Home Mathematics Optimal large time behavior of the 3D rate type viscoelastic fluids
Article Open Access

Optimal large time behavior of the 3D rate type viscoelastic fluids

  • Yangyang Chen and Yinghui Zhang EMAIL logo
Published/Copyright: October 29, 2025
Download Article (PDF)
Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

We investigate optimal decay estimates of solutions to the 3D Cauchy problem of the rate type viscoelastic fluids. The main novelty of this article involves three aspects: first, we show that the second-order and third-order spatial derivative of the global solution converges to zero at L 2 -rate ( 1 + t ) − 7 4 and ( 1 + t ) − 9 4 , respectively, which improve previous related decay results. Second, we prove that the decay rate of the m th order spatial derivative (where m ∈ [ 0 , 3 ] ) of the extra stress tensor of the field in L 2 is ( 1 + t ) − 5 4 − m 2 , which is faster than that of the velocity. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates, which coincide with the upper rates. Therefore, our results are precisely optimal in this sense.

Keywords: rate type viscoelastic fluids; optimal decay estimate; upper bound; lower bound
MSC 2020: 35B40; 35Q35; 74H40

1 Introduction and main results

We investigate the 3D Cauchy problem of the rate-type viscoelastic fluids, which describes the motion of viscoelastic fluids and takes the following form in R 3 :

(1.1) v t + v ⋅ ∇ v = div T + f , B t + v ⋅ ∇ B − β △ B + θ 1 ( B − I ) + θ 2 ( B 2 − B ) = ( W ( v ) B − B W ( v ) ) + a ( D ( v ) B + B D ( v ) ) , div v = 0 , v ( x , t ) ∣ t = 0 = v 0 ( x ) , B ( x , t ) ∣ t = 0 = B 0 ( x ) ,

where the Cauchy stress is given by

(1.2) T = 2 κ D ( v ) + 2 a [ ( 1 − α ) ( B − I ) + α ( B 2 − B ) ] − P I .

The terms on the right-hand side of equation ( 1.1 ) 2 adhere to the principle of material frame indifference. D ( v ) and W ( v ) denote the symmetric and antisymmetric parts of the velocity gradient ∇ v , respectively, which are defined as following:

D ( v ) = 1 2 [ ∇ v + ( ∇ v ) T ] , represents the rate of deformation tensor;

W ( v ) = 1 2 [ ∇ v − ( ∇ v ) T ] , represents the vorticity tensor.

Here, ( x , t ) ∈ R 3 × [ 0 , + ∞ ] . Let P : R 3 → R , v : R 3 → R 3 , B : R 3 → R > 0 3 × 3 and f : R 3 → R 3 represent the pressure, the velocity field, given density of the external body and the extra stress tensor of the field, respectively. And the known constants β , θ 1 , θ 2 ≥ 0 , κ > 0 , α ∈ ( 0 , 1 ) and a ∈ R of the aforemetnioned system. In addition, B represents the set of symmetric positive definite ( n × n ) -matrices.

1.1 History of the problem

Let’s give some necessary explanations about the aforementioned model. The system (1.1) is a significant type of model used to describe the motion of viscoelastic fluids. The model can be seen as a coupling of the Navier-Stokes (NS) equation with a rate type viscoelastic fluid model, which is completed with a diffusion term when β > 0 . By coupling the NS equation with the viscoelastic fluid model, the system (1.1) provides a more comprehensive framework for researching the behavior of viscoelastic fluids in various flow scenarios.

When B is an identity matrix or a = 0 in equation (1.2), the NS equation decouples from the system (1.1). The term B in (1.1)2 indicates that a rate type model is being considered. It is noteworthy that the values of a , α , θ 1 , and θ 2 correspond to different models, with these factors being categorized into three cases:

  1. Giesekus model, for a = α = θ 1 = 0 and θ 2 > 0 , refer to [3,10];

  2. Johnson-Segalman model, for α = 0 and a ∈ [ − 1 , 1 ] , refer to [15];

  3. Oldroyd-B model, for a = α = θ 2 = 0 and θ 1 > 0 , refer to [20].

It is worth mentioning that the viscoelastic rate-type fluids have been extensively studied by [7,21]. In addition, the incompressible viscoelastic rate-type models with stress diffusion are highly valuable as they describe non-Newtonian phenomena. There have been many studies on the mechanics of viscoelastic rate-type fluids with stress diffusion and their applications. Málek et al. introduced thermodynamically consistent models for viscoelastic fluids, including the variations of the compressible and incompressible Maxwell and Oldroyd-B models in [19]. Bathory et al. constructed a comprehensive existence theory that is applicable for all time and for large datasets, specifically within the framework of weak solutions in [2]. The global existence of smooth solutions for small data for viscoelastic rate-type models was established in [6,9,16,17].

Recently, Bulíček et al. [4] established the global existence and uniqueness of weak solution to the Cauchy question (1.1) with arbitrarily large initial data in a two-dimensional torus T 2 . Ai et al. [1] showed global existence of the solution to (1.1) with stress diffusion when the H 3 -norm of initial data is small enough. In addition, if the initial data belong to homogeneous Sobolev H ˙ − s or Besov spaces B ˙ 2 , ∞ − s   ( s ∈ ( 0 , 3 2 ) ) , they also proved the optimal decay rates of the solution and its higher-order derivatives. Wang and Wen obtained the global well-posedness for strong solution of for an Oldroyd-B model, along with time-decay estimates in Sobolev spaces in [23]. In addition, Huang et al. [11] established the global-in-time existence and obtained some time decay estimates for the incompressible diffusive Oldroyd-B model without diffusive properties or without viscous dissipation in three dimensions, which means that a = α = θ 2 = 0 , θ 1 > 0 , and κ = 0 or β = 0 in system (1.1). And then, Wang established the optimal decay estimate for the highest-order spatial derivatives for the incompressible diffusive Oldroyd-B model without viscous dissipation in [24].

However, to the best of our knowledge, up to now, there is no result on the optimal decay rate of the 3D Cauchy problem of the incompressible rate type viscoelastic system with stress diffusion property. The main motivation of this article is to give a definite answer to this issue. Our main novelty of this article can be outlined as follows: First, we show that the second-order and third-order spatial derivatives of the global solution converge to zero at L 2 -rate ( 1 + t ) − 7 4 and ( 1 + t ) − 9 4 , respectively, which improve the decay results in [1]. Second, we prove that the decay rate of the m th order spatial derivative (where m ∈ [ 0 , 3 ] ) of the extra stress tensor of the field in L 2 is ( 1 + t ) − 5 4 − m 2 , which is faster than that of the velocity. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates, which are the same as the upper rates. Therefore, our results are precisely optimal in this regard.

Our main purpose is to investigate the optimal upper and lower decay rates of the 3D viscoelastic fluids with stress diffusion. To do this, we first reformulate the system. By taking the change of variables by b = B − I and v = v , and making the extra stress tensor term disappear (i.e., f = 0 ) for brevity, we can reformulate the system (1.1) into:

(1.3) v t + v ⋅ ∇ v − κ △ v + ∇ P = 2 a div ( b + α b b ) , b t + v ⋅ ∇ b − β △ b + ( θ 1 + θ 2 ) b + θ 2 b b = a ( ∇ v + ( ∇ v ) ⊤ ) + a + 1 2 ( ( ∇ v b ) + ( ∇ v b ) ⊤ ) + a − 1 2 ( ( b ∇ v ) + ( b ∇ v ) ⊤ ) , div v = 0 ,

for ( x , t ) ∈ R 3 × R + . The initial condition is given by

(1.4) ( v , b ) ( x , 0 ) = ( v 0 ( x ) , b 0 ( x ) ) .

As in [1], we assume that β > 0 and θ 1 + θ 2 > 0 . Then, when the initial perturbation ( v 0 , b 0 ) is small un H N for any integer ( N ≥ 3 ), the unique global solution of the Cauchy problems (1.3) and (1.4) has been proved [1]. The results of the decay rate of the global solution and its derivative of the system can be seen in Lemmas 2.1 and 2.2.

1.2 Notations

Let’s introduce the notations frequently used in this article. We use L p to denote the usual Lebesgue space L p ( R 3 ) with norm ∥ ⋅ ∥ L p and H ℓ to denote Sobolev spaces H ℓ ( R 3 ) = W ℓ , 2 ( R 3 ) with norms ∥ ⋅ ∥ H ℓ . In addition, we denote ∥ ( a , b ) ∥ X ≜ ∥ a ∥ X + ∥ b ∥ X for simplicity. The notation a ≲ b means that a ≤ C b for a generic positive constant C > 0 only depends on the parameters coming from the problem. Moreover, we drop the x -dependence of differential operators, and one has ∇ f = ∇ x f = ( ∂ x 1 f , ∂ x 2 f , ∂ x 3 f ) and ∇ k to denote any partial derivative ∂ α with multi-index α , ∣ α ∣ = k .

For any f ∈ L 2 ( R 3 ) , we use ℱ to denote Fourier transform of f , where

ℱ ( f ) ( ξ ) ≜ f ˆ ( ξ ) = 1 ( 2 π ) 3 2 ∫ R 3 e − i x ξ f ( x ) d x ,

and ℱ − 1 is its inverse. Let’s define the low-frequency and high-frequency decomposition firstly. For any f ∈ L 2 ( R 3 ) , we define low-frequency part and the high-frequency part of f

f l = ℱ − 1 [ φ ( ξ ) f ˆ ] , f h = ℱ − 1 [ ( 1 − φ ( ξ ) ) f ˆ ] ,

where φ ( ξ ) is a smooth cut-off function satisfying

0 ⩽ φ ( ξ ) ⩽ 1 , φ 0 ( ξ ) ≜ 1 − φ ( ξ ) , φ ( ξ ) = 0 , ∣ ξ ∣ ≥ η , φ ( ξ ) = 1 , ∣ ξ ∣ ≤ η 2 ,

and the positive constant η is defined in the Lemma 3.3. And then for any f ∈ H 3 ( R ) , by Plancherel’s theorem, the following estimates hold

(1.5) ‖ ∇ k f l ‖ L 2 + ‖ ∇ k f l 1 ‖ L 2 + ‖ ∇ k f h ‖ L 2 + ‖ ∇ k f h 1 ‖ L 2 ≲ ‖ ∇ k f ‖ L 2 ,

for 0 ≤ k ≤ 3 , and

(1.6) ‖ ∇ k f h 1 ‖ L 2 ≲ ‖ ∇ k f h ‖ L 2 ≲ ‖ ∇ k + 1 f h ‖ L 2 ≲ ‖ ∇ k + 1 f ‖ L 2 ,

for 0 ≤ k ≤ 2 , where we set

φ 1 ( ξ ) ≜ 1 − ( 1 − φ ( ξ ) ) 2 , φ 2 ( ξ ) ≜ ( 1 − φ ( ξ ) ) 2 = ( φ 0 ( ξ ) ) 2

and

f l ( x ) ≜ φ ( ξ ) ( D ) f ( x ) , f h ( x ) ≜ φ 0 ( ξ ) ( D ) f ( x ) ,

f l 1 ( x ) ≜ φ 1 ( ξ ) ( D ) f ( x ) , f h 1 ( x ) ≜ φ 2 ( ξ ) ( D ) f ( x ) .

For any integer k ∈ [ 0,4 ] , the notation ∇ k ( v h , b h ) denotes the high-frequency part of ∇ k ( v , b ) , the notation ∇ k ( v l , b l ) denotes the low-frequency part of ∇ k ( v , b ) .

Let P be the Leray projector and Λ ≜ − △ , which can be represented as follows:

( P v ) j = ℱ − 1 δ j , k − ξ j ξ k ∣ ξ ∣ 2 ( v ˆ k ) , Λ f = ℱ − 1 ( ∣ ξ ∣ f ˆ ) ,

where τ j , k ≜ Λ − 1 P div b with ( τ ˆ j , k ) j = i δ j , k − ξ j ξ k ∣ ξ ∣ 2 ξ l ∣ ξ ∣ ( b ˆ ) l , k .

1.3 Main results

With the help of global-in-time existence of the strong solution in [1], our main results are concerned with the following optimal time-decay rates of the global solution, which are given in the following two theorems.

Theorem 1.1

Assume that ( v 0 , b 0 ) ∈ H 3 ( R 3 ) ⋂ L 1 ( R 3 ) . Then if there exists a small constant δ 1 > 0 , such that

∥ ( v 0 , b 0 ) ∥ H 3 ≤ δ 1 ,

then the Cauchy problem exist, and (1.3)–(1.4) have a unique global solution ( v , b ) such that for any t > 0 and 0 ≤ k , m ≤ 3 , the following optimal time decay rates hold

(1.7) ‖ ∇ k v ( t ) ‖ L 2 ≤ C 0 t − 3 4 − k 2 ,

and

(1.8) ‖ ∇ m b ( t ) ‖ L 2 ≤ C 1 t − 5 4 − m 2 ,

where the positive constants C 0 and C 1 depend only on ∥ ( v 0 , b 0 ) ∥ L 1 ∩ H 3 but independent of time t .

Theorem 1.2

Under the assumptions of Theorem 1.1 and inf 0 ≤ ∣ ξ ∣ ≤ η ∣ v ˆ 0 ∣ ≥ c ˜ > 0 . Then there exists a lower bound on the decay rates for the kth order derivatives of the strong solution ( v , b ) for any t > t 0 > 0 and 0 ≤ k , m ≤ 3 , such that

(1.9) ‖ ∇ k v ( t ) ‖ L 2 ≥ c 0 t − 3 4 − k 2

and

(1.10) ‖ ∇ m b ( t ) ‖ L 2 ≥ c 1 t − 5 4 − m 2 ,

where the positive constants c 0 and c 1 depends only on ∥ ( v 0 , b 0 ) ∥ L 1 ∩ H 3 and c ˜ but independent of time t, and the positive time t 0 depends only on β , κ , θ 1 , θ 2 , and ∥ ( v 0 , b 0 ) ∥ L 1 ∩ H 3 .

Remark 1.3

Under the assumption of a small initial perturbation, the global existence of problems (1.3) and (1.4) has been established in [1]. This article focuses on the upper and lower optimal time decay rates of the solution as well as all-orders spatial derivatives. We utilize the low-high frequency decomposition technique developed in [11] and [25] to establish our main result.

