Global Existence and Convergence Rates for the Strong Solutions in 𝐻² to the 3D Chemotaxis Model

Abstract

We are concerned with a 3D chemotaxis model arising from biology, which is a coupled hyperbolic-parabolic system. We prove the global existence of a strong solution when 𝐻²-norm of the initial perturbation around a constant state is sufficiently small. Moreover, if additionally, 𝐿¹-norm of the initial perturbation is bounded; the optimal convergence rates are also obtained for such a solution. The proofs are obtained by combining spectral analysis with energy methods.

1. Introduction

In this paper, we investigate global existence and optimal convergence rates of strong solutions to the following 3D chemotaxis model: 𝑣_𝑡−∇𝑓(𝑢)=0,𝑥∈ℝ³𝑢,𝑡>0,_𝑡−∇⋅(𝑢𝑣)=𝐷Δ𝑢,𝑥∈ℝ³,𝑡>0, (1) with initial data (𝑣,𝑢)(𝑥,0)=𝑣₀,𝑢₀(𝑥)⟶(0,𝑢),as|𝑥|⟶∞, (2) where 𝑢>0 is a positive constant. The system (1) is one of the models describing the chemotaxis phenomenon in biology and is closely related to the following system: 𝜕𝑝𝑝𝜕𝑡=𝐷∇⋅𝑝∇lnΦ(𝑤),𝜕𝑤,𝜕𝑡=Ψ𝑝,𝑤 (3) which is motivated by biological considerations and numerical computations carried out by Othmer and Stevens in [1] and Levine and Sleeman in [2]. Here, 𝑝(𝑥,𝑡) denotes the particle density and 𝑤(𝑥,𝑡) is the concentration of chemicals. 𝐷>0 is the diffusion rate of particles. The function Φ is commonly referred to as the chemotactic potential and Ψ denotes the chemical kinetics. Depending on the specific modeling goals, the kinetic function Ψ(𝑝,𝑤) has a wide variability. In this paper, we consider a class of nonlinear kinetic functions Ψ(𝑝,𝑤): Ψ𝑝,𝑤=𝛽𝑓𝑝𝑤, (4) where 𝛽 is a positive constant and 𝑓 is a smooth function satisfying 𝑓^′(𝑢)>0, (5) for all 𝑢 under consideration.

In fact, as in [3, 4], let Φ(𝑤)=𝑤^−𝛼 with 𝛼 being a positive constant and Ψ(𝑝,𝑤) is defined in (4); the system (3) can be rewritten in the following form: 𝑝_𝑡=𝐷Δ𝑝+𝐷𝛼∇⋅𝑝∇𝑤𝑤,𝑤_𝑡=𝛽𝑓𝑝𝑤. (6) Furthermore, by setting 𝑞=∇(ln𝑤)=∇𝑤𝑤, (7) we can rewrite the system (6) as 𝑝_𝑡,𝑞=𝐷Δ𝑝+𝐷𝛼∇⋅𝑝𝑞_𝑡=𝛽∇𝑓𝑝. (8) Finally, for positive constants 𝐴, 𝐵, and 𝑐₁ to be determined below, let 𝜏=𝐴𝑡, 𝜉=𝐵𝑥, 𝑢=𝑝, and 𝑣=𝑐₁𝑞; then the system (8) becomes 𝑣_𝜏=𝛽𝐵𝑐₁𝐴∇_𝜉𝑓(𝑢),𝑢_𝜏=𝐷𝐵²𝐴Δ_𝜉𝑢+𝐷𝛼𝐵𝐴𝑐₁∇_𝜉⋅(𝑢𝑣). (9) If we choose 𝛽𝐵𝑐₁𝐴𝐵=1,²𝐴=1,𝐷𝛼𝐵𝐴𝑐₁=1, (10) that is, 𝐴=𝐷𝛼𝛽>0,𝐵=𝐷𝛼𝛽>0,𝑐₁=𝐷𝛼𝛽>0, (11) then it is easy to see that 𝑢 and 𝑣 satisfy 𝑣_𝜏−∇_𝜉𝑓(𝑢)𝑢=0,_𝜏−∇_𝜉⋅(𝑢𝑣)=𝐷Δ_𝜉𝑢. (12) If we replace the variables (𝜏,𝜉) by (𝑥,𝑡), (12) is exactly (1).

To go directly to the theme of this paper, we now only review some former results which are closely related. For the one-dimensional version of the chemotaxis model (1), the existence and asymptotic behavior of smooth solutions have been studied by several authors. When the kinetic function is linear, that is, 𝑓(𝑢)=𝜆𝑢−𝜇 with 𝜆(>0) and 𝜇(≥0) being given constants, the corresponding system reads as follows: 𝑣_𝑡−𝑢_𝑥𝑢=0,_𝑡−(𝑢𝑣)_𝑥=𝑢_𝑥𝑥. (13) The initial boundary value problem and Cauchy problem for the system (13) were considered by [3, 5] and [4], respectively. In [3], they considered the initial boundary value problem for the system (13). When ‖𝑢₀−1‖²_𝐻2+‖𝑣₀‖²_𝐻2 is sufficiently small, they proved the global existence of smooth solutions to the system (13). The authors in [5] generalized the results of [3] to the arbitrarily large initial data case. In [4], the authors obtained the global existence of smooth solutions to the Cauchy problem for the system (13) with large initial data. Recently, the authors in [6–8] extended the results of [3–5] to the nonlinear kinetic function case, respectively. For high dimensions, the global well-posedness of smooth small solution to (1) with 𝑓(𝑢)=𝑢 was investigated in [5, 9] for initial-boundary value problem and Cauchy problem, respectively. For other related results, such as nonlinear stability of waves in one dimension and so on, please refer to [6–8, 10–31] and references therein.

