The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System

Abstract

The inviscid limit problem for the smooth solutions of the Boussinesq system is studied in this paper. We prove the 𝐻^𝑠 convergence result of this system as the diffusion and the viscosity coefficients vanish with the initial data belonging to 𝐻^𝑠. Moreover, the 𝐻^𝑠 convergence rate is given if we allow more regularity on the initial data.

1. Introduction and the Main Results

The two-dimensional Boussinesq system for the homogeneous incompressible fluids with diffusion and viscosity is given by 𝐮_𝑡+(𝐮⋅∇)𝐮+∇𝑝=𝜈Δ𝐮+𝜃𝐞₂,𝜃_𝑡+()𝑢|𝐮⋅∇𝜃=𝜅Δ𝜃,∇⋅𝐮=0,_𝑡=0=𝑢₀,𝜃|_𝑡=0=𝜃₀, (1) where the space variable 𝐱=(𝑥₁,𝑥₂) is in ℝ², 𝐮=(𝑢¹(𝑡,𝐱),𝑢²(𝑡,𝐱)) is the velocity, 𝑝=𝑝(𝑡,𝐱) denotes the scalar pressure and 𝜃=𝜃(𝑡,𝐱) the scalar temperature, 𝐞₂=(0,1), and 𝜅>0 and 𝜈>0 denote, respectively, the molecular diffusion and the viscosity. Such Boussinesq systems are simple models widely used in the modeling of oceanic and atmospheric motions, and these models also appear in many other physical problems; see [1, 2] for more discussions. It is also interesting to consider the system (1) without diffusion and viscosity namely (for the sake of convenience for our limit argument, we use different notation), 𝐯_𝑡+(𝐯⋅∇)𝐯+∇𝜋=𝜌𝐞₂,𝜌_𝑡+(𝐯⋅∇)𝜌=0,∇⋅𝐯=0,𝐯∣_𝑡=0=𝐯₀,𝜌∣_𝑡=0=𝜌₀. (2) Moreover, it is known that the two-dimensional viscous (resp., inviscid) Boussinesq equations are closely related to the three-dimensional axisymmetric Navier-Stokes equations (resp., Euler equations) with swirl. Therefore, the Boussinesq systems, especially in two-dimensional case, have been widely studied by many researchers, and we refer, for instance, to [3–9] and the references therein.

It is well known that the system (1) has a unique global in time regularity solution. Moreover, Hou and Li in [9] obtained the global existence of smooth solution for (1) even with the zero diffusivity case (i.e., 𝜈>0 and 𝜅=0). Meanwhile, Chae in [5] also proved global regularity for the 2D Boussinesq system (1) both with the zero diffusivity case (𝜈>0 and 𝜅=0) and the zero viscosity case (𝜈=0 and 𝜅>0). However, for the case 𝜈=0 and 𝜅=0, it is only known that smooth solution exists locally in time (see, e.g., [4]), and it is not known whether such smooth solutions can develop singularities in finite time. In fact, as well as the famous blow-up problem for the Navier-Stokes equations or Euler equations, the regularity or singularity question for the locally smooth solution of the system (2) appears also as an outstanding open problem in the mathematical fluid mechanics; see [10].

In this paper, we are interested in studying the limit behavior of the smooth solution for (1) as (𝜈,𝜅)→0; that is, we study the vanishing viscosity limit of solutions of (1). This type limit problem appears not only in the community of applied mathematics, but also in physical reality. A good example of this problem is the vanishing viscosity limit of solutions of the Navier-Stokes equations which appears as a singular limit especially in bounded domains due to the boundary layers effect, and we refer to [11–17] and the references therein. Most of the previous convergence results require some loss of derivatives; namely, if the initial data lies in the space 𝐻^𝑠, usually one can obtain the convergence results in 𝐻^𝑠′ with 𝑠^′<𝑠. In this literature, Masmoudi in [15] obtained the inviscid limit results for the Navier-Stokes equations without loss of derivatives. Inspired by [15], in this paper, we obtain the 𝐻^𝑠 convergence of the solution with the initial data belonging to the same space.

Now we state our main results of the paper.

Theorem 1. Let 𝑠>3, (𝐮₀^𝜈,𝜅,𝜃₀^𝜈,𝜅)∈𝐇^𝑠(ℝ²;ℝ²)×𝐻^𝑠(ℝ²), and (𝐯₀,𝜌₀)∈𝐇^𝑠(ℝ²;ℝ²)×𝐻^𝑠(ℝ²) satisfying ∇⋅𝐮₀^𝜈,𝜅=0, ∇⋅𝐯₀=0, and 𝐮₀^𝜈,𝜅⟶𝐯₀,𝜃₀^𝜈,𝜅⟶𝜌₀𝑖𝑛𝐻^𝑠, (3) as (𝜈,𝜅)→0. Assume that (𝐮^𝜈,𝜅,𝜃^𝜈,𝜅)∈𝐶(ℝ⁺;𝐇^𝑠×𝐻^𝑠) is the classical solution of (1) with initial data (𝐮₀^𝜈,𝜅,𝜃₀^𝜈,𝜅), and (𝐯,𝜌)∈𝐶([0,𝑇^∗);𝐇^𝑠×𝐻^𝑠) is the classical solution of the inviscid system (2) with initial data (𝐯₀,𝜌₀); here; 𝑇^∗ is the maximal existence time of the solution (𝐯,𝜌). Then, for any 𝑇₀∈(0,𝑇^∗) and for any 𝑡∈[0,𝑇₀], one has(1)(convergence rate in the 𝐻^𝑠−2 norm) 𝐮^𝜈,𝜅(𝑡)−𝐯(𝑡)_𝐇𝑠−2+𝜃^𝜈,𝜅(𝑡)−𝜌(𝑡)_𝐻𝑠−2≤𝐶₁𝜈+𝜅+𝐮₀^𝜈,𝜅−𝐯_{0𝐇𝑠−2}+𝜃₀^𝜈,𝜅−𝜌_{0𝐻𝑠−2}, (4)(2) (convergence rate in the 𝐻^𝑠1 norm with 𝑠−2<𝑠₁≤𝑠−1) 𝐮^𝜈,𝜅(𝑡)−𝐯(𝑡)_𝐇𝑠1+𝜃^𝜈,𝜅(𝑡)−𝜌(𝑡)_𝐻𝑠1≤𝐶₂𝜈^{(𝑠−𝑠1)/2}+𝜅^{(𝑠−𝑠1)/2}+𝐮₀^𝜈,𝜅−𝐯_0𝐇𝑠1+𝜃₀^𝜈,𝜅−𝜌_0𝐻𝑠1, (5)(3) (convergence rate in the 𝐻^𝑠2 norm with 𝑠−1<𝑠₂<𝑠) 𝐮^𝜈,𝜅(𝑡)−𝐯(𝑡)_𝐇𝑠2+𝜃^𝜈,𝜅(𝑡)−𝜌(𝑡)_𝐻𝑠2≤𝐶₃𝜈^{(𝑠−𝑠2)/2}+𝜅^{(𝑠−𝑠2)/2}+𝐮₀^𝜈,𝜅−𝐯₀^𝑠−𝑠2_𝐇𝑠−1+𝜃₀^𝜈,𝜅−𝜌₀^𝑠−𝑠2_𝐻𝑠−1, (6)(4) (convergence in the 𝐻^𝑠 norm) 𝐮^𝜈,𝜅−𝐯_{𝐶([0,𝑇0];𝐇𝑠)}+𝜃^𝜈,𝜅−𝜌_{𝐶([0,𝑇0];𝐻𝑠)}⟶0as(𝜈,𝜅)⟶0, (7)where the constants 𝐶_𝑖(𝑖=1,2,3) in (4)–(6) are dependent of 𝑇₀ and the 𝐻^𝑠 norm of the initial data (𝐯₀,𝜌₀), but independent of the parameters 𝜈 and 𝜅.