Remark 1.4

Comparing to the mail result (1.3) in [1], our new contributions lie in the following three respects: First, the decay rates in (1.7) imply that the decay rate of the second-order and third-order spatial derivatives of v are ( 1 + t ) − 7 4 and ( 1 + t ) − 9 4 , respectively, which are faster than the corresponding decay result in (1.3). Second, the decay rates in (1.8) imply that the k -order spatial derivative of b converges to zero at the L 2 -rate ( 1 + t ) − 5 4 − k 2 , which improves the decay result in (1.3). Finally, the lower bounds on the convergence rates of the solution, including its spatial derivatives of all orders in (1.9) and (1.10), are totally new as compared to [1]. It should be mentioned that the lower bounds for the decay rate coincide with the upper rate. Therefore, our results are precisely optimal in this regard.

Now, let’s sketch the strategy of the proofs of Theorems 1.1 and 1.2, and explain the main difficulties and techniques involved in the process.

To prove Theorem 1.1, we first rewrite system (1.1) into (1.3)–(1.4). Then, we make a careful spectral analysis of the corresponding linear system. Next, under Duhamel’s principle, Plancherel theorem, and some results of linear system analysis in [11], we can derive the optimal decay rates for the second-order and third-order spatial derivatives of the solution ( v , b ) . Our methods are based on the low-frequency and high-frequency decomposition and delicate energy estimates. To illustrate our difficulties, we choose the proof of the decay rate of the third-order spatial derivative of b for an example. In the derivation of third-order energy estimate in Lemma 4.6, we encounter the troublesome term ∫ R 3 ∇ 3 ( v ⋅ ∇ b ) ∇ 3 b d x . Due to the lack of the dissipation estimate on ∇ 3 v on the left hand (4.54), it seems impossible for us to handle the trouble term. To overcome this difficulty, we use low-frequency and high-frequency decomposition to the following crucial energy estimates:

d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b h ‖ L 2 2 ≲ ( 1 + t ) − 25 4

and

d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + α 3 ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) ≲ ( 1 + t ) − 25 4 .

which enable us to derive the third-order spatial derivative of the solution b as follows:

d d t ‖ ∇ 3 b ‖ L 2 2 + β ‖ ∇ 4 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b ‖ L 2 2 ≲ ( 1 + t ) − 11 2 ,

which together with Cauchy’s inequality, the Gagliardo-Nirenberg-Sobolev inequality, and Gronwall’s inequality implies the desired decay estimates of ∇ 3 b . For more details, we refer to the proofs from (4.46) to (4.67). Ultimately, by employing the Fourier splitting method, as detailed in the works of Schonbek in [12,13], Schonbek and Wiegner in [14], we are able to determine the time decay rate as given by equations (1.7) and (1.8).

To establish lower bounds on the convergence rates of the solution ( v , b ) in Theorem 1.2, we utilize Duhamel’s principle, Plancherel’s theorem, low-frequency and high-frequency decomposition, and the interpolation trick to arrive at

‖ ∇ k v ( t ) ‖ L 2 2 = ‖ ∣ ξ ∣ k v ˆ ( t ) ‖ L 2 2 ≥ ‖ ∇ k v ( t ) ‖ L ∣ ξ ∣ ≤ η 2 2 ≳ c 1 ( 1 + t ) − 3 2 − k

and

‖ ∇ k b ( t ) ‖ L 2 2 ≥ ‖ ∇ k τ ( t ) ‖ L 2 2 = ‖ ∣ ξ ∣ k τ ˆ ( t ) ‖ L 2 2 ≥ ‖ ∇ k τ ( t ) ‖ L ∣ ξ ∣ ≤ η 2 2 ≳ c 1 ( 1 + t ) − 5 2 − k .

2 Reformulation

For system (1.3), we can obtain the following lemma concerning the global existence of smooth solution to the Cauchy problems (1.3) and (1.4), which has been proved in [1].

Lemma 2.1

Assume the initial perturbation ( v 0 , b 0 ) in H N for any integer ( N ≥ 3 ). If there exists a constant δ 0 is small enough, such that if ∑ k = 0 N ‖ ∇ k ( v , b ) ‖ L 2 2 ≤ δ 0 . Then the Cauchy problems (1.3) and (1.4) admit a unique global solution ( v , b ) ∈ C ( [ 0 , ∞ ] ; H 3 ) satisfying

sup 0 ≤ t ≤ ∞ ∑ k = 0 3 ‖ ∇ k ( v , b ) ‖ L 2 2 + ∫ 0 ∞ ∑ k = 0 3 ‖ ∇ k b ‖ L 2 2 + ∑ k = 0 N ‖ ∇ k + 1 ( v , b ) ‖ L 2 2 d s ≲ ∑ k = 0 3 ‖ ∇ k ( v 0 , b 0 ) ‖ L 2 2 .

Moreover, for any N ≥ 3 , the aforementioned Cauchy problem has a unique global solution ( v , b ) satisfying:

sup 0 ≤ t ≤ ∞ ∑ k = 0 N ‖ ∇ k ( v , b ) ‖ L 2 2 + ∫ 0 ∞ ∑ k = 0 N ‖ ∇ k b ‖ L 2 2 + ∑ k = 0 N ‖ ∇ k + 1 ( v , b ) ‖ L 2 2 d s ≲ ∑ k = 0 N ‖ ∇ k ( v 0 , b 0 ) ‖ L 2 2 .

Lemma 2.2

Assuming that all conditions of Lemma 2.1 hold, let ( v , b ) represent the solution to system (1.3)–(1.4), with ( v 0 , b 0 ) ∈ L p for a given p ∈ [ 1 , 2 ] . If N ≥ k + 2 , there holds

‖ ∇ k ( v , b ) ( t ) ‖ H N − 2 ≲ ( 1 + t ) − k + s p 2 ,

where

s p ≜ 3 p − 3 2 .

Therefore, for the case that N = 3 , p = 1 , and k = 0 , one has

(2.1) ‖ ( v , b ) ( t ) ‖ L 2 ≲ ( 1 + t ) − 3 4 ,

for the case that N = 3 , p = 1 , and k = 1 , one has

(2.2) ‖ ∇ k ( v , b ) ( t ) ‖ H 2 ≲ ( 1 + t ) − 5 4 .

3 Analysis of the linearized system and linear estimates

To supplement the dissipation of b , we denote by

τ j , k ≜ Λ − 1 P div b .

Then, we apply the Leray projector P ≜ I d + ∇ ( △ ) − 1 div and the operator P div to (1.3)1 and (1.3)2 to obtain the following auxiliary system:

(3.1) ∂ t v + P ( v ⋅ ∇ v ) − κ △ v = 2 a P ( div ( b + α b b ) ) , ∂ t P div b + P div ( v ⋅ ∇ b ) − β △ P div b + ( θ 1 + θ 2 ) P div b + θ 2 P div b b = a P div ( ∇ v + ( ∇ v ) ⊤ ) + a + 1 2 P div ( ∇ v b + ( ∇ v b ) ⊤ ) + a − 1 2 P div ( b ∇ v + ( b ∇ v ) ⊤ ) , div v = 0 ,

Then, by applying the operator Λ − 1 to the equation (3.1)2 and denoting by

τ j , k ≜ Λ − 1 P div b ,

we have

( τ ˆ j , k ) j = i δ j , k − ξ j ξ k ∣ ξ ∣ 2 ξ l ∣ ξ ∣ ( b ˆ ) l , k .

Therefore, one has the following rewritten system, which is equivalent to (3.1)

(3.2) ∂ t v − κ △ v − 2 a Λ τ = N 1 , ∂ t τ − β △ τ + ( θ 1 + θ 2 ) τ + a Λ v = N 2 , div v = 0 ,

where, N 1 and N 2 denote the nonlinear terms, which are given by

(3.3) N 1 = − P ( v ⋅ ∇ v ) + 2 a α P ( div ( b b ) ) , N 2 = − Λ − 1 P div ( v ⋅ ∇ b ) − ( θ 1 + θ 2 ) Λ − 1 P div ( b b ) + a + 1 2 Λ − 1 P div ( ∇ v b + ( ∇ v b ) ⊤ ) + a − 1 2 Λ − 1 P div ( b ∇ v + ( b ∇ v ) ⊤ ) .

Now, we linearize the (1.3) into

(3.4) ∂ t v − κ △ v = 2 a P div b , ∂ t b − β △ b + ( θ 1 + θ 2 ) b = 2 a D v , ( v , τ ) ( x , 0 ) = ( v 0 ( x ) , b 0 ( x ) ) ,

where we denote D v = 1 2 ( ∇ v + ( ∇ v ) ⊤ ) by the deformation tensor. Thus, the corresponding linear system of (3.4) is given by

(3.5) ∂ t v − κ △ v − 2 a Λ τ = 0 , ∂ t τ − β △ τ + ( θ 1 + θ 2 ) τ + a Λ v = 0 , ( v , τ ) ( x , 0 ) = ( v 0 ( x ) , b 0 ( x ) ) .

We use the notations G ( ξ , t ) and A ( ξ , t ) to express the Green matrixes of equations (3.4) and (3.5), respectively . Therefore, their Fourier transforms can be expressed as follows.

Lemma 3.1

Let ( v , τ ) be the solution to the linear system (3.5). Then, it holds that

(3.6) v ˆ τ ˆ = A ˆ ( ξ , t ) v ˆ 0 τ ˆ 0 ,

where

(3.7) A ˆ ( ξ , t ) = ( G 1 − κ ∣ ξ ∣ 2 G 2 ) I 3 × 3 2 a ∣ ξ ∣ G 2 I 3 × 3 − a ∣ ξ ∣ G 2 I 3 × 3 ( − ( θ 1 + θ 2 + β ∣ ξ ∣ 2 ) G 2 + G 1 ) I 3 × 3 ,

with

(3.8) G 1 ( ξ , t ) = λ − e λ + t − λ + e λ − t λ − − λ + , G 2 ( ξ , t ) = e λ + t − e λ − t λ − − λ + .

and I 3 × 3 is an unit matrix.

Proof

Applying the semigroup theory to the linear system (3.5), for U = ( v , τ ) ⊤ , it holds that

(3.9) U t = A ( ξ , t ) U , U ∣ t = 0 = A ( ξ , t ) U 0 ,

where the operator A is defined by

(3.10) A ( ξ , t ) = κ Λ 2 2 a Λ − a Λ − ( θ 1 + θ 2 ) + β Λ 2 .

Let’s apply Fourier transform to the system (3.9), one has

(3.11) U ˆ t = A ˆ ( ξ , t ) U ˆ , U ˆ ∣ t = 0 = A ˆ ( ξ , t ) U ˆ 0 ,

where the operator A ˆ is defined by

A ˆ ( ξ , t ) = − κ ∣ ξ ∣ 2 2 a ∣ ξ ∣ − a ∣ ξ ∣ − ( θ 1 + θ 2 ) − β ∣ ξ ∣ 2 .

Form the characteristic determinant

det ( λ I − A ˆ ( ξ , t ) ) = 0 ,

which implies the following characteristic equation:

(3.12) λ 2 + ( ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 ) λ + [ κ β ∣ ξ ∣ 2 + κ ( θ 1 + θ 2 ) + 2 a 2 ] ∣ ξ ∣ 2 = 0 ,

whose roots λ ± satisfy

(3.13) λ ± = − [ ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 ] ± [ ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 ] 2 − 4 [ κ β ∣ ξ ∣ 2 + κ ( θ 1 + θ 2 ) + 2 a 2 ] ∣ ξ ∣ 2 2 , λ + + λ − = − ( β + κ ) ∣ ξ ∣ 2 − θ 1 − θ 2 , λ + λ − = [ κ β ∣ ξ ∣ 2 + κ ( θ 1 + θ 2 ) + 2 a 2 ] ∣ ξ ∣ 2 .

Therefore, it holds that

(3.14) e A t = e λ + t P 1 ( ξ ) + e λ − t P 2 ( ξ ) ,

where the projector P 1 ( ξ ) and P 2 ( ξ ) are computed as follows:

(3.15) P 1 ( ξ ) = A ( ξ ) − λ − I λ + − λ − and P 2 ( ξ ) = A ( ξ ) − λ + I λ − − λ + .

In a conclusion, we can represent the solution of the linear system (3.5) as follows:

(3.16) U ˆ ( ξ , t ) = ( e λ + t P 1 ( ξ ) + e λ − t P 2 ( ξ ) ) U ˆ 0 ( ξ )

In the view of (3.12)–(3.16), we arrive at

(3.17) v ˆ j ( t ) = λ − e λ + t − λ + e λ − t λ − − λ + + κ ∣ ξ ∣ 2 e λ + t − e λ − t λ − − λ + v ˆ 0 j + 2 a ∣ ξ ∣ e λ − t − e λ + t λ − − λ + τ ˆ 0 j .

In a similar way, one has

(3.18) τ ˆ j ( t ) = 2 a ∣ ξ ∣ e λ + t − e λ − t λ − − λ + v ˆ 0 j + ( θ 1 + θ 2 + β ∣ ξ ∣ 2 ) e λ + t − e λ − t λ − − λ + τ ˆ 0 j + λ − e λ + t − λ + e λ − t λ − − λ + τ ˆ 0 j .

We denote

(3.19) G 1 ( ξ , t ) = λ − e λ + t − λ + e λ − t λ − − λ + and G 2 ( ξ , t ) = e λ − t − e λ + t λ − − λ + .

Therefore, we arrive at

(3.20) v ˆ j ( t ) = ( G 1 ( ξ , t ) − κ ∣ ξ ∣ 2 G 2 ( ξ , t ) ) v ˆ 0 j + 2 a ∣ ξ ∣ G 2 ( ξ , t ) τ ˆ 0 j ,

and

(3.21) τ ˆ j ( t ) = − a ∣ ξ ∣ G 2 ( ξ , t ) v ˆ 0 j − ( θ 1 + θ 2 + β ∣ ξ ∣ 2 ) G 2 ( ξ , t ) τ ˆ 0 j + G 1 ( ξ , t ) τ ˆ 0 j .

Moreover, according to the definitions, we conclude that

(3.22) G 1 ( ξ , t ) = e λ + t + λ + G 2 ( ξ , t ) .