However, to our knowledge, so far there is no result on the optimal convergence rates of the strong solutions to the Cauchy problem (1)-(2). The main motivation of this paper is to give a positive answer to this question. In particular, we prove the global existence of a strong solution when 𝐻²-norm of the initial perturbation around a constant state is sufficiently small. Moreover, if in addition, 𝐿¹-norm of the initial perturbation is bounded, the optimal convergence rates are also obtained for such a solution. The proofs are based on energy methods and spectral analysis which have been developed in [32–35] and references therein.

Before stating our main results, we explain the notations and conventions used throughout this paper. We denote positive constants by 𝐶. Moreover, the character “𝐶” may differ in different places. 𝐿^𝑝=𝐿^𝑝(ℝ³)  (1≤𝑝≤∞) denotes the usual Lebesgue space with the norm 𝑔_𝐿𝑝=_ℝ3𝑔(𝑥)^𝑝𝑑𝑥^1/𝑝𝑔,1≤𝑝<∞,_𝐿∞=sup_ℝ3𝑔(𝑥). (14)𝐻^𝑙(ℝ³)  (𝑙≥0) denotes the usual 𝑙th-order Sobolev space with the norm 𝑔_𝑙=^𝑙_𝑗=0∇^𝑗𝑔^21/2, (15) where ‖⋅‖=‖⋅‖₀=‖⋅‖_𝐿².

Now, we are ready to state our main results.

Theorem 1. Assume that ∇×𝑣₀=0 and ‖(𝑣₀,𝑢₀−𝑢)‖₂ is sufficiently small; then there exists a unique globally strong solution (𝑣,𝑢) of the Cauchy problem (1)-(2) such that for any 𝑡∈[0,∞), satisfying 𝑣∈𝐶⁰0,∞;𝐻²ℝ³∩𝐶¹0,∞;𝐻¹ℝ³,𝑢−𝑢∈𝐶⁰0,∞;𝐻²ℝ³∩𝐶¹0,∞;𝐿²ℝ³,‖(𝑣,𝑢−𝑢)(𝑡)‖²₂+^𝑡₀‖∇𝑣(𝜏)‖²₁+‖∇𝑢(𝜏)‖²₂𝑑𝜏≤𝐶₁𝑣₀,𝑢₀−𝑢²₂, (16) where 𝐶₁ is a positive constant independent of 𝑡.
Moreover, if, in addition, ‖(𝑣₀,𝑢₀−𝑢)‖_𝐿1 is bounded, then there is a positive constant 𝐶₂ independent of 𝑡 such that for any 𝑡≥0, the solution (𝑣,𝑢) has the following decay properties: ‖∇(𝑣,𝑢)(𝑡)‖₁≤𝐶₂(1+𝑡)^−5/4,‖(𝑣,𝑢−𝑢)(𝑡)‖_𝐿^∞≤𝐶₂(1+𝑡)^−5/4,‖(𝑣,𝑢−𝑢)(𝑡)‖_𝐿^𝑞≤𝐶₂(1+𝑡)^{(−3/2)(1−1/𝑞)},2≤𝑞≤6,𝜕_𝑡(𝑣,𝑢)(𝑡)≤𝐶₂(1+𝑡)^−5/4. (17)(18)(19)(20)

Remark 2. As compared to the classic results in [9, 17, 30], where smallness conditions on 𝐻³-norm of the initial data were proposed, we are able to prove the global existence and convergence rates for the strong solutions to the Cauchy problem under only that 𝐻²-norm of the initial data is sufficiently small.

Remark 3. Similar ideas can be applied to study the initial boundary value problem. This will be reported in a forthcoming paper.

Remark 4. Here and in what follows, (𝑣,𝑢) denotes the vector (𝑣₁,𝑣₂,𝑣₃,𝑢) with 𝑣=(𝑣₁,𝑣₂,𝑣₃) and 𝑣∈𝐶⁰(0,∞;𝐻²(ℝ³))∩𝐶¹(0,∞;𝐻¹(ℝ³)) represents each component 𝑣_𝑖(𝑖=1,2,3) of the vector 𝑣 belonging to the function space 𝐶⁰(0,∞;𝐻²(ℝ³))∩𝐶¹(0,∞;𝐻¹(ℝ³)).

The proofs of Theorem 1 are based on energy methods and spectral analysis. The key point for the proof of global existence is to obtain some a priori estimates independent of 𝑡. Then, we can combine spectral analysis with the uniform a priori estimates to prove our desired decay estimates (17)–(20).

