On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces

Xiaochun Sun; Gaoting Xu; Yulian Wu

doi:10.1515/math-2023-0170

Article Open Access

On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces

Xiaochun Sun , Gaoting Xu and Yulian Wu

Published/Copyright: December 31, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence T * is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.

Keywords: Navier-Stokes equations; Coriolis force; existence; blow-up criterion; Lei-Lin-Gevrey spaces

MSC 2010: 35A01; 35Q35; 76D06

1 Introduction

In this article, we consider the 3D Navier-Stokes-Coriolis equations:

(1.1) ∂ t u + ν ( − Δ ) α u + Ω e 3 × u + ( u ⋅ ∇ ) u = − ∇ p , in R 3 × ( 0 , ∞ ) , div u = 0 , in R 3 × ( 0 , ∞ ) , u ∣ t = 0 = u 0 , in R 3 ,

where u = ( u 1 , u 2 , u 3 ) denotes the velocity field of the fluid and p is the pressure. The positive constant ν is the viscosity coefficient; the Coriolis parameter Ω ∈ R , which denotes twice the speed of rotation around the vertical unit vector e 3 = ( 0 , 0 , 1 ) ; α is a positive parameter; and ( − Δ ) α ( α > 0 ) denotes the fractional Laplacian, which is defined as:

ℱ ( ( − Δ ) α u ) ( t , ξ ) = ∣ ξ ∣ 2 α ℱ ( u ) ( t , ξ ) ,

where ℱ ( u ) is the Fourier transform of u with respect to spatial variable x .

When α = 1 and Ω ≠ 0 , equation (1.1) reduces to the classical Navier-Stokes equations with Coriolis force. Babin et al. [1,2] proved the global existence and regularity of the solution to the 3D rotating Navier-Stokes equations in the case ∣ Ω ∣ is large enough. Wang and Wu [3] proved the global well-posedness of the mild solution to the Navier-Stokes equations with the Coriolis force if the initial data are in χ 1 − 2 α spaces. Iwabuchi and Takada [4] proved the existence of global unique solutions to the Navier-Stokes equations with the Coriolis force in Sobolev spaces H ˙ s with 1 2 < s < 3 4 if the speed of rotation ∣ Ω ∣ is sufficiently large. Iwabuchi and Takada [5] also obtained the global in time existence and the uniqueness of the mild solution for small initial data in Fourier-Besov spaces ℱℬ 1 , 2 − 1 ( R 3 ) . Hieber and Shibata [6] proved that the Navier-Stokes equations with the Coriolis force possess a unique global mild solution for arbitrary speed of rotation provided the initial data u 0 is small enough in H 1 2 ( R 3 ) .

When α ≠ 1 and Ω = 0 , equation (1.1) reduces to the fractional Navier-Stokes equations. Ding and Sun [7] studied the uniqueness of the weak solution to the fractional Navier-Stokes equations. Sun and Liu [8] obtained the uniqueness of the weak solution to the fractional anisotropic Navier-Stokes system. Sun and Liu [9] studied the long time decay of the fractional Navier-Stokes equations in Sobolev-Gevrey spaces.

When α = 1 and Ω = 0 , equation (1.1) reduces to the classical Navier-Stokes equations. Benameur and Jlali [10] proved a global well-posedness of three-dimensional incompressible Navier-Stokes equations under initial data u 0 in the Lei-Lin-Gevrey space and proved blow-up criterion. Cannone and Wu [11] proved the global well-posedness for the 3D Navier-Stokes equations in critical Fourier-Herz spaces. Jlali [12] studied the long time decay of global solution to the 3D incompressible Navier-Stokes equations in Fourier-Lei-Lin spaces.

Besides, Cannone and Karch [13] proved the existence of singular solutions of the incompressible Navier-Stokes system with singular external forces. Guterres et al. [14] proved the results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. We can refer to [15] for more questions about Navier-Stokes equations.

In this article, we studied the existence and blow-up criterion of the solution to the 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces. Recently, many authors have made considerable progresses regarding the existence, uniqueness, and blow-up of solutions for the generalized magnetohydrodynamics equations in Lei-Lin and Lei-Lin-Gevrey spaces, which can be refer to [16–18]. This article is inspired by [18], and on this basis, our main results are as follows.

