( [ 0 , T ] ; L σ 2 ( R 3 ) ) ∩ L 2 ( [ 0 , T ] ; H ˙ 1 ( R 3 ) ) and ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ∈ L 1 ( [ 0 , T ] , L 1 ( R 3 ) ) .

  2. ∂ t u − Δ u + u . ∇ u + α ( e β ∣ u ∣ 2 − 1 ) 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 + ∫ 0 T ( ( u . ∇ ) u ; Φ ) L 2 + α ∫ 0 T ( ( e β ∣ u ∣ 2 − 1 ) u ; Φ ) L 2 = ( u 0 ; Φ ( 0 ) ) L 2 .

  3. 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 ) .)

Remark 1.2

  1. If ( u , p ) is a solution of the system ( S ) , then ( e β ∣ u ∣ 2 − 1 ) u ∈ L loc 1 ( R + , L 1 ( R 3 ) ) . Indeed: for T > 0 , consider the subsets A T 1 = { ( t , x ) ∈ [ 0 , T ] × R 3 ; 0 < ∣ u ( t , x ) ∣ < 1 } and A T 2 = { ( t , x ) ∈ [ 0 , T ] × R 3 ; ∣ u ( t , x ) ∣ ≥ 1 } . We have

    ∫ [ 0 , T ] × R 3 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ = ∫ A T 1 ∪ A T 2 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ ≤ ∫ A T 1 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ + ∫ A T 2 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ ≤ ∫ A T 1 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ ∣ u ∣ 2 + ∫ A T 2 ( e β ∣ u ∣ 2 − 1 ) 1 ∣ u ∣ ∣ u ∣ 2 ≤ M β ∫ A T 1 ∣ u ∣ 2 + ∫ A T 2 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 , M β = max 0 < X ≤ 1 ( e β X 2 − 1 ) X ≤ M β ∫ 0 T ∫ R 3 ∣ u ∣ 2 + ∫ 0 T ∫ R 3 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2

    ≤ M β T ‖ u ‖ L ∞ ( [ 0 , T ] , L 2 ( R 3 ) ) + ∫ [ 0 , T ] × R 3 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ≤ M β T ‖ u ‖ L ∞ ( [ 0 , T ] , L 2 ( R 3 ) ) + ‖ ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ‖ L 1 ( [ 0 , T ] , L 1 ( R 3 ) ) ,

  2. which implies the desired result.

  3. By this definition, the scalar pressure function p is well defined in L loc 1 ( R + , H − s ( R 3 ) ) , for all s > 3 / 2 . Indeed: Formally, if p exists, then

    p = ( − Δ ) − 1 ( div u . ∇ u + α div ( e β ∣ u ∣ 2 − 1 ) u ) .

    So, just prove that ( − Δ ) − 1 ( div ( u . ∇ u ) ) , ( − Δ ) − 1 ( α div ( e β ∣ u ∣ 2 − 1 ) u ) ∈ L loc 1 ( R + , H − s ( R 3 ) ) , for all s > 3 / 2 . For s > 3 / 2 , we have

    ‖ ( − Δ ) − 1 ( div ( u . ∇ u ) ) ( t ) ‖ H − s ( R 3 ) = ‖ ( − Δ ) − 1 ( div ( div ( u ⊗ u ) ) ) ( t ) ‖ H − s ( R 3 ) ≤ ‖ u ⊗ u ( t ) ‖ H − s ( R 3 ) ≤ ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − s ∣ ℱ ( u ⊗ u ) ( t , ξ ) ∣ 2 d ξ 1 / 2 ≤ c s ‖ ℱ ( u ⊗ u ) ( t ) ‖ L ∞ ( R 3 ) , c s = ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − s d ξ 1 / 2 ≤ c s ‖ u ⊗ u ( t ) ‖ L 1 ( R 3 ) ≤ c s ‖ u ( t ) ‖ L 2 ( R 3 ) 2 ∈ L loc 1 ( R + ) .

    On the other hand, we have

    ‖ ( − Δ ) − 1 ( div ( e β ∣ u ∣ 2 − 1 ) u ) ( t ) ‖ H − s ( R 3 ) ≤ ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − s ∣ ξ ∣ − 2 ∣ ℱ ( ( e β ∣ u ∣ 2 − 1 ) u ) ( t , ξ ) ∣ 2 d ξ 1 / 2 ≤ C s ‖ ℱ ( ( e β ∣ u ∣ 2 − 1 ) u ( t ) ) ‖ L ∞ ( R 3 ) , C s = ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − s ∣ ξ ∣ − 2 d ξ 1 / 2 ≤ C s ‖ ( e β ∣ u ∣ 2 − 1 ) u ( t ) ‖ L 1 ( R 3 ) ∈ L loc 1 ( R + ) .

    Then, p is well defined in L loc 1 ( R + , H − s ( R 3 ) ) , s > 3 / 2 .

  4. If ( u ( t , x ) , p ( t , x ) ) is a solution of ( S ) , then ( u ( t , x ) , p ( t , x ) + h ( t ) ) is also the solution of ( S ) . Then, if we look for p in the space L loc 1 ( R + , H − 2 ( R 3 ) ) , we obtain h = 0 and we can express p in terms of u .

In our case of exponential damping, we are trying to find more regularity of Leray solution in ∩ p L p ( R + , L p ( R 3 ) ) . In particular, we give a new energy estimate. Our main result is the following.

Theorem 1.3

Let u 0 ∈ L 2 ( R 3 ) be a divergence-free vector fields, then there is a global solution of ( S ) u ∈ L ∞ ( R + , L 2 ( R 3 ) ) ∩ C ( R + , H − 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ ℰ β . Moreover, for all t ≥ 0 ,

(1.1) ‖ u ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u ( z ) ‖ L 2 2 d z + 2 α ∫ 0 t ‖ ( e β ∣ u ( z ) ∣ 2 − 1 ) ∣ u ( z ) ∣ 2 ‖ L 1 d z ≤ ‖ u 0 ‖ L 2 2 .

Remark 1.4

  1. If ( u , p ) is a solution of ( S ) given by Theorem 1.3, then by (1.1) and Proposition 2.1(3), we obtain

    lim t → 0 ‖ u ( t ) − u 0 ‖ L 2 = 0 .

  2. The fact ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ∈ L 1 ( R + , L 1 ( R 3 ) ) implies that u ∈ ∩ 4 ≤ p < ∞ L p ( R + , L p ( R 3 ) ) . Indeed, we have

    ∫ 0 ∞ ‖ ( e β ∣ u ( t ) ∣ 2 − 1 ) ∣ u ( t ) ∣ 2 ‖ L 1 d t = ∑ k = 1 ∞ β k k ! ∫ 0 ∞ ‖ u ( t ) ‖ L 2 k + 2 2 k + 2 d t .