Of course, we can generalize the previous results to arbitrary spatial dimension case with 𝑠>3 replaced by 𝑠>(𝑛/2)+2. The important part of Theorem 1 is the convergence result (7). This result tells us that the 𝐻^𝑠 convergence can be maintained by the solution at its arbitrary existence time. We emphasize that 𝑇^∗ is not assumed to be small; indeed, the standard energy estimate yields that the classical solution (𝐯,𝜌) blows up at time 𝑇^∗ if and only if ‖𝐯(𝑡)‖_𝐇^𝑠+‖𝜌(𝑡)‖_𝐻𝑠→∞ as 𝑡↑𝑇^∗. Note that the rate of 𝐻^𝑠 convergence depends on how we regularize our initial data; see (75) in the next section. Moreover, if one allows more regularity on the initial data (𝐯₀,𝜌₀), then we can obtain the following 𝐻^𝑠 convergence rate.

Theorem 2. Suppose that the same assumptions as Theorem 1 hold. Moreover, one assumes that (𝐯₀,𝜌₀)∈𝐇^𝑠+𝛿×𝐻^𝑠+𝛿 with 𝛿∈(0,2]. Then, for any 𝑡∈[0,𝑇₀], there hold 𝐮^𝜈,𝜅(𝑡)−𝐯(𝑡)_𝐇𝑠+𝜃^𝜈,𝜅(𝑡)−𝜌(𝑡)_𝐻𝑠≤𝐶₄𝜈^𝛿/2+𝜅^𝛿/2+𝐮₀^𝜈,𝜅−𝐯_0𝐇𝑠+𝜃₀^𝜈,𝜅−𝜌_0𝐻𝑠,𝐮1≤𝛿≤2,_𝜈,𝜅(𝑡)−𝐯(𝑡)_𝐇𝑠+𝜃^𝜈,𝜅(𝑡)−𝜌(𝑡)_𝐻𝑠≤𝐶₅𝜈^𝛿/2+𝜅^𝛿/2+𝐮₀^𝜈,𝜅−𝐯_0𝐇𝑠+𝜃₀^𝜈,𝜅−𝜌_0𝐻𝑠+𝐶𝐮₀^𝜈,𝜅−𝐯_{0𝐇𝑠−2}+𝜃₀^𝜈,𝜅−𝜌_{0𝐻𝑠−2}^𝛿,0<𝛿<1, (8)(9) where the constants 𝐶₄ and 𝐶₅ depend only on 𝑇₀, ‖𝑣₀‖_𝐇^𝑠+𝛿, and ‖𝜌₀‖_{𝐻𝑠+𝛿}.

Finally, we end this section by setting some notations which will be used throughout the paper. For 𝑝∈[1,∞], ‖⋅‖_𝐿^𝑝 denotes the norm in the Lebesgue space 𝐿^𝑝(ℝ^𝑛). We set the operator 𝐽∶=(𝐼−Δ)^1/2, and for 𝑠∈ℝ, we denote by 𝐻^𝑠,𝑝(ℝ^𝑛) the nonhomogeneous Sobolev spaces with the norm defined as ‖⋅‖_𝐻^𝑠,𝑝∶=𝐽^𝑠⋅_𝐿𝑝. (10) If 𝑝=2, for brevity, we write 𝐻^𝑠(ℝ^𝑛) instead of 𝐻^𝑠,2(ℝ^𝑛). Obviously, 𝐻⁰(ℝ^𝑛)=𝐿²(ℝ^𝑛). In some places, we use the notation 𝐇^𝑠(ℝ^𝑛;ℝ^𝑘) to mean that this space consists of vector-valued functions 𝐟∶ℝ^𝑛→ℝ^𝑘 with each component of 𝐟 belonging to 𝐻^𝑠(ℝ^𝑛). If there is no confusion, the spaces 𝐇^𝑠(ℝ^𝑛) and 𝐇^𝑠(ℝ^𝑛;ℝ^𝑘) will be simply denoted by 𝐻^𝑠. For 𝐟,𝐠∈𝐿²(ℝ^𝑛;ℝ^𝑘), we denote by ⟨𝐟,𝐠⟩ the usual inner product of 𝐟 and 𝐠; namely, ⟨𝐟,𝐠⟩∶=_ℝ𝑛𝐟(𝐱)⋅𝐠(𝐱)𝑑𝐱. (11) For any Banach space 𝐁, the space 𝐿^𝑝(0,𝑇;𝐁) consists of all strongly measurable functions 𝐮∶[0,𝑇]→𝐁 equipped with the norm ‖𝐮‖_𝐿^𝑝_{(0,𝑇;𝐁)}∶=^𝑇₀‖𝐮(𝑡)‖^𝑃_𝐁𝑑𝑡^1/𝑝<∞ (12) for 1≤𝑝<∞, and ‖𝐮‖_𝐿^∞_{(0,𝑇;𝐁)}=esssup_{𝑡∈[0,𝑇]}‖𝐮(𝑡)‖_𝐁<∞. (13) And the space 𝐶([0,𝑇];𝐁) denotes the set of continuous functions 𝐮∶[0,𝑇]→𝐁 with ‖𝐮‖_{𝐶([0,𝑇];𝐁)}=max_{𝑡∈[0,𝑇]}‖𝐮(𝑡)‖_𝐁<∞. (14) In this paper, the letter 𝐶 is a generic constant and its value may change at each appearance. Moreover, every 𝐶 is independent of the parameters 𝜈 and 𝜅.

2. Proof of Theorem 1

In this section, we present the proof of Theorem 1. To this goal, we need the following calculus inequality, the proof of which can be found in [18, 19].