Theorem 1.1

Assume that a ≥ 0 , σ ≥ 1 , 1 − 2 α < s ≤ 0 , α ∈ ( 1 2 , 1 ] , then there exists a time T > 0 , such that for any u 0 ∈ χ a , σ s ( R 3 ) , Navier-Stokes-Coriolis equation (1.1) has a unique solution u ∈ C ( [ 0 , T ) , χ a , σ s ( R 3 ) ) ⋂ L 1 ( [ 0 , T * ) , χ a , σ s + 2 α ( R 3 ) ) . Moreover, u ∈ L 2 ( [ 0 , T * ) , χ a , σ s + α ( R 3 ) ) .

Theorem 1.2

Suppose that a ≥ 0 , σ ≥ 1 , 1 − 2 α < s ≤ 0 , α ∈ ( 1 2 , 1 ] , and 0 ≤ t < T * . Let u ∈ C ( [ 0 , T ) , χ a , σ s ( R 3 ) ) ⋂ L 1 ( [ 0 , T * ) , χ a , σ s + 2 α ( R 3 ) ) be a maximal solution of the Navier-Stokes-Coriolis system. If T * is finite, then

limsup t → T * ‖ u ( t ) ‖ χ a , σ s ( R 3 ) = ∞ .

2 Preliminaries

In this section, we recall some notations, definitions, and lemmas that will be used later.

The Fourier transform of f is defined by:
ℱ ( f ) ( ξ ) = f ^ ( ξ ) = ∫ R 3 e − i x ⋅ ξ f ( x ) d x , ξ ∈ R 3 , f ∈ L 1 ( R 3 ) ,
and the Fourier inverse transform of f is defined by:
ℱ − 1 ( f ) ( x ) = ( 2 π ) − 3 ∫ R 3 e i x ⋅ ξ f ( ξ ) d ξ , x ∈ R 3 , f ∈ L 1 ( R 3 ) .
Lei-Lin-Gevrey space is defined by:
χ a , σ s ( R 3 ) = f ∈ S ′ : f ^ ∈ L loc 1 , ‖ u ( t ) ‖ χ a , σ s ( R 3 ) = ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ∣ f ^ ( ξ ) ∣ d ξ < ∞ ,
where s ∈ R , a ≥ 0 , and σ ≥ 1 .
Assuming that ( X , ‖ ⋅ ‖ ) is a normed vector space and T > 0 , the space L T p ( X ) , 1 ≤ p ≤ ∞ , contains all measurable functions f : [ 0 , T ] → X for which the following norms are finite:
‖ f ‖ L T ∞ ( X ) = esssup t ∈ [ 0 , T ] ‖ f ‖ , L T p ( X ) = ∫ 0 T ‖ f ‖ p d t 1 p ( 1 ≤ p < ∞ ) .
C T ( X ) = { f : [ 0 , T ] → X continuous } is endowed with the norm ‖ ⋅ ‖ L T ∞ ( X ) .
The tensor product and the usual convolution, respectively, are given by:
f ⊗ g = ( g 1 f , g 2 f , g 3 f ) , u ∗ v ( x ) = ∫ R 3 u ( x − y ) v ( y ) d y ,
where f , g : R 3 → R 3 , and u , v : R 3 → R .

Lemma 2.1

[18] Let a ≥ 0 , σ ≥ 1 , s ≤ 0 , then χ a , σ s ( R 3 ) is a Banach space.

Lemma 2.2

[18] Let a ≥ 0 , γ ≥ 1 2 , and σ ≥ 1 . The following inequalities hold

‖ f ‖ χ a , σ s + 1 ≤ ‖ f ‖ χ a , σ s 1 − 1 2 γ ‖ f ‖ χ a , σ s + 2 γ 1 2 γ , if s ∈ R ;
‖ f ‖ χ a , σ 0 ≤ ‖ f ‖ χ a , σ s 1 + s 2 γ ‖ f ‖ χ a , σ s + 2 γ − s 2 γ , if − 2 γ ≤ s ≤ 0 .

Lemma 2.3

[18] Assume that f , g ∈ χ a , σ s + 1 ( R 3 ) ⋂ χ a σ , σ 0 ( R 3 ) , with a ≥ 0 , σ ≥ 1 , and s ≥ − 1 . Then, f g ∈ χ a , σ s + 1 ( R 3 ) . Moreover, the inequality holds

‖ f g ‖ χ a , σ s + 1 ( R 3 ) ≤ 2 s − 2 π − 3 ( ‖ f ‖ χ a σ , σ 0 ( R 3 ) ‖ g ‖ χ a , σ s + 1 ( R 3 ) + ‖ f ‖ χ a , σ s + 1 ( R 3 ) ‖ g ‖ χ a σ , σ 0 ( R 3 ) ) .