The rest of this paper is organized as follows. In Section 2, we reformulate the problem. In Section 3, we give some elementary facts on the decay-in-time estimates on (𝑣,𝑢) for the linearized system. Sections 4 and 5 are dedicated to prove Theorem 1.

2. Reformulated System

In this section, we will first reformulate the problem. Set 𝜆=𝑓^′(𝑢)𝑢,𝜆₁=𝑢𝑓^′(𝑢),𝜆₂=𝑢𝑓^′(𝑢). (21)

Taking change of variables (𝑣,𝑢)→(𝜆𝑣,𝑢+𝑢) and linearizing the system around (0,𝑢), we can reformulate the Cauchy problem (1)-(2) as 𝑣_𝑡−𝜆₁∇𝑢=𝐹₁,𝑢_𝑡−𝜆₁∇⋅𝑣−𝐷Δ𝑢=𝐹₂,(𝑣,𝑢)(𝑥,0)=𝑣₀,𝑢₀(𝑥)⟶(0,0)as|𝑥|⟶∞, (22) where 𝐹₁=𝜆₂𝑓^′(𝑢+𝑢)−𝑓^′(𝑢)𝐹∇𝑢,₂=𝜆∇⋅(𝑢𝑣). (23) Here and in the sequel, for the notational simplicity, we still denote the reformulated variables by (𝑣,𝑢).

To prove the global existence of a solution to (22), we will combine the local existence result together with a priori estimates. To begin with, we state the following local existence, the proof of which can be found in [4, 36–38].

Proposition 5 (local existence). Assume that (𝑣₀,𝑢₀)∈𝐻²(ℝ³). Then, there exists a sufficiently small positive constant 𝑡₀ depending only on ‖(𝑣₀,𝑢₀)‖₂ such that the Cauchy problem (22) admits a unique solution (𝑣,𝑢)(𝑥,𝑡)∈𝐶([0,𝑡₀],𝐻²(ℝ³)) satisfying sup_{𝑡∈0,𝑡0}‖(𝑣,𝑢)(⋅,𝑡)‖₂≤2𝑣₀,𝑢₀₂. (24)

Proposition 6 (a priori estimate). Let ∇×𝑣₀=0 and (𝑣₀,𝑢₀)∈𝐻²(ℝ³). Assume that the Cauchy problem (22) has a solution (𝑣,𝑢)(𝑥,𝑡) on ℝ³×[0,𝑇] for some 𝑇>0 in the same function class as in Proposition 5. Then, there exist a small constant 𝛿>0 and a constant 𝐶₃, which are independent of T, such that if sup_{0≤𝑡≤𝑇}‖(𝑣,𝑢)(𝑡)‖₂≤𝛿, (25) then for any 𝑡∈[0,𝑇], it holds that ‖(𝑣,𝑢)(𝑡)‖²₂+^𝑡₀‖∇𝑣(𝜏)‖²₁+‖∇𝑢(𝜏)‖²₂𝑑𝜏≤𝐶₃𝑣₀,𝑢₀−𝑢²₂. (26) Moreover, if, in addition, ‖(𝑣₀,𝑢₀)‖_𝐿1 is bounded, then there is a positive constant 𝐶₄ independent of 𝑇 such that for any 𝑡∈[0,𝑇], the solution (𝑣,𝑢) has the following decay properties: ‖∇(𝑣,𝑢)(𝑡)‖₁≤𝐶₄(1+𝑡)^−5/4,‖(𝑣,𝑢)(𝑡)‖_𝐿^∞≤𝐶₄(1+𝑡)^−5/4,‖(𝑣,𝑢)(𝑡)‖_𝐿^𝑞≤𝐶₄(1+𝑡)^{(−3/2)(1−1/𝑞)},2≤𝑞≤6,𝜕_𝑡(𝑣,𝑢)(𝑡)≤𝐶₄(1+𝑡)^−5/4. (27)(28)(29)(30)

Theorem 1 follows from Propositions 5 and 6 and standard continuity arguments. The proof of Proposition 6 will be given in Sections 4 and 5.

3. Linear Decay Estimates

In this section, we consider the Cauchy problem for the linearized equations corresponding to (22)₁-(22)₂: 𝑣_𝑡−𝜆₁𝑢∇𝑢=0,_𝑡−𝜆₁()=𝑣∇⋅𝑣−𝐷Δ𝑢=0,𝑣,𝑢)(𝑥,0₀,𝑢₀(𝑥)⟶(0,0)as|𝑥|⟶∞. (31) The solutions (𝑣,𝑢) of the linear system (31) can be expressed as (𝑣,𝑢)^𝑡(𝑡)=𝐺(𝑡)∗𝑣₀,𝑢₀^𝑡,𝑡≥0. (32) Here, 𝐺(𝑡)∶=𝐺(𝑥,𝑡) is the Green's matrix for the system (31).

To derive the large time behavior of the solutions, we first give an explicit expression for the Fourier transform 𝐺(𝜉,𝑡) of Green's matrix 𝐺(𝑥,𝑡).