Lemma 3. Assume that 𝑠>0 and 𝑝∈(1,+∞). If 𝑓,𝑔∈𝒮(ℝ^𝑛), the Schwartz class, then 𝐽^𝑠𝑓𝑔−𝑓𝐽^𝑠𝑔_𝐿𝑝≤𝐶∇𝑓_𝐿𝑝1𝑔_{𝐻2𝑠−1,𝑝}+𝑓_{𝐻3𝑠,𝑝}𝑔_𝐿𝑝4,𝐽^𝑠(𝑓𝑔)_𝐿𝑝≤𝐶𝑓_𝐿𝑝1𝑔_{𝐻2𝑠,𝑝}+𝑓_{𝐻3𝑠,𝑝}𝑔_𝐿𝑝4 (15)(16) with 𝑝₂,𝑝₃∈(1,+∞) such that 1𝑝=1𝑝₁+1𝑝₂=1𝑝₃+1𝑝₄. (17)

Of course, Lemma 3 also holds when 𝑓 and 𝑔 are replaced by vector-valued functions. Using (15) and (16), we have the following result.

Lemma 4. Let 𝑠>1+(𝑛/2). Then,(1)for any 𝐟,𝐠∈𝐇^𝑠(ℝ^𝑛;ℝ^𝑛)∩𝐇^𝑟(ℝ^𝑛;ℝ^𝑛) with ∇⋅𝐟=0, there hold 𝐽^𝑟((𝐟⋅∇)𝐠),𝐽^𝑟𝐠≤𝐶‖𝐟‖_𝐻^𝑠‖𝐠‖²_𝐻^𝑟𝐽,1<𝑟≤𝑠,_𝑟((𝐟⋅∇)𝐠),𝐽^𝑟𝐠≤𝐶‖𝐟‖_𝐻^𝑠‖𝐠‖²_𝐻^𝑟+𝐶‖𝐟‖_𝐻^𝑟‖𝐠‖_𝐻^𝑟‖𝐠‖_𝐻^𝑠,𝑟>𝑠; (18)(19)(2)for any 𝐟∈𝐇^𝑠(ℝ^𝑛;ℝ^𝑛), 𝐠,𝐡∈𝐇^𝑟(ℝ^𝑛;ℝ^𝑛) with 𝑟∈[0,𝑠−1], there holds 𝐽^𝑟((𝐠⋅∇)𝐟),𝐽^𝑟𝐡≤𝐶‖𝐟‖_𝐻^𝑠‖𝐠‖_𝐻^𝑟‖𝐡‖_𝐻^𝑟. (20)

Proof. Using the divergence free condition and the communicator estimate (15), one sees 𝐽^𝑟((𝐟⋅∇)𝐠),𝐽^𝑟𝐠=𝐽^𝑟((𝐟⋅∇)𝐠)−𝐟⋅𝐽^𝑟∇𝐠,𝐽^𝑟𝐠≤𝐶‖∇𝐟‖_𝐿^∞𝐽^𝑟−1∇𝐠_𝐿2+𝐽^𝑟𝐟_𝐿𝑝‖∇𝐠‖_𝐿^𝑞𝐽^𝑟𝐠_𝐿2≤𝐶‖𝐟‖_𝐻^𝑠‖𝐠‖_𝐻^𝑟+‖𝐟‖_𝐻^𝑟,𝑝‖𝐠‖_𝐻^1,𝑞‖𝐠‖_𝐻^𝑟, (21) where 𝑝 and 𝑞 satisfy 1/𝑝+1/𝑞=1/2(1<𝑝<∞). Since 𝑠>1+(𝑛/2) and 𝑟∈(1,𝑠], we can always choose 𝑝,𝑞 such that 𝐻^𝑠↪𝐻^𝑟,𝑝 and 𝐻^𝑟−1↪𝐿^𝑞. In the case 𝑟>𝑠, we choose 𝑝=2 and 𝑞=∞. Hence, the estimates (18) and (19) follow immediately. For the estimate (20), the case 𝑟=0 is treated by the Hölder inequality, and for 𝑟>0, we use (16); then, 𝐽^𝑟((𝐠⋅∇)𝐟),𝐽^𝑟𝐡≤𝐶𝐽^𝑟𝐠_𝐿2‖∇𝐟‖_𝐿^∞+‖𝐠‖_𝐿^𝑝𝐽^𝑟∇𝐟_𝐿𝑞𝐽^𝑟𝐡_𝐿2≤𝐶‖𝐠‖_𝐻^𝑟‖𝐟‖_𝐻^𝑠+‖𝐠‖_𝐿^𝑝‖𝐟‖_𝐻^𝑟+1,𝑞‖𝐡‖_𝐻^𝑟. (22) With the condition, we can choose 𝑝 and 𝑞 satisfying 𝐻^𝑟↪𝐿^𝑝, 𝐻^𝑠↪𝐻^𝑟+1,𝑞, and 1/𝑝+1/𝑞=1/2(1<𝑝<∞), and thus (20) follows.

To prove Theorem 1, we first establish the uniform bounds for the solutions of (1) with the bound independent of 𝜈 and 𝜅.

Lemma 5. With the same hypotheses as Theorem 1, then there exist 𝑇>0 and 𝐶>0 such that for sufficiently small 𝜈, 𝜅>0, there holds 𝐮^𝜈,𝜅_{𝐶([0,𝑇];𝐻𝑠)}+𝜃^𝜈,𝜅_{𝐶([0,𝑇];𝐻𝑠)}≤𝐶 (23) with 𝑇 and 𝐶 both depending only on the 𝐻^𝑠 norm of (𝐯₀,𝜌₀) and not depending on 𝜈 and 𝜅.

