On the Local Well-Posedness of the Cauchy Problem for a Modified Two-Component Camassa-Holm System in Besov Spaces

Abstract

We consider the Cauchy problem for an integrable modified two-component Camassa-Holm system with cubic nonlinearity. By using the Littlewood-Paley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally well-posed in Besov spaces 𝐵^𝑠_𝑝,𝑟 with 1≤𝑝, 𝑟≤+∞ and 𝑠>max{2+(1/𝑝),5/2}.

1. Introduction

The following modified Camassa-Holm equation with cubic nonlinearity 𝑚_𝑡+𝑢²−𝑢²_𝑥𝑚_𝑥=0,𝑚=𝑢−𝑢_𝑥𝑥 (1) was proposed as a integrable system by Fuchssteiner [1] and Olver and Rosenau [2] by applying the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [3] from the two-dimensional Euler equations, where the variables 𝑢(𝑡,𝑥) and 𝑚(𝑡,𝑥) represent, respectively, the velocity of the fluid and its potential density. Qiao also [3] obtained the cuspons and 𝑊/𝑀-shape-peaks solitons of (1). In [4], it was shown that (1) admits a Lax pair and hence can be solved by the method of inverse scattering. Fu et al. [5] showed that the Cauchy problem of (1) is locally well-posed in a range of the Besov spaces. They determined the blow-up scenario and the lower bound of the maximal time of existence. They also described a blow-up mechanism for solutions with certain initial profiles and the nonexistence of the smooth traveling wave solutions was also demonstrated. In addition, they obtained the persistence properties of the strong solutions for (1). Gui et al. [6] investigated the formation of singularities and showed that singularities of the solutions occur only in the form of wave breaking. They obtained a new wave-breaking mechanism for solutions with certain initial profiles. It was proved that the peaked functions of the form 𝑢(𝑡,𝑥)=±3𝑐2𝑒^{−|𝑥−𝑐𝑡|},𝑐∈ℝ⁺ (2) are global weak solutions to (1) [6].

Recently, Song et al. [7] suggested a new integrable two-component vision of (1) as follows: 𝑚_𝑡=[𝑚𝑎(𝑢,𝑣)]_𝑥,ℎ_𝑡=[ℎ𝑎(𝑢,𝑣)]_𝑥, (3) with 𝑚=𝑢−𝑢_𝑥𝑥, ℎ=𝑣−𝑣_𝑥𝑥, and 𝑎(𝑢,𝑣)=𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣. Apparently, it reduces to (1) when 𝑣=𝑢. They showed that this system has Lax-pair and is also geometrically integrable. As a consequence of geometric integrability, its conservation laws were constructed by expanding the pseudopotential. Finally, the cuspons and 𝑊/𝑀-shape solitons of system (3) were obtained.

In this paper, we are interesting in the local well-posedness of the following Cauchy problem for (3) 𝑚_𝑡=[𝑚𝑎(𝑢,𝑣)]_𝑥ℎ,𝑥∈ℝ,𝑡≥0,_𝑡=[ℎ𝑎(𝑢,𝑣)]_𝑥𝑢(),𝑥∈ℝ,𝑡≥0,0,𝑥=𝑢₀(𝑥),𝑣(0,𝑥)=𝑣₀(𝑥),𝑥∈ℝ, (4) with 𝑚=𝑢−𝑢_𝑥𝑥, ℎ=𝑣−𝑣_𝑥𝑥, and 𝑎(𝑢,𝑣)=𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣. Although Kato’s theory is a useful method to obtain the local well-posedness of the Cauchy problem in Sobolev space 𝐻^𝑠(ℝ) for lots of equations, such as the Camassa-Holm equation [8], the Degasperis-Procesi equation [9], and the Novikov equation [10]. However, it seems to be unapplicable to the Cauchy problem (4). Fortunately, by using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem (4) is locally well-posed in the Besov space 𝐵^𝑠_𝑝,𝑟 with 1≤𝑝, 𝑟≤+∞, and 𝑠>max{1+(1/𝑝),5/2}. As a corollary, it is locally well-posed in the Sobolev space 𝐻^𝑠(ℝ) with 𝑠>5/2. The Littlewood-Paley decomposition and nonhomogeneous Besov spaces which were introduced in [11] have been used to establish the well-posedness of the Euler equations, the Navier-Stokes equations, the Camassa-Holm equation, the Novikov equation, the two-component Camassa-Holm system, the two-component Degasperis-Procesi system, and the coupled Camassa-Holm equations; see [12–20].

The rest of the paper is organized as follows. In Section 2, we recall some basic results on Besov spaces and linear transport equation. In Section 3, we establish the local well-posedness of the Cauchy problem (4) in a range of Besov spaces. Some open problems are presented in Section 4.

Notation. Since all spaces of functions are over ℝ, we drop ℝ in our notations of function spaces for simplicity. For a given Banach space 𝐸, we denote its norm by ‖⋅‖_𝐸. We denote ℱ𝑢 or 𝑢 the Fourier transform of the function 𝑢. Let 𝒞^𝑘([0,𝑇];𝐻) denote the space of all 𝑘 times continuously differentiable functions 𝑢(𝑥,𝑡) defined on [0,𝑡] with values in space 𝐻. 𝐿^∞([0,𝑇];𝐻) denotes the space of functions 𝑢(𝑥,𝑡) which belong to 𝐻 as a function of 𝑥 for every fixed 𝑡(0≤𝑡≤𝑇) and sup_{0≤𝑡≤𝑇}‖𝑢(⋅,𝑡)‖_𝐻<∞ and 𝐿¹([0,𝑇];𝐻) the space of functions 𝑢(𝑥,𝑡) which belong to 𝐻 as a function of 𝑥 for every fixed 𝑡(0≤𝑡≤𝑇) and ∫^𝑇₀‖𝑢(⋅,𝑡)‖_𝐻𝑑𝑡<∞. 𝒮^′ denotes the tempered distribution spaces.

2. Preliminaries

In this section, we recall some results on the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces, and the linear transport equation which can be seen in [10–20].

Proposition 1 (Littlewood-Paley decomposition). Let ℬ≜{𝜉∈ℝ,|𝜉|≤4/3} and 𝒞≜{𝜉∈ℝ,3/4≤|𝜉|≤8/3}. There exist two radial functions 𝜒∈𝒞^∞_𝑐(ℬ) and 𝜑∈𝒞^∞_𝑐(𝒞) such that 𝜒(𝜉)+_𝑞≥0𝜑2^−𝑞𝜉=1,∀𝜉∈ℝ^𝑑,𝑞−𝑞^′≥2⟹Supp𝜑2^−𝑞⋅∩Supp𝜑2^−𝑞′⋅(⋅)2=𝜙,𝑞≥1⟹Supp𝜒∩Supp𝜑^−𝑞⋅1=𝜙,3≤𝜒(𝜉)²+_𝑞≥0𝜑2^−𝑞𝜉²≤1,∀𝜉∈ℝ^𝑑. (5) Furthermore, let ℎ≜ℱ𝜑, ℎ≜ℱ⁻¹𝜒 and define the dyadic operators Δ_𝑞 and low-frequency cut-off operators 𝑆_𝑞 as follows: Δ_𝑞𝑓≜𝜑2^−𝑞𝐷𝑓=2^𝑞𝑑_ℝ𝑑ℎ2^𝑞𝑦𝑓Δ𝑥−𝑦𝑑𝑦,𝑓𝑜𝑟𝑞≥0,₋₁𝑓≜𝑆₀𝑓,Δ_𝑞𝑆𝑓≜0,𝑓𝑜𝑟𝑞≤−2,_𝑞𝑓≜𝜒2^−𝑞𝐷𝑓=_{−1≤𝑘≤𝑞−1}Δ_𝑘𝑓=2^𝑞𝑑_ℝ𝑑ℎ2^𝑞𝑦𝑓𝑥−𝑦𝑑𝑦. (6) Then, for all 𝑢∈𝒮^′, the nonhomogeneous Littlewood-Paley decomposition of a distribution 𝑢 is defined 𝑢=_𝑞∈ℤΔ_𝑞𝑢. (7)