Lemma 7. The Fourier transform 𝐺 of Green's matrix for the linear system (31) is given by 𝐺()=𝜆𝜉,𝑡⁺𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋𝐼𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉^𝑡𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉𝜆₊𝑒^𝜆+𝑡−𝜆₋𝑒^𝜆−𝑡𝜆₊−𝜆₋, (33) where 𝜆_±𝜉𝐷=−2𝜉²±12𝐷²𝜉⁴−4𝜆²₁𝜉². (34) The representation above holds for |𝜉|≠±2𝜆₁/𝐷.

Proof. The proof is in spirit of Hoff and Zumbrun [39, 40]. Applying the Fourier transform to the system (31), we have 𝑣_𝑡̂=𝑖𝜆₁𝜉𝑢,𝑢_𝑡=−𝐷𝜉²𝑢+𝑖𝜆₁𝜉⋅𝑣.̂ (35)(36) From (35)-(36), we obtain the following initial value problem for 𝑢(𝜉,𝑡): 𝑢_𝑡𝑡+𝐷𝜉²𝑢_𝑡+𝜆²₁𝜉²𝑢()=𝑢𝑢=0,𝜉,0₀(𝜉),𝑢_𝑡(𝜉,0)=−𝐷𝜉²𝑢₀(𝜉)+𝑖𝜆₁𝜉⋅𝑣₀̂(𝜉). (37) The 𝜆_±(𝜉) defined in (34) are exactly the eigenvalues of the ODE (37); hence for 𝜆₊≠𝜆₋, 𝑢(𝜉,𝑡)=𝐴(𝜉)𝑒^{𝜆−(𝜉)𝑡}+𝐵(𝜉)𝑒^{𝜆+(𝜉)𝑡}. (38) The initial conditions in (37) give 𝐴=−𝜆₋𝑢₀−𝑖𝜆₁𝜉⋅𝑣₀̂𝜆₊−𝜆₋𝜆,𝐵=⁺𝑢₀+𝑖𝜆₁𝜉⋅𝑣₀̂𝜆₊−𝜆₋. (39) Substituting (39) into (38), by a straightforward computation, we obtain 𝑢()=𝜆𝜉,𝑡⁺𝑒^𝜆+𝑡−𝜆₋𝑒^𝜆−𝑡𝜆₊−𝜆₋𝑢₀+𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉⋅𝑣₀̂. (40) To compute 𝑣̂, we substitute (40) into (35) to obtain 𝑣(̂)=𝜆𝜉,𝑡⁺𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋𝑣₀̂+𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉𝑢₀. (41)
Hence, (40)-(41) give (33) and the proof of Lemma 7 is completed.

To derive the long-time decay rate of solutions in 𝐿²-framework, we need to verify the approximation of Green's function 𝐺(𝜉,𝑡) for both lower frequency and high frequency. In terms of the definition of the eigenvalues (34), we are able to obtain that it holds for |𝜉|≪1 that 𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋∼sin(𝑏𝑡)𝑏𝑒^{(−𝐷/2)|𝜉|2𝑡},𝜉𝜆≪1,₊𝑒^𝜆+𝑡−𝜆₋𝑒^𝜆−𝑡𝜆₊−𝜆₋∼()+𝐷cos𝑏𝑡2sin(𝑏𝑡)𝑏𝜉²𝑒^{(−𝐷/2)|𝜉|2𝑡},𝜉𝜆≪1,₊𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋∼()−𝐷cos𝑏𝑡2sin(𝑏𝑡)𝑏𝜉²𝑒^{(−𝐷/2)|𝜉|2𝑡},𝜉≪1, (42) where 1𝑏=24𝜆²₁𝜉²−𝐷²𝜉⁴∼𝑂𝜉,𝜉≪1. (43)

Next, we deal with the high frequency |𝜉|≫1. From the definitions of the eigenvalues (34), we can analyze the eigenvalues for |𝜉|≫1. Indeed, we have the leading orders of the eigenvalues for |𝜉|≫1 as 𝜆₊𝜆∼−²¹𝐷+𝑂𝜉⁻¹,𝜆₋∼−𝐷𝜉²+𝜆²₁𝐷+𝑂𝜉⁻¹. (44) This approximation gives the leading order terms of the elements of Green's function as follows: 𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋=1𝜆₊−𝜆₋𝑒^𝜆+𝑡×1−𝑒^{(𝜆−−𝜆+)𝑡}∼𝑂(1)𝑒^−𝜗𝑡,𝜉𝜆≥𝑅,₊𝑒^𝜆+𝑡−𝜆₋𝑒^𝜆−𝑡𝜆₊−𝜆₋=𝜆₊𝜆₊−𝜆₋𝑒^𝜆+𝑡1−𝑒^{(𝜆−−𝜆+)𝑡}+𝑒^𝜆−𝑡∼𝑂(1)𝑒^−𝜗𝑡,𝜉𝜆≥𝑅,₊𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋=12𝑒^𝜆+𝑡1+𝑒^{(𝜆−−𝜆+)𝑡}+𝐷𝜉²𝜆₊−𝜆₋𝑒^𝜆+𝑡×1−𝑒^{(𝜆−−𝜆+)𝑡}∼𝑂(1)𝑒^−𝜗𝑡,𝜉≥𝑅, (45) where 𝜗 and 𝑅 are some positive constants.