Thus, we have completed the proof of Lemma 3.1.□

Lemma 3.2

Let ( v , b ) be the solution to the system (3.4). Then, it holds that

(3.23) v ˆ b ˆ = G ˆ ( ξ , t ) v ˆ 0 b ˆ 0 ,

where

(3.24) v ˆ j ( t ) = ( G 1 ( ξ , t ) − κ ∣ ξ ∣ 2 G 2 ( ξ , t ) ) v ˆ 0 j + 2 a i ξ l G 1 ( ξ , t ) δ j , k − ξ j ξ k ∣ ξ ∣ 2 b ˆ 0 l , p ,

and

(3.25) b ˆ j , k ( ξ , t ) = e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t b ˆ 0 j , k + i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) G 2 ( ξ , t ) + ∣ ξ ∣ l ∣ ξ ∣ 2 b ˆ 0 l , p ξ k δ j , p − ξ j ξ p ∣ ξ ∣ 2 + ξ j δ k , p − ξ k ξ p ∣ ξ ∣ 2 × ( G 3 + κ ∣ ξ ∣ 2 G 2 − e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t ) ,

where

G 3 ( ξ , t ) = λ − e λ − t − λ + e λ + t λ − − λ + .

Proof

By applying Fourier transform to the equation (3.4)2, one has

b ˆ t j , k + ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) b ˆ j , k = i a ( ξ k v ˆ j + ξ j v ˆ k ) ,

which implies

(3.26) b ˆ j , k = e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t b ˆ 0 j , k + i a e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z ( ξ k v ˆ j + ξ j v ˆ k ) d z .

Similarly, we have

(3.27) τ ˆ 0 j = i δ j , p − ξ j ξ p ∣ ξ ∣ 2 ξ l ∣ ξ ∣ b ˆ 0 l , p .

It can be concluded that

(3.28) ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z ( ξ k v ˆ j + ξ j v ˆ k ) d z = ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z λ − e λ + z − λ + e λ − z λ − − λ + + κ ∣ ξ ∣ 2 e λ + z − e λ − z λ − − λ + d z ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) + 2 a ∣ ξ ∣ ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z e λ − z − e λ + z λ − − λ + d z ( ξ k τ ˆ 0 j + ξ j τ ˆ 0 k ) ≜ H 1 + H 2 .

First, by using Newton-Leibniz formula and (3.21), one obtains

(3.29) H 1 = ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z λ − e λ + z − λ + e λ − z λ − − λ + + κ ∣ ξ ∣ 2 e λ + z − e λ − z λ − − λ + d z ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) , = ∫ 0 t e ( ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 − κ ∣ ξ ∣ 2 ) z λ − e λ + z − λ + e λ − z + κ ∣ ξ ∣ 2 e λ + z − e λ − z κ ∣ ξ ∣ 2 λ − − λ + d z ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) , = ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) ∫ 0 t e − ( λ + + λ − + κ ∣ ξ ∣ 2 ) z e λ + z ( λ − + κ ∣ ξ ∣ 2 ) − e λ − z ( λ + + κ ∣ ξ ∣ 2 ) λ − − λ + d z = ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) ∫ 0 t e − ( λ − + κ ∣ ξ ∣ 2 ) z λ − + κ ∣ ξ ∣ 2 λ − − λ + d z − ∫ 0 t e − ( λ + + κ ∣ ξ ∣ 2 ) z λ + + κ ∣ ξ ∣ 2 λ − − λ + d z = ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) − 1 λ − + κ ∣ ξ ∣ 2 λ − + κ ∣ ξ ∣ 2 λ − − λ + ( e − ( λ − + κ ∣ ξ ∣ 2 ) t − 1 ) + ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) 1 λ + + κ ∣ ξ ∣ 2 λ + + κ ∣ ξ ∣ 2 λ − − λ + ( e − ( λ + + κ ∣ ξ ∣ 2 ) t − 1 ) = ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e − κ ∣ ξ ∣ 2 t e − λ + t − e − λ − t λ − − λ + .

Similarly, for the term H 2 , it holds that

(3.30) H 2 = 2 a ∣ ξ ∣ ∫ 0 t e ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) z e λ − z − e λ + z λ − − λ + d z ( ξ k τ ˆ 0 j + ξ j τ ˆ 0 k ) = 1 a ∣ ξ ∣ ( ξ k τ ˆ 0 j + ξ j τ ˆ 0 k ) ( λ + + κ ∣ ξ ∣ 2 ) ( λ − + κ ∣ ξ ∣ 2 ) ∫ 0 t e − ( λ + + λ − + κ ∣ ξ ∣ 2 ) z e λ − z − e λ + z λ − − λ + d z = ξ k τ ˆ 0 j + ξ j τ ˆ 0 k a ∣ ξ ∣ λ + e − λ − t − λ − e − λ + t λ − − λ + e − κ ∣ ξ ∣ 2 t + e − λ − t − e − λ + t λ − − λ + e − κ ∣ ξ ∣ 2 t κ ∣ ξ ∣ 2 + 1 ,

where we have used the fact that λ + λ − + κ ∣ ξ ∣ 2 ( λ + + λ − ) + κ 2 ∣ ξ ∣ 4 = 2 a 2 ∣ ξ ∣ 2 . This directly leads to

(3.31) i a e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t H 1 = i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e − ( κ ∣ ξ ∣ 2 + β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t e − λ + t − e − λ − t λ − − λ + = i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e λ − t − e λ + t λ − − λ + ,

and

(3.32) i a e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t H 2 = i ∣ ξ ∣ ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e ( λ + + λ − ) t λ + e − λ − t − λ − e − λ + t λ − − λ + + e − λ − t − e − λ + t λ − − λ + κ ∣ ξ ∣ 2 + i ∣ ξ ∣ ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t = i ∣ ξ ∣ ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) λ + e λ + t − λ − e λ − t λ − − λ + + e λ + t − e λ − t λ − − λ + κ ∣ ξ ∣ 2 + e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t .

Therefore, we conclude that

(3.33) b ˆ j , k = e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t b ˆ 0 j , k + i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e λ − t − e λ + t λ − − λ + + i ∣ ξ ∣ ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) λ + e λ + t − λ − e λ − t λ − − λ + + e λ + t − e λ − t λ − − λ + κ ∣ ξ ∣ 2 + e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t = e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t b ˆ 0 j , k + i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) e λ − t − e λ + t λ − − λ + − ξ l ∣ ξ ∣ 2 b ˆ 0 l , p ξ k δ j , p − ξ j ξ p ∣ ξ ∣ 2 + ξ j δ k , p − ξ k ξ p ∣ ξ ∣ 2 × λ + e λ + t − λ − e λ − t λ − − λ + + e λ + t − e λ − t λ − − λ + κ ∣ ξ ∣ 2 + e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t = e − ( β ∣ ξ ∣ 2 + θ 1 + θ 2 ) t b ˆ 0 j , k + i a ( ξ k v ˆ 0 j + ξ j v ˆ 0 k ) G 2 ( ξ , t ) + ξ l ∣ ξ ∣ 2 b ˆ 0 l , p ξ k δ j , p − ξ j ξ p ∣ ξ ∣ 2 + ξ j δ k , p − ξ k ξ p ∣ ξ ∣ 2 × ( G 3 + κ ∣ ξ ∣ 2 G 2 ) − ξ l ∣ ξ ∣ 2 b ˆ 0 l , p ξ k δ j , p − ξ j ξ p ∣ ξ ∣ 2 + ξ j δ k , p − ξ k ξ p ∣ ξ ∣ 2 × e ( λ + + λ − + κ ∣ ξ ∣ 2 ) t ,

and

(3.34) v ˆ j ( t ) = ( G 1 ( ξ , t ) − κ ∣ ξ ∣ 2 G 2 ( ξ , t ) ) v ˆ 0 j + 2 a ∣ ξ ∣ G 1 ( ξ , t ) τ ˆ 0 j = ( G 1 ( ξ , t ) − κ ∣ ξ ∣ 2 G 2 ( ξ , t ) ) v ˆ 0 j + 2 a i ξ l G 1 ( ξ , t ) δ j , k − ξ j ξ k ∣ ξ ∣ 2 b ˆ 0 l , p .

Thus, the proof of Lemma 3.2 has been completed.□

Lemma 3.3

There exist positive constants η ( a , β , κ , θ 1 , θ 2 ) , η i ( a , β , κ , θ 1 , θ 2 ) , and μ i ( a , β , κ , θ 1 , θ 2 ) , such that

(3.35) ∣ G 1 ∣ + ∣ G 2 ∣ + ∣ G 3 ∣ + ∣ G ∣ + ∣ A ∣ ≤ μ 1 e − η 1 ∣ ξ ∣ 2 t ,

for any ∣ ξ ∣ ≤ η ≪ 1 , t > 0 , and η is a positive constant. And

(3.36) ∣ G 1 ∣ + ∣ G 2 ∣ ≥ μ 2 e − η 2 ∣ ξ ∣ 2 t , ∣ G 3 ∣ ≤ μ 3 ∣ ξ ∣ 2 e − η 1 ∣ ξ ∣ 2 t + e − θ 1 + θ 2 2 t ,

for any ∣ ξ ∣ ≤ η ≪ 1 , t > t 1 ( a , β , κ , θ 1 , θ 2 ) , and η is a positive constant.

Proof

From equation (3.13), it is clear that there exists a positive constant η such that, for ∣ ξ ∣ ≤ η ≪ 1 , the spectral has the following Taylor series expansion:

(3.37) λ + = 1 2 ( β + κ ) − 2 κ − 4 a 2 θ 1 + θ 2 ∣ ξ ∣ 2 + O ( ∣ ξ ∣ 4 ) , λ − = − ( θ 1 + θ 2 ) + O ( ∣ ξ ∣ 2 ) .

Define the operator △ ( ξ ) by

(3.38) △ ( ξ ) = [ ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 ] 2 − 4 [ κ β ∣ ξ ∣ 2 + κ ( θ 1 + θ 2 ) + 2 a 2 ] ∣ ξ ∣ 2 .

It is clear that

(3.39) λ − ≤ − 1 2 ( θ 1 + θ 2 ) , △ ( ξ ) ≤ [ ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 ] 2 , △ ( ξ ) ≥ 1 2 ( θ 1 + θ 2 ) 2 ,

and

(3.40) λ + = − 2 [ κ β ∣ ξ ∣ 2 + κ ( θ 1 + θ 2 ) + 2 a 2 ] ∣ ξ ∣ 2 ( β + κ ) ∣ ξ ∣ 2 + θ 1 + θ 2 + △ ( ξ ) ≤ − κ ( θ 1 + θ 2 ) + 2 a 2 ( β + κ ) ∣ η ∣ 2 + θ 1 + θ 2 ∣ ξ ∣ 2 .

Then, by defining η 1 ≜ min { κ ( θ 1 + θ 2 ) + 2 a 2 ( β + κ ) ∣ η ∣ 2 + θ 1 + θ 2 , 1 2 ( θ 1 + θ 2 ) } , we have

λ − ≤ λ + ≤ − η 1 ∣ ξ ∣ 2 .

From equations (3.13) and (3.38), one obtains

G 2 ( ξ , t ) = e λ − t − e λ + t λ − − λ + = e λ + t ( 1 − e ( λ − − λ + ) t ) λ + − λ − ≤ e λ + t 1 2 ( θ 1 + θ 2 ) 2 ,

G 1 ( ξ , t ) = e λ + t − λ + G 2 ( ξ , t ) ≤ η 1 e λ + t 1 2 ( θ 1 + θ 2 ) 2 + e λ + t ,

and

G 3 ( ξ , t ) = λ + G 2 ( ξ , t ) + e λ − t ≤ η 1 e λ + t 1 2 ( θ 1 + θ 2 ) 2 + e λ − t .

With equations (3.20), (3.21), (3.33), and (3.34) in hand, we arrive at the equation (3.35). On the other hand, one has

(3.41) λ + ≥ − 1 2 ( β + κ ) + 2 κ + 4 a 2 θ 1 + θ 2 ∣ ξ ∣ 2 .

By defining η 2 ≜ 1 2 ( β + κ ) + 2 κ + 4 a 2 θ 1 + θ 2 , we have λ + ≥ − η 2 ∣ ξ ∣ 2 . Then, we arrive at

G 1 ( ξ , t ) = e λ + t ( λ + − λ − ) e ( λ − − λ + ) t λ + − λ − ≥ e λ + t ,

G 2 ( ξ , t ) = e λ + t ( 1 − e ( λ − − λ + ) t ) λ + − λ − ≥ e λ + t [ ( β + κ ) ∣ η ∣ 2 + θ 1 + θ 2 ] 2 ,

and

G 3 ( ξ , t ) = λ + λ + − λ − e λ + t + λ − λ + − λ − e λ − t ≤ η 1 ∣ ξ ∣ 2 1 2 ( θ 1 + θ 2 ) 2 e λ + t + θ 1 + θ 2 1 2 ( θ 1 + θ 2 ) 2 e λ − t .

Therefore, we can obtain equation (3.36) immediately, and the proof of Lemma 3.3 is completed.□

4 The proof of Theorem 1.1

Provided that the initial perturbation ( v 0 , b 0 ) is small in H 3 and bounded in the L 1 , we utilize the Fourier splitting method to deduce the optimal decay rates for the second-order and third-order spatial derivatives of the solution ( v , b ) . To begin with, we deduce the time-decay estimate for the low-frequency part of the solution ( v , b ) to system (1.3).

Lemma 4.1

Assume that all conditions of Theorem 1.1 are in force. Then, it holds that

(4.1) ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ∣ v ˆ ∣ 2 d ξ 1 2 ≲ ( 1 + t ) − 3 4 − k 2 ∥ ( v 0 , b 0 ) ∥ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 3 4 − k 2 T 1 ( z ) T 2 ( z ) L 1 d z + ∫ t 2 t ( 1 + t − z ) − k 2 T 1 ( z ) T 2 ( z ) L 2 d z ,

where

(4.2) T 1 = − P ( v ⋅ ∇ v ) + 2 a α P ( div ( b b ) ) , T 2 = − v ⋅ ∇ b − ( θ 1 + θ 2 ) b b + a + 1 2 ( ∇ v b + ( ∇ v b ) ⊤ ) + a − 1 2 ( b ∇ v + ( b ∇ v ) ⊤ ) .