Definition 2 (Besov space). Let 𝑠∈ℝ and 1≤𝑝, 𝑟≤∞. The inhomogenous Besov space 𝐵^𝑠_𝑝,𝑟(ℝ) (𝐵^𝑠_𝑝,𝑟 for short) is defined by 𝐵^𝑠_𝑝,𝑟≜𝑓∈𝒮^′(ℝ),𝑓_{𝐵𝑠𝑝,𝑟}<∞, (8) where 𝑓_{𝐵𝑠𝑝,𝑟}≜_𝑞∈ℤ2^𝑞𝑠𝑟Δ_𝑞𝑓^𝑟_𝐿𝑝^1/𝑟,for𝑟<∞,sup_𝑞∈ℤ2^𝑞𝑠Δ_𝑞𝑓_𝐿𝑝,for𝑟=∞. (9) If 𝑠=∞, 𝐵^∞_𝑝,𝑟≜⋂_𝑠∈ℝ𝐵^𝑠_𝑝,𝑟.

Proposition 3 (properties of Besov space). Let 𝑠∈ℝ and 1≤𝑝,𝑟,𝑝_𝑗,𝑟_𝑗≤∞, 𝑗=1,2; then the following properties hold.
(1) Topological properties: 𝐵^𝑠_𝑝,𝑟 is a Banach space which is continuously embedded in 𝒮^′.
(2) Density: 𝒞^∞_𝑐 is dense in 𝐵^𝑠_𝑝,𝑟⇔1≤𝑝, 𝑟<∞.
(3) Sobolev embeddings: if 𝑝₁≤𝑝₂ and 𝑟₁≤𝑟₂, then 𝐵^𝑠_𝑝1,𝑟1↪𝐵^{𝑠−𝑑((1/𝑝1)−(1/𝑝2}_𝑝⁾⁾_2,𝑟2. If 𝑠₁<𝑠₂, 1≤𝑝≤+∞ and 1≤𝑟₁,𝑟₂≤+∞, then the embedding 𝐵^𝑠2_𝑝,𝑟2↪𝐵^𝑠1_𝑝,𝑟1 is locally compact.
(4) Algebraic properties: for 𝑠>0, 𝐵^𝑠_𝑝,𝑟∩𝐿^∞ is an algebra. Moreover, 𝐵^𝑠_𝑝,𝑟 is an algebra ⇔𝐵^𝑠_𝑝,𝑟↪𝐿^∞⇔𝑠>𝑑/𝑝 or (𝑠≥𝑑/𝑝 and 𝑟=1).
(5) Fatou property: if (𝑢^(𝑛))_𝑛∈ℕ is a bounded sequence of 𝐵^𝑠_𝑝,𝑟 which tends to 𝑢 in 𝒮^′, then 𝑢∈𝐵^𝑠_𝑝,𝑟 and ‖𝑢‖_𝐵^𝑠_𝑝,𝑟≤liminf_𝑛→∞𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}. (10)
(6) Complex interpolation: if 𝑢∈𝐵^𝑠_𝑝,𝑟∩𝐵^̃𝑠_𝑝,𝑟 and 𝜃∈[0,1], 1≤𝑝,𝑟≤∞, then 𝑢∈𝐵^{̃𝑠𝜃𝑠+(1−𝜃)}_𝑝,𝑟 and ‖𝑢‖_𝐵^{̃𝑠𝜃𝑠+(1−𝜃)}_𝑝,𝑟≤‖𝑢‖^𝜃_𝐵^𝑠_𝑝,𝑟‖𝑢‖_𝐵^1−𝜃_{̃𝑠𝑝,𝑟}. (11)
(7) Generalized derivatives: let 𝑓∈𝒞^∞(ℝ^𝑑) and 𝑓 be a homogeneous function of degree 𝑚∈ℝ away from a neighborhood of the origin. There exists a constant 𝐶 depending only on 𝑓 and such that 𝑓(𝐷)𝑢_{𝐵𝑠−𝑚𝑝,𝑟}≤𝐶‖𝑢‖_𝐵^𝑠_𝑝,𝑟. (12)
(8) 1-D Moser-type estimates:(i)for 𝑠>0,𝑓𝑔_{𝐵𝑠𝑝,𝑟}≤𝐶𝑓_{𝐵𝑠𝑝,𝑟}𝑔_𝐿∞+𝑔_{𝐵𝑠𝑝,𝑟}𝑓_𝐿∞, (13)(ii)for 𝑠₁≤1/𝑝, 𝑠₂>(1/𝑝)(𝑠₂≥1/𝑝 if 𝑟=1) and 𝑠₁+𝑠₂>0,𝑓𝑔_𝐵^𝑠_1𝑝,𝑟≤𝐶𝑓_𝐵^𝑠_1𝑝,𝑟𝑔_𝐵^𝑠_2𝑝,𝑟, (14)where 𝐶^′𝑠 are constants independent of 𝑓 and 𝑔.