With the help of the formula (33) for the Fourier transform 𝐺 of Green's matrix and the asymptotical analysis on its elements, we are able to establish the 𝐿² time decay rate. Indeed, we have the 𝐿² time decay rate of the global solution to the Cauchy problem for the linear problem (31) as follows.

Lemma 8. Assume that (𝑣,𝑢) is the solution of the linear problem (31) with the initial data (𝑣₀,𝑢₀)∈𝐻²(ℝ³)∩𝐿¹(ℝ³); then ∇^𝑘(𝑣,𝑢)(𝑡)≤𝐶(1+𝑡)^{−3/4−𝑘/2}×𝑣₀,𝑢_0𝐿1+∇^𝑘𝑣₀,𝑢₀, (46) for 0≤𝑘≤2.

Proof. From (33), (42), and (45), we have 𝑣(̂)=𝜆𝜉,𝑡⁺𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋𝑣₀̂+𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉𝑢₀∼𝑂(1)𝑒^{−(𝐷/2)|𝜉|2𝑡}𝑣₀̂+𝑢₀,𝜉𝑂(1)𝑒≪1,_−𝜗𝑡𝑣₀̂+𝑢₀,𝜉≫1, (47) with 𝜗>0 being a constant here and below, and 𝑢()=𝜆𝜉,𝑡⁺𝑒^𝜆+𝑡−𝜆₋𝑒^𝜆−𝑡𝜆₊−𝜆₋𝑢₀+𝑖𝜆₁𝑒^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋𝜉⋅𝑣₀̂∼𝑂(1)𝑒^{−(𝐷/2)|𝜉|2𝑡}𝑣₀̂+𝑢₀,𝜉𝑂(1)𝑒≪1,_−𝜗𝑡𝑣₀̂+𝑢₀,𝜉≫1, (48) where here and below 𝑅>0 denotes a small but fixed constant.
Therefore, to prove (46), we only need to deal with the case for 𝑣 since the case for 𝑢 can be proved similarly. By virtue of the pointwise estimate (47), the Parseval theorem, and Hausdorff-Young's inequality, for 0≤𝑘≤3, we have ∇^𝑘𝑣(𝑡)²=_{|𝜉|≤𝑅}𝜉^2𝑘𝑣(̂𝜉,𝑡)²+𝑑𝜉_{|𝜉|≥𝑅}𝜉^2𝑘𝑣(̂𝜉,𝑡)²𝑑𝜉≤𝐶_{|𝜉|≤𝑅}𝑒^{−𝐷|𝜉|2𝑡}𝜉^2𝑘𝑣₀̂(𝜉)²+𝑢₀(𝜉)²𝑑𝜉+𝐶_{|𝜉|≥𝑅}𝑒^−𝜗𝑡𝜉^2𝑘𝑣₀̂(𝜉)²+𝑢₀(𝜉)²()𝑑𝜉≤𝐶1+𝑡^{−3/2−𝑘}𝑣₀̂,𝑢₀²_𝐿∞+𝐶𝑒^−𝜗𝑡∇^𝑘𝑣₀,𝑢₀²≤𝐶(1+𝑡)^{−3/2−𝑘}𝑣₀,𝑢₀²_𝐿1+∇^𝑘𝑣₀,𝑢₀². (49) Thus, (49) gives (46) immediately, and the proof of Lemma 8 is completed.

4. A Priori Estimates

Throughout this section and the next section, we assume that all conditions of Proposition 6 are satisfied. Moreover, we make a priori assumption: sup_{0≤𝑡≤𝑇}‖(𝑣,𝑢)(𝑡)‖₂≤𝛿, (50) where 𝛿 is a sufficiently small positive constant.

In the following, a series of lemmas on the energy estimates are given. Firstly, we will obtain the lower-order energy estimate for (𝑣,𝑢) in the following lemma.

Lemma 9. There exists a positive constant 𝐷₁, which is sufficiently large and independent of 𝛿, such that 𝑑𝐷𝑑𝑡¹‖(𝑣,𝑢)(𝑡)‖²+⟨∇𝑢,𝑣⟩(𝑡)+𝐶‖∇(𝑣,𝑢)(𝑡)‖²≤𝐶∇²𝑢(𝑡)², (51) for any 0≤𝑡≤𝑇.