Proof

First, we obtain the estimates for the nonlinear terms T 1 and T 2 as follows:

(4.3) ‖ T 1 ‖ L 1 = ‖ P ( v ⋅ ∇ v ) ‖ L 1 + ‖ 2 a α P ( div ( b b ) ) ‖ L 1 ≲ ‖ v ‖ L 2 ‖ ∇ v ‖ L 2 + 2 ∣ a ∣ α ‖ b ‖ L 2 ‖ div b ‖ L 2 ,

and

(4.4) ‖ T 2 ‖ L 1 = ‖ v ⋅ ∇ b ‖ L 1 + ( θ 1 + θ 2 ) ‖ b b ‖ L 1 + a + 1 2 ( ∇ v b + ( ∇ v b ) ⊤ ) + a − 1 2 ( b ∇ v + ( b ∇ v ) ⊤ ) L 1 ≲ ‖ v ‖ L 2 ‖ ∇ b ‖ L 2 + ‖ b ‖ L 2 ‖ ∇ v ‖ L 2 + ‖ b ‖ L 2 2 .

By using integration by parts and Lemma 7.1, we have

(4.5) ‖ T 1 ‖ L 2 = ‖ P ( v ⋅ ∇ v ) ‖ L 2 + ‖ 2 a α P ( div ( b b ) ) ‖ L 2 ≲ ‖ v ‖ L 6 ‖ ∇ v ‖ L 3 + 2 ∣ a ∣ α ‖ b ‖ L 6 ‖ div b ‖ L 3 ≲ ‖ ∇ v ‖ H 1 ‖ ∇ v ‖ L 2 + 2 ∣ a ∣ α ‖ ∇ b ‖ H 1 ‖ ∇ b ‖ L 2 .

Similarly, one has

(4.6) ‖ T 2 ‖ L 2 = ‖ v ⋅ ∇ b ‖ L 2 + ( θ 1 + θ 2 ) ‖ b b ‖ L 2 + a + 1 2 ( ∇ v b + ( ∇ v b ) ⊤ ) + a − 1 2 ( b ∇ v + ( b ∇ v ) ⊤ ) L 2 ≲ ‖ v ‖ L 6 ‖ ∇ b ‖ L 3 + ‖ b ‖ L 6 ‖ ∇ v ‖ L 3 + ‖ b ‖ L 6 ‖ b ‖ L 3 ≲ ‖ ∇ v ‖ H 1 ‖ ∇ b ‖ L 2 + ‖ ∇ b ‖ H 1 ‖ ∇ v ‖ L 2 + ‖ b ‖ H 1 ‖ ∇ b ‖ L 2 .

From Duhamel’s principle, we have

(4.7) v b ( t ) = G ( ξ , t ) * ( v 0 , b 0 ) + ∫ 0 t G ( t − z ) * ( T 1 , T 2 ) ⊤ ( z ) d z .

Denoting that U 0 = ( v 0 , b 0 ) ⊤ , we have from Lemma 7.1 that

(4.8) ∥ ∣ ξ ∣ k G ( ξ , t ) U ˆ 0 ∥ L ∣ ξ ∣ ≤ η 2 ≲ ‖ U ˆ 0 ‖ L ∣ ξ ∣ ≤ η q ∥ ∣ ξ ∣ k e − η ∣ ξ ∣ 2 t ∥ L ∣ ξ ∣ ≤ η 2 q q − 2 ≲ ‖ U 0 ‖ L p ∫ ∣ ζ ∣ ≤ η 2 t ζ t − 1 2 2 k e − 2 η ∣ ζ ∣ 2 q q − 2 t − 3 2 d ζ q − 2 2 q ≲ ‖ U 0 ‖ L p ( 1 + t ) − 3 2 1 2 − 1 q − k 2 ,

where ζ 2 ≜ ξ 2 t , 1 p + 1 q = 1 and q ⩾ 2 . Therefore, we arrive at

(4.9) ∥ ∣ ξ ∣ k ( v ˆ , b ˆ ) ∥ L ∣ ξ ∣ ≤ η 2 = ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ k ∣ G ∣ ∣ U ˆ 0 ∣ d ξ 1 2 + ∫ 0 t G ( t − z ) * ( T 1 , T 2 ) ⊤ ( z ) d z ≲ ‖ ( v ˆ 0 , b ˆ 0 ) ‖ L ξ q ( 1 + t ) − 3 2 × ( 1 2 − 1 q ) − k 2 + ∫ 0 t ( 1 + t − z ) − 3 2 1 2 − 1 q − k 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L p d z ≲ ‖ ( v 0 , b 0 ) ‖ L p ( 1 + t ) − 3 2 × ( 1 2 − 1 q ) − k 2 + ∫ 0 t ( 1 + t − z ) − 3 2 1 2 − 1 q − k 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L p d z ≲ ( 1 + t ) − 3 4 − k 2 ‖ ( v 0 , b 0 ) ‖ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 3 4 − k 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 1 d z + ∫ t 2 t ( 1 + t − z ) − k 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 2 d z .

Thus, the proof of Lemma 4.1 has been completed.□

Next, we deduce the dissipation estimate of v .

Lemma 4.2

Assume that all conditions of Theorem 1.1 are in force. Then, it holds that

(4.10) 1 2 d d t ∑ k = 2 3 ‖ Λ k − 1 v h ‖ L 2 2 + κ 2 ‖ Λ k v h ‖ L 2 2 ≲ D 1 δ 1 ( ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) , for k = 2 ; ε ( a ) ‖ ∇ 2 v ‖ L 2 2 + C ( 1 + t ) − 3 2 ‖ ∇ 4 v h ‖ L 2 2 + D 2 δ 1 ( ‖ ∇ 2 b ‖ H 1 2 + ‖ ∇ 3 v h ‖ L 2 2 ) , for k = 3 ,

where D 1 and D 2 are positive constants.

Proof

By multiplying Λ k − 1 φ 0 ( D ) (3.2)1 by Λ k − 1 v h , where k = 2 , 3, and by adding them up and integrating the resultant equation on R 3 , we have

(4.11) 1 2 d d t ∑ k = 2 3 ‖ Λ k − 1 v h ‖ L 2 2 + κ ‖ Λ k v h ‖ L 2 2 = 2 a ∑ k = 2 3 ⟨ Λ k − 1 Λ τ h , Λ k − 1 v h ⟩ + ∑ k = 2 3 ⟨ Λ k − 1 N 1 h , Λ k − 1 v h , ⟩ = 2 a ∑ k = 2 3 ⟨ Λ k τ h , Λ k − 1 v h ⟩ + ∑ k = 2 3 ⟨ Λ k − 1 P ( v ⋅ ∇ v ) h , Λ k − 1 v h ⟩ + 2 a α ∑ k = 2 3 ⟨ Λ k − 1 P div ( b b ) h , Λ k − 1 v h ⟩ ≜ ∑ i = 1 3 N 1 , i .

For the three terms on the right-hand side of the aforementioned equality, one has the following estimations:

(4.12) ∣ N 1 , 1 ∣ ≲ 2 a ‖ ∇ k v h ‖ L 2 ‖ ∇ k − 1 τ h ‖ L 2

and

(4.13) ∣ N 1 , 2 ∣ ≲ ‖ v ‖ L ∞ ‖ ∇ v ‖ L 2 ‖ ∇ 2 v h ‖ L 2 , for k = 2 ; ‖ v ‖ L ∞ ‖ ∇ v ‖ L 2 ‖ ∇ 4 v h ‖ L 2 , for k = 3 ; ≲ ‖ ∇ v ‖ H 1 2 ‖ ∇ 2 v h ‖ L 2 , for k = 2 ; ‖ ∇ v ‖ H 1 2 ‖ ∇ 4 v h ‖ L 2 , for k = 3 ,

and

(4.14) ∣ N 1,3 ∣ ≲ 2 a α ∑ k = 2 3 ⟨ Λ k − 1 div ( b b ) h , Λ k − 1 v h ⟩ ≲ ∑ k = 2 3 ‖ ∇ k ( b b ) ‖ L 6 5 ‖ ∇ k − 1 v h ‖ L 6 ≲ ( ‖ b ‖ L 3 ‖ ∇ 2 b ‖ L 2 + ‖ ∇ b ‖ L 3 ‖ ∇ b ‖ L 2 ) ‖ ∇ 2 v h ‖ L 2 , for k = 2 ; ( ‖ b ‖ L 3 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ b ‖ L 3 ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 3 v h ‖ L 2 , for k = 3 ; ≲ δ 1 ( ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) , for k = 2 ; δ 1 ( ‖ ∇ 2 b ‖ H 1 2 + ‖ ∇ 3 v h ‖ L 2 2 ) , for k = 3 .

By substituting (4.12)–(4.14) into (4.11), we conclude that

(4.15) 1 2 d d t ∑ k = 2 3 ‖ Λ k − 1 v h ‖ L 2 2 + κ ‖ Λ k v h ‖ L 2 2 ≲ κ 2 ‖ ∇ k v h ‖ L 2 2 + δ 1 ( ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) , for k = 2 ; ‖ ∇ 2 b ‖ L 2 ‖ ∇ 3 v h ‖ L 2 + δ 1 ‖ ∇ 4 v h ‖ L 2 2 + δ 1 ( ‖ ∇ 2 b ‖ H 1 2 + ‖ ∇ 3 v h ‖ L 2 2 ) , for k = 3 ,

which implies that

(4.16) d d t ∑ k = 2 3 ‖ Λ k − 1 v h ‖ L 2 2 + κ ‖ Λ k v h ‖ L 2 2 ≲ D 1 δ 1 ‖ ∇ 2 v h ‖ L 2 2 + δ 1 ‖ ∇ 2 b ‖ L 2 2 , for k = 2 ; ‖ ∇ 2 b ‖ L 2 ‖ ∇ 3 v h ‖ L 2 + δ 1 ‖ ∇ 4 v h ‖ L 2 2 + δ 1 ( ‖ ∇ 2 b ‖ H 1 2 + ‖ ∇ 3 v h ‖ L 2 2 ) , for k = 3 ,

where ε is small sufficiently. We define the following temporal energy functional:

(4.17) E 1 ( t ) = ‖ ∇ k b ‖ L 2 2 + ‖ ∇ k v ‖ L 2 2 , E 1 h ( t ) = ‖ ∇ k b h ‖ L 2 2 + ‖ ∇ k v h ‖ L 2 2 , E 2 ( t ) = ‖ ∇ k b ‖ L 2 2 + ‖ ∇ k v ‖ L 2 2 + ‖ ∇ k − 1 v h ‖ L 2 2 E 2 h ( t ) = ‖ ∇ k b h ‖ L 2 2 + ‖ ∇ k v h ‖ L 2 2 + ‖ ∇ k − 1 v h ‖ L 2 2 .

Then, it is clear that

R ̲ E 1 ( t ) ≤ E 2 ( t ) ≤ R ¯ E 1 ( t )

and

W ̲ E 1 h ( t ) ≤ E 2 h ( t ) ≤ W ¯ E 1 h ( t ) ,

where R ̲ , W ̲ , R ¯ , and W ¯ are positive constants. Thus, the proof of Lemma 4.2 has been completed.□

Based on the aforementioned dissipation estimate of v , it is not difficult for us to obtain the following second-order and third-order total energy estimates.

Lemma 4.3

Based on the aforementioned dissipation supplement, it holds that

d d t ( ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ v h ‖ L 2 2 ) + κ 2 ‖ ∇ 2 v h ‖ L 2 2 + κ 2 ‖ ∇ 3 v ‖ L 2 2 + β 2 ‖ ∇ 3 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 2 b ‖ L 2 2 ≲ D 1 δ 1 ‖ ∇ 2 v h ‖ L 2 2 + D 3 δ 1 ‖ ∇ 2 v ‖ L 2 2 ,

for k = 2 . And for the case k = 3 , one has

d d t ( ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) + κ 2 ‖ ∇ 3 v h ‖ L 2 2 + κ 2 ‖ ∇ 4 v ‖ L 2 2 + β 2 ‖ ∇ 4 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 3 b ‖ L 2 2 ≲ ‖ ∇ 2 b ‖ L 2 ‖ ∇ 3 v h ‖ L 2 + D 4 δ 1 ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 2 b ‖ H 1 ‖ ∇ 2 b ‖ L 2 ‖ ∇ 4 v ‖ L 2 .

Here, D i ( i = 1 , 2, 3, 4) are positive constants.

Proof

Multiply ∇ k v and ∇ k b by ∇ k (1.3)1 and ∇ k (1.3)2, respectively, adding them up and then integrating the resultant equation on R 3 , where k = 2 , 3 . Then, by using integration by parts, one has

(4.18) 1 2 d d t ( ‖ ∇ k v ‖ L 2 2 + ‖ ∇ k b ‖ L 2 2 ) + κ ‖ ∇ k + 1 v ‖ L 2 2 + β ‖ ∇ k + 1 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ k b ‖ L 2 2 = − ∫ R 3 ∇ k ( v ⋅ ∇ v ) ∇ k v d x − ∫ R 3 ∇ k ( v ⋅ ∇ b ) ∇ k b d x − θ 2 ∫ R 3 ∇ k ( b b ) ∇ k b d x + 2 a α ∫ R 3 ∇ k div ( b b ) ∇ k v d x + 2 a ∫ R 3 ∇ k div b ∇ k v d x + a ∫ R 3 ∇ k ( ∇ v + ( ∇ v ) ⊤ ) ∇ k b d x + a + 1 2 ∫ R 3 ∇ k ( ( ∇ v b ) + ( ∇ v b ) ⊤ ) ∇ k b d x + a − 1 2 ∫ R 3 ∇ ( ( b ∇ v ) + ( b ∇ v ) ⊤ ) ∇ k b d x ≜ ∑ i = 1 8 N 2 , i .