Lemma 4 (a prior estimates for 1-D linear transport equation). Let 1≤𝑝, 𝑟≤∞, and 𝑠>−min{1/𝑝,1−(1/𝑝)}. Assume that 𝑓₀∈𝐵^𝑠_𝑝,𝑟, 𝐹∈𝐿¹([0,𝑇];𝐵^𝑠_𝑝,𝑟) and 𝜕_𝑥𝑣∈𝐿¹([0,𝑇];𝐵^𝑠−1_𝑝,𝑟) if 𝑠>1+(1/𝑝) or 𝜕_𝑥𝑣∈𝐿¹(0,𝑇;𝐵^1/𝑝_𝑝,𝑟∩𝐿^∞) otherwise. If 𝑓∈𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)∩𝒞([0,𝑇];𝒮^′) solves the following 1-D linear transport equation 𝑓_𝑡+𝑣𝑓_𝑥𝑓()=𝐹,𝑥,0=𝑓₀, (15) then there exists a constant 𝐶 depending only on 𝑠, 𝑝, and 𝑟 such that the following statements hold:
(1) If 𝑟=1 or 𝑠≠1+(1/𝑝), 𝑓_{𝐵𝑠𝑝,𝑟}≤𝑓_{0𝐵𝑠𝑝,𝑟}+^𝑡₀‖𝐹(𝜏)‖_𝐵^𝑠_𝑝,𝑟𝑑𝜏+𝐶^𝑡₀𝑉^′(𝜏)𝑓(𝜏)_{𝐵𝑠𝑝,𝑟}𝑑𝜏 (16) or 𝑓_{𝐵𝑠𝑝,𝑟}≤𝑒^{𝐶𝑉(𝑡)}𝑓_{0𝐵𝑠𝑝,𝑟}+^𝑡₀𝑒^{−𝐶𝑉(𝜏)}‖𝐹(𝜏)‖_𝐵^𝑠_𝑝,𝑟𝑑𝜏, (17) where 𝑉(𝑡)=^𝑡0𝑣_𝑥(𝜏)_𝐵^1/𝑝_{𝑝,𝑟∩𝐿∞}1𝑑𝜏,𝑖𝑓𝑠<1+𝑝_𝑡0,𝑣_𝑥(𝜏)_{𝐵𝑠−1𝑝,𝑟}1𝑑𝜏,𝑖𝑓𝑠>1+𝑝. (18)
(2) If 𝑠≤1+(1/𝑝), 𝑓_0,𝑥∈𝐿^∞((0,𝑇)×ℝ), and 𝐹_𝑥∈𝐿¹([0,𝑇];𝐿^∞), then 𝑓(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑓_𝑥(𝑡)_𝐿∞≤𝑒^{𝐶𝑉(𝑡)}𝑓_{0𝐵𝑠𝑝,𝑟}+𝑓_{0,𝑥𝐿∞}+^𝑡₀𝑒^{−𝐶𝑉(𝜏)}‖𝐹(𝜏)‖_𝐵^𝑠_𝑝,𝑟+𝐹_𝑥(𝜏)_𝐿∞𝑑𝜏, (19) with 𝑉(𝑡)=∫^𝑡₀‖𝑣_𝑥(𝜏)‖_𝐵^1/𝑝_{𝑝,𝑟∩𝐿∞}𝑑𝜏.
(3) If 𝑓=𝑣, then, for all 𝑠>0, estimate (17) holds true with 𝑉(𝑡)=∫^𝑡₀‖𝑣_𝑥(𝜏)‖_𝐿∞𝑑𝜏.
(4) If 𝑟<∞, then 𝑓∈𝒞([0,𝑇];𝐵^𝑠_𝑝,𝑟).
(5) If 𝑟=∞, then 𝑓∈𝒞([0,𝑇];𝐵^𝑠′_𝑝,1) for all 𝑠^′<𝑠.

Lemma 5 (existence and uniqueness). Let 𝑝, 𝑟, 𝑠, 𝑓₀, and 𝐹 be as in the statement of Lemma 4. Assume that 𝑣∈𝐿^𝜌([0,𝑇];𝐵^−𝑀_∞,∞) for some 𝜌>1 and 𝑀>0 and 𝑣_𝑥∈𝐿¹([0,𝑇];𝐵^𝑠−1_𝑝,𝑟) if 𝑠>1+(1/𝑝) or 𝑠=1+(1/𝑝) and 𝑟=1 and 𝑣_𝑥∈𝐿¹([0,𝑇];𝐵^1/𝑝_𝑝,𝑟∩𝐿^∞) if 𝑠<1+(1/𝑝). Then the problem (15) has a unique solution 𝑓∈𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)∩(⋂_𝑠′<𝑠𝒞([0,𝑇];𝐵^𝑠′_𝑝,1)) and the inequalities of Lemma 4 can hold true. Moreover, if 𝑟<+∞, then 𝑓∈𝒞([0,𝑇];𝐵^𝑠_𝑝,𝑟).

3. Local Well-Posedness

In this section, we discuss the local well-posedness of the Cauchy problem (4).

For 𝑇>0, 𝑠∈ℝ, and 1≤𝑝≤∞, we denote 𝐸^𝑠_𝑝,𝑟(𝑇)≜𝒞[0,𝑇];𝐵^𝑠_𝑝,𝑟𝒞¹[0,𝑇];𝐵^𝑠−1_𝑝,𝑟𝐸,for1≤𝑟<∞,_𝑠𝑝,∞(𝑇)≜𝐿^∞[0,𝑇];𝐵^𝑠_𝑝,∞Lip[0,𝑇];𝐵^𝑠−1_𝑝,∞. (20)

Our main local well-posedness result is the following theorem.

Theorem 6. Suppose that 1≤𝑝, 𝑟≤+∞, 𝑠>max{2+(1/𝑝),(5/2)}, and (𝑢₀,𝑣₀)∈𝐵^𝑠_𝑝,𝑟×𝐵^𝑠_𝑝,𝑟. Then, there exists a time 𝑇>0 such that the Cauchy problem (4) has a unique solution (𝑢,𝑣)∈𝐸^𝑠_𝑝,𝑟(𝑇)² and the map (𝑢₀,𝑣₀)→(𝑢,𝑣) is continuous from a neighborhood of (𝑢₀,𝑣₀) in 𝐵^𝑠_𝑝,𝑟×𝐵^𝑠_𝑝,𝑟 into 𝐸^𝑠′_𝑝,𝑟(𝑇)² for every 𝑠^′<𝑠 when 𝑟=+∞ and 𝑠^′=𝑠 whereas 𝑟<+∞.