  3. The fact u ∈ L ∞ ( R + , L 2 ( R 3 ) ) ∩ C ( R + , H − 2 ( R 3 ) ) implies that u ∈ C b ( R + , H − r ( R 3 ) ) , for all r > 0 .

The remainder of our paper is organized as follows. In Section 2, we provide some notations, definitions, and preliminary results. Section 3 is devoted to prove Theorem 1.3, and this proof is done in two steps. In the first, we give a general result of equicontinuity, and in the second, we apply the Friedritch method to construct a global solution of ( S ) , and the only delicate point is to show the convergence of the nonlinear part ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) to ( e β ∣ u ∣ 2 − 1 ) u . To do this, we approximate ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) by ∑ k = 1 m β k k ! ∣ u φ ( n ) ∣ 2 k u φ ( n ) , and we use the convergence in L loc p ( R + , L loc p ( R 3 ) ) given by technical lemmas.

2 Notations and preliminary results

2.1 Notations

  • For R > 0 : B R = { x ∈ R 3 ; ∣ x ∣ < R } .

  • For m ∈ N : P m ( X ) = ∑ k = 1 m X k k ! . Clearly, for each R > 0 , we have P m ( β R 2 ) ↗ ( e β R 2 − 1 ) , as m → + ∞ .

  • For R > 0 , the Friedritch operator J R is defined by

    J R ( D ) f = ℱ − 1 ( 1 B R ( ξ ) f ^ ) .

  • The Leray projector P : ( L 2 ( R 3 ) ) 3 → ( L 2 ( R 3 ) ) 3 is defined by

    ℱ ( P f ) = f ^ ( ξ ) − f ^ ( ξ ) ⋅ ξ ∣ ξ ∣ ξ ∣ ξ ∣ = M ( ξ ) f ^ ( ξ ) ; M ( ξ ) = δ k , l − ξ k ξ l ∣ ξ ∣ 2 1 ≤ k , l ≤ 3 .

  • For R > 0 , we define the operator A R ( D ) by

    A R ( D ) f = P J R ( D ) f = ℱ − 1 ( M ( ξ ) 1 B R ( ξ ) f ^ ) .

  • If p , q ∈ ( 1 , ∞ ) and Ω 1 ⊂ R n , Ω 2 ⊂ R m are two open subsets, and we define L p ( Ω 1 , L q ( Ω 2 ) ) by: f ∈ L p ( Ω 1 , L q ( Ω 2 ) ) if f : Ω 1 × Ω 2 → C , ( x 1 , x 2 ) → f ( x 1 , x 2 ) is measurable and

    ‖ f ‖ L p ( Ω 1 , L q ( Ω 2 ) ) = ‖ ‖ f ‖ L q ( Ω 2 ) ‖ L p ( Ω 1 )

    is well defined and finite. Particularly, if p = q , then L p ( Ω 1 , L q ( Ω 2 ) ) = L p ( Ω 1 × Ω 2 ) and

    ‖ f ‖ L p ( Ω 1 , L p ( Ω 2 ) ) = ‖ f ‖ L p ( Ω 1 × Ω 2 ) .

2.2 Preliminary results

In this section, we collect some classical results, and we provide some lemmas, which are well suited to the study of the system ( S ) .

Proposition 2.1

[8] Let H be Hilbert space.

  1. 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 .

  2. 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 ‖ .

  3. 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 2.2

[9] Let s 1 and s 2 be two real numbers and d ∈ N .

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

    ‖ f g ‖ H ˙ s 1 + s 2 − d 2 ≤ C 1 ( ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 + ‖ f ‖ H ˙ s 2 ‖ g ‖ H ˙ s 1 ) .

  2. If s 1 , s 2 < d / 2 and s 1 + s 2 > 0 , there exists a constant C 2 = C 2 ( d , s 1 , s 2 ) such that: if f ∈ H ˙ s 1 ( R d ) and g ∈ H ˙ s 2 ( R d ) , then f ⋅ g ∈ H ˙ s 1 + s 2 − d 2 ( R d ) and

    ‖ f g ‖ H ˙ s 1 + s 2 − d 2 ≤ C 2 ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 .

Lemma 2.3

For m ∈ N and β > 0 , there is c m , β > 0 such that

( P m ( β z 2 ) ) 2 ≤ c m , β ( e β z 2 − 1 ) z 2 , ∀ z ∈ R .

Proof

It suffices to prove that ( P m ( β z 2 ) ) 2 ( e β z 2 − 1 ) − 1 z − 2 is bounded on ( 0 , ∞ ) .□

Lemma 2.4

For m ∈ N and β > 0 , there is C m , β > 0 such that

∣ P m ( β ∣ x ∣ 2 ) x − P m ( β ∣ y ∣ 2 ) y ∣ ≤ C m , β ( P m ( β ∣ x ∣ 2 ) + P m ( β ∣ y ∣ 2 ) ) ∣ x − y ∣ , ∀ x , y ∈ R 3 .

Proof

Suppose that ∣ y ∣ ≤ ∣ x ∣ . We have

P m ( β ∣ x ∣ 2 ) x − P m ( β ∣ y ∣ 2 ) y = P m ( β ∣ x ∣ 2 ) ( x − y ) + ∑ k = 1 m β k k ! ( ∣ x ∣ 2 k − ∣ y ∣ 2 k ) y .

For k ∈ { 1 , … , m } , we have

∣ ∣ x ∣ 2 k − ∣ y ∣ 2 k ∣ ⋅ ∣ y ∣ ≤ 2 k ∣ x − y ∣ ⋅ ( ∣ x ∣ 2 k − 1 + ∣ y ∣ 2 k − 1 ) ⋅ ∣ y ∣ ≤ 2 m ∣ x − y ∣ ⋅ ( ∣ x ∣ 2 k − 1 ∣ y ∣ + ∣ y ∣ 2 k ) ≤ 2 m ∣ x − y ∣ ⋅ ( ∣ x ∣ 2 k + ∣ y ∣ 2 k ) ,

which implies the desired result.□

Lemma 2.5

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 .

Proof

Using the fact s > d / 2 , we obtain

∫ R d ( 1 + ∣ ξ ∣ 2 ) − s ∣ f ^ ( ξ ) ∣ 2 d ξ ≤ ∫ R d ( 1 + ∣ ξ ∣ 2 ) − s d ξ ‖ f ^ ‖ L ∞ ( R d ) 2 ≤ ∫ R d ( 1 + ∣ ξ ∣ 2 ) − s d ξ ‖ f ‖ L 1 ( R d ) 2 ,

which ends the proof.□

Lemma 2.6

Let ε ∈ ( 0 , 1 ) and d ∈ N such that d ≥ 2 , then there is a constant C > 0 such that If ( f , g ) ∈ H − ε 2 ( R d ) × H 1 − ε 2 ( R d ) , then f g ∈ H 1 − ε − d 2 ( R d ) and

‖ f g ‖ H 1 − ε − d 2 ( R d ) ≤ C ‖ f ‖ H − ε 2 ( R d ) ‖ g ‖ H 1 − ε 2 ( R d ) .