For the eight terms on the right-hand side of the aforementioned equality, one has the following estimations from the equation (1.3)3:

(4.19) ∣ N 2,1 ∣ = ∑ k = 2 3 ( ⟨ ∇ k ( v ⋅ ∇ v ) , ∇ k v ⟩ − ⟨ ( v ⋅ ∇ ) ∇ k v , ∇ k v ⟩ ) = ∑ k = 2 3 ‖ ∇ k ( v ⋅ ∇ v ) − ( v ⋅ ∇ ) ∇ k v ‖ L 2 ‖ ∇ k v ‖ L 2 = ∑ k = 2 3 ‖ [ ∇ k , v ] ∇ v ‖ L 2 ‖ ∇ k v ‖ L 2 ≲ ∑ k = 2 3 ‖ ∇ v ‖ L ∞ ‖ ∇ k − 1 ∇ v ‖ L 2 2 + ‖ ∇ v ‖ L ∞ ‖ ∇ k v ‖ L 2 2 ≲ ‖ ∇ v ‖ H 1 ‖ ∇ k v ‖ L 2 2 ≲ δ 1 ‖ ∇ k v ‖ L 2 2 ,

where the commutator in Lemma 7.2 is defined by

[ ∇ k , v ] ∇ v = ∇ k ( v ⋅ ∇ v ) − v ⋅ ∇ k + 1 v .

for k = 2 , 3 . Similarly, one has

(4.20) ∣ N 2,2 ∣ = ∑ k = 2 3 ( ⟨ ∇ k ( v ⋅ ∇ b ) , ∇ k b ⟩ − ⟨ ( v ⋅ ∇ ) ∇ k b , ∇ k b ⟩ ) ≲ δ 1 ( ‖ ∇ k v ‖ L 2 2 + ‖ ∇ k b ‖ L 2 2 ) ,

where we have used the commutator:

[ ∇ k , v ] ∇ b = ∇ k ( v ⋅ ∇ b ) − v ⋅ ∇ k + 1 b

and

(4.21) ∣ N 2 , 3 ∣ = θ 2 ∑ k = 2 3 ‖ ∇ k ( b b ) ∇ k b ‖ L 1 ≲ ∑ k = 2 3 ‖ ∇ k ( b b ) ‖ L 6 5 ‖ ∇ k b ‖ L 6 ≲ δ 1 ( ‖ ∇ k b ‖ L 2 2 + ‖ ∇ k + 1 b ‖ L 2 2 )

and

(4.22) ∣ N 2,4 ∣ = 2 a ∣ α ∣ ∑ k = 2 3 ‖ ∇ k div ( b b ) ∇ k v ‖ L 1 ≲ ∑ k = 2 3 ‖ ∇ k div ( b b ) ‖ L 6 5 ‖ ∇ k v ‖ L 6 ≲ ( ‖ b ‖ L 3 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ b ‖ L 2 ‖ ∇ 2 b ‖ L 3 ) ‖ ∇ 2 v ‖ L 6 , for k = 2 ; ( ‖ b ‖ L 3 ‖ ∇ 4 b ‖ L 2 + ‖ ∇ b ‖ L 3 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ 2 b ‖ L 3 ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 3 v ‖ L 6 , for k = 3 , ≲ δ 1 ‖ ∇ 3 ( v , b ) ‖ L 2 2 , for k = 2 ; δ 1 ‖ ∇ 4 ( v , b ) ‖ L 2 2 , for k = 3 ,

where we have used the following fact that

‖ b ‖ L 3 ‖ ∇ 3 b ‖ L 2 ‖ ∇ 2 v ‖ L 6 ≲ ‖ b ‖ L 2 1 − r ‖ ∇ 3 b ‖ L 2 r ‖ ∇ α 1 b ‖ L 2 ‖ ∇ 3 b ‖ L 2 1 − r ‖ ∇ 3 v ‖ L 2 ≲ ‖ b 1 2 ‖ L 2 ‖ ∇ 3 b ‖ L 2 ‖ ∇ 3 v ‖ L 2 ≲ ‖ b ‖ L 2 1 2 ‖ ∇ b ‖ L 2 1 2 ‖ ∇ 3 b ‖ L 2 ‖ ∇ 3 v ‖ L 2 ≲ δ 1 ( ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 3 b ‖ L 2 2 ) ,

where r = 1 and α 1 = 1 2 . Similarly, one has

‖ b ‖ L 3 ‖ ∇ 4 b ‖ L 2 ‖ ∇ 3 v ‖ L 6 ≲ δ 1 ( ‖ ∇ 4 v ‖ L 2 2 + ‖ ∇ 4 b ‖ L 2 2 ) .

By Using integration by parts, it is clear that

(4.23) ∣ N 2,5 ∣ + ∣ N 2,6 ∣ = 0 .

For the last two terms, we have

(4.24) ∣ N 2,7 ∣ + ∣ N 2,8 ∣ ≲ a ∑ k = 3 2 ‖ ∇ k ( ∇ v b ) ∇ k b ‖ L 1 ≲ δ 1 ( ‖ ∇ k + 1 v ‖ L 2 2 + ‖ ∇ k + 1 b ‖ L 2 2 ) .

Substituting (4.19)–(4.24) into (4.18), we conclude that

(4.25) d d t ( ‖ ∇ k v ‖ L 2 2 + ‖ ∇ k b ‖ L 2 2 ) + κ ‖ ∇ k + 1 v ‖ L 2 2 + β ‖ ∇ k + 1 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ k b ‖ L 2 2 ≲ δ 1 ( ‖ ∇ 2 ( v , b ) ‖ L 2 2 + ‖ ∇ 3 ( v , b ) ‖ L 2 2 ) , for k = 2 ; δ 1 ( ‖ ∇ 3 ( v , b ) ‖ L 2 2 + ‖ ∇ 4 ( v , b ) ‖ L 2 2 ) , for k = 3 ; ≲ D 3 δ 1 ‖ ∇ 2 v ‖ L 2 2 , for k = 2 ; D 4 δ 1 ‖ ∇ 3 v ‖ L 2 2 , for k = 3 .

Next, by summing the equations (4.16) and (4.25) up, we arrive at

(4.26) d d t ( ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ v h ‖ L 2 2 ) + κ 2 ‖ ∇ 2 v h ‖ L 2 2 + κ 2 ‖ ∇ 3 v ‖ L 2 2 + β 2 ‖ ∇ 3 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 2 b ‖ L 2 2 ≲ D 1 δ 1 ‖ ∇ 2 v h ‖ L 2 2 + D 3 δ 1 ‖ ∇ 2 v ‖ L 2 2 ,

for k = 2 . And for the case k = 3 , one obtains

(4.27) d d t ( ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) + κ 2 ‖ ∇ 3 v h ‖ L 2 2 + κ 2 ‖ ∇ 4 v ‖ L 2 2 + β 2 ‖ ∇ 4 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 3 b ‖ L 2 2 ≲ ‖ ∇ 2 b ‖ L 2 ‖ ∇ 3 v h ‖ L 2 + D 4 δ 1 ‖ ∇ 3 v ‖ L 2 2 .

Thus, the proof of Lemma 4.3 has been completed.□

Next, we are in the position to deduce the optimal decay rates for the second-order and third-order spatial derivatives of the solution ( v , b ) .

Lemma 4.4

Assume that all conditions of Theorem 1.1 are in force. Then, it holds that

(4.28) ‖ ∇ 2 ( v , b ) ( t ) ‖ L 2 ≲ ( 1 + t ) − 7 4

and

(4.29) ‖ ∇ 3 ( v , b ) ( t ) ‖ L 2 ≲ ( 1 + t ) − 9 4 .

Proof

Equation (4.26) implies that

(4.30) d d t ( ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ v h ‖ L 2 2 ) + κ 4 ‖ ∇ 2 v h ‖ L 2 2 + κ 2 ‖ ∇ 3 v ‖ L 2 2 + β 2 ‖ ∇ 3 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 2 b ‖ L 2 2 ≲ ( D 1 + D 3 ) δ 1 ‖ ∇ 2 v l ‖ L 2 2 ,

where we have used the simple fact that

(4.31) ‖ ∇ k v ‖ L 2 2 ≤ ‖ ∇ k v l ‖ L 2 2 + ‖ ∇ k v h ‖ L 2 2 .

Thus, we have

(4.32) d d t ( ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ v h ‖ L 2 2 ) + α 1 ( ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ 2 b ‖ L 2 2 ) ≲ ( D 1 + D 3 ) δ 1 + κ 4 ‖ ∇ 2 v l ‖ L 2 2 ,

where α 1 ≜ min { θ 1 + θ 2 2 , κ 4 } . Then, by taking α 2 ≜ δ 1 ( D 1 + D 2 ) + κ 4 , we have

(4.33) d d t E 2 ( t ) + α 1 E 2 ( t ) ≤ α 2 ‖ ∇ 2 v l ‖ L 2 2 ,

for k = 2 . By using Lemma 4.1, equations (2.1) and (2.2), we obtain

(4.34) ∥ ∣ ξ ∣ 2 ∣ v ˆ ∣ ∥ L ∣ ξ ∣ ≤ η 2 ≲ ( 1 + t ) − 3 4 − 2 2 ‖ ( v 0 , b 0 ) ‖ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 3 4 − 2 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 1 d z + ∫ t 2 t ( 1 + t − z ) − 2 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 2 d z ≲ ( 1 + t ) − 7 4 ∥ U 0 ∥ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 7 4 ( 1 + t ) − 3 4 × 2 d z + ∫ t 2 t ( 1 + t − z ) − 2 2 ( 1 + t ) − 3 4 ( 1 + t ) − 5 4 d z ≲ ( 1 + t ) − 7 4 + ∫ 0 t 2 ( 1 + t ) − 13 4 d z + ∫ t 2 t ( 1 + t − z ) − 3 d z ≲ ( 1 + t ) − 7 4 .

Combining the aforementioned relations and using Gronwall’s inequality, we have

(4.35) ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v ‖ L 2 2 ≤ 2 ( ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v ‖ L 2 2 + ‖ ∇ v h ‖ L 2 2 ) ≤ e − 2 α 1 t ( ‖ ∇ 2 b 0 ‖ L 2 2 + ‖ ∇ 2 v 0 ‖ L 2 2 + ‖ ∇ v 0 h ‖ L 2 2 ) + α 2 ∫ 0 t e − 2 α 1 ( t − z ) ∥ ∣ ξ ∣ 2 ∣ v ˆ ( z ) ∣ ∥ L ∣ ξ ∣ ≤ η 2 d z ≲ ( 1 + t ) − 7 2 .

Therefore, we obtain the following second-order energy estimates of solution, i.e.,

(4.36) ‖ ∇ 2 ( v , b ) ( t ) ‖ L 2 ≲ ( 1 + t ) − 7 4 .

In a similar way, by using equations (4.27), (4.31), and (4.36), we obtain the following fact that

d d t ( ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) + κ 2 ‖ ∇ 3 v h ‖ L 2 2 + κ 2 ‖ ∇ 4 v ‖ L 2 2 + β 2 ‖ ∇ 4 b ‖ L 2 2 + θ 1 + θ 2 2 ‖ ∇ 3 b ‖ L 2 2 ≲ ( D 2 + D 4 ) δ 1 ‖ ∇ 3 v l ‖ L 2 2 .

Therefore, one have

d d t E 2 ( t ) + α 1 E 2 ( t ) ≤ α 3 ‖ ∇ 3 v l ‖ L 2 2 ,

for k = 3 . Here, α 1 = min { θ 1 + θ 2 2 , κ 4 } . And then, we taking α 3 ≜ δ 1 ( D 2 + D 4 ) + κ 4 . On the other hand, we can obtain

∥ ∣ ξ ∣ 3 ∣ v ˆ ∣ ∥ L ∣ ξ ∣ ≤ η 2 ≲ ( 1 + t ) − 3 4 − 3 2 ‖ ( v 0 , b 0 ) ‖ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 3 4 − 3 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 1 d z + ∫ t 2 t ( 1 + t − z ) − 3 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 2 d z ≲ ( 1 + t ) − 9 4 + ∫ 0 t 2 ( 1 + t − z ) − 9 4 − 3 4 × 2 d z ∫ t 2 t ( 1 + t − z ) − 3 2 ( 1 + t ) − 4 2 d z ≲ ( 1 + t ) − 9 4 .

Thus, we obtain the third-order energy estimate of solution:

‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ 3 v ‖ L 2 2 ≤ 2 ( ‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ 3 v ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) ≤ e − 2 α 1 t ( ‖ ∇ 3 b 0 ‖ L 2 2 + ‖ ∇ 3 v 0 ‖ L 2 2 + ‖ ∇ 2 v 0 h ‖ L 2 2 ) + α 3 ∫ 0 t e − 2 α 1 ( t − z ) ∥ ∣ ξ ∣ 3 ∣ v ˆ ( z ) ∣ ∥ L ∣ ξ ∣ ≤ η 2 d z ≲ ( 1 + t ) − 9 2 ,

i.e.,

‖ ∇ 3 ( v , b ) ( t ) ‖ L 2 ≲ ( 1 + t ) − 9 4 .

Therefore, the proof of Lemma 4.4 has been completed.□

Lemma 4.5

Assume that all conditions of Theorem 1.1 are in force. Then, it holds that

(4.37) ‖ ∇ k b ( t ) ‖ L 2 ≲ ( 1 + t ) − 5 4 − k 2 ,

for any t > 0 and k = 0 , 1, 2.

Proof

Multiplying ∇ k b by ∇ k (1.3)2, adding them up, and then integrating the resultant equation on R 3 . Then using integration by parts, we obtain

(4.38) 1 2 d d t ‖ ∇ k b ‖ L 2 2 + β ‖ ∇ k + 1 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ k b ‖ L 2 2 = − θ 2 ∫ R 3 ∇ k ( b b ) ∇ k b d x + a ∫ R 3 ∇ k ( ∇ v + ( ∇ v ) ⊤ ) ∇ k b d x − ∫ R 3 ∇ k ( v ⋅ ∇ b ) ∇ k b d x + a + 1 2 ∫ R 3 ∇ k ( ( ∇ v b ) + ( ∇ v b ) ⊤ ) ∇ k b d x + a − 1 2 ∫ R 3 ∇ k ( b ∇ v + ( b ∇ v ) ⊤ ) ∇ k b d x ≜ ∑ i = 1 5 N 3 , i .