Proof. We divide the proof into seven steps. In the following, we denote 𝐶>0 a generic constant only depending on 𝑝, 𝑟, and 𝑠.
Step 1 (approximate solution). We use a standard iterative process to construct a solution. Starting from 𝑢₀∶=0, 𝑣₀∶=0 by induction, we define a sequence of smooth functions (𝑢^(𝑛),𝑣^(𝑛))_𝑛∈ℕ∈𝒞(ℝ⁺;𝐵^∞_𝑝,𝑟)² by solving the following linear transport equations: 𝜕_𝑡+−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)(𝑡)=𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥,𝜕_𝑡+−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥ℎ^(𝑛+1)(𝑡)=ℎ^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥,𝑢^(𝑛+1)_𝑡=0=𝑢₀^(𝑛+1)=𝑆_𝑛+1𝑢₀,𝑣^(𝑛+1)_𝑡=0=𝑣₀^(𝑛+1)=𝑆_𝑛+1𝑣₀, (21) with 𝑎(𝑢^(𝑛),𝑣^(𝑛))=𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛).
Since all the data belong to 𝐵^∞_𝑝,𝑟, Lemma 5 enables us to show by induction that, for all 𝑛∈ℕ, system (21) has a global solution which belongs to 𝒞(ℝ⁺;𝐵^∞_𝑝,𝑟)².
Step 2 (uniform bounds). Applying (17) to the first equation in (21) we have 𝑚^(𝑛+1)(𝑡)_{𝐵𝑠−2𝑝,𝑟}≤𝑚_{0𝐵𝑠−2𝑝,𝑟}𝑒^{𝐶∫𝑡0‖𝜕𝑥[−𝑎(𝑢(𝑛),𝑣(𝑛))](𝜏)‖𝐵𝑠−3𝑝,𝑟𝑑𝜏}+^𝑡₀𝑒^{𝐶∫𝑡𝜏‖𝜕𝑥[−𝑎(𝑢(𝑛),𝑣(𝑛))](𝜏′)‖𝐵𝑠−3𝑝,𝑟𝑑𝜏′}×𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥(𝜏)_{𝐵𝑠−2𝑝,𝑟}𝑑𝜏. (22) Since 𝐵^𝑠−2_𝑝,𝑟 is an algebra when 𝑠>max{2+(1/𝑝),5/2}, we derive from Proposition 3(7) that 𝜕_𝑥−𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}=𝐶𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}²,𝑚^(𝑛)[𝑎(𝑢^(𝑛),𝑣^(𝑛))]_{𝑥𝐵𝑠−2𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−2𝑝,𝑟}[𝑎(𝑢^(𝑛),𝑣^(𝑛))]_{𝑥𝐵𝑠−2𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑎(𝑢^(𝑛),𝑣^(𝑛))_{𝐵𝑠−1𝑝,𝑟}=𝐶𝑚^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}. (23) It follows from (22)-(23) that 𝑢^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}≤𝑢_{0𝐵𝑠𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛)(𝜏)²_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}𝑑𝜏. (24) Similarly, we have 𝑣^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}≤𝑣_{0𝐵𝑠𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)(𝜏)²_{𝐵𝑠𝑝,𝑟}𝑑𝜏. (25) Combining (24) and (25) we arrive at 𝑢^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}≤𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛)(𝜏)²_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)(𝜏)²_{𝐵𝑠𝑝,𝑟}≤𝑢𝑑𝜏_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}³𝑑𝜏. (26) Choose a 𝑇>0 such that 4𝐶(‖𝑢₀‖_{𝐵𝑠𝑝,𝑟}+‖𝑣₀‖_{𝐵𝑠𝑝,𝑟})²𝑇<1 and suppose that for all 𝑡∈[0,𝑇], 𝑢^(𝑛)(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)(𝑡)_{𝐵𝑠𝑝,𝑟}≤𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/2. (27) Then we have 𝐶^𝑡₀𝑢^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)(𝜏)_{𝐵𝑠𝑝,𝑟}²𝑑𝜏≤𝐶^𝑡₀𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏1𝑑𝜏=−4ln1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡,𝐶^𝑡_𝜏𝑢^(𝑛)𝜏^′_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)𝜏^′_{𝐵𝑠𝑝,𝑟}²𝑑𝜏^′≤𝐶^𝑡_𝜏𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏^′𝑑𝜏^′1=−4ln1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡+14ln1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏, (28) which along with (26) leads to 𝑢^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+1)(𝑡)_{𝐵𝑠𝑝,𝑟}≤𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/4+𝐶1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/4×^𝑡₀1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏^1/4×𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}³1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏^3/2=𝑢𝑑𝜏_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/4×1+𝐶^𝑡₀𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝜏^5/4=𝑢𝑑𝜏_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/4×11+1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/4=𝑢−1_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^1/2. (29) Therefore, we conclude that (𝑢^(𝑛),𝑣^(𝑛))_𝑛∈ℕ∈𝒞([0,𝑇];𝐵^𝑠_𝑝,𝑟)².
If 𝑠>3+(1/𝑝), then 𝐵^𝑠−3_𝑝,𝑟 is an algebra. Hence, according to Proposition 3(7) we have −𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝐵𝑠−3𝑝,𝑟}𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−3𝑝,𝑟}𝑚^(𝑛+1)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛+1)_{𝐵𝑠𝑝,𝑟},𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−3𝑝,𝑟}𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−3𝑝,𝑟}𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}. (30) If max{2+(1/𝑝),5/2}<𝑠≤3+(1/𝑝), using the Moser-type estimate (14), we obtain −𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑚^(𝑛+1)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛+1)_{𝐵𝑠𝑝,𝑟},𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑚^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}. (31) Thus, using the first equation in (21) and combining (30)-(31) with (29), we have 𝜕_𝑡𝑚^(𝑛+1)(𝑡)_{𝐵𝑠−3𝑝,𝑟}≤−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}+𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛+1)_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{𝐵𝑠𝑝,𝑟}≤𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}³1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^3/2. (32) In a similar way, we can obtain that 𝜕_𝑡ℎ^(𝑛+1)(𝑡)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}³1−4𝐶𝑢_{0𝐵𝑠𝑝,𝑟}+𝑣_{0𝐵𝑠𝑝,𝑟}²𝑡^3/2. (33) Thus, (32) and (33) imply that (𝜕_𝑡𝑢^(𝑛+1),𝜕_𝑡𝑣^(𝑛+1))∈𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)². Therefore 𝑢^(𝑛),𝑣^(𝑛)_𝑛∈ℕ∈𝐸^𝑠_𝑝,𝑟(𝑇)². (34)