Proof. From the first equation of (1), we have 𝐽^𝑠𝐮^𝜈,𝜅_𝑡+𝐽^𝑠𝐮^𝜈,𝜅⋅∇𝐮^𝜈,𝜅+∇𝐽^𝑠𝑝^𝜈,𝜅=𝜈Δ𝐽^𝑠𝐮^𝜈,𝜅+𝐽^𝑠𝜃^𝜈,𝜅𝐞₂. (24) Multiplying this equation by 𝐽^𝑠𝐮^𝜈,𝜅 and integrating the result, while noting that _ℝ2∇𝐽^𝑠𝑝^𝜈,𝜅⋅𝐽^𝑠𝐮^𝜈,𝜅𝑑𝑥=−_ℝ2𝐽^𝑠𝑝^𝜈,𝜅⋅𝐽^𝑠∇⋅𝐮^𝜈,𝜅𝑑𝑥=0,_ℝ2𝐽^𝑠𝐮^𝜈,𝜅⋅∇𝐮^𝜈,𝜅⋅𝐽^𝑠𝐮^𝜈,𝜅𝑑𝑥≤𝐶𝐽^𝑠𝐮^𝜈,𝜅3_𝐿2, (25) where we have used the estimate (18) in the previous inequality, then we obtain 12𝑑𝐽𝑑𝑡^𝑠𝐮^𝜈,𝜅2_𝐿2+𝜈∇𝐽^𝑠𝐮^𝜈,𝜅2_𝐿2≤𝐶𝐽^𝑠𝐮^𝜈,𝜅3_𝐿2+𝐽^𝑠𝐮^𝜈,𝜅_𝐿2𝐽^𝑠𝜃^𝜈,𝜅_𝐿2, (26) which gives 𝑑𝐮𝑑𝑡^{𝜈,𝜅𝐻𝑠}≤𝐶𝐮^𝜈,𝜅2_𝐻𝑠+𝜃^𝜈,𝜅_𝐻𝑠. (27) Using the same argument to the second equation of (1), we can obtain 12𝑑𝐽𝑑𝑡^𝑠𝜃^𝜈,𝜅2_𝐿2+𝜅∇𝐽^𝑠𝜃^𝜈,𝜅2_𝐿2≤𝐶𝐽^𝑠𝐮^𝜈,𝜅_𝐿2𝐽^𝑠𝜃^𝜈,𝜅2_𝐿2. (28) Then, we have 𝑑𝜃𝑑𝑡^{𝜈,𝜅𝐻𝑠}≤𝐶𝐮^𝜈,𝜅_𝐻𝑠𝜃^𝜈,𝜅_𝐻𝑠. (29) Hence, it concludes from the estimates (27) and (29) that 𝑑𝜑𝑑𝑡^𝜈,𝜅(𝑡)≤𝐶𝜑^𝜈,𝜅(𝑡)²,𝜑^𝜈,𝜅(𝑡)∶=𝐮^𝜈,𝜅(𝑡)_𝐻𝑠+𝜃^𝜈,𝜅(𝑡)_𝐻𝑠+1. (30) Solving this ODE gives 𝜑^𝜈,𝜅(𝑡)≤𝜑^𝜈,𝜅(0)1−𝐶𝜑^𝜈,𝜅(0)𝑡. (31) Since (3) holds, we may assume 𝜑^𝜈,𝜅(0)≤𝐶₀ for small 𝜈 and 𝜅 that here, 𝐶₀ depends only on ‖𝐯₀‖_𝐻𝑠 and ‖𝜌₀‖_𝐻𝑠. Hence, we finally arrive at 𝜑^𝜈,𝜅(𝑡)≤𝐶₀1−𝐶𝐶₀𝑡. (32) The estimate (23) follows from the previous inequality provided that we select 𝑇 such that 𝑇<1/𝐶𝐶₀ (e.g., we can choose 𝑇=1/2𝐶𝐶₀). The proof of Lemma 5 is complete.

Remark 6. From the proof of Lemma 5, we also see the solution of system (2) satisfying ‖𝐯‖_{𝐶([0,𝑇];𝐻}^𝑠₎+𝜌_{𝐶([0,𝑇];𝐻𝑠)}≤𝐶, (33) where 𝑇 and 𝐶 are the same as (23).

Remark 7. For fixed 𝑇₀∈(0,𝑇^∗), without loss of generality, we may assume that the time 𝑇 determined by Lemma 5 satisfies 𝑇<𝑇₀. Indeed, as will be seen in the proof of Theorem 1, no matter how small the 𝑇 is, we can always use bootstrap argument to extend the interval [0,𝑇] into our desired time interval [0,𝑇₀].
In the following, we define 𝐰^𝜈,𝜅∶=𝐮^𝜈,𝜅−𝐯 and 𝜒^𝜈,𝜅∶=𝜃^𝜈,𝜅−𝜌. Considering the equations, for (𝐮^𝜈,𝜅,𝜃^𝜈,𝜅) and (𝐯,𝜌), we see (𝐰^𝜈,𝜅,𝜒^𝜈,𝜅) satisfying 𝐰_𝑡^𝜈,𝜅+(𝐯⋅∇)𝐰^𝜈,𝜅+𝐰^𝜈,𝜅⋅∇𝐮^𝜈,𝜅+∇𝑝^𝜈,𝜅−𝜋=𝜈Δ𝐰^𝜈,𝜅+𝜈Δ𝐯+𝜒^𝜈,𝜅𝐞₂,𝜒_𝑡^𝜈,𝜅+(𝐯⋅∇)𝜒^𝜈,𝜅+𝐰^𝜈,𝜅⋅∇𝜃^𝜈,𝜅=𝜅Δ𝜒^𝜈,𝜅+𝜅Δ𝜌,∇⋅𝐰^𝜈,𝜅𝐰=0,_𝜈,𝜅|_𝑡=0=𝐮₀^𝜈,𝜅−𝐯₀,𝜒^𝜈,𝜅|_𝑡=0=𝜃₀^𝜈,𝜅−𝜌₀. (34) For the sake of convenience, we often omit the superscripts 𝜈 and 𝜅 in the succeeding arguments; hence, 𝐰 means 𝐰^𝜈,𝜅, 𝜃 stands for 𝜃^𝜈,𝜅, and so on.

Proof of Theorem 1. We split the proof into several steps.

Step 1. We first show that (4) holds on [0,𝑇]. By using (18) and (20) with 𝑟=𝑠−2, we can obtain the 𝐻^𝑠−2 energy for 𝐰 as 12𝑑‖𝐰‖𝑑𝑡^{2𝐻𝑠−2}+𝜈‖∇𝐰‖²_𝐻𝑠−2≤𝐶‖𝐯‖_𝐻^𝑠+‖𝐮‖_𝐻^𝑠‖𝐰‖²_𝐻𝑠−2+𝜈‖Δ𝐯‖_𝐻^𝑠−2‖𝐰‖_𝐻^𝑠−2+𝜒_𝐻𝑠−2‖𝐰‖_𝐻^𝑠−2≤𝐶‖𝐰‖²_𝐻𝑠−2+𝐶𝜈‖𝐰‖_𝐻^𝑠−2+𝜒_𝐻𝑠−2‖𝐰‖_𝐻^𝑠−2, (35) where we have used the uniform estimates (23) and (33) in the last step. Hence, we get 𝑑‖𝐰‖𝑑𝑡^𝐻𝑠−2≤𝐶‖𝐰‖_𝐻^𝑠−2+𝐶𝜈+𝜒_𝐻𝑠−2. (36) Similarly, the 𝐻^𝑠−2 energy estimate for 𝜒 can be written as 12𝑑𝜒𝑑𝑡^{2𝐻𝑠−2}+𝜅∇𝜒²_𝐻𝑠−2≤𝐶‖Δ𝐯‖_𝐻^𝑠𝜒²_𝐻𝑠−2+‖𝐰‖_𝐻^𝑠−2‖𝜃‖_𝐻^𝑠𝜒_𝐻𝑠−2+𝜅Δ𝜌_𝐻𝑠−2𝜒_𝐻𝑠−2≤𝐶𝜒²_𝐻𝑠−2+𝐶‖𝐰‖_𝐻^𝑠−2𝜒_𝐻𝑠−2+𝐶𝜅𝜒_𝐻𝑠−2 (37) which gives 𝑑𝜒𝑑𝑡^𝐻𝑠−2≤𝐶𝜒_𝐻𝑠−2+𝐶‖𝐰‖_𝐻^𝑠−2+𝐶𝜅. (38) Therefore, one has 𝑑𝜙𝑑𝑡¹(𝑡)≤𝐶𝜙₁(𝑡)𝜙+𝐶𝜈+𝐶𝜅,₁(𝑡)∶=‖𝐰(𝑡)‖_𝐻^𝑠−2+𝜒(𝑡)_𝐻𝑠−2. (39) Then, the Gronwall inequality yields that for all 𝑡∈[0,𝑇](⊂[0,𝑇₀]), 𝜙₁(𝑡)≤𝑒^𝐶𝑡𝜙₁(0)+𝐶(𝜈+𝜅)𝑡𝑒^𝐶𝑡≤𝐶₁𝜙₁(0)+𝜈+𝜅, (40) where 𝐶₁∶=𝑒^𝐶𝑇0(1+𝐶𝑇₀) with 𝐶 depending on ‖𝑣₀‖_𝐻𝑠 and ‖𝜌₀‖_𝐻𝑠.