By applying Lemma 7.1, one has

(4.39) ∣ N 3,1 ∣ = θ 2 ∑ k = 0 2 ‖ ∇ k ( b b ) ∇ k b ‖ L 1 ≲ ‖ b b ‖ L 6 5 ‖ b ‖ L 6 + ‖ ∇ ( b b ) ‖ L 6 5 ‖ ∇ b ‖ L 6 + ‖ ∇ 2 ( b b ) ‖ L 6 5 ‖ ∇ 2 b ‖ L 6 ≲ ‖ b ‖ L 3 ‖ b ‖ L 2 ‖ b ‖ L 6 + ‖ b ‖ L 3 ‖ ∇ b ‖ L 2 ‖ ∇ b ‖ L 6 + ‖ ∇ b ‖ L 2 ‖ b ‖ L 3 ‖ ∇ b ‖ L 6 + ‖ b ‖ L 3 ‖ ∇ 2 b ‖ L 2 ‖ ∇ 2 b ‖ L 6 + ‖ ∇ b ‖ L 2 ‖ ∇ b ‖ L 3 ‖ ∇ 2 b ‖ L 6 + ‖ ∇ 2 b ‖ L 2 ‖ b ‖ L 3 ‖ ∇ 2 b ‖ L 6 ≲ ‖ b ‖ H 1 ‖ b ‖ L 2 ‖ ∇ b ‖ L 2 + ‖ b ‖ H 1 ‖ ∇ b ‖ L 2 ‖ ∇ 2 b ‖ L 2 + ‖ b ‖ H 1 ‖ ∇ 2 b ‖ L 2 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ b ‖ L 2 ‖ ∇ b ‖ H 1 ‖ ∇ 3 b ‖ L 2 ≲ ( 1 + t ) − 11 4 + ( 1 + t ) − 15 4 + ( 1 + t ) − 19 4 ≲ ( 1 + t ) − 5 2 − k ,

(4.40) ∣ N 3,2 ∣ ≲ ∑ k = 0 2 ‖ ∇ k ∇ v ∇ k b ‖ L 2 2 ≲ ‖ ∇ v ‖ L 2 2 ‖ b ‖ L ∞ 2 + ‖ ∇ 2 v ‖ L 2 2 ‖ ∇ b ‖ L ∞ 2 + ‖ ∇ 3 v ‖ L 2 2 ‖ ∇ 2 b ‖ L ∞ 2 ≲ ‖ ∇ v ‖ L 2 2 ‖ ∇ b ‖ L 2 2 + ‖ ∇ 2 v ‖ L 2 2 ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 3 v ‖ L 2 2 ‖ ∇ 3 b ‖ L 2 2 ≲ ( 1 + t ) − 5 2 − k ,

and

(4.41) ∣ N 3,3 ∣ ≲ ‖ v ‖ L ∞ 2 ‖ ∇ 3 b ‖ L 2 2 + ‖ ∇ v ‖ L ∞ 2 ‖ ∇ 2 b ‖ L 2 2 + ‖ ∇ 2 v ‖ L ∞ 2 ‖ ∇ b ‖ L 2 2 ≲ ( 1 + t ) − 5 2 − k .

For the last two terms, we obtain

(4.42) ∣ N 3,4 ∣ + ∣ N 3,5 ∣ ≲ ∑ k = 0 2 ‖ ∇ k ∇ v ∇ k b ‖ L 2 2 ≲ ( 1 + t ) − 5 2 − k .

By substituting (4.39)–(4.42) into (4.38), we conclude that

(4.43) 1 2 d d t ‖ ∇ k b ‖ L 2 2 + β ‖ ∇ k + 1 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ k b ‖ L 2 2 ≲ ( 1 + t ) − 5 2 − k .

By applying Gronwall’s inequality, we have

(4.44) 1 2 d d t ‖ ∇ k b ‖ L 2 2 ≲ e − ( θ 1 + θ 2 ) t ‖ ∇ k b 0 ‖ L 2 2 + ∫ 0 t e − ( θ 1 + θ 2 ) ( t − z ) ( 1 + z ) − 5 2 − k d z ≲ ( 1 + t ) − 5 2 − k ,

for any t > 0 and k = 0 , 1, 2.

Thus, the proof of Lemma 4.5 has been completed.□

Lemma 4.6

Assume that all conditions of Theorem 1.1 are in force. Then, it holds that

(4.45) ‖ ∇ 3 b ( t ) ‖ L 2 ≲ ( 1 + t ) − 11 4 ,

for any t > 0 .

Proof

By multiplying ∇ 3 φ 0 ( D ) (1.3)1 and ∇ 3 φ 0 ( D ) (1.3)2 by ∇ 3 v h and ∇ 3 b h , respectively, adding them up, and then integrating the resultant equation on R 3 , we have

(4.46) 1 2 d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b h ‖ L 2 2 = − κ ⟨ ∇ 3 ( v ⋅ ∇ v ) h , ∇ 3 v h ⟩ − β ⟨ ∇ 3 ( v ⋅ ∇ b ) h , ∇ 3 b h ⟩ + ⟨ ∇ 3 div ( b b ) h , ∇ 3 v h ⟩ − θ 2 ⟨ ∇ 3 ( b b ) h , ∇ 3 b h ⟩ + 2 a ⟨ ∇ 3 div ( b ) h , ∇ 3 v h ⟩ + a ⟨ ∇ 3 ( ∇ v + ( ∇ v ) ⊤ ) h , ∇ 3 b h ⟩ × a + 1 2 ⟨ ∇ 3 ( ( ∇ v b ) + ( ∇ v b ) ⊤ ) h , ∇ 3 b h ⟩ + a − 1 2 ⟨ ∇ 3 ( ( b ∇ v ) + ( b ∇ v ) ⊤ ) H , ∇ 3 b h ⟩ ≜ ∑ i = 1 8 N 4 , i .

For the eight terms on the right-hand side of the aforementioned equality, one has the following estimates:

(4.47) ∣ N 4,1 ∣ ≤ κ ⟨ ℱ ( ∇ 3 ( ( v ⋅ ∇ v ) h ) ) , ℱ ( ∇ 3 v h ) ⟩ = κ ⟨ ∇ 3 ( v ⋅ ∇ v l 1 ) , ∇ 3 v h 1 ⟩ + κ ⟨ ∇ 3 ( v ⋅ ∇ v h 1 ) , ∇ 3 v h 1 ⟩ − κ ⟨ ( v ⋅ ∇ ) ∇ 3 v h 1 , ∇ 3 v h 1 ⟩ ≜ I 1 , 1 + I 1 , 2 + I 1,3 .

By making careful computation and using integration by parts, it holds that

(4.48) ∣ I 1 , 1 ∣ ≲ ( ‖ ∇ 3 v ‖ L 2 ‖ ∇ v l 1 ‖ L ∞ + ‖ ∇ 2 v ‖ L 2 ‖ ∇ 2 v l 1 ‖ L ∞ ) ‖ ∇ 3 v h 1 ‖ L 2 + ( ‖ ∇ v ‖ L 2 ‖ ∇ 3 v l 1 ‖ L ∞ + ‖ v ‖ L ∞ ‖ ∇ 4 v l 1 ‖ L 2 ) ‖ ∇ 3 v h 1 ‖ L 2 ≲ ( ‖ ∇ 3 v ‖ L 2 ‖ ∇ 2 v l 1 ‖ L 2 + ‖ ∇ 2 v ‖ L 2 ‖ ∇ 3 v l 1 ‖ L 2 + ‖ ∇ v ‖ L 2 ‖ ∇ 4 v l 1 ‖ L 2 ) ‖ ∇ 3 v h 1 ‖ L 2 ≲ ( 1 + t ) − 25 4 + ( 1 + t ) − 7 2 ‖ ∇ 4 v l 1 ‖ L 2

and

(4.49) ∣ I 1 , 2 + I 1,3 ∣ ≲ ( ‖ ∇ 3 v ‖ L 2 ‖ ∇ v h 1 ‖ L ∞ + ‖ ∇ 2 v ‖ L 2 ‖ ∇ 2 v h 1 ‖ L ∞ + ‖ ∇ v ‖ L ∞ ‖ ∇ 3 v h 1 ‖ L 2 ) ‖ ∇ 3 v h 1 ‖ L 2 ≲ ( 1 + t ) − 25 4 .

By using integration by parts, Cauchy inequality, equation (3.1) and (3.2), we obtain

(4.50) ∣ N 4,2 ∣ ≤ β ⟨ ∇ 2 ( v ⋅ ∇ b ) h , ∇ 4 b h ⟩ ≲ β ( ‖ ∇ 2 v ‖ L 2 ‖ ∇ b ‖ L ∞ + ‖ ∇ v ‖ L 2 ‖ ∇ 2 b ‖ L ∞ + ‖ v ‖ L ∞ ‖ ∇ 3 b ‖ L 2 ) ‖ ∇ 4 b h ‖ L 2 ≲ β ( ‖ ∇ 2 v ‖ L 2 ‖ ∇ 2 b ‖ L 2 + ‖ ∇ v ‖ L 2 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ v ‖ L 2 ‖ ∇ 3 b ‖ L 2 ) ‖ ∇ 4 b h ‖ L 2 ≲ ( 1 + t ) − 7 2 ‖ ∇ 4 b h ‖ L 2 ≲ ε ( β ) ( 1 + t ) − 7 + C β ( ε ) ‖ ∇ 4 b h ‖ L 2 2 .

In a similar way, one has the following estimations:

(4.51) ∣ N 4,3 ∣ ≲ ( ‖ ∇ 3 b ‖ L 2 ‖ b ‖ L ∞ + ‖ ∇ 2 b ‖ L 2 ‖ ∇ b ‖ L ∞ + ‖ ∇ b ‖ L ∞ ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 3 div b h ‖ L 2 ≲ ( 1 + t ) − 7 + C ∣ a ∣ α ( ε ) ‖ ∇ 4 b h ‖ L 2 2 ,

(4.52) ∣ N 4,4 ∣ ≲ ‖ ∇ 3 ( b b ) ‖ L 6 5 ‖ ∇ 3 b h ‖ L 6 ≲ ( ‖ ∇ 3 b ‖ L 2 ‖ b ‖ L 3 + ‖ ∇ 2 b ‖ L 2 ‖ ∇ b ‖ L 3 + ‖ ∇ b ‖ L 3 ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 4 b h ‖ L 2 ≲ ( 1 + t ) − 7 + C θ 2 ( ε ) ‖ ∇ 4 b h ‖ L 2 2 ,

and

(4.53) ∣ N 4,7 ∣ + ∣ N 4,8 ∣ ≤ ∣ a ∣ ⟨ ∇ 2 ( ∇ v b ) , ∇ 4 b h ⟩ ≲ ( ‖ ∇ v ‖ L 2 ‖ ∇ 2 b ‖ L ∞ + ‖ ∇ 2 v ‖ L 2 ‖ ∇ b ‖ L ∞ + ‖ ∇ 3 v ‖ L 3 ‖ b ‖ L ∞ ) ‖ ∇ 4 b h ‖ L 2 ≲ ( 1 + t ) − 7 + C a ( ε ) ‖ ∇ 4 b h ‖ L 2 2 .

It is noted that N 4,5 + N 4,6 = 0 . By substituting (4.47)–(4.53) into (4.46), it holds that there exists a sufficiently small positive constant ε such that

(4.54) 1 2 d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b h ‖ L 2 2 ≲ ( 1 + t ) − 25 4 + ( 1 + t ) − 7 2 ‖ ∇ 4 v l 1 ‖ L 2 .

From Lemma 4.1, it holds that

(4.55) ∥ ∣ ξ ∣ 4 ∣ v ˆ ∣ ∥ L ∣ ξ ∣ ≤ η 2 ≲ ( 1 + t ) − 3 4 − 4 2 ∥ ( v 0 , b 0 ) ∥ L 1 + ∫ 0 t 2 ( 1 + t − z ) − 3 4 − 4 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 1 d z + ∫ t 2 t ( 1 + t − z ) − 4 2 ∥ ( T 1 , T 2 ) ⊤ ( z ) ∥ L 2 d z ≲ ( 1 + t ) − 11 4 + ∫ 0 t 2 ( 1 + t ) − 17 4 d z + ∫ t 2 t ( 1 + t − z ) − 12 4 d z ≲ ( 1 + t ) − 11 4 ,

which implies that

∥ ∇ 4 v ∥ L ∣ ξ ∣ ≤ η 2 = ∥ ∣ ξ ∣ 4 ∣ v ˆ ∣ ∥ L ∣ ξ ∣ ≤ η 2 ≲ ( 1 + t ) − 11 4 .

Substituting (4.57) into (4.56) yields the following estimation immediately:

(4.56) d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b h ‖ L 2 2 ≲ ( 1 + t ) − 25 4 ,

which implies that

‖ ∇ 4 v h ‖ L 2 2 ≲ ( 1 + t ) − 25 4 .

From equation (4.27), Lemmas 4.4 and 4.5, it holds that

(4.57) 1 2 d d t ‖ ∇ 2 v h ‖ L 2 2 + κ 2 ‖ ∇ 3 v h ‖ L 2 2 ≲ ( 1 + t ) − 3 2 ‖ ∇ 4 v h ‖ L 2 2 .

By combining (4.56) and (4.57), it holds that

(4.58) d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 + ‖ ∇ 2 v h ‖ L 2 2 ) + κ ‖ ∇ 4 v h ‖ L 2 2 + β ‖ ∇ 4 b h ‖ L 2 2 + α 3 ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) ≲ ( 1 + t ) − 25 4 ,

where α 3 = min { κ , θ 1 + θ 2 } . Thus, we obtain

(4.59) d d t ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) + α 3 ( ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ) ≲ ( 1 + t ) − 25 4 .

From Gronwall’s inequality, we have

(4.60) ‖ ∇ 3 ( v h , b h ) ( t ) ‖ L 2 2 ≲ ‖ ∇ 3 v h ‖ L 2 2 + ‖ ∇ 3 b h ‖ L 2 2 ≲ e − 2 α 3 t ( ‖ ∇ 3 v 0 h ‖ L 2 2 + ‖ ∇ 3 b 0 h ‖ L 2 2 ) + ∫ 0 t e − 2 α 3 ( t − z ) ( 1 + z ) − 25 4 d z ≲ ( 1 + t ) − 25 4 .

Now, we are in the position to estimate the third-order spatial derivative of b . Similar to the estimation of (4.46), we multiply ∇ 3 b by ∇ 3 (1.3)2, and then integrating the resultant equation on R 3 , we have from integration by parts that

(4.61) 1 2 d d t ‖ ∇ 3 b ‖ L 2 2 + β ‖ ∇ 4 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b ‖ L 2 2 = − θ 2 ∫ R 3 ∇ 3 ( b b ) ∇ 3 b d x + a ∫ R 3 ∇ 3 ( ∇ v + ( ∇ v ) ⊤ ) ∇ 3 b d x − ∫ R 3 ∇ 3 ( v ⋅ ∇ b ) ∇ 3 b d x + a + 1 2 ∫ R 3 ∇ 3 ( ( ∇ v b ) + ( ∇ v b ) ⊤ ) ∇ 3 b d x + a − 1 2 ∫ R 3 ∇ 3 ( ( b ∇ v ) + ( b ∇ v ) ⊤ ) ∇ 3 b d x ≜ ∑ i = 1 5 N 5 , i .