Proof

We have

‖ f g ‖ H 1 − ε − d 2 2 ≤ ‖ f g ‖ H ˙ 1 − ε − d 2 2 ≤ ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η f ^ ( η ) g ^ ( ξ − η ) 2 ≤ ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η ∣ f ^ ( ξ − η ) ∣ ∣ g ^ ( η ) ∣ 2 ≤ ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η ( 1 + ∣ ξ − η ∣ 2 ) − ε / 4 ∣ f ^ ( ξ − η ) ∣ ( 1 + ∣ ξ − η ∣ 2 ) ε / 4 ∣ g ^ ( η ) ∣ 2 .

By using the elementary inequality

( 1 + ∣ ξ − η ∣ 2 ) ε 4 ≤ ( 1 + 2 ∣ ξ ∣ 2 + 2 ∣ η ∣ 2 ) ε 4 ≤ 6 ε 4 ( 1 + ∣ ξ ∣ ε 2 + ∣ η ∣ ε 2 ) ,

we obtain

‖ f g ‖ H 1 − ε − d 2 2 ≤ C ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η ( 1 + ∣ ξ − η ∣ 2 ) − ε 4 ∣ f ^ ( ξ − η ) ∣ ( 1 + ∣ ξ ∣ ε 2 + ∣ η ∣ ε / 2 ) ∣ g ^ ( η ) ∣ 2 ≤ C ( I 1 + I 2 + I 3 ) ,

with

I 1 = ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η ( 1 + ∣ ξ − η ∣ 2 ) − ε 4 ∣ f ^ ( ξ − η ) ∣ ∣ g ^ ( η ) ∣ 2 = ‖ u 1 v 1 ‖ H ˙ 1 − ε − d 2 2 , u 1 = ℱ − 1 ( ( 1 + ∣ ξ ∣ 2 ) − ε 4 ∣ f ^ ( ξ ) ∣ ) , v 1 = ℱ − 1 ( ∣ g ^ ( ξ ) ∣ ) , I 2 = ∫ ξ ∣ ξ ∣ 2 1 − ε 2 − d 2 ∫ η ( 1 + ∣ ξ − η ∣ 2 ) − ε 4 ∣ f ^ ( ξ − η ) ∣ ∣ g ^ ( η ) ∣ d η 2 d ξ = ‖ u 2 v 2 ‖ H ˙ 1 − ε 2 − d 2 2 , u 2 = ℱ − 1 ( ( 1 + ∣ ξ ∣ 2 ) − ε 4 ∣ f ^ ( ξ ) ∣ ) , v 2 = ℱ − 1 ( ∣ g ^ ( ξ ) ∣ ) ,

and

I 3 = ∫ ξ ∣ ξ ∣ 2 1 − ε − d 2 ∫ η ( 1 + ∣ ξ − η ∣ 2 ) − ε 4 ∣ f ^ ( ξ − η ) ∣ ∣ η ∣ ε 2 ∣ g ^ ( η ) ∣ 2 = ‖ u 3 v 3 ‖ H ˙ 1 − ε − d 2 2 , u 3 = ℱ − 1 ( 1 + ∣ ξ ∣ 2 ) − ε 4 f ^ ( ξ ) , v 3 = ℱ − 1 ∣ ξ ∣ ε 2 ∣ g ^ ( ξ ) ∣ .

By applying Lemma 2.2 with the following choices:

( s 1 , t 1 ) = ( 0 , 1 − ε ) , ( s 2 , t 2 ) = 0 , 1 − ε 2 , ( s 3 , t 3 ) = ( 0 , 1 − ε ) ,

we obtain

‖ u i v i ‖ H ˙ s i + t i − d 2 ( R d ) ≤ C ‖ u i ‖ H ˙ s i ( R d ) ‖ v i ‖ H ˙ t i ( R d ) , i ∈ { 1 , 2 , 3 } .

Then

I 1 ≤ C ‖ u 1 ‖ H ˙ 0 2 ‖ v 1 ‖ H ˙ 1 − ε 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H ˙ 1 − ε 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H 1 − ε 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H 1 − ε 2 2 , I 2 ≤ C ‖ u 2 ‖ H ˙ 0 2 ‖ v 2 ‖ H ˙ 1 − ε 2 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H 1 − ε 2 2 , I 3 ≤ C ‖ u 3 ‖ H ˙ 0 2 ‖ v 3 ‖ H ˙ 1 − ε 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H ˙ 1 − ε 2 2 ≤ C ‖ f ‖ H − ε 2 2 ‖ g ‖ H 1 − ε 2 2 ,

which ends the proof of the Lemma.□

Lemma 2.7

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 ) , then there is a subsequence ( f h ( n ) ) and f ∈ L 2 ( Ω ) ∩ L p ( Ω ) such that

lim n → ∞ ( f h ( n ) ∣ g ) L 2 = ( f ∣ g ) L 2 , ∀ g ∈ L 2 ( Ω ) , ‖ f ‖ L p ≤ liminf n → ∞ ‖ f h ( n ) ‖ L p , ‖ f ‖ L 2 ≤ liminf n → ∞ ‖ f h ( n ) ‖ L 2 .

Proof

Let q the conjugate exponent of Hölder inequality

1 p + 1 q = 1 .

The sequence ( f n ) is bounded in the Hilbert space L 2 ( Ω ) , and then there is a subsequence ( f h ( n ) ) and f ∈ L 2 ( Ω ) such that

lim n → ∞ ( f h ( n ) ∣ g ) L 2 = ( f ∣ g ) L 2 , g ∈ L 2 ( Ω ) .

By Hölder inequality, we obtain

∣ ( f h ( n ) ∣ g ) L 2 ∣ ≤ ‖ f h ( n ) ‖ L p ‖ g ‖ L q , ∀ n ∈ N ,

which implies

∣ ( f ∣ g ) L 2 ∣ ≤ ( liminf n → ∞ ‖ f h ( n ) ‖ L p ) ⋅ ‖ g ‖ L q .

Particularly, for g ∈ L 1 ( Ω ) ∩ L ∞ ( Ω ) , we have

(2.1) ∣ ( f ∣ g ) L 2 ∣ ≤ ( liminf n → ∞ ‖ f h ( n ) ‖ L p ) ⋅ ‖ g ‖ L q .