Proof. Multiplying (22)₁-(22)₂ by 𝑣,𝑢, respectively, then summing up and integrating the resultant equation, we obtain 12𝑑𝑑𝑡‖(𝑣,𝑢)‖²+𝐷‖∇𝑢‖²=𝑣,𝐹₁+𝑢,𝐹₂. (52) Applying mean value theorem, Hölder’s inequality, and Sobolev’s inequality, it is clear that the two terms on the right-hand side of (52) can be estimated as follows: 𝑣,𝐹₁+𝑢,𝐹₂≤𝐶_ℝ3||‖𝑣‖𝑣𝑢∇𝑢𝑑𝑥≤𝐶_𝐿³‖𝑢‖_𝐿⁶‖‖‖𝑣‖∇𝑢≤𝐶₁‖∇𝑢‖²≤𝐶𝛿‖∇𝑢‖². (53) Combining (52) with (53) and using the fact that 𝛿 is sufficiently small, we have 𝑑𝑑𝑡‖(𝑣,𝑢)‖²+𝐶‖∇𝑢‖²≤0. (54)
Next, we will estimate ‖∇𝑣‖². Multiplying (22)₂ by ∇⋅𝑣 and integrating the resulting equation over ℝ³, we have 𝜆₁‖∇⋅𝑣‖²=𝑢_𝑡,∇⋅𝑣+⟨−𝐷Δ𝑢,∇⋅𝑣⟩+𝐹₂,∇⋅𝑣, (55) where from (22)₁, the first term on the right-hand side can be written as 𝑢_𝑡=𝑑,∇⋅𝑣⟨⟩−𝑑𝑡𝑢,∇⋅𝑣𝑢,∇⋅𝑣_𝑡𝑑=−⟨⟩+𝑑𝑡∇𝑢,𝑣∇𝑢,𝑣_𝑡𝑑=−⟨⟩+𝑑𝑡∇𝑢,𝑣∇𝑢,𝜆₁∇𝑢+𝐹₁. (56) Then, it follows from (55)-(56), (23), Hölder’s inequality, and Young’s inequality that 𝜆₁‖∇⋅𝑣‖²+𝑑⟨⟩=𝑑𝑡∇𝑢,𝑣∇𝑢,𝜆₁∇𝑢+𝐹₁+⟨−𝐷Δ𝑢,∇⋅𝑣⟩+𝐹₂∇,∇⋅𝑣≤𝐶²𝑢²+𝜆₁4‖∇⋅𝑣‖²+𝐶𝛿‖∇⋅𝑣‖²+𝐶‖∇𝑢‖². (57) This together with the fact that 𝛿 is sufficiently small implies that 𝑑⟨⟩+𝜆𝑑𝑡∇𝑢,𝑣¹2‖∇⋅𝑣‖²≤𝐶‖∇𝑢‖²+∇²𝑢². (58) Taking the curl for (22)₁ and noting that ∇×𝑣₀=0, we have ∇×𝑣=0. (59) Since Δ𝑣=∇(∇⋅𝑣)−∇×(∇×𝑣), we have from (59) that Δ𝑣=∇(∇⋅𝑣),‖∇𝑣‖≤𝐶‖∇⋅𝑣‖. (60) Combining (58) with (60) yields 𝑑⟨⟩‖‖𝑑𝑡∇𝑢,𝑣+𝐶∇𝑣²≤𝐶‖∇𝑢‖²+∇²𝑢². (61)
Finally, multiplying (54) by 𝐷₁ which is suitably large and adding it to (61), one has (51) since 𝛿>0 is sufficiently small. This completes the proof of Lemma 9.

The following lemma is concerned with the higher-order energy estimate on (𝑣,𝑢).

Lemma 10. There exists a positive constant 𝐷₂, which is sufficiently large and independent of 𝛿, such that 𝑑𝐷𝑑𝑡²‖∇(𝑣,𝑢)(𝑡)‖²₁+⟨𝑡)∇Δ𝑢,∇⋅𝑣⟩(+𝐶²𝑣(𝑡)²+‖∇²𝑢(𝑡)‖²₁≤𝐶𝛿‖∇(𝑣,𝑢)(𝑡)‖², (62) for any 0≤𝑡≤𝑇.