Therefore, by using integration by parts and Lemma 4.4, we obtain

(4.62) ∣ N 5,1 ∣ = θ 2 ∫ R 3 ∇ 3 ( b b ) ∇ 3 b d x ≲ ( ‖ b ‖ L ∞ ‖ ∇ 3 b ‖ L 2 + ‖ ∇ b ‖ L ∞ ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 3 b ‖ L 2 ≲ ( ‖ ∇ b ‖ H 1 ‖ ∇ 3 b ‖ L 2 + ‖ ∇ 2 b ‖ H 1 ‖ ∇ 2 b ‖ L 2 ) ‖ ∇ 3 b ‖ L 2 ≲ ( 1 + t ) − 25 4 .

With the equations (4.55) and (4.60) in hand, one obtains

(4.63) ∣ N 5,2 ∣ ≤ ∣ a ∣ ∫ R 3 ∇ 3 ( ∇ v l + ( ∇ v l ) ⊤ ) ∇ 3 b d x + ∣ a ∣ ∫ R 3 ∇ 2 ( ∇ v h + ( ∇ v h ) ⊤ ) ∇ 4 b d x ≲ ‖ ∇ 3 b ‖ L 2 ‖ ∇ 4 v l ‖ L 2 + ‖ ∇ 4 b ‖ L 2 ‖ ∇ 3 v h ‖ L 2 ≲ θ 1 + θ 2 2 ‖ ∇ 3 b ‖ L 2 2 + β 2 ‖ ∇ 4 b ‖ L 2 2 + C ‖ ∇ 4 v l ‖ L 2 2 + C ‖ ∇ 3 v h ‖ L 2 ≲ θ 1 + θ 2 2 ‖ ∇ 3 b ‖ L 2 2 + β 2 ‖ ∇ 4 b ‖ L 2 2 + ( 1 + t ) − 11 2 .

For the term N 5,3 , one obtains

(4.64) ∣ N 5,3 ∣ ≤ ∫ R 3 ∇ 2 ( v ⋅ ∇ b ) ∇ 4 b d x ≲ ( ‖ v ‖ L ∞ ‖ ∇ 3 b ‖ L 2 + ‖ ∇ v ‖ L ∞ ‖ ∇ 2 b ‖ L 2 + ‖ ∇ 2 v ‖ L ∞ ‖ ∇ b ‖ L 2 ) ‖ ∇ 4 b ‖ L 2 ≲ ( 1 + t ) − 7 ‖ ∇ 4 b ‖ L 2 2

and

(4.65) ∣ N 5,4 ∣ + ∣ N 5,5 ∣ ≤ 2 ∣ a ∣ ∫ R 3 ∇ 2 ( b ∇ v ) ∇ 4 b d x ≲ ( ‖ b ‖ L ∞ ‖ ∇ 3 v ‖ L 2 ) + ( ‖ ∇ b ‖ L ∞ ‖ ∇ 2 v ‖ L 2 ) + ( ‖ ∇ 2 b ‖ L ∞ ‖ ∇ v ‖ L 2 ) ‖ ∇ 4 b ‖ L 2 ≲ ( 1 + t ) − 7 + C a ( ε ) ‖ ∇ 4 b ‖ L 2 2 .

Substituting (4.62)–(4.65) into (4.61), it holds that there exists a sufficiently small positive constant ε such that

(4.66) d d t ‖ ∇ 3 b ‖ L 2 2 + β ‖ ∇ 4 b ‖ L 2 2 + ( θ 1 + θ 2 ) ‖ ∇ 3 b ‖ L 2 2 ≲ ( 1 + t ) − 11 2 .

By applying Gronwall’s inequality, we have

(4.67) d d t ‖ ∇ 3 b ‖ L 2 2 ≲ e − ( θ 1 + θ 2 ) t ‖ ∇ 3 b 0 ‖ L 2 2 + ∫ 0 t e − ( θ 1 + θ 2 ) ( t − z ) ( 1 + z ) − 11 2 d z ≲ ( 1 + t ) − 11 2 ,

for all t > 0 .

Thus, the proof of Theorem 1.1 has been completed.□

5 The proof of Theorem 1.2

In this section, we devote ourselves to proving the lower decay estimates to the Cauchy problems (1.3)–(1.4) for the zero-order to the third-order spatial derivatives of the solution ( v , b ) .

Proof

Applying Duhamel’s principle, Plancherel’s theorem, we obtain

(5.1) ‖ ∇ k v ( t ) ‖ L 2 2 = ‖ ∣ ξ ∣ k v ˆ ( t ) ‖ L 2 2 = ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( v ˆ ( t ) ) 2 d ξ + ∫ ∣ ξ ∣ ≥ η ∣ ξ ∣ 2 k ( v ˆ ( t ) ) 2 d ξ = ∇ k ( A 1 ( t ) ∗ U 0 ) + ∫ 0 t A 1 ( t − z ) ∗ N 1 N 2 ( z ) L 2 2 ≥ ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( A ˆ 1 ( t ) ∗ U ˆ 0 ) 2 d ξ − ∫ 0 t ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k A ˆ 1 2 ( t − z ) N ˆ 1 ( z ) N ˆ 2 ( z ) 2 d ξ d z ,

where A ( t ) = A 1 ( t ) A 2 ( t ) . For the first term on the right-hand side of the aforementioned inequality, it holds that

(5.2) ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( A ˆ 1 ( t ) ∗ U ˆ 0 ) 2 d ξ = ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( G 1 ( ξ , t ) v ˆ 0 j ) 2 + κ ∣ ξ ∣ 2 k + 4 ( G 2 ( ξ , t ) v ˆ 0 j ) 2 − 2 a ∣ ξ ∣ 2 k + 2 ( G 1 ( ξ , t ) τ ˆ 0 j ) 2 d ξ ≳ inf ∣ ξ ∣ ≤ η ∣ v ˆ 0 ∣ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k e − 2 η 2 ∣ ξ ∣ 2 t d ξ + κ inf ∣ ξ ∣ ≤ η ∣ v ˆ 0 ∣ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k + 4 e − 2 η 2 ∣ ξ ∣ 2 t d ξ − 2 a inf ∣ ξ ∣ ≤ η ∣ v ˆ 0 ∣ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k + 2 e − 2 η 2 ∣ ξ ∣ 2 t d ξ ≳ c ˜ 0 ∫ ∣ ζ ∣ ≤ t − 1 2 η t − 1 2 ζ 2 k t − 3 2 e − 2 η 2 ∣ ζ ∣ 2 d ζ − c ˜ 0 κ ∫ ∣ ζ ∣ ≤ t − 1 2 η t − 1 2 ζ 2 k + 4 t − 3 2 e − 2 η 2 ∣ ζ ∣ 2 d ζ − 2 a ∫ ∣ ζ ∣ ≤ t − 1 2 η t − 1 2 ζ 2 k + 2 t − 3 2 e − 2 η 2 ∣ ζ ∣ 2 d ζ ≳ c ˜ ( 1 + t ) − 3 2 − k − ( 1 + t ) − 7 2 − k − ( 1 + t ) − 5 2 − k ≳ c ˜ 0 ( 1 + t ) − 3 2 − k .

For the second term on the right-hand side of the aforementioned inequality, we have

(5.3) ∫ 0 t ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k A ˆ 1 2 ( t − z ) N ˆ 1 ( z ) N ˆ 2 ( z ) 2 d ξ d z ≲ ∫ 0 t ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( G 1 ( ξ , t ) N ˆ 1 ) 2 − κ ∣ ξ ∣ 2 k + 4 ( G 2 ( ξ , t ) N ˆ 1 ) 2 + 2 a ∣ ξ ∣ 2 k + 2 ( G 1 ( ξ , t ) N ˆ 2 ) 2 d ξ d z ≲ ∫ 0 t 2 ( ∥ ∣ ξ ∣ − 1 N ˆ 1 ( z ) ∥ L ∞ 2 + ∣ 2 ∣ a ∣ ∥ N ˆ 2 ( z ) ∥ L ∞ 2 ) × ∥ ∣ ξ ∣ k + 1 e − η 2 ∣ ξ ∣ 2 ( t − z ) ∥ L 2 2 d z + ∫ t 2 t N ˆ 1 ( z ) N ˆ 2 ( z ) L 2 2 ( 1 + t − z ) − k d z ≲ ∫ 0 t 2 ( 1 + z ) − 3 ( 1 + t − z ) − 5 2 − k d z + ∫ t 2 t ( 1 + z ) − 5 ( 1 + t − z ) − k d z ≲ ( 1 + t ) − 5 2 − k ,

where we have used the facts that

∥ ∣ ξ ∣ − 1 N ˆ 1 ( z ) ∥ L ∞ = ∥ ∇ − 1 N 1 ( z ) ^ ∥ L ∞ ≲ ∥ ∇ − 1 N 1 ( z ) ∥ L 1 ≲ ∥ ∇ − 1 ( − P ( v ⋅ ∇ v ) + 2 a α P ( div ( b b ) ) ) ∥ L 1 ≲ ‖ ∇ − 1 ( ∇ ⋅ ( v v ) ) ‖ L 1 2 + 2 ∣ a ∣ α ‖ ∇ − 1 ( div ( b b ) ) ‖ L 1 ≲ ‖ v ‖ L 2 ‖ v ‖ L 2 + 2 ∣ a ∣ α ‖ b ‖ L 2 ‖ b ‖ L 2 ≲ ( 1 + t ) − 3 2 ,

and

∥ N ˆ 2 ( z ) ∥ L ∞ ≲ ∥ N 2 ( z ) ∥ L 1 ≲ ‖ v ‖ L 2 ‖ ∇ b ‖ L 2 + ‖ b ‖ L 2 ‖ ∇ v ‖ L 2 + ‖ b ‖ L 2 2 ≲ ( 1 + t ) − 5 2 ,

and

N ˆ 1 ( z ) N ˆ 2 ( z ) L 2 ≲ ‖ ∇ v ‖ H 1 ‖ ∇ v ‖ L 2 + 2 ∣ a ∣ α ‖ ∇ b ‖ H 1 ‖ ∇ b ‖ L 2 + ‖ ∇ v ‖ H 1 ‖ ∇ b ‖ L 2 + ‖ ∇ b ‖ H 1 ‖ ∇ v ‖ L 2 + ‖ b ‖ H 1 ‖ ∇ b ‖ L 2 ≲ ( 1 + t ) − 5 2 + ( 1 + t ) − 3 .

With the equations (5.1)–(5.3) in hand, we arrive at

(5.4) ‖ ∇ k v ( t ) ‖ L 2 2 ≳ c 1 ( 1 + t ) − 3 2 − k ,

which proves (1.9) immediately. Next, we make full use of the structure of the equations (3.14), (3.21), and (3.23) to obtain the following fact that

(5.5) − ( θ 1 + θ 2 + β ∣ ξ ∣ 4 ) ∣ ξ ∣ 2 k G 2 ( ξ , t ) = ( λ + + λ − + κ ∣ ξ ∣ 2 ) G 2 ( ξ , t ) = κ ∣ ξ ∣ 2 G 2 ( ξ , t ) + G 3 − G 1

Thus, one has

(5.6) ‖ ∇ k τ ( t ) ‖ L 2 2 ≥ ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( A ˆ 2 ( t ) ∗ U ˆ 0 ) 2 d ξ − ∫ 0 t ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k A ˆ 2 2 ( t − z ) N ˆ 1 ( z ) N ˆ 2 ( z ) 2 d ξ d z ≳ ∫ ∣ ξ ∣ ≤ η − a ∣ ξ ∣ 2 k + 2 ( G 2 v ˆ 0 j ) 2 + κ ∣ ξ ∣ 2 k + 4 ( G 2 τ ˆ 0 j ) 2 + ∣ ξ ∣ 2 k ( G 3 τ ˆ 0 j ) 2 d ξ − ∫ 0 t ∣ ξ ∣ k A ˆ 2 ( t − z ) N ˆ 1 ( z ) N ˆ 2 ( z ) L 2 2 d z ,

which together with the following estimates:

(5.7) ∫ ∣ ξ ∣ ≤ η − a ∣ ξ ∣ 2 k + 2 ( G 2 v ˆ 0 j ) 2 + κ ∣ ξ ∣ 2 k + 4 ( G 2 τ ˆ 0 j ) 2 + ∣ ξ ∣ 2 k ( G 3 τ ˆ 0 j ) 2 d ξ ≳ ∣ a ∣ inf ∣ ξ ∣ ≤ η ∣ v ˆ 0 ∣ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k + 2 e − 2 η 2 ∣ ξ ∣ 2 t d ξ − κ ‖ τ ˆ 0 ∣ ‖ L ∞ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k + 4 e − 2 η 2 ∣ ξ ∣ 2 t d ξ − ‖ τ ˆ 0 ∣ ‖ L ∞ 2 ∫ ∣ ξ ∣ ≤ η ∣ ξ ∣ 2 k ( ∣ ξ ∣ 2 e − η 1 ∣ ξ ∣ 2 t + e − θ 1 + θ 2 2 ) 2 d ξ ≳ ∣ a ∣ c 0 ( 1 + t ) − 5 2 − k − ‖ τ 0 ‖ L 1 2 ( 1 + t ) − 7 2 − k ≳ ( 1 + t ) − 5 2 − k ,

and

(5.8) ∫ 0 t ∣ ξ ∣ k A ˆ 2 ( t − z ) N ˆ 1 ( z ) N ˆ 2 ( z ) L 2 2 d z ≲ ∫ 0 t 2 ( ∥ ∣ ξ ∣ − 1 N ˆ 1 ( z ) ∥ L ∞ 2 + ∥ N ˆ 2 ( z ) ∥ L ∞ 2 ) × ∣ ξ ∣ k + 1 ( e − η 2 ∣ ξ ∣ 2 ( t − z ) + e − θ 1 + θ 2 2 ( t − z ) ) L 2 2 d z + ∫ t 2 t N ˆ 1 ( z ) N ˆ 2 ( z ) L 2 2 ( 1 + t − z ) − k d z ≲ ∫ 0 t 2 ( 1 + z ) − 3 ( 1 + t − z ) − 7 2 − k d z + ∫ t 2 t ( 1 + z ) − 5 ( ( 1 + t − z ) − k + e − ( θ 1 + θ 2 ) ( t − z ) ) d z ≲ ( 1 + t ) − 7 2 − k

imply that

‖ ∇ k τ ( t ) ‖ L 2 2 ≳ c 2 ( 1 + t ) − 5 2 − k .