Lemma 2.4

[18] Assume that a ≥ 0 , σ ≥ 1 , T > 0 , and 1 − 2 α ≤ s ≤ 0 , where α ∈ ( 1 2 , 1 ] . Then,

∫ 0 T ‖ f ⊗ g ‖ χ a , σ s + 1 ( R 3 ) d τ ≤ C s T s − 1 2 α + 1 ‖ f ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) 1 + s 2 α ‖ g ‖ L T ∞ ( χ a , σ s ( R 3 ) ) 1 − 1 2 α ‖ f ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) − s 2 α ‖ g ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) 1 2 α + ‖ f ‖ L T ∞ ( χ a , σ s ( R 3 ) ) 1 − 1 2 α ‖ g ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) 1 + s 2 α ‖ f ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) 1 2 α ‖ g ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) − s 2 α .

Remark 1

Lemma 2.8 in [18] can be taken as α = β .

Lemma 2.5

[18] Let a ≥ 0 , σ ≥ 1 , γ ∈ R , s ∈ R , and f ∈ S ′ such that ∇ ⋅ f = 0 , then

∫ 0 t ‖ e − ( t − τ ) ( − △ ) γ P ( u ⋅ ∇ ) u ‖ χ a , σ s ( R 3 ) d τ ≤ ∫ 0 t ‖ u ⊗ u ‖ χ a , σ s + 1 ( R 3 ) d τ .

Proof

Helmholtz’s projector P satisfies

(2.1) ℱ [ P ( u ) ] ( ξ ) = u ^ ( ξ ) − u ^ ( ξ ) ⋅ ξ ∣ ξ ∣ 2 ξ , ∀ ξ ∈ R 3 .

□ ∫ 0 t ‖ e − ( t − τ ) ( − △ ) γ P ( u ⋅ ∇ ) u ‖ χ a , σ s d τ = ∫ 0 t ∫ R 3 e − ( t − τ ) ∣ ξ ∣ 2 γ ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ∣ P ( u ⋅ ∇ ) u ^ ∣ d ξ d τ ≤ ∫ 0 t ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ∣ ( u ⋅ ∇ ) u ^ ∣ d ξ d τ ≤ ∫ 0 t ∫ R 3 ∣ ξ ∣ s + 1 e a ∣ ξ ∣ 1 σ ∣ u ⊗ u ^ ∣ d ξ d τ = ∫ 0 t ‖ u ⊗ u ‖ χ a , σ s + 1 d τ .

Lemma 2.6

[19] Let ( X , ‖ ⋅ ‖ ) be a Banach space and B : X × X → X be a bilinear continuous operator, and there exists a positive constant C such that

‖ B ( x , y ) ‖ ≤ C ‖ x ‖ ‖ y ‖ , ∀ x , y ∈ X .

Then, for each x 0 ∈ X that satisfies 4 C ‖ x 0 ‖ < 1 , one has that equation

a = x 0 + B ( a , a )

admits a solution x = a ∈ X . Moreover, x obeys the inequality ‖ x ‖ ≤ 2 ‖ x 0 ‖ , and it is the only one such that ‖ x ‖ ≤ 1 2 C .

3 Proof of main results

First, we consider the following linear generalized problem:

(3.1) ∂ t u + ν ( − Δ ) α u + Ω e 3 × u + ∇ P = 0 , in R 3 × ( 0 , ∞ ) , div u = 0 , in R 3 × ( 0 , ∞ ) , u ∣ t = 0 = u 0 , in R 3 .