Proof. Applying ∇ to (22)₁-(22)₂ and multiplying them by ∇𝑣, ∇𝑢, respectively, and then integrating them over ℝ³, we obtain 12𝑑‖∇(𝑑𝑡𝑣,𝑢)‖²+𝐷∇²𝑢²=∇𝑣,∇𝐹₁+∇𝑢,∇𝐹₂∶=𝐽₁+𝐽₂. (63) Next, we estimate the terms 𝐽₁ and 𝐽₂ one by one. To begin with, by using (23), (50), Hölder’s inequality, Cauchy’s inequality, and Sobolev’s inequality, we can estimate the term 𝐽₁ as follows: 𝐽₁≤𝐶‖∇𝑢‖‖∇𝑢‖_𝐿⁶‖∇𝑣‖_𝐿³+‖𝑢‖_𝐿^∞∇²𝑢‖∇‖∇𝑣‖≤𝐶∇𝑢‖²𝑢‖∇𝑣‖₁+‖𝑢‖₂∇²𝑢‖∇‖∇𝑣≤𝐶𝛿²𝑢²+‖∇(𝑣,𝑢)‖². (64) Using similar arguments, we also have the following estimate for the term 𝐽₂: 𝐽₂≤𝐶𝛿∇²𝑢²+‖∇(𝑣,𝑢)‖². (65) Substituting (64) and (65) into (63) and noting that 𝛿 is sufficiently small, we have 𝑑‖∇(𝑑𝑡𝑣,𝑢)‖²+𝐶∇²𝑢²≤𝐶𝛿‖∇(𝑣,𝑢)‖². (66) For higher-order derivatives of (𝑣,𝑢), we can apply the similar arguments used in obtaining (66) to get 𝑑∇𝑑𝑡²(𝑣,𝑢)²+𝐶∇³𝑢²≤𝐶𝛿∇²(𝑣,𝑢)². (67) Combining (66) and (67) and using the fact that 𝛿 is sufficiently small, we have 𝑑‖∇(𝑑𝑡𝑣,𝑢)‖²₁+𝐶∇²𝑢²₁≤𝐶𝛿‖∇(𝑣,𝑢)‖²+∇²𝑣². (68)
Next, we estimate ‖∇²𝑣‖². To begin with, applying ∇ to (22)₂ and then multiplying by ∇(∇⋅𝑣), we have from Cauchy’s inequality that 𝜆₁2‖∇(∇⋅𝑣)‖²≤∇𝑢_𝑡,∇(∇⋅𝑣)+𝐶∇³𝑢²+𝐶∇𝐹₂². (69) By integrating by parts several times, we estimate the first term on the right-hand side of (69) as follows: ∇𝑢_𝑡(𝑑,∇∇⋅𝑣)=−⟨⟩+𝑑𝑡Δ𝑢,∇⋅𝑣Δ𝑢,∇⋅𝑣_𝑡𝑑=−⟨⟩+𝑑𝑡Δ𝑢,∇⋅𝑣Δ𝑢,𝜆₁Δ𝑢+∇⋅𝐹₁𝑑≤−⟨⟩∇𝑑𝑡Δ𝑢,∇⋅𝑣+𝐶²𝑢². (70) From (23), (50), Hölder’s inequality, Cauchy’s inequality, and Sobolev’s inequality, we have ∇𝐹₂≤𝐶‖∇𝑣‖_𝐿³‖∇𝑢‖_𝐿⁶+‖𝑣‖_𝐿^∞∇²𝑢+‖𝑢‖_𝐿^∞∇²𝑣≤𝐶𝛿∇²(𝑣,𝑢). (71) Substituting (70) and (71) into (69) yields 𝑑⟨𝑡)‖∇(𝑑𝑡Δ𝑢,∇⋅𝑣⟩(+𝐶∇⋅𝑣)‖²≤𝐶∇²𝑢²₁+𝛿∇²𝑣². (72) Due to (60), we have ∇²𝑣²≤𝐶‖∇(∇⋅𝑣)‖². (73) Substituting (73) into (72) and using the fact that 𝛿 is sufficiently small, we obtain 𝑑⟨𝑡)∇𝑑𝑡Δ𝑢,∇⋅𝑣⟩(+𝐶²𝑣²≤𝐶∇²𝑢²₁. (74)
Finally, since 𝛿 is sufficiently small, multiplying (68) by a suitably large positive constant 𝐷₂ and adding it to (74) give (62), and this completes the proof of Lemma 10.

5. Global Existence and Convergence Rate

In this section, we devote ourselves to prove Proposition 6. To begin with, we give the following lemma which is concerned with the a priori decay-in-time estimates on (𝑣,𝑢).

Lemma 11. Let 𝐾₀=‖(𝑣₀,𝑢₀)‖_{𝐻2∩𝐿1}, then ‖∇(𝑣,𝑢)(𝑡)‖≤𝐶𝐾₀(1+𝑡)^−5/4+𝐶𝛿^𝑡₀(1+𝑡−𝜏)^−5/4‖∇(𝑣,𝑢)(𝜏)‖₁𝑑𝜏, (75) for any 0≤𝑡≤𝑇.

Proof. From Duhamel's principle and Lemma 8, we have ‖∇(𝑣,𝑢)(𝑡)‖≤𝐶𝐾₀(1+𝑡)^−5/4+𝐶𝛿^𝑡₀(1+𝑡−𝜏)^−5/4×𝐹₁,𝐹₂(𝜏)_{𝐻1∩𝐿1}𝑑𝜏. (76) From (23), (50), Hölder’s inequality, and Sobolev’s inequality, we have 𝐹₁,𝐹₂(𝑡)_{𝐻1∩𝐿1}≤𝐶𝛿‖∇(𝑣,𝑢)(𝑡)‖₁. (77) Therefore, (75) follows from (76) and (77) immediately, and this completes the proof of Lemma 11.

Now, we are in position to prove Proposition 6.

5.1. Proof of Proposition 6

We will do it in two steps.

Step  1. Since 𝛿>0 is suitably small, from Lemmas 9 and 10, we can choose a suitably large positive constant 𝐷₃ such that 𝑑𝐷𝑑𝑡³‖(𝑣,𝑢)(𝑡)‖²₂+⟨⟩+⟨⟩‖(𝑡∇𝑢,∇⋅𝑣Δ𝑢,∇⋅𝑣+𝐶∇𝑣)‖²₂+‖∇𝑢(𝑡)‖²₂≤0, (78) for any 0≤𝑡≤𝑇. Notice that the expression under 𝑑/𝑑𝑡 in (78) is equivalent to ‖(𝑣,𝑢)(𝑡)‖²₂. Hence, by integrating (78) directly in time, we obtain (26).

Step  2. Define the temporal energy functional 𝔼(𝑡)=𝐷₂‖∇(𝑣,𝑢)(𝑡)‖²₁+⟨Δ𝑢,∇⋅𝑣⟩(𝑡), (79) for any 0≤𝑡≤𝑇, where it should be mentioned that 𝔼(𝑡) is equivalent to ‖∇(𝑣,𝑢)‖²₁ since 𝐷₂ is sufficiently large.