Step 2. We show that (5) holds on [0,𝑇]. Applying the estimates (18) and (20) with 𝑟=𝑠₁, then we can easily obtain the 𝐻^𝑠1 energy for 𝐰 as 12𝑑‖𝐰‖𝑑𝑡^2𝐻𝑠1+𝜈‖∇𝐰‖²_𝐻^𝑠₁≤𝐶‖𝐯‖_𝐻^𝑠+‖𝐮‖_𝐻^𝑠‖𝐰‖²_𝐻^𝑠₁+𝜈‖∇𝐯‖_𝐻^𝑠−1‖∇𝐰‖_𝐻1^{2𝑠−𝑠+1}+𝜒_𝐻𝑠1‖𝐰‖_𝐻^𝑠₁≤𝐶‖𝐰‖²_𝐻^𝑠₁+𝐶𝜈‖∇𝐰‖_𝐻1^{2𝑠−𝑠+1}+𝜒_𝐻𝑠1‖𝐰‖_𝐻^𝑠₁. (41) Since 𝑠₁∈(𝑠−2,𝑠−1], we have 𝑠₁−1<2𝑠₁−𝑠+1≤𝑠₁. By the Gagliardo-Nirenberg interpolation inequality and Young's inequality, we have 𝐶‖∇𝐰‖_𝐻1^{2𝑠−𝑠+1}≤𝐶‖∇𝐰‖^𝑠−𝑠1_𝐻⁻¹_𝑠1−1‖∇𝐰‖^{2−𝑠+𝑠1}_𝐻𝑠1≤𝐶‖𝐰‖^{2(𝑠−𝑠1−1)/(𝑠−𝑠1)}_𝐻𝑠1+‖∇𝐰‖²_𝐻^𝑠₁. (42) Inserting this inequality into (41) and using Young's inequality again, we thus get 12𝑑‖‖𝑑𝑡∇𝐰²_𝐻^𝑠₁≤𝐶‖∇𝐰‖²_𝐻^𝑠₁+𝐶𝜈‖𝐰‖^{2(𝑠−𝑠1−1)/(𝑠−𝑠1)}_𝐻𝑠1+𝜒_𝐻𝑠1‖𝐰‖_𝐻^𝑠₁≤𝐶‖𝐰‖²_𝐻^𝑠₁+𝐶𝜈^𝑠−𝑠1+𝐶𝜒²_𝐻𝑠1. (43) With similar argument as abovementioned, we can also obtain the 𝐻^𝑠1 energy for 𝜒 as 12𝑑𝜒𝑑𝑡^2𝐻𝑠1≤𝐶𝜒²_𝐻𝑠1+𝐶‖𝐰‖²_𝐻^𝑠₁+𝐶𝜅^𝑠−𝑠1. (44) The previous two estimates give 𝑑𝜙𝑑𝑡²(𝑡)≤𝐶𝜙₂(𝑡)+𝐶𝜈^𝑠−𝑠1+𝐶𝜅^𝑠−𝑠1,𝜙₂(𝑡)∶=‖𝐰(𝑡)‖²_𝐻^𝑠₁+𝜒(𝑡)²_𝐻𝑠1, (45) and by the Gronwall inequality we get 𝜙₂(𝑡)≤𝑒^𝐶𝑡𝜙₂(0)+𝐶𝜈^𝑠−𝑠1+𝜅^𝑠−𝑠1𝑡𝑒^𝐶𝑡≤𝐶₂𝜙₂(0)+𝜈^𝑠−𝑠1+𝜅^𝑠−𝑠1,∀𝑡∈[0,𝑇], (46) which implies that the estimate (5) holds on [0,𝑇] with 𝐶₂∶=𝑒^𝐶𝑇0(1+𝐶𝑇₀).