The solution of this equation can be given by the generalized Stokes-Coriolis semigroup T Ω , α ( t ) , which has the following explicit representation:

T Ω , α ( t ) u = ℱ − 1 cos Ω ξ 3 ∣ ξ ∣ t e − ν t ∣ ξ ∣ 2 α I + sin Ω ξ 3 ∣ ξ ∣ t e − ν t ∣ ξ ∣ 2 α R ( ξ ) ∗ u = ℱ − 1 cos Ω ξ 3 ∣ ξ ∣ t I + sin Ω ξ 3 ∣ ξ ∣ t R ( ξ ) ∗ ( e − ν t ( − Δ ) α u ) ,

where divergence free vector field u ∈ S ( R 3 ) , I is the unit matrix in M 3 × 3 ( R ) , and R ( ξ ) is skew-symmetric matrix defined by:

R ( ξ ) ≔ 1 ∣ ξ ∣ 0 ξ 3 − ξ 2 − ξ 3 0 ξ 1 ξ 2 − ξ 1 0 , ξ ∈ R 3 \ { 0 } .

Now, we prove our main results by using lemmas.

Proof of Theorem 1.1

First, we know that the solution of equation (1.1) is as follows:

(3.2) u ( t ) = T Ω , α u 0 − ∫ 0 t T Ω , α ( t − τ ) P div ( u ⊗ u ) ( τ ) d τ ,

where P = I − ∇ ( − △ ) − 1 div is the Leray-Hopf projection. So, the solution of equation (1.1) can be rewritten as:

u ( t ) = T Ω , α u 0 + B ( u , u ) ,

where

B ( u , v ) ( t ) = − ∫ 0 t T Ω , α ( t − τ ) P div ( u ⊗ v ) ( τ ) d τ .

Moreover, T > 0 , χ T = C ( [ 0 , T ) , χ a , σ s ( R 3 ) ) ⋂ L 1 ( [ 0 , T * ) , χ a , σ s + 2 α ( R 3 ) ) is endowed with the norm:

‖ u ‖ χ T = ‖ u ‖ L T ∞ ( χ a , σ s ( R 3 ) ) + ‖ u ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) .

Step 1. Estimate of u in χ a , σ s ( R 3 ) .

Using the norm definition, the Hölder inequality, and the boundedness of multiplier T Ω , α , we have

‖ T Ω , α u 0 ‖ χ a , σ s ( R 3 ) = ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ∣ T Ω , α u 0 ^ ∣ d ξ = ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ℱ ℱ − 1 cos Ω ξ 3 ∣ ξ ∣ t I + sin Ω ξ 3 ∣ ξ ∣ t R ( ξ ) * e − ν ( − △ ) α t u 0 d ξ = ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ cos Ω ξ 3 ∣ ξ ∣ t I + sin Ω ξ 3 ∣ ξ ∣ t R ( ξ ) e − ν t ∣ ξ ∣ 2 α ∣ u 0 ^ ∣ d ξ ≤ ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ e − ν t ∣ ξ ∣ 2 α ∣ u 0 ^ ∣ d ξ ≤ ‖ u 0 ‖ χ a , σ s ( R 3 ) .

By boundedness of (2.1) and multipliers T Ω , α , we obtain

‖ B ( u , v ) ‖ χ a , σ s ( R 3 ) ≤ ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ∫ 0 t ∣ ℱ [ T Ω , α ( t − τ ) P div ( u ⊗ v ) ( τ ) ] ∣ d τ d ξ ≤ ∫ R 3 ∣ ξ ∣ s + 1 e a ∣ ξ ∣ 1 σ ∫ 0 t cos Ω ξ 3 ∣ ξ ∣ ( t − τ ) I + sin Ω ξ 3 ∣ ξ ∣ ( t − τ ) R ( ξ ) e − ν ( t − τ ) ∣ ξ ∣ 2 α ∣ u ⊗ v ^ ∣ d τ d ξ ≤ ∫ R 3 ∣ ξ ∣ s + 1 e a ∣ ξ ∣ 1 σ ∫ 0 T ∣ u ⊗ v ^ ∣ d τ d ξ ≤ ∫ 0 T ‖ u ⊗ v ^ ‖ χ a , σ s + 1 ( R 3 ) d τ .