For k ∈ N , put the following functions:

g ( x ) = f ( x ) ¯ ∣ f ( x ) ∣ ∣ f ( x ) ∣ q − 1 , if f ( x ) ≠ 0 , 0 , if F ( x ) = 0

and

g k ( x ) = 1 A k ( x ) g ( x ) ,

where

A k = { x ∈ Ω ; ∣ g ( x ) ∣ ≤ k } ∩ B k .

Clearly, g k ∈ L 1 ( Ω ) ∩ L ∞ ( Ω ) and 1 A k ⟶ 1 Ω almost every where. Applying inequality (2.1) to g k , we obtain

∫ Ω 1 A k ( x ) ∣ f ( x ) ∣ p d x ≤ ( liminf n → ∞ ‖ f h ( n ) ‖ L p ) ∫ Ω 1 A k ( x ) ∣ f ( x ) ∣ p d x 1 / q

and

∫ Ω 1 A k ( x ) ∣ f ( x ) ∣ p d x 1 / p ≤ liminf n → ∞ ‖ f h ( n ) ‖ L p .

As A k ⊂ A k + 1 and

1 A k ( x ) ∣ f ( x ) ∣ p ⟶ ∣ f ( x ) ∣ p , a.e x ∈ Ω ,

then Monotone Convergence Theorem implies that f ∈ L p ( Ω ) and

‖ f ‖ L p ≤ liminf n → ∞ ‖ f h ( n ) ‖ L p ,

which ends the proof of the first inequality.

The last result is given by Proposition 2.1.□

Remark 2.8

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 .

Lemma 2.9

Let T > 0 and ( f n ) be a bounded sequence in L T 2 ( H 1 ( R 3 ) ) such that

f n → f strongly in C ( [ 0 , T ] , H loc − 4 ( R 3 ) ) .

Then

f n ⟶ f strongly in L 2 ( [ 0 , T ] , L loc 2 ( R 3 ) ) .

Proof

Combining the inclusion C ( [ 0 , T ] , H loc − 4 ( R 3 ) ) ↪ L 2 ( [ 0 , T ] , H loc − 4 ( R 3 ) ) and the interpolation

L 2 ( [ 0 , T ] , H loc − 4 ( R 3 ) ∩ H 1 ( R 3 ) ) ↪ L 2 ( [ 0 , T ] , L loc 2 ( R 3 ) ) ,

we obtain the desired result.□

Lemma 2.10

Let T > 0 and p 0 ∈ ( 2 , + ∞ ) . If ( f n ) is a bounded sequence in L T 2 ( H 1 ( R 3 ) ) ∩ L T p 0 ( L p 0 ( R 3 ) ) such that

f n → f strongly in C ( [ 0 , T ] , H loc − 4 ( R 3 ) ) ,

then f ∈ L T p 0 ( L p 0 ( R 3 ) ) and

f n ⟶ f strongly in L p ( [ 0 , T ] , L loc p ( R 3 ) ) , ∀ 2 ≤ p < p 0 .

Proof

By Lemma 2.9, we obtain

f n ⟶ f strongly in L 2 ( [ 0 , T ] , L loc 2 ( R 3 ) ) .

By interpolation, between L p 0 ( [ 0 , T ] , L p 0 ( R 3 ) ) and L 2 ( [ 0 , T ] , L loc 2 ( R 3 ) ) , we obtain

f n → f strongly in L p ( [ 0 , T ] , L loc p ( R 3 ) ) , ∀ p ∈ [ 2 , p 0 ) .

It remains to show that f ∈ L T p 0 ( L p 0 ( R 3 ) ) . For m ∈ N , put

Ω m = ] 0 , T [ × B m and g m , n = 1 Ω m f n .

Clearly, Ω m is an open subset of R 4 and the sequence ( g m , n ) n is bounded in

L 2 ( [ 0 , T ] , L 2 ( B m ) ) ∩ L p 0 ( [ 0 , T ] , L p 0 ( B m ) ) = L 2 ( Ω m ) ∩ L p 0 ( Ω m ) .

Then, by Lemma 2.7 and Remark 2.8, we obtain 1 Ω m f ∈ L 2 ( Ω m ) ∩ L p 0 ( Ω m ) and

‖ 1 Ω m f ‖ L p 0 ( Ω m ) ≤ liminf n → ∞ ‖ g n , m ‖ L p 0 ( Ω m ) .

But ‖ g n , m ‖ L p 0 ( Ω m ) = ‖ f n ‖ L p 0 ( Ω m ) ≤ ‖ f n ‖ L T p 0 ( L p 0 ( R 3 ) ) , then

‖ 1 Ω m f ‖ L p 0 ( Ω m ) ≤ liminf n → ∞ ‖ f n ‖ L T p 0 ( L p 0 ( R 3 ) ) .

By applying the monotonic convergence theorem, we obtain f ∈ L T p 0 ( L p 0 ( R 3 ) ) and

‖ f ‖ L T p 0 ( L p 0 ( R 3 ) ) ≤ liminf n → ∞ ‖ f n ‖ L T p 0 ( L p 0 ( R 3 ) ) .

3 Proof of Theorem 1.3

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

Then:

  • For m ∈ N , there is a constant C = C ( m , α , β ) such that

    (3.1) ∫ 0 ∞ ∫ R 3 ( P m ( β ∣ F n ( t , x ) ∣ 2 ) ) 2 d t d x ≤ C ‖ f 0 ‖ L 2 2 .

  • For all T , R > 0 and s > 3 / 2 , we have

    (3.2) ∫ 0 T ‖ ( e β ∣ F n ∣ 2 − 1 ) ∣ F n ∣ ‖ H − s ≤ σ 3 , s M β , R T + 1 2 R α ‖ f 0 ‖ L 2 2 , ∀ n ∈ N ,

    where M β , R = sup 0 < r ≤ R e β r 2 − 1 r .

  • For every ε > 0 , there is δ = δ ( ε , α , β , ν 1 , ν 2 , ν 3 , r 1 , r 2 , r 3 , ‖ f 0 ‖ L 2 ) > 0 such that for all t 1 , t 2 ∈ R + , we have

    (3.3) ( ∣ 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 ) .

Remark 3.2

As ‖ F n ( t ) ‖ L 2 ≤ ‖ f 0 ‖ L 2 for all ( t , n ) ∈ R + × N , then by interpolation between H − s 0 ( R 3 ) and H 0 ( R 3 ) = L 2 ( R 3 ) , we obtain for all σ > 0 , for each ε > 0 , there is δ > 0 such that

( ∣ t 2 − t 1 ∣ < δ ⇒ ‖ F n ( t 2 ) − F n ( t 1 ) ‖ H − σ < ε ) , ∀ n ∈ N .