Step 3. To prove (7), we need to regularize the initial data. Define 𝜓(𝑥)∈𝐶^∞₀(ℝ²) by 𝜓(𝑥)=𝑐₀−1exp1−|𝑥|²,|𝑥||𝑥|<1,0,≥1, (47) where the constant 𝑐₀ is selected so that ∫_ℝ2𝜓(𝑥)𝑑𝑥=1. Let 𝜓_𝜖(𝑥)=𝜖⁻²𝜌(𝜖⁻¹𝑥), and define the mollification 𝒥_𝜖𝑓 of 𝑓∈𝐿¹_loc(ℝ²) by (𝒥_𝜖𝑓)(𝑥)=(𝜓_𝜖∗𝑓)(𝑥). By this definition, one can see 𝒥_𝜖𝑓∈𝐶^∞; moreover, if 𝑓∈𝐻^𝑠, we have 𝒥_𝜖𝑓→𝑓 in 𝐻^𝑠 as 𝜖→0 and Lim_𝜖→0𝒥_𝜖𝑓−𝑓_𝑠−𝑘≤𝐶_𝑠𝑘𝜖^𝑘𝑓_𝑠,𝒥_𝜖𝑓_𝑠+𝑘≤𝐶_𝑠𝑘𝜖^𝑘𝑓_𝑠. (48) For the proof of these properties, see Lemma 3.5 in [20]. Now let (𝐯^𝜖,𝜌^𝜖) be the solution of the inviscid system (2) with initial data (𝒥_𝜖𝐯₀,𝒥_𝜖𝜌₀); namely, (𝐯^𝜖,𝜌^𝜖) solves the equations 𝐯^𝜖_𝑡+𝐯^𝜖⋅∇𝐯^𝜖+∇𝜋^𝜖=𝜌^𝜖𝐞₂,𝜌^𝜖_𝑡+𝐯^𝜖⋅∇𝜌^𝜖=0,∇⋅𝐯^𝜖𝐯=0,_𝜖|_𝑡=0=𝒥_𝜖𝐯₀,𝜌^𝜖|_𝑡=0=𝒥_𝜖𝜌₀. (49) So, the 𝐻^𝑠 energy for 𝐯^𝜖 and 𝜌^𝜖 can be written as 12𝑑𝐯𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝐯^𝜖3_𝐻𝑠+𝐯^𝜖_𝐻𝑠𝜌^𝜖_𝐻𝑠,12𝑑𝜌𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝐯^𝜖_𝐻𝑠𝜌^𝜖2_𝐻𝑠. (50) With the same discussion as Lemma 5, we know that there exist 𝑇^′>0 and 𝐶>0 both only depending on the 𝐻^𝑠 norm of (𝐯₀,𝜌₀) such that 𝐯^𝜖_{𝐶([0,𝑇′];𝐻𝑠)}+𝜌^𝜖_{𝐶([0,𝑇′];𝐻𝑠)}≤𝐶. (51) Moreover, taking the 𝐻^𝑠+𝑘 energy for 𝐯^𝜖 and 𝜌^𝜖 and using (19), then for 𝑘∈ℕ⁺, 12𝑑𝐯𝑑𝑡^{𝜖2𝐻𝑠+𝑘}≤𝐶𝐯^𝜖_𝐻𝑠𝐯^𝜖2_{𝐻𝑠+𝑘}+𝐯^𝜖_{𝐻𝑠+𝑘}𝜌^𝜖_{𝐻𝑠+𝑘}≤𝐶𝐯^𝜖2_{𝐻𝑠+𝑘}+𝐶𝜌^𝜖2_{𝐻𝑠+𝑘},12𝑑𝜌𝑑𝑡^{𝜖2𝐻𝑠+𝑘}≤𝐶𝐯^𝜖_𝐻𝑠𝜌^𝜖2_{𝐻𝑠+𝑘}+𝐶𝐯^𝜖_{𝐻𝑠+𝑘}𝜌^𝜖_{𝐻𝑠+𝑘}𝜌^𝜖_𝐻𝑠≤𝐶𝐯^𝜖2_{𝐻𝑠+𝑘}+𝐶𝜌^𝜖2_{𝐻𝑠+𝑘}. (52)(53) Using (48), we deduce from the Previous energy estimate that 𝐯^𝜖_{𝐶([0,𝑇′];𝐻𝑠+𝑘)}+𝜌^𝜖_{𝐶([0,𝑇′];𝐻𝑠+𝑘)}≤𝐶𝑒^𝐶𝑇0𝜖^𝑘,𝑘∈ℕ⁺ (54) where 𝐶 depends only on 𝑘, ‖𝐯₀‖_𝐻𝑠, and ‖𝜌₀‖_𝐻𝑠. Without loss of generality, in the following, we may assume 𝑇^′=𝑇, where that 𝑇 is determined by Lemma 5.

Step 4. Set 𝐯^𝜖=𝐯^𝜖−𝐯 and 𝜌^𝜖=𝜌^𝜖−𝜌; then, from (2) and (49), we know that (𝐯^𝜖,𝜌^𝜖) satisfies 𝐯^𝜖_𝑡+(𝐯⋅∇)𝐯^𝜖+𝐯^𝜖⋅∇𝐯^𝜖+∇𝜋^𝜖−𝜋=𝜌^𝜖𝐞₂,𝜌^𝜖_𝑡+(𝐯⋅∇)𝜌^𝜖+𝐯^𝜖⋅∇𝜌^𝜖=0. (55) Using (18) with 𝑟=𝑠, we have 𝐽^𝑠𝐯^𝜖⋅∇𝐯^𝜖,𝐽^𝑠𝐯^𝜖≤𝐶𝐯^𝜖_𝐻𝑠𝐯^𝜖2_𝐻𝑠. (56) By the Cauchy-Schwarz inequality and (16), one has 𝐽^𝑠𝐯^𝜖⋅∇𝐯^𝜖,𝐽^𝑠𝐯^𝜖≤𝐶𝐽^𝑠𝐯^𝜖_𝐿2∇𝐯^𝜖_𝐿∞+𝐯^𝜖_𝐿∞𝐽^𝑠∇𝐯^𝜖_𝐿2𝐽^𝑠𝐯^𝜖_𝐿2≤𝐶𝐯^𝜖_𝐻𝑠𝐯^𝜖_𝐻𝑠+𝐯^𝜖_𝐿∞𝐯^𝜖_𝐻𝑠+1𝐯^𝜖_𝐻𝑠. (57) With these two estimates, it is easy to obtain the following 𝐻^𝑠 energy estimate for 𝐯^𝜖: 12𝑑𝐯𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝐯^𝜖_𝐻𝑠+‖𝐯‖_𝐻^𝑠𝐯^𝜖2_𝐻𝑠+𝐶𝐯^𝜖_𝐿∞𝐯^𝜖_𝐻𝑠+1𝐯^𝜖_𝐻𝑠+𝜌^𝜖_𝐻𝑠𝐯^𝜖_𝐻𝑠≤𝐶𝐯^𝜖2_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐯^𝜖_𝐿∞𝐯^𝜖_𝐻𝑠+𝜌^𝜖_𝐻𝑠𝐯^𝜖_𝐻𝑠, (58) where we have used (54) and the uniform estimate (51) in the last step. Similarly, we can obtain the following 𝐻^𝑠 energy estimate for 𝜌^𝜖: 12𝑑𝜌𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶‖𝐯‖_𝐻^𝑠+𝜌^𝜖_𝐻𝑠𝜌^𝜖2_𝐻𝑠+𝐶𝐯^𝜖_𝐿∞𝜌^𝜖_𝐻𝑠+1𝜌^𝜖_𝐻𝑠≤𝐶𝜌^𝜖2_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐯^𝜖_𝐿∞𝜌^𝜖_𝐻𝑠. (59) Now, we have to estimate ‖𝐯^𝜖‖_𝐿^∞. From (55), we take the 𝐻^𝑠−2 energy for (𝐯^𝜖,𝜌^𝜖) and obtain (using (18) and (20)) 12𝑑𝐯𝑑𝑡^{𝜖2𝐻𝑠−2}≤𝐶𝐯^𝜖2_𝐻𝑠−2+𝐶𝜌^𝜖_𝐻𝑠−2𝐯^𝜖_𝐻𝑠−2,12𝑑𝜌𝑑𝑡^{𝜖2𝐻𝑠−2}≤𝐶𝜌^𝜖2_𝐻𝑠−2+𝐶𝜌^𝜖_𝐻𝑠−2𝐯^𝜖_𝐻𝑠−2; (60)(61) Then, the Gronwall inequality and (48) yield 𝐯^𝜖(𝑡)_𝐻𝑠−2+𝜌^𝜖(𝑡)_𝐻𝑠−2≤𝑒^𝐶𝑇0𝐯^𝜖(0)_𝐻𝑠−2+𝜌^𝜖(0)_𝐻𝑠−2≤𝐶𝑒^𝐶𝑇0𝜖², (62) for all 𝑡∈[0,𝑇], which in turn by Sobolev embedding theorem gives ‖𝐯^𝜖(𝑡)‖_𝐿^∞+‖𝜌^𝜖(𝑡)‖_𝐿∞≤𝐶𝑒^𝐶𝑇0𝜖² since 𝑠>3. Inserting this estimate into (58) and (59), we can see 𝑑𝜙𝑑𝑡³≤𝐶𝑒^𝐶𝑇0𝜙₃+𝐶𝑒^𝐶𝑇0𝜖,𝜙₃∶=𝐯^𝜖_𝐻𝑠+𝜌^𝜖_𝐻𝑠, (63) where we have used the relation 𝑒^𝐶𝑇0⋅𝑒^𝐶𝑇0=𝑒^2𝐶𝑇0=∶𝑒^𝐶𝑇0 in the last step since the value of 𝐶 at each appearance may be different. Hence, the Gronwall inequality gives 𝜙₃(𝑡)≤𝑒^{𝐶𝑇0𝑒0𝐶𝑇}1+𝐶𝑇₀𝑒^𝐶𝑇0𝜙₃(0),[].+𝜖∀𝑡∈0,𝑇 (64)