By applying Lemma 2.4 and Young’s inequality, we deduce

‖ B ( u , v ) ‖ χ a , σ s ( R 3 ) ≤ C s T s − 1 2 α + 1 ‖ u ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) 1 + s 2 α ‖ v ‖ L T ∞ ( χ a , σ s ( R 3 ) ) 1 − 1 2 α ‖ u ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) − s 2 α ‖ v ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) 1 2 α + ‖ u ‖ L T ∞ ( χ a , σ s ( R 3 ) ) 1 − 1 2 α ‖ v ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) 1 + s 2 α ‖ u ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) 1 2 α ‖ v ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) − s 2 α ≤ C s , α T s − 1 2 α + 1 ( ‖ u ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) + ‖ u ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) ) ( ‖ v ‖ L T ∞ ( χ a , σ s ( R 3 ) ) + ‖ v ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) ) + ( ‖ v ‖ L T ∞ ( χ a σ , σ s ( R 3 ) ) + ‖ v ‖ L T 1 ( χ a σ , σ s + 2 α ( R 3 ) ) ) ( ‖ u ‖ L T ∞ ( χ a , σ s ( R 3 ) ) + ‖ u ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) ) ≤ C s , α T s − 1 2 α + 1 ‖ u ‖ χ T ‖ v ‖ χ T .

Step 2. Estimate of u in L 1 ( [ 0 , T * ) , χ a , σ s + 2 α ( R 3 ) ) .

Using the norm definition and the boundedness of multiplier T Ω , α , we obtain

‖ T Ω u 0 ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) = ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∫ 0 T cos Ω ξ 3 ∣ ξ ∣ t I + sin Ω ξ 3 ∣ ξ ∣ t R ( ξ ) e − ν t ∣ ξ ∣ 2 α ∣ u 0 ^ ∣ d t d ξ ≤ ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∣ u 0 ^ ∣ ∫ 0 T e − ν t ∣ ξ ∣ 2 α d t d ξ ≤ 1 ν ‖ u 0 ‖ χ a , σ s ( R 3 ) .

Applying Minkowski’s inequality and Lemma 2.4, we obtain

‖ B ( u , v ) ‖ L T 1 ( χ a , σ s + 2 α ( R 3 ) ) ≤ ∫ 0 T ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∫ 0 t ∣ ℱ [ T Ω , α ( t − τ ) P div ( u ⊗ v ) ( τ ) ] ∣ d τ d ξ d t ≤ ∫ 0 T ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∫ 0 t e − ν ( t − τ ) ∣ ξ ∣ 2 α ∣ u ⊗ v ^ ∣ d τ d ξ d t ≤ ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∫ 0 T ∣ u ⊗ v ^ ∣ ∫ τ T e − ν ( t − τ ) ∣ ξ ∣ 2 α d t d τ d ξ ≤ 1 ν ∫ R 3 ∣ ξ ∣ s + 2 α e a ∣ ξ ∣ 1 σ ∫ 0 T ∣ u ⊗ v ^ ∣ d τ d ξ ≤ 1 ν ∫ 0 T ‖ u ⊗ v ^ ‖ χ a , σ s + 1 ( R 3 ) d τ ≤ 1 ν C s , α T s − 1 2 α + 1 ‖ u ‖ χ T ‖ v ‖ χ T .

Therefore,

‖ T Ω , α u 0 ‖ χ T ≤ 1 + 1 ν ‖ u 0 ‖ χ a , σ s .

B : χ T × χ T → χ T is a continuous bilinear operator that satisfies the following inequality:

‖ B ( u , v ) ‖ χ T ≤ C s , α , ν T s − 1 2 α + 1 ‖ u ‖ χ T ‖ v ‖ χ T .

Thereby, we can apply Lemma 2.6 to obtain a unique solution u ∈ χ T for equation (1.1).

Finally, we show that the solution obtained above for equation (1.1) also belongs to L 2 ( [ 0 , T * ) , χ a , σ s + α ( R 3 ) ) .

Using the definition, we obtain

‖ T Ω , α u 0 ‖ L T 2 ( χ a , σ s + α ( R 3 ) ) ≤ ∫ R 3 ∫ 0 T ( ∣ ξ ∣ s + α e a ∣ ξ ∣ 1 σ ∣ T Ω , α u 0 ^ ∣ ) 2 d t 1 2 d ξ ≤ ∫ R 3 ∣ ξ ∣ s + α e a ∣ ξ ∣ 1 σ ∣ u 0 ^ ( ξ ) ∣ ∫ 0 T e − 2 ν t ∣ ξ ∣ 2 α d t 1 2 d ξ = ∫ R 3 ∣ ξ ∣ s + α e a ∣ ξ ∣ 1 σ ∣ u 0 ^ ( ξ ) ∣ 1 − e − 2 ν T ∣ ξ ∣ 2 α 2 ν ∣ ξ ∣ 2 α 1 2 d ξ ≤ 1 2 ν ‖ u 0 ‖ χ a , σ s ( R 3 ) .