Proof of Proposition 3.1

  • First, (3.1) is given by ( E 2 ) and Lemma 2.3.

  • To prove (3.2), beginning by noting that the function h β : [ 0 , + ∞ ) ⟶ R + , y ⟼ e β y 2 − 1 y if t > 0 0 if t = 0 is continuous, and M β , R = sup y ∈ [ 0 , R ] h β ( y ) is well defined and finite.

For t ∈ [ 0 , T ] , put the following subset of R 3

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

Then, for x ∈ R 3 , we have

x ∈ X n , R , t ⇒ ( e β ∣ F n ( t , x ) ∣ 2 − 1 ) ∣ F n ( t , x ) ∣ = h β ( ∣ F n ( z , x ) ∣ ) ∣ F n ( t , x ) ∣ 2 ≤ M β , R ∣ F n ( t , x ) ∣ 2 , x ∈ X n , R , t c ⇒ ( e β ∣ F n ( t , x ) ∣ 2 − 1 ) ∣ F n ( t , x ) ∣ ≤ 1 R ( e β ∣ F n ( t , x ) ∣ 2 − 1 ) ∣ F n ( t , x ) ∣ 2 .

By Lemma 2.5, we obtain

D n ( T ) ≔ ∫ 0 T ‖ ( e β ∣ F n ( z ) ∣ 2 − 1 ) ∣ F n ( z ) ∣ ‖ H − s ≤ σ 3 , s ∫ 0 T ‖ ( e β ∣ F n ( z ) ∣ 2 − 1 ) ∣ F n ( z ) ∣ ‖ L 1 d z .

Combining this inequality with the aforementioned inequalities, we obtain

1 σ 3 , s D n ( T ) = ∫ 0 T ∫ R 3 ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ = ∫ 0 T ∫ X n , R , z ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ + ∫ 0 T ∫ X n , R , z c ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ ≤ M β , R ∫ 0 T ∫ X n , R , z ∣ F n ( z , x ) ∣ 2 + 1 R ∫ 0 T ∫ X n , R , z c ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ 2

≤ M β , R ∫ 0 T ∫ R 3 ∣ F n ( z , x ) ∣ 2 + 1 R ∫ 0 T ∫ R 3 ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ 2 ≤ M β , R ∫ 0 T ‖ F n ( z ) ‖ L 2 2 d z + 1 R ∫ 0 T ‖ ( e β ∣ F n ( z ) ∣ 2 − 1 ) ∣ F n ( z ) ∣ 2 ‖ L 1 ( R 3 ) .

By using ( E 2 ) , we obtain the desired result.

  • To prove (3.3), integrate ( E 1 ) over [ t 1 , t 2 ] ⊂ R + , we obtain

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

    with

    I n , 1 ( t 1 , t 2 ) = ∑ k = 1 3 ν k ∫ t 1 t 2 ‖ ∣ D k ∣ 2 r k F n ‖ H − s 0 , I n , 2 ( t 1 , t 2 ) = ∫ t 1 t 2 ‖ A n ( D ) div ( F n ⊗ F n ) ‖ H − s 0 , I n , 3 ( t 1 , t 2 ) = ∫ t 1 t 2 ‖ α A n ( D ) [ ( e β ∣ F n ∣ 2 − 1 ) 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 n , k ( t 1 , t 2 ) < ε / 3 , k ∈ { 1 , 2 , 3 } .

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

    I n , 1 ( t 1 , t 2 ) ≤ ∑ 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 ( z ) ‖ H 0 d z ≤ ∑ k = 1 3 ν k ‖ f ^ 0 ‖ L 2 ( t 2 − t 1 ) .

Then if ∣ t 2 − t 1 ∣ < δ 1 ≔ ε 3 ∑ k = 1 3 ν k ‖ f ^ 0 ‖ L 2 + 3 , we obtain I n , 1 ( t 1 , t 2 ) < ε / 3 .

  • Estimate I n , 2 ( t 1 , t 2 ) : By Lemma 2.5, we obtain

    I n , 2 ( t 1 , t 2 ) ≤ ∫ t 1 t 2 ‖ div ( F n ⊗ F n ) ( z ) ‖ H − 3 d z ≤ ∫ t 1 t 2 ‖ ( F n ⊗ F n ) ( z ) ‖ H − 2 d z ≤ σ 2 , 3 ∫ t 1 t 2 ‖ ( F n ⊗ F n ) ( z ) ‖ L 1 d z ≤ σ 2 , 3 ∫ t 1 t 2 ‖ F n ( z ) ‖ L 2 2 d z ≤ σ 2 , 3 ‖ f 0 ‖ L 2 2 ( t 2 − t 1 ) .

Then if ∣ t 2 − t 1 ∣ < δ 2 ≔ ε 3 σ 2 , 3 ‖ f 0 ‖ L 2 2 + 3 , we obtain I n , 2 ( t 1 , t 2 ) < ε / 3 .

  • Estimate of I n , 3 ( t 1 , t 2 ) : Let R > 0 (to fixed later) and t 1 < t 2 ∈ R + . By Lemma 2.5, we obtain

    ∫ t 1 t 2 ‖ ( e β ∣ F n ( z ) ∣ 2 − 1 ) ∣ F n ( z ) ∣ ‖ H − 3 ≤ σ 3 , 3 ∫ t 1 t 2 ‖ ( e β ∣ f ( z ) ∣ 2 − 1 ) ∣ f ( z ) ∣ ‖ L 1 d z .

Combining this inequality with the above inequalities, we obtain

I 3 ( t 1 , t 2 ) α σ 3 , 3 = ∫ t 1 t 2 ∫ R 3 ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ = ∫ t 1 t 2 ∫ X n , R , z ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ + ∫ t 1 t 2 ∫ X n , R , z c ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ ≤ M β , R ∫ t 1 t 2 ∫ X n , R , z ∣ F n ( z , x ) ∣ 2 + 1 R ∫ t 1 t 2 ∫ X n , R , z c ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ 2 ≤ M β , R ∫ t 1 t 2 ∫ R 3 ∣ F n ( z , x ) ∣ 2 + 1 R ∫ t 1 t 2 ∫ R 3 ( e β ∣ F n ( z , x ) ∣ 2 − 1 ) ∣ F n ( z , x ) ∣ 2 ≤ M β , R ∫ t 1 t 2 ‖ F n ( z ) ‖ L 2 2 d z + 1 R ∫ t 1 t 2 ‖ ( e β ∣ F n ( z ) ∣ 2 − 1 ) ∣ F n ( z ) ∣ 2 ‖ L 1 ( R 3 ) .