Step 3 (convergence). Now we are going to show that (𝑢^(𝑛),𝑣^(𝑛))_𝑛∈ℕ is a Cauchy sequence in 𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)². In fact, according to (21), for all 𝑛,𝑙∈ℕ, we obtain that 𝜕_𝑡+−𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)𝜕_𝑥𝑚^{(𝑛+𝑙+1)}−𝑚^(𝑛+1)(𝑡)=𝑔₁𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),𝑚^(𝑛),𝑚^(𝑛+1),𝑚^(𝑛+𝑙),𝜕_𝑡+−𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)𝜕_𝑥ℎ^{(𝑛+𝑙+1)}−ℎ^(𝑛+1)(𝑡)=𝑔₂𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),ℎ^(𝑛),ℎ^(𝑛+1),ℎ^(𝑛+𝑙), (35) where 𝑔₁𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),𝑚^(𝑛),𝑚^(𝑛+1),𝑚^(𝑛+𝑙)=𝑚^(𝑛+𝑙)−𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥+𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)+𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥𝑚^(𝑛+𝑙),𝑔₂𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),ℎ^(𝑛),ℎ^(𝑛+1),ℎ^(𝑛+𝑙)=ℎ^(𝑛+𝑙)−ℎ^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥+𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥ℎ^(𝑛+1)+𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥ℎ^(𝑛+𝑙). (36) According to (17), for every 𝑡∈[0,𝑇], the following inequality holds: 𝑚^{(𝑛+𝑙+1)}(𝑡)−𝑚^(𝑛+1)(𝑡)_{𝐵𝑠−3𝑝,𝑟}≤𝑚₀^{(𝑛+𝑙+1)}−𝑚₀^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}𝑒^{𝐶∫𝑡0‖𝜕𝑥[−𝑎(𝑢(𝑛+𝑙),𝑣(𝑛+𝑙))](𝜏)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏‖𝜕𝑥[−𝑎(𝑢(𝑛+𝑙),𝑣(𝑛+𝑙))](𝜏′)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏′}×𝑔₁𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),𝑚^(𝑛),𝑚^(𝑛+1),𝑚^(𝑛+𝑙)(𝜏)_{𝐵𝑠−3𝑝,𝑟}𝑑𝜏, (37) is equivalent to 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀^{(𝑛+𝑙+1)}−𝑢₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0‖𝜕𝑥[𝑎(𝑢(𝑛+𝑙),𝑣(𝑛+𝑙))](𝜏)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏}+^𝑡₀𝑒^{𝐶∫𝑡𝜏‖𝜕𝑥[𝑎(𝑢(𝑛+𝑙),𝑣(𝑛+𝑙))](𝜏′)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏′}×𝑔₁𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),𝑚^(𝑛),𝑚^(𝑛+1),𝑚^(𝑛+𝑙)(𝜏)_{𝐵𝑠−3𝑝,𝑟}𝑑𝜏. (38) If max{2+(1/𝑝),5/2}<𝑠≤3+(1/𝑝), using the Moser-type estimate (14), we obtain 𝜕_𝑥−𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)_{𝐵𝑠−4𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}=𝐶𝑢_𝑥^(𝑛+𝑙)𝑣_𝑥^(𝑛+𝑙)−𝑢^(𝑛+𝑙)𝑣^(𝑛+𝑙)+𝑢^(𝑛+𝑙)𝑣_𝑥^(𝑛+𝑙)−𝑢_𝑥^(𝑛+𝑙)𝑣^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥^(𝑛+𝑙)_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}+𝑢^(𝑛+𝑙)_{𝐵𝑠−2𝑝,𝑟}𝑣^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}+𝑢^(𝑛+𝑙)_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}+𝑢_𝑥^(𝑛+𝑙)_{𝐵𝑠−2𝑝,𝑟}𝑣^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}²,𝑚^(𝑛+𝑙)−𝑚^(𝑛)𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑚^(𝑛+𝑙)−𝑚^(𝑛)_{𝐵𝑠−3𝑝,𝑟}𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢_𝑥^(𝑛)𝑣_𝑥^(𝑛)−𝑢^(𝑛)𝑣^(𝑛)+𝑢^(𝑛)𝑣_𝑥^(𝑛)−𝑢_𝑥^(𝑛)𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑣^(𝑛)_{‖𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)2_{‖𝐵𝑠𝑝,𝑟},𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝜕_𝑥𝑚^(𝑛+1)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_𝑥𝑣_𝑥^(𝑛+𝑙)+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_𝑥𝑢_𝑥^(𝑛)−𝑢^(𝑛+𝑙)−𝑢^(𝑛)𝑣^(𝑛+𝑙)−𝑣^(𝑛+𝑙)−𝑣^(𝑛)𝑢^(𝑛)+𝑢^(𝑛+𝑙)−𝑢^(𝑛)𝑣_𝑥^(𝑛+𝑙)+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_𝑥𝑢^(𝑛)−𝑢^(𝑛+𝑙)−𝑢^(𝑛)_𝑥𝑣^(𝑛+𝑙)−𝑣^(𝑛+𝑙)−𝑣^(𝑛)𝑢_𝑥^(𝑛)_{𝐵𝑠−2𝑝,𝑟}𝑚^(𝑛+1)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑣^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛+1)_{‖𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟},𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)_𝑥𝑚^(𝑛+𝑙)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙)−𝑎𝑢^(𝑛),𝑣^(𝑛)_{𝑥𝐵𝑠−3𝑝,𝑟}𝑚^(𝑛+1)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_𝑥𝑣_𝑥^(𝑛+𝑙)+𝑣^(𝑛+𝑙)−𝑣^(𝑛)𝑢_𝑥^(𝑛)−𝑢^(𝑛+𝑙)−𝑢^(𝑛)𝑣^(𝑛+𝑙)−𝑣^(𝑛+𝑙)−𝑣^(𝑛)𝑢^(𝑛)+𝑢^(𝑛+𝑙)−𝑢^(𝑛)𝑣_𝑥^(𝑛+𝑙)+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_𝑥𝑢^(𝑛)−𝑢^(𝑛+𝑙)−𝑢^(𝑛)_𝑥𝑣^(𝑛+𝑙)−𝑣^(𝑛+𝑙)−𝑣^(𝑛)𝑢_𝑥^(𝑛)_{𝐵𝑠−2𝑝,𝑟}×𝑚^(𝑛+𝑙)_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑣^(𝑛+𝑙)_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)_{𝐵𝑠𝑝,𝑟}𝑢^(𝑛+𝑙)_{‖𝐵𝑠𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{‖𝐵𝑠𝑝,𝑟}. (39)(40)(41)(42) If 𝑠>3+(1/𝑝), then 𝐵^𝑠−3_𝑝,𝑟 is an algebra, and we can verify that (39)–(42) also hold true. Thus, we have 𝑔₁𝑢^(𝑛),𝑣^(𝑛),𝑢^(𝑛+𝑙),𝑣^(𝑛+𝑙),𝑚^(𝑛),𝑚^(𝑛+1),𝑚^(𝑛+𝑙)(𝜏)_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{‖𝐵𝑠𝑝,𝑟}. (43) Plugging (39) and (43) into (38) we get 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀^{(𝑛+𝑙+1)}−𝑢₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{‖𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{‖𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏. (44) Similarly, we can derive that 𝑣^{(𝑛+𝑙+1)}(𝑡)−𝑣^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑣₀^{(𝑛+𝑙+1)}−𝑣₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑣^(𝑛+1)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)2_{‖𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+1)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}𝑑𝜏. (45) Combining (44) and (45), we have 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}+𝑣^{(𝑛+𝑙+1)}(𝑡)−𝑣^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀^{(𝑛+𝑙+1)}−𝑢₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}+𝑣₀^{(𝑛+𝑙+1)}−𝑣₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}×𝑒^{𝐶∫𝑡0(‖𝑢(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟+‖𝑣(𝑛+𝑙)(𝜏′)‖𝐵𝑠𝑝,𝑟)2𝑑𝜏′}×𝑢^(𝑛+𝑙)−𝑢^(𝑛)_{𝐵𝑠−1𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)_{𝐵𝑠−1𝑝,𝑟}×𝑢^(𝑛)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+1)2_{𝐵𝑠𝑝,𝑟}+𝑢^(𝑛+𝑙)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛)2_{‖𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+1)2_{𝐵𝑠𝑝,𝑟}+𝑣^(𝑛+𝑙)2_{‖𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏. (46) Since 𝑢₀^{(𝑛+𝑙+1)}−𝑢₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}=𝑆_𝑛+𝑙+1𝑢₀−𝑆_𝑛+1𝑢_{0𝐵𝑠−1𝑝,𝑟}=^𝑛+𝑙_𝑞=𝑛+1Δ_𝑞𝑢_{0𝐵𝑠−1𝑝,𝑟}≤𝐶2^−𝑛𝑢_{0‖𝐵𝑠−1𝑝,𝑟},𝑣₀^{(𝑛+𝑙+1)}−𝑣₀^(𝑛+1)_{𝐵𝑠−1𝑝,𝑟}=𝑆_𝑛+𝑙+1𝑢₀−𝑆_𝑛+1𝑣_{0𝐵𝑠−1𝑝,𝑟}=^𝑛+𝑙_𝑞=𝑛+1Δ_𝑞𝑣_{0𝐵𝑠−1𝑝,𝑟}≤𝐶2^−𝑛𝑣_{0‖𝐵𝑠−1𝑝,𝑟}, (47) which can be seen on page 2142 of [20], and (𝑢^(𝑛),𝑣^(𝑛))_𝑛∈ℕ is uniformly bounded in 𝐸^𝑠_𝑝,𝑟(𝑇)², then there exists a constant 𝐶_𝑇 independent of 𝑛 and 𝑙 such that, for all 𝑡∈[0,𝑇], 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}+𝑣^{(𝑛+𝑙+1)}(𝑡)−𝑣^(𝑛+1)(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝐶_𝑇2^−𝑛+^𝑡₀𝑢^(𝑛+𝑙)−𝑢^(𝑛)(𝜏)_{𝐵𝑠−1𝑝,𝑟}+𝑣^(𝑛+𝑙)−𝑣^(𝑛)(𝜏)_{𝐵𝑠−1𝑝,𝑟}𝑑𝜏. (48) Arguing by induction with respect to the index 𝑛, we can see that 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐿∞𝑇(𝐵𝑠−1𝑝,𝑟)}+𝑣^{(𝑛+𝑙+1)}(𝑡)−𝑣^(𝑛+1)(𝑡)_{𝐿∞𝑇(𝐵𝑠−1𝑝,𝑟)}≤𝑇𝐶_𝑇^𝑛+1()!