Applying Minkowski’s inequality and Lemma 2.4, we obtain

‖ B ( u , u ) ‖ L T 2 ( χ a , σ s + α ( R 3 ) ) ≤ ∫ 0 T ∫ R 3 ∣ ξ ∣ s + α + 1 e a ∣ ξ ∣ 1 σ ∫ 0 t ∣ ℱ [ T Ω , α ( t − τ ) P div ( u ⊗ u ) ( τ ) ] ∣ d τ d ξ 2 d t 1 2 ≤ ∫ R 3 ∫ 0 T ∫ τ T ∣ ξ ∣ s + α + 1 e a ∣ ξ ∣ 1 σ e − ν ( t − τ ) ∣ ξ ∣ 2 α ∣ u ⊗ u ^ ∣ 2 d t 1 2 d τ d ξ = ∫ R 3 ∣ ξ ∣ s + α + 1 e a ∣ ξ ∣ 1 σ ∫ 0 T ∣ u ⊗ u ^ ∣ 1 − e − 2 ν ( T − τ ) ∣ ξ ∣ 2 α 2 ν ∣ ξ ∣ 2 α 1 2 d τ d ξ ≤ 1 2 ν ∫ 0 T ‖ u ⊗ u ^ ‖ χ a , σ s + 1 ( R 3 ) d τ ≤ C s , ν T s − 1 2 α + 1 ‖ u ‖ χ T ‖ u ‖ χ T .

Thus, we conclude that u ∈ L 2 ( [ 0 , T * ) , χ a , σ s + α ( R 3 ) ) .□

Proof of Theorem 1.2

Suppose by contradiction that Theorem 1.2 does not hold, i.e.,

(3.3) limsup t → T * ‖ u ( t ) ‖ χ a , σ s ( R 3 ) < ∞ .

In other words, there exists a constant C > 0 such that

(3.4) ‖ u ( t ) ‖ χ a , σ s ( R 3 ) ≤ C , ∀ t ∈ [ 0 , T * ) .

By applying the Fourier transform and taking the scalar product in C 3 of the first equation of (1.1) with u ^ ( t ) , one has

⟨ ∂ t u ^ , u ^ ⟩ + ν ⟨ ( − Δ ) α u ^ , u ^ ⟩ + ⟨ ( u ⋅ ∇ ) u ^ , u ^ ⟩ + ⟨ Ω e 3 × u ^ , u ^ ⟩ + ⟨ ∇ p ^ , u ^ ⟩ = 0 ,

we all know that

⟨ Ω e 3 × u ^ , u ^ ⟩ = Ω ⟨ ( − u 2 ^ , u 1 ^ , 0 ) ( u 1 ^ , u 2 ^ , u 3 ^ ) ⟩ = Ω [ − u 2 ^ u 1 ^ ¯ + u 1 ^ u 2 ^ ¯ ] = 2 Ω i Im ( u 1 ^ u 2 ^ ¯ ) .

Hence, we have

Re ⟨ Ω e 3 × u ^ , u ^ ⟩ = 0 .

Therefore, we can write

1 2 ∂ t ∣ u ^ ∣ 2 + ν ∣ ξ ∣ 2 α ∣ u ^ ∣ 2 ≤ ∣ ⟨ ( u ⋅ ∇ ) u ^ , u ^ ⟩ ∣ .

For ε > 0 arbitrary,

1 2 ∂ t ∣ u ^ ∣ 2 = 1 2 ∂ t ( ∣ u ^ ∣ 2 + ε ) = ∣ u ^ ∣ 2 + ε ∂ t ∣ u ^ ∣ 2 + ε .

Thus, we can obtain

∣ u ^ ∣ 2 + ε ∂ t ∣ u ^ ∣ 2 + ε + ν ∣ ξ ∣ 2 α ∣ u ^ ∣ 2 ≤ ∣ ⟨ ( u ⋅ ∇ ) u ^ , u ^ ⟩ ∣ , ∂ t ∣ u ^ ∣ 2 + ν ∣ ξ ∣ 2 α ∣ u ^ ∣ 2 ∣ u ^ ∣ 2 + ε ≤ ∣ ( u ⋅ ∇ ) u ^ ∣ .