By using ( E 2 ) , we obtain

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

Then, with the choices R = R ε = 3 σ 3 , 3 ‖ f 0 ‖ L 2 2 + 3 ε and δ 3 = ε 6 α σ 3 , 3 M β , R ε ‖ f 0 ‖ L 2 2 + 6 , we obtain

∣ t 1 − t 2 ∣ < δ 3 ⇒ I n , 3 ( t 1 , t 2 ) < ε / 3 .

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

3.2 Step 2

In this step, we construct a global solution of ( S ) , where we use a method inspired by [9]. For this, consider the approximate system with the parameter n ∈ N :

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

  • By Cauchy-Lipschitz theorem, we obtain a unique solution u n ∈ C 1 ( R + , L 2 ( R 3 ) ) of ( S n ) . Moreover, J n u n = u n and

    (3.5) ‖ u n ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u n ‖ L 2 2 + 2 α ∫ 0 t ‖ ( e β ∣ u n ∣ 2 − 1 ) ∣ u n ∣ 2 ‖ L 1 ≤ ‖ u 0 ‖ L 2 2 .

  • By inequality (3.5), we obtain ( u n ) is bounded in L loc 2 ( R + , H 1 ( R 3 ) ) :

    (3.6) ∫ 0 T ( ‖ u n ‖ L 2 2 + ‖ ∇ u n ‖ L 2 2 ) ≤ ‖ u 0 ‖ L 2 2 T + ‖ u 0 ‖ L 2 2 2 = M T , ∀ T ≥ 0 .

  • By using inequality (3.5) and the fact that

    (3.7) ( e β ∣ u n ∣ 2 − 1 ) ∣ u n ∣ 2 = ∑ k = 1 ∞ β k k ! ∣ u n ∣ 2 k + 2 ,

    we obtain for all k ∈ N , ( u n ) is bounded in L 2 k + 2 ( R + × R 3 ) and

    (3.8) ∫ 0 ∞ ∫ R 3 ∣ u n ∣ 2 k + 2 ≤ k ! 2 α β k .

  • By applying Proposition 3.1 and Remark 2.8, we obtain

    (3.9) 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 (3.5)–(3.9) 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 ) ) such that for all q ∈ N , we have

(3.10) lim n → ∞ ‖ θ q ( u φ ( n ) − u ) ‖ L ∞ ( [ 0 , T q ] , H − 4 ) = 0 .

  • Combining inequalities (3.7)–(3.10) and applying Lemma 2.10, we obtain

    (3.11) u φ ( n ) → u strongly in L loc p ( R + , L loc p ( R 3 ) ) , ∀ p ∈ [ 2 , ∞ ) .

  • The sequence ( u n ) is bounded in the Hilbert space L 2 ( R + , H ˙ 1 ( R 3 ) ) , then

    u φ ( n ) → u weakly in L 2 ( R + , H ˙ 1 ( R 3 ) ) .

Particularly u ∈ L 2 ( R + , H ˙ 1 ( R 3 ) ) and

(3.12) ∫ 0 t ‖ ∇ u ‖ L 2 2 ≤ liminf n → ∞ ∫ 0 t ‖ ∇ u φ ( n ) ‖ L 2 2 , ∀ t ≥ 0 .

  • Prove that for all T > 0 and k ∈ N ,

    (3.13) ∫ 0 T ∫ R 3 ∣ u ∣ 2 k + 2 ≤ liminf n → ∞ ∫ 0 T ∫ R 3 ∣ u φ ( n ) ∣ 2 k + 2 .

For this take m ∈ N and put Ω m = ] 0 , T [ × B m . Applying Lemma 2.10 to the sequence ( 1 Ω m u φ ( n ) ) with p 0 = 2 k + 4 and p = 2 k + 2 , with taking into account (3.8), we obtain

∫ 0 T ∫ B m ∣ u ∣ 2 k + 2 ≤ liminf n → ∞ ∫ 0 T ∫ B m ∣ u φ ( n ) ∣ 2 k + 2 .

Then

∫ 0 T ∫ B m ∣ u ∣ 2 k + 2 ≤ liminf n → ∞ ∫ 0 T ∫ R 3 ∣ u φ ( n ) ∣ 2 k + 2 .

By using monotonic convergence theorem, we obtain the desired result.

  • Prove that for all T > 0 ,

    (3.14) ∫ 0 T ∫ R 3 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ≤ liminf n → ∞ ∫ 0 T ∫ R 3 ( e β ∣ u φ ( n ) ∣ 2 − 1 ) ∣ u φ ( n ) ∣ 2 .

For this, take m ∈ N and by (3.13), we obtain

∫ 0 T ∫ R 3 P m ( β ∣ u ∣ 2 ) ∣ u ∣ 2 ≤ liminf n → ∞ ∫ 0 T ∫ R 3 P m ( β ∣ u φ ( n ) ∣ 2 ) ∣ u φ ( n ) ∣ 2 .

Then

∫ 0 T ∫ R 3 P m ( β ∣ u ∣ 2 ) ∣ u ∣ 2 ≤ liminf n → ∞ ∫ 0 T ∫ R 3 ( e β ∣ u φ ( n ) ∣ 2 − 1 ) ∣ u φ ( n ) ∣ 2 .

Monotonic convergence theorem gives the desired result.

  • Combining the aforementioned inequalities, we obtain for all t ≥ 0 ,

    (3.15) ‖ u ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u ( z ) ‖ L 2 2 d z + 2 α ∫ 0 t ‖ ( e β ∣ u ( z ) ∣ 2 − 1 ) ∣ u ( z ) ∣ 2 ‖ L 1 d z ≤ ‖ u 0 ‖ L 2 2 .

It remains to show that u is a solution of the system ( S ) . To do this, we must verify that

(3.16) lim n → ∞ J φ ( n ) ( u φ ( n ) i u φ ( n ) j ) = u i u j in D ′ ( [ 0 , ∞ ) × R 3 ) ,

(3.17) lim n → ∞ J φ ( n ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] = ( e β ∣ u ∣ 2 − 1 ) u in D ′ ( [ 0 , ∞ ) × R 3 ) .

  • Let Φ in C 0 ∞ ( R + × R 3 ) . There is an integer q ∈ N such that

    supp ( Φ ) ⊂ [ 0 , T q ) × B ( 0 , q ) .

We start by proving (3.16). For this, we write

Φ u φ ( n ) i u φ ( n ) j − Φ u i u j = θ q Φ u φ ( n ) i u φ ( n ) j − θ q Φ u i u j = θ q ( u φ ( n ) i − u i ) Φ u φ ( n ) j + θ q ( u φ ( n ) j − u j ) Φ u i .