𝑢𝑛+1^{(𝑙)𝐿∞𝑇(𝐵𝑠𝑝,𝑟)}+𝑣^(𝑙)_{𝐿∞𝑇(𝐵𝑠𝑝,𝑟)}+𝐶_𝑇^𝑛_𝑘=02^{−(𝑛−𝑘)}𝑇𝐶_𝑇^𝑘.𝑘! (49) Since ‖𝑢^(𝑙)‖_𝐿^∞_𝑇(𝐵^𝑠_𝑝,𝑟) and ‖𝑣^(𝑙)‖_𝐿^∞_𝑇(𝐵^𝑠_𝑝,𝑟) are bounded independently of 𝑙, we conclude that there exists some new constant 𝐶^′_𝑇 independent of 𝑛 and 𝑙 such that 𝑢^{(𝑛+𝑙+1)}(𝑡)−𝑢^(𝑛+1)(𝑡)_{𝐿∞𝑇(𝐵𝑠−1𝑝,𝑟)}+𝑣^{(𝑛+𝑙+1)}(𝑡)−𝑣^(𝑛+1)(𝑡)_{𝐿∞𝑇(𝐵𝑠−1𝑝,𝑟)}≤𝐶^′_𝑇2^−𝑛. (50) Hence (𝑢^(𝑛),𝑣^(𝑛))_𝑛∈ℕ is a Cauchy sequence in 𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)² and converges to some function (𝑢,𝑣)∈𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)².
Step 4 (existence of solution in 𝐸^𝑠_𝑝,𝑟(𝑇)²). We will prove that (𝑢,𝑣) belongs to 𝐸^𝑠_𝑝,𝑟(𝑇)² and satisfies (4). Since (𝑢^(𝑛),𝑣^(𝑛)) in uniformly bounded in 𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)², the Fatou property for the Besov spaces (Proposition 3(5)) guarantees that (𝑢,𝑣) also belongs to 𝐿^∞([0,𝑇);𝐵^𝑠_𝑝,𝑟)².
On the other hand, as (𝑢^(𝑛),𝑣^(𝑛)) converges to (𝑢,𝑣) in 𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)², we conclude that the convergence holds in 𝒞([0,𝑇];𝐵^𝑠′_𝑝,𝑟)² for any 𝑠^′<𝑠. In fact, if 𝑠^′≤𝑠−1, then 𝑢^(𝑛)−𝑢_{𝐵𝑠′𝑝,𝑟}+𝑣^(𝑛)−𝑣_{𝐵𝑠′𝑝,𝑟}≤𝐶𝑢^(𝑛)−𝑢_{𝐵𝑠−1𝑝,𝑟}+𝐶𝑣^(𝑛)−𝑣_{𝐵𝑠−1𝑝,𝑟}. (51) If 𝑠−1<𝑠^′<𝑠, then Proposition 3(6) gives us 𝑢^(𝑛)−𝑢_{𝐵𝑠′𝑝,𝑟}+𝑣^(𝑛)−𝑣_{𝐵𝑠′𝑝,𝑟}≤𝐶𝑢^(𝑛)−𝑢^𝜃_{𝐵𝑠−1𝑝,𝑟}𝑢^(𝑛)−𝑢_𝐵^1−𝜃_{𝑠𝑝,𝑟}+𝐶𝑣^(𝑛)−𝑣^𝜃_{𝐵𝑠−1𝑝,𝑟}𝑣^(𝑛)−𝑣_𝐵^1−𝜃_{𝑠−1𝑝,𝑟}, (52) where 𝜃=𝑠−𝑠^′.
Passing to the limit in (21) and we can easily conclude that (𝑢,𝑣) is indeed a solution to the Cauchy problem (4). Thanks to the fact that (𝑢,𝑣)∈𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)², the right-hand side of the equations 𝑚_𝑡−𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣𝑚_𝑥=𝑚𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣_𝑥,ℎ_𝑡−𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣ℎ_𝑥=ℎ𝑢_𝑥𝑣_𝑥−𝑢𝑣+𝑢𝑣_𝑥−𝑢_𝑥𝑣_𝑥 (53) belongs to 𝐿^∞([0,𝑇);𝐵^𝑠−2_𝑝,𝑟)². In particular, for the case 𝑟<+∞, Lemma 5 enables us to conclude that (𝑢,𝑣)∈𝒞([0,𝑇];𝐵^𝑠′_𝑝,𝑟)² for any 𝑠^′≤𝑠. Finally, using (53) again, we see that (𝜕_𝑡𝑢,𝜕_𝑡𝑣)∈𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟)² if 𝑟<+∞, and in 𝐿^∞([0,𝑇);𝐵^𝑠−1_𝑝,𝑟))² otherwise. Consequently, (𝑢,𝑣) belongs to 𝐸^𝑠_𝑝,𝑟(𝑇)².
Step 5 (uniqueness of solution). Let (𝑢⁽¹⁾,𝑣⁽¹⁾) and (𝑢⁽²⁾,𝑣⁽²⁾) be two given solutions of the Cauchy problem (4) with the initial data (𝑢₀⁽¹⁾,𝑣₀⁽¹⁾),(𝑢₀⁽²⁾,𝑣₀⁽²⁾)∈𝐵^𝑠_𝑝,𝑟×𝐵^𝑠_𝑝,𝑟 satisfying (𝑢⁽¹⁾,𝑣⁽¹⁾),(𝑢⁽²⁾,𝑣⁽²⁾)∈𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)∩𝒞([0,𝑇];𝒮^′)².
Denote 𝑢⁽¹²⁾≜𝑢⁽¹⁾−𝑢⁽²⁾, 𝑣⁽¹²⁾≜𝑣⁽¹⁾−𝑣⁽²⁾, 𝑚⁽¹²⁾≜𝑚⁽¹⁾−𝑚⁽²⁾ and ℎ⁽¹²⁾≜ℎ⁽¹⁾−ℎ⁽²⁾. It is obvious that 𝑢⁽¹²⁾,𝑣⁽¹²⁾∈𝐿^∞([0,𝑇];𝐵^𝑠_𝑝,𝑟)∩𝒞([0,𝑇];𝒮^′), which implies that 𝑢⁽¹²⁾,𝑣⁽¹²⁾∈𝒞([0,𝑇];𝐵^𝑠−1_𝑝,𝑟), and 𝑢⁽¹²⁾, 𝑣⁽¹²⁾, 𝑚⁽¹²⁾ and ℎ⁽¹²⁾ solves the transport equations 𝑚⁽¹²⁾𝜕_𝑡+−𝑎𝑢⁽¹⁾,𝑣⁽¹⁾𝑚⁽¹²⁾_𝑥=𝑓₁𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,𝑚⁽²⁾,𝑚⁽¹²⁾,ℎ⁽¹²⁾𝜕_𝑡+−𝑎𝑢⁽¹⁾,𝑣⁽¹⁾ℎ⁽¹²⁾_𝑥=𝑓₂𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,ℎ⁽²⁾,ℎ⁽¹²⁾,𝑚⁽¹²⁾_𝑡=0=𝑚₀⁽¹²⁾≜𝑚₀⁽¹⁾−𝑚₀⁽²⁾,ℎ⁽¹²⁾_𝑡=0=ℎ₀⁽¹²⁾≜ℎ₀⁽¹⁾−ℎ₀⁽²⁾, (54) where 𝑓₁𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,𝑚⁽²⁾,𝑚⁽¹²⁾=𝑎𝑢⁽¹⁾,𝑣⁽¹⁾−𝑎𝑢⁽²⁾,𝑣⁽²⁾𝑚⁽²⁾_𝑥+𝑎𝑢⁽¹⁾,𝑣⁽¹⁾𝑚⁽¹²⁾,𝑓₂𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,ℎ⁽²⁾,ℎ⁽¹²⁾=𝑎𝑢⁽¹⁾,𝑣⁽¹⁾−𝑎𝑢⁽²⁾,𝑣⁽²⁾ℎ⁽²⁾_𝑥+𝑎𝑢⁽¹⁾,𝑣⁽¹⁾ℎ⁽¹²⁾. (55) Applying (17) to the first equation in (54), we obtain that 𝑚⁽¹²⁾(𝑡)_{𝐵𝑠−3𝑝,𝑟}≤𝑚₀⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}𝑒^{𝐶∫𝑡0‖𝜕𝑥[−𝑎(𝑢(1),𝑣(1))](𝜏)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏}+^𝑡₀𝑒^{𝐶∫𝑡𝜏‖𝜕𝑥[−𝑎(𝑢(1),𝑣(1))](𝜏′)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏′}×𝑓₁𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,𝑚⁽²⁾,𝑚⁽¹²⁾(𝜏)_{𝐵𝑠−3𝑝,𝑟}𝑑𝜏, (56) which is equivalent to 𝑢⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0‖𝜕𝑥[−𝑎(𝑢(1),𝑣(1))](𝜏)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏}+^𝑡₀𝑒^{𝐶∫𝑡𝜏‖𝜕𝑥[−𝑎(𝑢(1),𝑣(1))](𝜏′)‖𝐵𝑠−4𝑝,𝑟𝑑𝜏′}×𝑓₁𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,𝑚⁽²⁾,𝑚⁽¹²⁾(𝜏)_{𝐵𝑠−3𝑝,𝑟}𝑑𝜏. (57) Since 𝐵^𝑠−2_𝑝,𝑟 is an algebra when 𝑠>max{2+(1/𝑝),5/2}, we get 𝑎𝑢⁽¹⁾,𝑣⁽¹⁾−𝑎𝑢⁽²⁾,𝑣⁽²⁾𝑚⁽²⁾_{𝑥𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢⁽¹⁾,𝑣⁽¹⁾−𝑎𝑢⁽²⁾,𝑣⁽²⁾𝑚⁽²⁾_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢_𝑥⁽¹²⁾𝑣_𝑥⁽¹⁾+𝑣_𝑥⁽¹²⁾𝑢_𝑥⁽²⁾−𝑢⁽¹²⁾𝑣⁽¹⁾−𝑣⁽¹²⁾𝑢⁽²⁾+𝑢⁽¹²⁾𝑣_𝑥⁽¹⁾+𝑣_𝑥⁽¹²⁾𝑢⁽²⁾−𝑢_𝑥⁽¹²⁾𝑣⁽¹⁾−𝑣⁽¹²⁾𝑢_𝑥⁽²⁾_{𝐵𝑠−2𝑝,𝑟}𝑚⁽²⁾_{𝐵𝑠−2𝑝,𝑟}≤𝐶𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}𝑢⁽²⁾_{𝐵𝑠𝑝,𝑟}𝑢⁽²⁾_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}. (58) If max{2+(1/𝑝),5/2}<𝑠≤3+(1/𝑝), using the Moser-type estimate (14), we have 𝜕_𝑥−𝑎𝑢⁽¹⁾,𝑣⁽¹⁾_{𝐵𝑠−4𝑝,𝑟}≤𝐶𝑎𝑢⁽¹⁾,𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}=𝐶𝑢_𝑥⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢⁽¹⁾𝑣⁽¹⁾+𝑢⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢_𝑥⁽¹⁾𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟},𝑎(𝑢⁽¹⁾,𝑣⁽¹⁾)𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎(𝑢⁽¹⁾,𝑣⁽¹⁾)_{𝐵𝑠−2𝑝,𝑟}𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}=𝐶𝑢_𝑥⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢⁽¹⁾𝑣⁽¹⁾+𝑢⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢_𝑥⁽¹⁾𝑣⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}+𝑢_𝑥⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−2𝑝,𝑟}𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}. (59)(60) If 