Therefore, by noticing the fact that ‖ ∇ k τ ( t ) ‖ L 2 ≲ ‖ ∇ k b ( t ) ‖ L 2 , we prove (1.10) immediately.

Thus, the proof of Theorem 1.2 has been completed.□

6 Conclusions

In this article, we studied the optimal decay estimates of solutions to the 3D Cauchy problem of the rate type viscoelastic fluids. More precisely, we demonstrated that the decay rate of k ( 0 ≤ k ≤ 3 ) th-order spatial derivative of the velocity v is t − 3 4 + k 2 in L 2 , and the decay rate of m ( 0 ≤ m ≤ 2 ) th order spatial derivative of the extra stress tensor b is t − 3 5 + m 2 . And then, we showed the lower bounds on the convergence rates, which coincide with the upper rates.

7 Analytic tools

For the convenience of the reader, we provide some several tools frequently used throughout this article.

Lemma 7.1

[22] (Gagliardo-Nirenberg-Sobolev inequality) Let 2 ≤ p ≤ ∞ , 0 ≤ s , l ≤ k and 0 ≤ θ ≤ 1 , then

∥ ∇ s f ∥ L p ≲ ∥ ∇ k f ∥ L r θ ∥ ∇ l f ∥ L q 1 − θ ,

where θ is determined by

s 3 − 1 p = k 3 − 1 r θ + l 3 − 1 q ( 1 − θ ) .

Particularly, when p = 3 , q = r = 2 , s = l = 0 , k = 1 , one obtains

∥ f ∥ L 3 ≲ ∥ f ∥ H 1 ,

and then

∥ f ∥ L p ≲ ∥ f ∥ H 1 , 2 ≤ q ≤ 6 .

When p = ∞ , q = r = 2 , s = 0 , l = 1 , k = 2 , we obtain

(7.1) ∥ f ∥ L ∞ ≲ ∥ ∇ f ∥ L 2 1 2 ∥ ∇ 2 f ∥ L 2 1 2 ≲ ∥ ∇ f ∥ H 1 .

Lemma 7.2

[5] Assume that there exits functions f and g be smooth functions in L 1 ∩ H r for any r ≥ 1 , and define the commutator, which satisfies

[ ∇ r , f ] g = ∇ r ( f g ) − f ∇ r g .

Then, we have

‖ [ ∇ r , f ] g ‖ L r 0 ≲ ‖ ∇ f ‖ L p 1 ‖ ∇ r − 1 g ‖ L q 1 + ‖ ∇ r f ‖ L p 2 ‖ g ‖ L q 2 .

In addition, for any positive generic constant r 1 , it holds that

‖ ∇ r 1 ( f g ) ‖ L r 0 ≲ ‖ ∇ r 1 f ‖ L p 1 ‖ g ‖ L q 1 + ‖ ∇ r 1 g ‖ L p 2 ‖ f ‖ L q 2 ,

where 1 ≤ p 1 , p 2 , q 2 , q 2 ≤ ∞ , and satisfy

1 r 0 = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 .

Lemma 7.3

[8] Assume that there exist a 1 > 0 and a 2 ∈ [ 0 , a 1 ] . Then, we have

∫ 0 t ( 1 + t − z ) − a 1 ( 1 + z ) − a 2 d z ≤ C ( a 1 , a 2 ) ( 1 + t ) − a 2 .

We also need the following lemma concerning the estimation of the low-frequency part and high-frequency part of f .

Lemma 7.4

[18] Assume that there exits f ∈ H N ( R 3 )   ( N is a n i n t e g e r ) . Then, it holds that

‖ ∇ k f l ‖ L 2 ≤ ‖ ∇ k − 1 f l ‖ L 2 , for k ≥ 1 ;

‖ ∇ k f h ‖ L 2 ≤ ‖ ∇ k + 1 f h ‖ L 2 , for k ≥ 1 ;

‖ ∇ k f h ‖ L 2 ≤ ‖ ∇ k f l ‖ L 2 + ‖ ∇ k f h ‖ L 2 , for k ≥ 0 .

  1. Funding information: This work was partially supported by National Natural Science Foundation of China #12271114, Guangxi Natural Science Foundation #2024GXNSFDA010071, Center for Applied Mathematics of Guangxi (Guangxi Normal University), and the Key Laboratory of Mathematical Model and Application (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.

  2. Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: No data were used for the research described in the article.

References

[1] C. Ai, Z. Tan, and J. Zhou, Global existence and decay estimate of solution to rate type viscoelastic fluids, J. Differential Equations 377 (2023), 188–220, https://doi.org/10.1016/j.jde.2023.08.039. Search in Google Scholar

[2] M. Bathory, M. Bulíček, and J. Málek, Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion, Adv. Nonlinear. Anal. 10 (2020), no. 1, 501–521, https://doi.org/10.1515/anona-2020-0144. Search in Google Scholar

[3] M. Bulíček, T. Los, Y. Lu, and J. Málek, On planar flows of viscoelastic fluids of Giesekus type, Nonlinearity 35 (2022), no. 12, 6557, https://doi.org/10.1088/1361-6544/ac9a2c. Search in Google Scholar

[4] M. Bulíček, J. Málek, and C. Rodriguez, Global well-posedness for two-dimensional flows of viscoelastic rate-type fluids with stress diffusion, J. Non-Newton. Fluid. Mech. 24 (2022), no. 3, 61, https://doi.org/10.1007/s00021-022-00696-1. Search in Google Scholar

[5] Q. Chen, G. Wu, and Y. Zhang, Optimal large time behavior of the compressible bipolar Navier-Stokes-Poisson system, Commun. Math. Sci. 21 (2023), no. 2, 323–349, https://doi.org/10.4310/CMS.2023.v21.n2.a2. Search in Google Scholar

[6] P. Constantin and M. Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Ration. Mech. An. 206 (2012), no. 3, 725–740, https://doi.org/10.1007/s00205-012-0537-0. Search in Google Scholar

[7] M. Dostalík, V. Pruša, and T. Skrivan, On diffusive variants of some classical viscoelastic rate-type models, In: AIP Conference Proceedings, vol. 2107. AIP Publishing, 2019, https://doi.org/10.1063/1.5109493. Search in Google Scholar

[8] R. Duan, S. Ukai, T. Yang, and H. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Math. Mod. Meth. Appl. S. 17 (2007), no. 5, 737–758, https://doi.org/10.1142/S021820250700208X. Search in Google Scholar

[9] A. W. El-Kareh and L. G. Leal, Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion, J. Non-Newton. Fluid. Mech. 33 (1989), no. 3, 257–287, https://doi.org/10.1016/0377-0257(89)80002-3. Search in Google Scholar

[10] H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility, J. Non-newton. Fluid. Mech. 11 (1982), no. 1–2, 69–109, https://doi.org/10.1016/0377-0257(82)85016-7. Search in Google Scholar

[11] J. Huang, Y. Wang, H. Wen, and R. Zíi, Optimal time-decay estimates for an Oldroyd-B model with zero viscosity, J. Differential Equations 306 (2022), 456–491, https://doi.org/10.1016/j.jde.2021.10.046. Search in Google Scholar

[12] M. Schonbek, Optimal time-decay estimates for an Oldroyd-B model with zero viscosity, Comm. Partial Differential Equations 5 (1980), no. 5, 449–473, https://doi.org/10.1080/03605308608820443. Search in Google Scholar

[13] M. Schonbek, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal, 88 (1985), 209–222, https://doi.org/10.1007/BF00752111. Search in Google Scholar

[14] M. Schonbek and M. Wiegner, On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect.A 126 (1996), no. 3, 677–685, https://doi.org/10.1017/S0308210500022976. Search in Google Scholar

[15] M. Johnson Jr. and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, J. Non-newton. Fluid. Mech. 2 (1977), no. 3, 255–270, https://doi.org/10.1016/0377-0257(77)80003-7. Search in Google Scholar

[16] F. Lin, C. Liu, and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Commun. Pure Appl. Math. 60 (2007), 838–866, https://doi.org/10.1002/cpa.20159. Search in Google Scholar

[17] Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal. 37 (2005), 797–814, https://doi.org/10.1137/040618813. Search in Google Scholar

[18] Z. Luo and Y. Zhang, Optimal large-time behavior of solutions to the full compressible Navier-Stokes equations with large initial data, J. Evol. Equ. 22 (2022), no. 4, 83, https://doi.org/10.1007/s00028-022-00841-3. Search in Google Scholar

[19] J. Málek, V. Prüša, T. Skřivan, and E. Süli, Thermodynamics of viscoelastic rate-type fluids with stress diffusion, Phys. Fluids. 30 (2018), no. 2, 023101, https://doi.org/10.1063/1.5018172. Search in Google Scholar

[20] J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A 200 (1950), no. 1063, 523–541, https://doi.org/10.1098/rspa.1950.0035. Search in Google Scholar

[21] K. R. Rajagopal and A. R. Srinivasa, A thermodynamic frame work for rate type fluid models, J. Non-newton. Fluid. Mech. 88 (2000), no. 3, 207–227, https://doi.org/10.1016/S0377-0257(99)00023-3. Search in Google Scholar

[22] V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in Rxn, Adv. Math. 261 (2014), 274–332, https://doi.org/10.1016/j.aim.2014.04.012. Search in Google Scholar

[23] W. Wang and H. Wen, The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Models Methods Appl. Sci. 30 (2020), no. 1, 139–179, https://doi.org/10.1142/S0218202520500049. Search in Google Scholar

[24] Y. Wang, Optimal time-decay estimates for a diffusive Oldroyd-B model, Z. Angew. Math. Phys. 74 (2023), no. 1, Paper No. 3, 13pp., https://doi.org/10.1007/s00033-022-01902-w. Search in Google Scholar

[25] G. Wu, L. Yao, and Y. Zhang, Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model, Math. Ann. 389 (2024), no. 4, 3379–3415, https://doi.org/10.1007/s00208-023-02732-5. Search in Google Scholar
Received: 2024-06-17
Revised: 2025-06-05
Accepted: 2025-09-12
Published Online: 2025-10-29

© 2025 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
  27. Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
  28. Existence of positive radial solutions of general quasilinear elliptic systems
  29. Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
  30. Sharp viscous shock waves for relaxation model with degeneracy
  31. Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
  32. Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
  33. Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
  34. Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
  35. Singularity for the macroscopic production model with Chaplygin gas
  36. Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
  37. Global dynamics of population-toxicant models with nonlocal dispersals
  38. α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
  39. High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
  40. On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
  41. Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
  42. On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
  43. Remark on the analyticity of the fractional Fokker-Planck equation
  44. Continuous dependence on initial data for damped fourth-order wave equation with strain term
  45. Unilateral problems for quasilinear operators with fractional Riesz gradients
  46. Boundedness of solutions to quasilinear elliptic systems
  47. Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
  48. Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
  49. Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
  50. Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
  51. Shape of extremal functions for weighted Sobolev-type inequalities
  52. One-dimensional boundary blow up problem with a nonlocal term
  53. Doubling measure and regularity to K-quasiminimizers of double-phase energy
  54. General solutions of weakly delayed discrete systems in 3D
  55. Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
  56. Optimal large time behavior of the 3D rate type viscoelastic fluids
  57. Local well-posedness for the two-component Benjamin-Ono equation
  58. Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
  59. Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
  60. Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
  61. On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
  62. Normal forms of piecewise-smooth monodromic systems
  63. Fractional Dirichlet problems with singular and non-locally convective reaction
  64. Sharp forced waves of degenerate diffusion equations in shifting environments
  65. Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
  66. Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
  67. Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
  68. Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
  69. Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
  70. Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
  71. Generalized quasi-linear fractional Wentzell problems
  72. Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
  73. Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
  74. Review Article
  75. Existence and stability of contact discontinuities to piston problems
https://doi.org/10.1515/anona-2025-0117
Keywords for this article
rate type viscoelastic fluids; optimal decay estimate; upper bound; lower bound
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Incompressible limit for the compressible viscoelastic fluids in critical space
  3. Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
  4. Intervals of bifurcation points for semilinear elliptic problems
  5. On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
  6. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
  7. Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
  8. Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
  9. Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
  10. Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
  11. Existence results for non-coercive problems
  12. Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
  13. Geometric characterization of generalized Hajłasz-Sobolev embedding domains
  14. Subharmonic solutions of first-order Hamiltonian systems
  15. Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
  16. Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
  17. Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
  18. Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
  19. Homoclinic solutions in periodic partial difference equations
  20. Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
  21. Properties of minimizers for L2-subcritical Kirchhoff energy functionals
  22. Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
  23. Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
  24. Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
  25. Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
  26. Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
  27. Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
  28. Existence of positive radial solutions of general quasilinear elliptic systems
  29. Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
  30. Sharp viscous shock waves for relaxation model with degeneracy
  31. Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
  32. Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
  33. Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
  34. Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
  35. Singularity for the macroscopic production model with Chaplygin gas
  36. Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
  37. Global dynamics of population-toxicant models with nonlocal dispersals
  38. α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
  39. High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
  40. On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
  41. Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
  42. On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
  43. Remark on the analyticity of the fractional Fokker-Planck equation
  44. Continuous dependence on initial data for damped fourth-order wave equation with strain term
  45. Unilateral problems for quasilinear operators with fractional Riesz gradients
  46. Boundedness of solutions to quasilinear elliptic systems
  47. Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
  48. Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
  49. Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
  50. Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
  51. Shape of extremal functions for weighted Sobolev-type inequalities
  52. One-dimensional boundary blow up problem with a nonlocal term
  53. Doubling measure and regularity to K-quasiminimizers of double-phase energy
  54. General solutions of weakly delayed discrete systems in 3D
  55. Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
  56. Optimal large time behavior of the 3D rate type viscoelastic fluids
  57. Local well-posedness for the two-component Benjamin-Ono equation
  58. Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
  59. Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
  60. Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
  61. On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
  62. Normal forms of piecewise-smooth monodromic systems
  63. Fractional Dirichlet problems with singular and non-locally convective reaction
  64. Sharp forced waves of degenerate diffusion equations in shifting environments
  65. Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
  66. Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
  67. Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
  68. Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
  69. Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
  70. Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
  71. Generalized quasi-linear fractional Wentzell problems
  72. Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
  73. Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
  74. Review Article
  75. Existence and stability of contact discontinuities to piston problems
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 17.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/anona-2025-0117/html?lang=en
Scroll to top button