Using Lemma 2.6, we obtain, for ε 0 ∈ ( 0 , 1 ) ,

‖ θ q ( u φ ( n ) i − u i ) Φ u φ ( n ) j ‖ L 2 ( R + , H − ε 0 − 1 2 ) ≤ C ‖ θ q ( u φ ( n ) i − u i ) ‖ L ∞ [ 0 , T q ] , H − ε 0 2 ‖ Φ u φ ( n ) j ‖ L 2 [ 0 , T q ] , H 1 − ε 0 2 ≤ C ‖ θ q ( u φ ( n ) i − u i ) ‖ L ∞ [ 0 , T q ] , H − ε 0 2 ‖ Φ u φ ( n ) j ‖ L 2 ( [ 0 , T q ] , H 1 ) ≤ C ′ ‖ θ q ( u φ ( n ) i − u i ) ‖ L ∞ [ 0 , T q ] ; H − ε 0 2 ‖ u φ ( n ) j ‖ L 2 ( [ 0 , T q ] ; H 1 ) ≤ C ′ M T q ‖ θ q ( u φ ( n ) i − u i ) ‖ L ∞ [ 0 , T q ] ; H − ε 0 2 ,

where M T q is given by (3.6). Then, equation (3.10) implies

lim n → ∞ ‖ θ q ( u φ ( n ) i − u i ) Φ u φ ( n ) j ‖ L 2 ( R + , H − ε 0 − 1 2 ) = 0 .

In a similar way, we show that

lim n → ∞ θ q ( u φ ( n ) j − u j ) Φ u i = 0 in L 2 ( R + , H − ε 0 − 1 2 ( R 3 ) ) .

Then, we obtain

lim n → ∞ Φ u φ ( n ) i u φ ( n ) j = Φ u i u j in L 2 ( R + , H − ε 0 − 1 2 ( R 3 ) ) .

Using the fact that ( u n ) is bounded in L ∞ ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) , we obtain

‖ ( I d − J φ ( n ) ) u i ( t ) u j ( t ) ‖ H − ε 0 − 1 2 ≤ ‖ ( I d − J φ ( n ) ) u i ( t ) u j ( t ) ‖ H ˙ − ε 0 − 1 2 ≤ ( φ ( n ) ) − ε 0 ‖ u i ( t ) u j ( t ) ‖ H ˙ − 1 2 ≤ C ( φ ( n ) ) − ε 0 ‖ u ( t ) ‖ L 2 ‖ u ( t ) ‖ H ˙ 1 .

Then, we obtain

‖ ( I d − J φ ( n ) ) u i ( t ) u j ( t ) ‖ L 2 R + , H − ε 0 − 1 2 ≤ C ( φ ( n ) ) − ε 0 ‖ u ‖ L ∞ ( R + , L 2 ) ‖ u ‖ L 2 ( R + , H ˙ 1 ) ,

which implies (3.16).

  • Now, we want to prove (3.17). Let ε > 0 and R 0 > 0 such that

    (3.18) 1 R 0 < ε 4 ‖ u 0 ‖ L 2 2 ‖ Φ ‖ L ∞ ( R + × R 3 ) + 4 .

Let m 0 ∈ N such that

(3.19) ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 = ∑ k = m 0 + 1 ∞ β k k ! R 0 2 k + 1 < ε 4 α ‖ Φ ‖ L 1 ( R + × R 3 ) + 4 .

We have

∫ 0 ∞ ∫ R 3 J φ ( n ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] Φ − ∫ 0 ∞ ∫ R 3 ( e β ∣ u ∣ 2 − 1 ) u Φ = ∑ i = 1 3 L n i + S ,

with

L n 1 = ∫ 0 ∞ ∫ R 3 [ P m 0 ( β ∣ u φ ( n ) ∣ 2 ) u φ ( n ) − P m 0 ( β ∣ u ∣ 2 ) u ] Φ , L n 2 = ∫ 0 ∞ ∫ R 3 ( I d − J φ ( n ) ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] Φ , L n 3 = ∫ 0 ∞ ∫ R 3 [ ( e β ∣ u φ ( n ) ∣ 2 − 1 − P m 0 ( β ∣ u φ ( n ) ∣ 2 ) ) u φ ( n ) ] Φ , S = ∫ 0 ∞ ∫ R 3 [ ( e β ∣ u ∣ 2 − 1 − P m 0 ( β ∣ u ∣ 2 ) ) u ] Φ .

  1. Estimate of term L n 1 : Using Lemma 2.4, we obtain

    ∣ L n 1 ∣ ≤ C m 0 , β ∫ 0 T q ∫ B ( 0 , q ) [ P m 0 ( β ∣ u φ ( n ) ∣ 2 ) + P m 0 ( β ∣ u ∣ 2 ) ] θ q ∣ u φ ( n ) − u ∣ ∣ Φ ∣ ≤ C m 0 , β ‖ Φ ‖ L ∞ ( R + × R 3 ) ∫ 0 T q ∫ B ( 0 , q ) [ P m 0 ( β ∣ u φ ( n ) ∣ 2 ) + P m 0 ( β ∣ u ∣ 2 ) ] θ q ∣ u φ ( n ) − u ∣ .

Applying Cauchy-Schwarz and using Lemma 2.3, we obtain

∣ L n 1 ∣ ≤ C m 0 , β ‖ Φ ‖ L ∞ ( R + × R 3 ) ( ‖ [ P m 0 ( β ∣ u φ ( n ) ∣ 2 ) ‖ L 2 ( [ 0 , T q ] × B ( 0 , q ) ) + ‖ P m 0 ( β ∣ u ∣ 2 ) ] ‖ L 2 ( [ 0 , T q ] × B ( 0 , q ) ) ) ‖ θ q ( u φ ( n ) − u ) ‖ L 2 ( [ 0 , T q ] × B ( 0 , q ) ) ≤ c m 0 , β C m 0 , β ‖ Φ ‖ L ∞ ( R + × R 3 ) ( ‖ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) ∣ u φ ( n ) ∣ 2 ‖ L 1 ( [ 0 , T q ] , L 1 ( R 3 ) ) 1 / 2 + ‖ ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ‖ L 1 ( [ 0 , T q ] , L 1 ( R 3 ) ) 1 / 2 ) ‖ θ q ( u φ ( n ) − u ) ‖ L 2 ( [ 0 , T q ] × B ( 0 , q ) ) .

Energy inequalities (3.5)–(3.15) give

∣ L n 1 ∣ ≤ c m 0 , β C m 0 , β ‖ Φ ‖ L ∞ ( R + × R 3 ) 2 ‖ u 0 ‖ L 2 α ‖ θ q ( u φ ( n ) − u ) ‖ L 2 ( [ 0 , T q ] , L 2 ) .

Then, the convergence result (3.10) gives an integer n 1 ∈ N such that: for all n ≥ n 1 , we have

(3.20) ∣ L n 1 ∣ < ε / 4 .