By virtue of Lemma 10, we have 𝑑𝔼(𝑡)∇𝑑𝑡+𝐶²(𝑣,𝑢)(𝑡)²≤𝐶𝛿‖∇(𝑣,𝑢)(𝑡)‖². (80) Adding ‖∇(𝑣,𝑢)(𝑡)‖² to both sides of (80) yields 𝑑𝔼(𝑡)𝑑𝑡+𝐷₄𝔼(𝑡)≤𝐶‖∇(𝑣,𝑢)(𝑡)‖², (81) where 𝐷₄ is a positive constant independent of 𝛿. As [7, 17], we define 𝔾(𝑡)=sup_{0≤𝜏≤𝑡}(1+𝜏)^5/2𝔼(𝜏). (82) Notice that ‖∇(𝑣,𝑢)(𝜏)‖₁≤𝐶𝔼(𝜏)≤𝐶(1+𝜏)^−5/4𝔾(𝑡),0≤𝜏≤𝑡≤𝑇. (83) Then, from (75) and (83), we have ‖∇(𝑣,𝑢)(𝑡)‖≤𝐶𝐾₀(1+𝑡)^−5/4+^𝑡₀(1+𝑡−𝜏)^−5/4(1+𝜏)^−5/4𝔾(𝑡)()𝑑𝜏≤𝐶1+𝑡^−5/4𝐾₀+𝛿𝔾(𝑡), (84) which together with Gronwall's inequality and (81) implies 𝔼(𝑡)≤𝔼(0)𝑒^−𝐷4𝑡+𝐶^𝑡₀𝑒^{−𝐷4(𝑡−𝜏)}‖∇(𝑣,𝑢)(𝜏)‖²(0)𝑒𝑑𝜏≤𝔼^−𝐷4𝑡+𝐶𝐾²₀+𝛿²𝔾(𝑡)×^𝑡₀𝑒^{−𝐷4(𝑡−𝜏)}(1+𝜏)^−5/2()𝑑𝜏≤𝐶1+𝑡^−5/2𝔼(0)+𝐾²₀+𝛿²𝔾(𝑡). (85) Since 𝔾 is nondecreasing, we have from (85) that 𝔾(𝑡)≤𝐶𝔼(0)+𝐾²₀+𝛿²𝔾(𝑡), (86) for any 0≤𝑡≤𝑇, which together with the fact that 𝛿>0 is small enough implying that 𝔾(𝑡)≤𝐶𝔼(0)+𝐾²₀≤𝐶𝐾²₀. (87) Thus, (82) and (87) yield ‖∇(𝑣,𝑢)(𝑡)‖₁≤𝐶𝐾₀(1+𝑡)^−5/4,0≤𝑡≤𝑇, (88) which together with Sobolev’s inequality gives ‖(𝑣,𝑢)(𝑡)‖_𝐿^∞≤𝐶‖∇(𝑣,𝑢)(𝑡)‖₁≤𝐶𝐾₀(1+𝑡)^−5/4,0≤𝑡≤𝑇. (89) Therefore, we have proved (27) and (28).

Next, we turn to prove (29). To begin with, by using Sobolev's inequality and (27), we have ‖(𝑣,𝑢)(𝑡)‖_𝐿⁶≤𝐶‖∇(𝑣,𝑢)(𝑡)‖≤𝐶𝐾₀(1+𝑡)^−5/4. (90) On the other hand, applying Duhamel's principle and Lemma 8 again, we have from (26) and (28) that ‖(𝑣,𝑢)(𝑡)‖≤𝐶𝐾₀(1+𝑡)^−3/4+𝐶^𝑡₀(1+𝑡−𝜏)^−3/4𝐹₁,𝐹₂(𝜏)_{𝐿2∩𝐿1}𝑑𝜏≤𝐶𝐾₀(1+𝑡)^−3/4+𝐶𝛿^𝑡₀(1+𝑡−𝜏)^−3/4‖∇(𝑣,𝑢)(𝜏)‖₁𝑑𝜏≤𝐶𝐾₀(1+𝑡)^−3/4+𝐶𝐾₀𝛿^𝑡₀(1+𝑡−𝜏)^−3/4(1+𝜏)^−5/4𝑑𝜏≤𝐶𝐾₀(1+𝑡)^−3/4, (91) for any 0≤𝑡≤𝑇.

Hence, by the interpolation, it holds that for any 2≤𝑞≤6, ‖(𝑣,𝑢)(𝑡)‖_𝐿^𝑞≤‖(𝑣,𝑢)(𝑡)‖^𝜃‖(𝑣,𝑢)(𝑡)‖_𝐿^1−𝜃₆≤𝐶₀(1+𝑡)^{(−3/2)(1−1/𝑞)}, (92) for any 0≤𝑡≤𝑇, where 𝜃=(6−𝑞)/2𝑞. Thus, we have proved (29). Therefore, (30) follows from (22)-(23) and (27)–(29) and this completes the proof of Proposition 6.

Acknowledgments

The second author was partially supported by the National Natural Science Foundation of China nos. 11301172 and 11226170, China Postdoctoral Science Foundation funded project no. 2012M511640, and Hunan Provincial Natural Science Foundation of China no. 13JJ4095.