Step 5. Let 𝐔^𝜖=𝐮−𝐯^𝜖, Θ^𝜖=𝜃−𝜌^𝜖, and recall that here 𝐮=𝐮^𝜈,𝜅 and 𝜃=𝜃^𝜈,𝜅; so, one can deduce from (2) and (55) that (𝐔^𝜖,Θ^𝜖) solves 𝐔^𝜖_𝑡+(𝐮⋅∇)𝐔^𝜖+𝐔^𝜖⋅∇𝐯^𝜖+∇𝑝−𝜋^𝜖=𝜈Δ𝐔^𝜖+𝜈Δ𝐯^𝜖+Θ^𝜖𝐞₂,Θ^𝜖_𝑡+(𝐮⋅∇)Θ^𝜖+𝐔^𝜖⋅∇𝜌^𝜖=𝜅ΔΘ^𝜖+𝜅Δ𝜌^𝜖. (65) Using the same reasonings that lead to (58), we have the 𝐻^𝑠 energy estimate for 𝐔^𝜖 as 12𝑑𝐔𝑑𝑡^{𝜖2𝐻𝑠}+𝜈∇𝐔^𝜖2_𝐻𝑠≤𝐶‖𝐮‖_𝐻^𝑠+𝐯^𝜖_𝐻𝑠𝐔^𝜖2_𝐻𝑠+𝐶𝐯^𝜖_𝐻𝑠+1𝐔^𝜖_𝐿∞𝐔^𝜖_𝐻𝑠+𝜈𝐯^𝜖_𝐻𝑠+2𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠𝐔^𝜖_𝐻𝑠≤𝐶𝐔^𝜖2_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐔^𝜖_𝐿∞𝐔^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜈𝜖⁻²𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠𝐔^𝜖_𝐻𝑠, (66) which yields 𝑑𝐔𝑑𝑡^𝜖𝐻𝑠≤𝐶𝐔^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐔^𝜖_𝐿∞+𝐶𝑒^𝐶𝑇0𝜈𝜖⁻²+Θ^𝜖_𝐻𝑠. (67) Similarly, we can obtain the 𝐻^𝑠 energy for Θ^𝜖 as 𝑑Θ𝑑𝑡^𝜖𝐻𝑠≤𝐶Θ^𝜖_𝐻𝑠+𝐶𝐔^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐔^𝜖_𝐿∞+𝐶𝑒^𝐶𝑇0𝜅𝜖⁻². (68) Now, we should estimate ‖𝐔^𝜖‖_𝐿^∞. Note that 𝐔^𝜖=𝐰−𝐯^𝜖; so, 𝐔^𝜖_𝐿∞≤‖𝐰‖_𝐿^∞+𝐯^𝜖_𝐿∞. (69) By (40), (62), and Sobolev embedding theorem, we have ‖𝐰‖_𝐿^∞_(0,𝑇;𝐿^∞₎≤𝐶‖𝐰‖_𝐿^∞_(0,𝑇;𝐻^𝑠−2₎≤𝐶₁𝜙₁(0),𝐯+𝜈+𝜅_{𝜖𝐿∞(0,𝑇;𝐿∞)}≤𝐶𝐯^𝜖_{𝐿∞(0,𝑇;𝐻𝑠−2)}≤𝐶𝑒^𝐶𝑇0𝜖², (70) which gives 𝐔^𝜖_𝐿∞≤𝐶^∗𝜙₁(0)+𝜈+𝜅+𝜖²,𝐶^∗∶=max𝐶₁,𝐶𝑒^𝐶𝑇0. (71) Inserting this estimate into (67) and (68), one has 𝑑𝜙𝑑𝑡⁴≤𝐶𝜙₄+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐔^𝜖_𝐿∞+𝐶𝑒^𝐶𝑇0(𝜈+𝜅)𝜖⁻²≤𝐶𝜙₄+𝐶𝑒^𝐶𝑇0𝜖⁻¹𝐶^∗𝜙₁(0)+𝜈+𝜅+𝜖²+𝐶𝑒^𝐶𝑇0(𝜈+𝜅)𝜖⁻²≤𝐶^∗∗𝜙₄+𝐶^∗∗𝜖+𝐶^∗∗𝜖⁻¹𝜙₁(0)+𝐶^∗∗𝜖⁻²(𝜈+𝜅), (72) where 𝜙₄∶=‖𝐔^𝜖‖_𝐻^𝑠+‖Θ^𝜖‖_𝐻^𝑠 and 𝐶^∗∗∶=max{𝐶,𝐶𝑒^𝐶𝑇0𝐶^∗,𝐶𝑒^𝐶𝑇0}. By the Gronwall inequality, one obtains 𝜙₄(𝑡)≤𝑒^{𝐶∗∗𝑇0}1+𝐶^∗∗𝑇₀×𝜙₄(0)+𝜖+𝜖⁻¹𝜙₁(0)+𝜖⁻²(𝜈+𝜅). (73) Recall that 𝐰=𝐔^𝜖+𝐯^𝜖,𝜒=Θ^𝜖+𝜌^𝜖; (74) therefore, it follows from (64) and (73) that ‖𝐰(𝑡)‖_𝐻^𝑠+𝜒(𝑡)_𝐻𝑠≤𝜙₃(𝑡)+𝜙₄(𝑡)≤𝐶𝜙₃(0)+𝜙₄(0)+𝜖+𝜖⁻¹𝜙₁(0)+𝜖⁻²()𝐶𝜙𝜈+𝜅≤2₃(0)+‖𝐰(0)‖_𝐻^𝑠+𝜒(0)_𝐻𝑠+𝜖+𝜖⁻¹𝜙₁(0)+𝜖⁻²(𝜈+𝜅) (75) for all 𝑡∈[0,𝑇], where we use 𝜙₄(0)≤𝜙₁(0)+𝜙₃(0) in the last step and 𝐶∶=max𝑒^{𝐶∗∗𝑇0}1+𝐶^∗∗𝑇₀,𝑒^{𝐶𝑇0𝑒0𝐶𝑇}1+𝐶𝑇₀𝑒^𝐶𝑇0. (76) Note that (3) gives ‖𝐰(0)‖_𝐻^𝑠+𝜒(0)_𝐻𝑠=𝐮₀^𝜈,𝜅−𝐯_0𝐻𝑠+𝜃₀^𝜈,𝜅−𝜌_0𝐻𝑠⟶0 (77) as (𝜈,𝜅)→0, and the property of the operator 𝒥_𝜖 yields 𝜙₃(0)=𝐯^𝜖(0)_𝐻𝑠+𝜌^𝜖(0)_𝐻𝑠=𝐯^𝜖(0)−𝐯(0)_𝐻𝑠+𝜌^𝜖(0)−𝜌(0)_𝐻𝑠=𝒥_𝜖𝐯₀−𝐯_0𝐻𝑠+𝒥_𝜖𝜌₀−𝜌_0𝐻𝑠⟶0 (78) as 𝜖→0. Now, we choose 𝜖=𝜖(𝜈,𝜅)>0 satisfying the following properties: (1)lim_{(𝜈,𝜅)→0}𝜖=0, (2)lim_{(𝜈,𝜅)→0}𝜖⁻¹𝜙₁(0)=lim_{(𝜈,𝜅)→0}𝜖⁻¹(‖𝐮₀^𝜈,𝜅−𝐯₀‖_𝐻^𝑠−2+‖𝜃₀^𝜈,𝜅−𝜌₀‖_𝐻^𝑠−2)=0, (3)lim_{(𝜈,𝜅)→0}𝜖⁻²(𝜈+𝜅)=0. Hence, combining the previous convergence results, it is easy to obtain from (75) that lim_{()𝜈,𝜅→0}𝐰^𝜈,𝜅_{𝐶([0,𝑇];𝐻𝑠)}+𝜒^𝜈,𝜅_{𝐶([0,𝑇];𝐻𝑠)}=0. (79)