  1. Estimate of term L n 2 : we have

    ∣ L n 2 ∣ = ∣ C ∫ 0 T q ∫ R 3 ( I d − J φ ( n ) ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] Φ ∣ = ∣ C ∫ 0 T q ( ( I d − J φ ( n ) ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] ∣ Φ ) L 2 ∣ ≤ C ∫ 0 T q ‖ ( I d − J φ ( n ) ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] ‖ H − 4 ‖ Φ ( t ) ‖ H 4 ≤ C φ ( n ) ∫ 0 T q ‖ [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] ‖ H − 3 ‖ Φ ‖ L ∞ ( R + , H 4 ) ≤ C φ ( n ) ‖ [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] ‖ L 1 ( [ 0 , T q ] , H − 3 ) ‖ Φ ‖ L ∞ ( R + , H 4 ) .

By using Proposition 2.1(3.2) with the choice ( R = 1 , s = 3 , T = T q ) , we obtain

∣ L n 2 ∣ ≤ C φ ( n ) M β , 1 T q + 1 2 α σ 3 , 3 ‖ u 0 ‖ L 2 2 ‖ Φ ‖ L ∞ ( R + , H 4 ) ,

which implies that there is an integer n 2 ∈ N such that

(3.21) ∣ L n 2 ∣ < ε / 4 , ∀ n ≥ n 2 .

  1. Estimate of term L n 3 : Put the following set

    Y n , R 0 = { ( t , x ) ∈ [ 0 , T q ] × B ( 0 , q ) ; ∣ u φ ( n ) ( t , x ) ∣ ≤ R 0 } .

We have

∣ L n 3 ∣ ≤ ∫ Y n , R 0 [ ( e β ∣ u φ ( n ) ∣ 2 − 1 − P m 0 ( β ∣ u φ ( n ) ∣ 2 ) ) ∣ u φ ( n ) ∣ ] ∣ Φ ∣ + ∫ Y n , R 0 c [ ( e β ∣ u φ ( n ) ∣ 2 − 1 − P m 0 ( β ∣ u φ ( n ) ∣ 2 ) ) ∣ u φ ( n ) ∣ ] ∣ Φ ∣ ≤ [ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ] ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 R 0 ∫ Y n , R 0 c [ ( e β ∣ u φ ( n ) ∣ 2 − 1 − P m 0 ( β ∣ u φ ( n ) ∣ 2 ) ) ∣ u φ ( n ) ∣ 2 ] ‖ Φ ‖ L ∞ ( R + × R 3 ) ≤ [ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ] ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 R 0 ∫ R + × R 3 [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) ∣ u φ ( n ) ∣ 2 ] ‖ Φ ‖ L ∞ ( R + × R 3 ) ≤ [ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ] ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 R 0 ‖ u 0 ‖ L 2 2 ‖ Φ ‖ L ∞ ( R + × R 3 ) ,

Using the choices (3.18)–(3.19), we obtain

(3.22) ∣ L n 3 ∣ < ε / 4 , ∀ n ∈ N .

  1. Estimate of term S : Put the following subset of R 4 ,

    Z R 0 = { ( t , x ) ∈ [ 0 , T q ] × B ( 0 , q ) ; ∣ u ( t , x ) ∣ ≤ R 0 } .

We have

∣ S ∣ ≤ ∫ Z R 0 [ ( e β ∣ u ∣ 2 − 1 − P m 0 ( β ∣ u ∣ 2 ) ) ∣ u ∣ ] ∣ Φ ∣ + ∫ Z R 0 c [ ( e β ∣ u ∣ 2 − 1 − P m 0 ( β ∣ u ∣ 2 ) ) ∣ u ∣ ] ∣ Φ ∣ ≤ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 R 0 ∫ Z R 0 c ( e β ∣ u ∣ 2 − 1 − P m 0 ( β ∣ u ∣ 2 ) ) ∣ u ∣ 2 ‖ Φ ‖ L ∞ ( R + × R 3 ) ≤ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 R 0 ∫ R + × R 3 ( e β ∣ u ∣ 2 − 1 ) ∣ u ∣ 2 ‖ Φ ‖ L ∞ ( R + × R 3 ) ≤ ( e β R 0 2 − 1 − P m 0 ( β R 0 2 ) ) R 0 ‖ Φ ‖ L 1 ( R + × R 3 ) + 1 2 α R 0 ‖ u 0 ‖ L 2 2 ‖ Φ ‖ L ∞ ( R + × R 3 ) .

By the choices (3.18)–(3.19), we obtain

(3.23) ∣ S ∣ < ε / 4 .

Combining (3.20), (3.21), (3.22), and (3.23), we obtain

∫ 0 ∞ ∫ R 3 J φ ( n ) [ ( e β ∣ u φ ( n ) ∣ 2 − 1 ) u φ ( n ) ] Φ − ∫ 0 ∞ ∫ R 3 ( e β ∣ u ∣ 2 − 1 ) u Φ < ε , ∀ n ≥ max ( n 1 , n 2 ) ,

which implies (3.17), and the proof of Theorem 1.3 is finished.

Acknowledgements

The author is deeply grateful to the referee for his suggestions that helped him to improve this paper.

  1. Funding information: This Project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (14-MAT730-02).

  2. Conflict of interest: The author states no conflict of interest.

References

[1] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213–231. 10.1002/mana.3210040121Search in Google Scholar

[2] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, Singapore, World Scientific, 1997. 10.1142/3538Search in Google Scholar

[3] F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal. 166 (2003), 359–376. 10.1007/s00205-002-0234-5Search in Google Scholar

[4] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), 193–248. 10.1007/BF02547354Search in Google Scholar

[5] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model, Comm. Math. Phys. 238 (2003), no. 1–2, 211–223. 10.1007/s00220-003-0859-8Search in Google Scholar

[6] D. Bresch, B. Desjardins, and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differ. Equ. 28 (2003), no. 3–4, 843–868. 10.1081/PDE-120020499Search in Google Scholar

[7] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes with damping, J. Math. Anal. Appl. 343 (2008), 799–809. 10.1016/j.jmaa.2008.01.041Search in Google Scholar

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

[9] J.-Y. Chemin, About Navier-Stokes Equations, Publications of Jaques-Louis Lions Laboratoiry, Paris VI University, R96023, 1996. Search in Google Scholar

[10] H. Bahouri, J. Y Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, Book Series: Grundlehren der mathematischen Wissenschaften, Vol. 343, Springer, Berlin, Heidelberg, 2011. 10.1007/978-3-642-16830-7Search in Google Scholar
Received: 2021-05-01
Revised: 2021-11-22
Accepted: 2022-06-23
Published Online: 2022-08-12

© 2022 Jamel Benameur, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0050
Keywords for this article
Navier-Stokes equations; Friedrich method; weak solution
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  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
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