𝑠>3+(1/𝑝), then 𝐵^𝑠−3_𝑝,𝑟 is an algebra. We directly obtain that 𝜕_𝑥−𝑎𝑢⁽¹⁾,𝑣⁽¹⁾_{𝐵𝑠−4𝑝,𝑟}≤𝐶𝑎𝑢⁽¹⁾,𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}=𝐶𝑢_𝑥⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢⁽¹⁾𝑣⁽¹⁾+𝑢⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢_𝑥⁽¹⁾𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟},𝑎𝑢⁽¹⁾,𝑣⁽¹⁾𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑎𝑢⁽¹⁾,𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}=𝐶𝑢_𝑥⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢⁽¹⁾𝑣⁽¹⁾+𝑢⁽¹⁾𝑣_𝑥⁽¹⁾−𝑢_𝑥⁽¹⁾𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}+𝑢_𝑥⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠−3𝑝,𝑟}𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑣⁽¹⁾_{𝐵𝑠𝑝,𝑟}𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}. (61)(62) It follows from (58), (60), and (62) that 𝑓₁𝑢⁽¹⁾,𝑢⁽²⁾,𝑣⁽¹⁾,𝑣⁽²⁾,𝑚⁽²⁾,𝑚⁽¹²⁾_{𝐵𝑠−3𝑝,𝑟}≤𝐶𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}. (63) Inserting (63) and (59) (or (61)) into (57), we find that 𝑢⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏)𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏′)𝑑𝜏′}×𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}𝑑𝜏. (64) For the component 𝑣 in (54), we derive the similar result as follows: 𝑣⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑣₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}𝑒^{𝐶∫𝑡0(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏)𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏′)𝑑𝜏′}×𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏. (65) Then, it follows from (64) and (65) that 𝑢⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑒^{𝐶∫𝑡0(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏)𝑑𝜏}+𝐶^𝑡₀𝑒^{𝐶∫𝑡𝜏(‖𝑢(1)‖2𝐵𝑠𝑝,𝑟+‖𝑢(2)‖2𝐵𝑠𝑝,𝑟+‖𝑣(1)‖2𝐵𝑠𝑝,𝑟+‖𝑣(2)‖2𝐵𝑠𝑝,𝑟)(𝜏′)𝑑𝜏′}×𝑢⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏. (66) Applying Gronwall’s lemma, we end up with 𝑢⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}+𝑣⁽¹²⁾(𝑡)_{𝐵𝑠−1𝑝,𝑟}≤𝑢₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹²⁾_{𝐵𝑠−1𝑝,𝑟}×exp𝐶^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏. (67)
Step 6 (continuity with respect to the initial data in 𝐸^𝑠′_𝑝,𝑟(𝑇)² with 𝑠^′<𝑠). When 𝑠^′=𝑠−1, the continuity with respect to the initial data can be obtained from Step 5.
When 𝑠^′<𝑠−1, by using Proposition 3(6) and (67) we have 𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)_{𝐵𝑠′𝑝,𝑟}+𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)_{𝐵𝑠′𝑝,𝑟}≤𝐶𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)^𝜃1_{𝐵𝑠1𝑝,𝑟}𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)^1−𝜃1_{𝐵𝑠−1𝑝,𝑟}+𝐶𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)^𝜃1_{𝐵𝑠1𝑝,𝑟}𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)^1−𝜃1_{𝐵𝑠−1𝑝,𝑟}≤𝐶𝑢⁽¹⁾(𝑡)_{𝐵𝑠1𝑝,𝑟}+𝑢⁽²⁾(𝑡)_{𝐵𝑠1𝑝,𝑟}^𝜃1×𝑢₀⁽¹⁾−𝑢₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}^1−𝜃1×exp𝐶1−𝜃₁^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑣𝑑𝜏+𝐶⁽¹⁾(𝑡)_{𝐵𝑠1𝑝,𝑟}+𝑣⁽²⁾(𝑡)_{𝐵𝑠1𝑝,𝑟}^𝜃1×𝑢₀⁽¹⁾−𝑢₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}^1−𝜃1×exp𝐶1−𝜃₁^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑢𝑑𝜏≤𝐶⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^𝜃1+𝑣⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^𝜃1×𝑢₀⁽¹⁾−𝑢₀^(2)𝜃2_{𝐵𝑠′𝑝,𝑟}𝑢₀⁽¹⁾−𝑢₀^(2)1−𝜃2_{𝐵𝑠𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀^(2)𝜃2_{𝐵𝑠′𝑝,𝑟}𝑣₀⁽¹⁾−𝑣₀^(2)1−𝜃2_{𝐵𝑠𝑝,𝑟}^1−𝜃1×exp𝐶1−𝜃₁^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑢𝑑𝜏≤𝐶⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^𝜃1+𝑣⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^𝜃1×𝑢₀⁽¹⁾−𝑢₀^(2)𝜃2_{𝐵𝑠′𝑝,𝑟}𝑢₀^(1)1−𝜃2_{𝐵𝑠𝑝,𝑟}+𝑢₀^(2)1−𝜃2_{𝐵𝑠𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀^(2)𝜃2_{𝐵𝑠′𝑝,𝑟}𝑣₀^(1)1−𝜃2_{𝐵𝑠𝑝,𝑟}+𝑣₀^(2)1−𝜃2_{𝐵𝑠𝑝,𝑟}^1−𝜃1×exp𝐶1−𝜃₁^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏, (68) where 𝑠₁<𝑠^′, 𝑠^′=𝜃₁𝑠₁+(1−𝜃₁)(𝑠−1) and 𝑠−1=𝜃₂𝑠^′+(1−𝜃₂)𝑠.
When 𝑠−1<𝑠^′<𝑠, by using Proposition 3(6) and (67) again, we have 𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)_{𝐵𝑠′𝑝,𝑟}+𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)_{𝐵𝑠′𝑝,𝑟}≤𝐶𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)^𝜃3_{𝐵𝑠−1𝑝,𝑟}𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡)^1−𝜃3_{𝐵𝑠𝑝,𝑟}+𝐶𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)^𝜃1_{𝐵𝑠−1𝑝,𝑟}𝑣⁽¹⁾(𝑡)−𝑣⁽²⁾(𝑡)^1−𝜃3_{𝐵𝑠𝑝,𝑟}≤𝐶𝑢⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^1−𝜃3×𝑢₀⁽¹⁾−𝑢₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}^𝜃3×exp𝐶𝜃₃^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑣𝑑𝜏+𝐶⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^1−𝜃3×𝑢₀⁽¹⁾−𝑢₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀⁽²⁾_{𝐵𝑠−1𝑝,𝑟}^𝜃3×exp𝐶𝜃₃^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑢𝑑𝜏≤𝐶⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^1−𝜃3+𝑣⁽¹⁾(𝑡)_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾(𝑡)_{𝐵𝑠𝑝,𝑟}^1−𝜃3×𝑢₀⁽¹⁾−𝑢₀⁽²⁾_{𝐵𝑠′𝑝,𝑟}+𝑣₀⁽¹⁾−𝑣₀⁽²⁾_{𝐵𝑠′𝑝,𝑟}^𝜃3×exp𝐶𝜃₃^𝑡₀𝑢⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑢⁽²⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽¹⁾²_{𝐵𝑠𝑝,𝑟}+𝑣⁽²⁾²_{𝐵𝑠𝑝,𝑟}(𝜏)𝑑𝜏, (69) where 𝑠^′=𝜃₃(𝑠−1)+(1−𝜃₃)𝑠.
Step 7 (continuity with respect to the initial data in 𝐸^𝑠_𝑝,𝑟(𝑇)²). A standard use of a sequence of viscosity approximate solutions (𝑢_𝜀,𝑣_𝜀)_𝜀>0 for the Cauchy problem (4) which converges uniformly in 𝐸^𝑠_𝑝,𝑟(𝑇)² gives the continuity of the solution (𝑢,𝑣) in 𝐸^𝑠_𝑝,𝑟(𝑇)².
Consequently, we complete the proof of the theorem.