By integrating from [ 0 , t ] , we conclude

∣ u ( t ) ^ ∣ 2 + ε + ν ∫ 0 t ∣ ξ ∣ 2 α ∣ u ^ ∣ 2 ∣ u ^ ∣ 2 + ε d τ ≤ ∣ u 0 ^ ∣ 2 + ε + ∫ 0 t ∣ ( u ⋅ ∇ ) u ^ ∣ d τ .

Passing to the limit as ε → 0

∣ u ( t ) ^ ∣ + ν ∣ ξ ∣ 2 α ∫ 0 t ∣ u ^ ∣ d τ ≤ ∣ u 0 ^ ∣ + ∫ 0 t ∣ ( u ⋅ ∇ ) u ^ ∣ d τ .

Multiplying by ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ and integrating over ξ , we deduce

‖ u ‖ χ a , σ s ( R 3 ) + ν ∫ 0 t ‖ u ‖ χ a , σ s + 2 α ( R 3 ) d τ ≤ ‖ u 0 ‖ χ a , σ s ( R 3 ) + ∫ 0 t ‖ u ⋅ ∇ u ‖ χ a , σ s ( R 3 ) d τ .

Therefore, applying Lemmas 2.4 and 2.5, we can obtain

‖ u ‖ χ a , σ s ( R 3 ) + ν ∫ 0 t ‖ u ‖ χ a , σ s + 2 α ( R 3 ) d τ ≤ ‖ u 0 ‖ χ a , σ s ( R 3 ) + ∫ 0 t ‖ u ⊗ u ‖ χ a , σ s + 1 ( R 3 ) d τ ≤ ‖ u 0 ‖ χ a , σ s ( R 3 ) + C s ∫ 0 t ‖ u ‖ χ a , σ s ( R 3 ) 2 + s − 1 2 α ‖ u ‖ χ a , σ s + 2 α ( R 3 ) 1 − s 2 α d τ .

As a consequence, by (3.4), we have

‖ u ‖ χ a , σ s ( R 3 ) + ν ∫ 0 t ‖ u ‖ χ a , σ s + 2 α ( R 3 ) d τ ≤ ‖ u 0 ‖ χ a , σ s ( R 3 ) + C s C 2 + s − 1 2 α ∫ 0 t ‖ u ‖ χ a , σ s + 2 α ( R 3 ) 1 − s 2 α d τ .

Hence, for all t ∈ [ 0 , T * ) , we obtain

(3.5) ∫ 0 T * ‖ u ‖ χ a , σ s + 2 α ( R 3 ) d τ ≤ C a , σ , s , α , u 0 , T * .

Assume that ( k n ) is a sequence such that k n → T * , where k n ≤ k m , for all m , n ∈ N with m ≤ n . Let us prove that

limsup n , m → ∞ ‖ u ( k m ) − u ( k n ) ‖ χ a , σ s ( R 3 ) = 0 .

According to (3.2), we can obtain

u ( k m ) − u ( k n ) = [ T Ω , α ( k m ) − T Ω , α ( k n ) ] u 0 − ∫ 0 k m [ T Ω , α ( k m − τ ) − T Ω , α ( k n − τ ) ] P ( u ⋅ ∇ ) u d τ + ∫ k m k n [ T Ω , α ( k n − τ ) ] P ( u ⋅ ∇ ) u d τ ≕ I 1 + I 2 + I 3 .

Observe that

‖ I 1 ‖ χ a , σ s ( R 3 ) ≤ ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ( e − k m ∣ ξ ∣ 2 α − e − T * ∣ ξ ∣ 2 α ) ∣ u 0 ^ ( ξ ) ∣ d ξ .

Consequently, by the dominated convergence theorem, we have

limsup n , m → ∞ ‖ I 1 ‖ χ a , σ s ( R 3 ) = 0 .

Furthermore,

‖ I 2 ‖ χ a , σ s ( R 3 ) ≤ ∫ 0 k m ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ ( e − ( k m − τ ) ∣ ξ ∣ 2 α − e − ( k n − τ ) ∣ ξ ∣ 2 α ) ∣ u ⋅ ∇ u ^ ∣ d ξ d τ ≤ ∫ 0 T * ∫ R 3 ∣ ξ ∣ s e a ∣ ξ ∣ 1 σ [ 1 − e − ( T * − k m ) ∣ ξ ∣ 2 α ] ∣ u ⋅ ∇ u ^ ∣ d ξ d τ ≤ ∫ 0 T * ‖ u ⋅ ∇ u ‖ χ a , σ s d τ .