When 𝑝=𝑟=2, the Besov space 𝐵^𝑠_𝑝,𝑟(ℝ) coincides with the Sobolev space 𝐻^𝑠(ℝ). So we have the following corollary.

Corollary 7. Suppose that 𝑠>5/2 and (𝑢₀,𝑣₀)∈𝐻^𝑠×𝐻^𝑠. Then there exists a time 𝑇>0 such that the Cauchy problem (4) has a unique solution (𝑢,𝑣)∈𝐻^𝑠×𝐻^𝑠 and the map (𝑢₀,𝑣₀)→(𝑢,𝑣) is continuous from a neighborhood of (𝑢₀,𝑣₀) in 𝐵^𝑠_𝑝,𝑟×𝐵^𝑠_𝑝,𝑟 into 𝒞[0,𝑇];𝐻^𝑠𝒞¹[0,𝑇];𝐻^𝑠−12. (70)

4. Open Problems

In this paper, we use some standard a priori estimates for linear transport equation to prove the local well-posedness of the Cauchy problem for a modified two-component Camassa-Holm system with cubic nonlinearity in Besov spaces with minimal regularity assumptions on the initial data.

There are some interesting but challenging problems which are still open. The first is whether the index 𝑠=max{2+(1/𝑝),5/2} is sharp for local well-posedness of the Cauchy problem (4). The second is whether the solution can exist globally in time under certain assumptions on the initial data. The third is whether the solution can blow up in finite time with special initial profiles. We will work on these problems in the near future.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (nos. 11171135, 10420130638), the Nature Science Foundation of Jiangsu (no. BK 2010329), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), and the Taizhou Social Development Project (no. 2011213).