Using Lemma 2.4, Lemma 2.5, and (3.4), one deduces

∫ 0 T * ‖ u ⋅ ∇ u ‖ χ a , σ s d τ ≤ C s T * 1 + 1 − s 2 α ‖ u ‖ L T * ∞ ( χ a , σ s ( R 3 ) ) 2 + s − 1 2 α ‖ u ‖ L T * 1 ( χ a , σ s + 2 α ( R 3 ) ) 1 − s 2 α ≤ C s T * 1 + 1 − s 2 α C 2 + s − 1 2 α C a , σ , s , α , u o , T * 1 − s 2 α .

As a result, by the dominated convergence theorem, we deduce

limsup n , m → ∞ ‖ I 2 ‖ χ a , σ s ( R 3 ) = 0 .

Finally, using Hölder’s inequality and (3.5), we obtain

‖ I 3 ‖ χ a , σ s ( R 3 ) ≤ ∫ k m k n ‖ T Ω , α ( k n − τ ) P ( u ⋅ ∇ ) u ‖ χ a , σ s ( R 3 ) d τ ≤ C s ∫ k m T * ‖ u ‖ χ a , σ s ( R 3 ) 2 + s − 1 2 α ‖ u ‖ χ a , σ s + 2 α ( R 3 ) 1 − s 2 α d τ ≤ C a , σ , s , α , u 0 , T * ∫ k m T * ‖ u ‖ χ a , σ s + 2 α ( R 3 ) 1 − s 2 α d τ ≤ C a , σ , s , α , u 0 , T * ( T * − k m ) 1 − 1 − s 2 α .

Consequently, by taking n , m → ∞ , we obtain

limsup n , m → ∞ ‖ I 3 ‖ χ a , σ s ( R 3 ) = 0 .

Thus, we show that u ( k n ) is a Cauchy sequence in the Banach space χ a , σ s . There is u 1 ∈ χ a , σ s such that

limsup n → ∞ ‖ u ( k n ) − u 1 ‖ χ a , σ s ( R 3 ) = 0 .

The aforementioned limit does not depend on k n , and choose ( ρ n ) n ∈ N ⊆ ( 0 , T * ) such that ρ n ⟶ T * ,

limsup n → ∞ ‖ u ( ρ n ) − u 2 ‖ χ a , σ s ( R 3 ) = 0 ,

for u 2 ∈ χ a , σ s ( R 3 ) . Define ( ς n ) n ∈ N ⊆ ( 0 , T * ) by ς 2 n = k n , and ς 2 n − 1 = ρ n , for all n ∈ N . It is easy to check that ς n ⟶ T * . There is u 3 ∈ χ a , σ s ( R 3 ) such that

limsup n → ∞ ‖ u ( ς n ) − u 3 ‖ χ a , σ s ( R 3 ) = 0 .

Consequently, we infer u 1 = u 2 = u 3 . We proved that the solution u can be extended beyond t = T * , by the existence and uniqueness of u ¯ ∈ C T ¯ ( χ a , σ s ( R 3 ) ) ( T ¯ > 0 ) , for the GNSC System (1.1). u ˜ ∈ C T ¯ + T * ( χ a , σ s ( R 3 ) ) given by:

(3.6) u ˜ ( t ) = u ( t ) , t ∈ [ 0 , T * ) ; u ¯ ( t − T * ) , t ∈ [ T * , T ¯ + T * ] ,

solves (1.1) in [ 0 , T ¯ + T * ] ; this is a contradiction. Thus, we have

□ limsup t → T * ‖ u ( t ) ‖ χ a , σ s ( R 3 ) = ∞ .

Acknowledgements

The authors would like to thank the anonymous referee and editor for their valuable comments and suggestions, which greatly helped us improve the presentation of this article.

Funding information: This work was supported by the National Natural Science Foundation of China (11601434).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state that there is no conflict of interest.

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Received: 2023-08-01

Revised: 2023-11-28

Accepted: 2023-12-10

Published Online: 2023-12-31

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

Articles in the same Issue

https://doi.org/10.1515/math-2023-0170

Keywords for this article

Navier-Stokes equations; Coriolis force; existence; blow-up criterion; Lei-Lin-Gevrey spaces

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