Step 6. By now, we have proved that (4), (5), and (7) hold on the time interval [0,𝑇]. Now our aim is to show that these three results still hold on [0,𝑇₀]. Define 𝑇₁∶=𝑇; now choose (𝐮^𝜈,𝜅(𝑇₁),𝜃^𝜈,𝜅(𝑇₁)) and (𝐯(𝑇₁),𝜌(𝑇₁)) as the new initial data, and one can see that the limit relation (3) still holds in the time 𝑇₁. Moreover, from Lemma 3, we know that ‖𝐮^𝜈,𝜅(𝑇₁)‖_𝐻𝑠, ‖𝜃^𝜈,𝜅(𝑇₁)‖_𝐻𝑠, ‖𝐯(𝑇₁)‖_𝐻𝑠, and ‖𝜌(𝑇₁)‖_𝐻𝑠 depend only on the 𝐻^𝑠 norm of the initial data (𝐯₀,𝜌₀). Then, we repeat the previous argument and find a positive sequence {𝑇_𝑘}^∞_𝑘=1 such that (4), (5), and (7) hold on [0,𝑇₁+⋯+𝑇_𝑘]. We assert that ∑^∞_𝑘=1𝑇_𝑘=𝑇^∗. Indeed, if ∑^∞_𝑘=1𝑇_𝑘=𝑇<𝑇^∗, and then the blow-up criterion implies that we can still extend [0,𝑇] to some bigger interval, so we can continue this procedure as long as ‖𝐯(𝑡)‖_𝐻^𝑠+‖𝜌(𝑡)‖_𝐻𝑠<∞, and by the blow up criterion again, we get our assertion. Since 𝑇₀<𝑇^∗, after finite times iteration, we obtain the convergence results (4), (5), and (7).
Finally, since (7) holds, we have ‖𝐮^𝜈,𝜅−𝐯‖_{𝐶([0,𝑇0];𝐻}^𝑠₎+‖𝜃^𝜈,𝜅−𝜌‖_{𝐶([0,𝑇0];𝐻𝑠)}≤𝐶; then, the convergence result (6) follows from (5) and the following interpolation inequality: 𝑓_𝐻𝑠′≤𝐶𝑓^{𝑠−𝑠′}_𝐻𝑠−1𝑓^{1−(𝑠−𝑠′)}_𝐻𝑠,𝑠^′∈[𝑠−1,𝑠]. (80) Therefore, we finish the proof of Theorem 1.

3. The 𝐻^𝑠 Convergence Rate with Some Loss of Derivatives

In this section, we will prove Theorem 2, and we still use the same notations that are used in the proof of Theorem 1.

Proof of Theorem 2. Since (7) holds, without loss of generality, we may assume that ‖𝐮‖_{𝐶([0,𝑇0];𝐻}^𝑠₎+‖𝜃‖_{𝐶([0,𝑇0];𝐻}^𝑠₎≤𝐶 (81) for small 𝜈 and 𝜅. By the extra regularity of the initial data (𝐯₀,𝜌₀), using the same reasonings that lead to (54) and the same extension method as Step 6 in the proof of Theorem 1, then we obtain 𝐯^𝜖_{𝐶([0,𝑇0];𝐻𝑠+𝛿)}+𝜌^𝜖_{𝐶([0,𝑇0];𝐻𝑠+𝛿)}𝐯≤𝐶,_{𝜖𝐶([0,𝑇0];𝐻𝑠+2)}+𝜌^𝜖_{𝐶([0,𝑇0];𝐻𝑠+2)}≤𝐶𝑒^𝐶𝑇0𝜖^𝛿−2. (82)(83) Now the proof is divided into two cases.
Case 1 (0≤𝛿≤2). In this case, the estimate (82) implies 𝐯^𝜖_{𝐶([0,𝑇0];𝐻𝑠+1)}+𝜌^𝜖_{𝐶([0,𝑇0];𝐻𝑠+1)}≤𝐶. (84) Using the estimates ‖𝐯^𝜖‖_𝐿^∞≤𝐶‖𝐯^𝜖‖_𝐻^𝑠 and (84), we deduce from the first inequality of (58) that 12𝑑𝐯𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝐯^𝜖2_𝐻𝑠+𝜌^𝜖_𝐻𝑠𝐯^𝜖_𝐻𝑠. (85) Similarly, one can infer from the first inequality of (59) that 12𝑑𝜌𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝜌^𝜖2_𝐻𝑠+𝐶𝜌^𝜖_𝐻𝑠𝐯^𝜖_𝐻𝑠. (86) The previous two inequalities together with (48) give 𝐯^𝜖(𝑡)_𝐻𝑠+𝜌^𝜖(𝑡)_𝐻𝑠≤𝐶𝑒^𝐶𝑇0𝜖^𝛿,∀𝑡∈0,𝑇₀. (87) On the other hand, from (81) and the first inequality of (66), we can get 12𝑑𝐔𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶𝐔^𝜖2_𝐻𝑠+𝜈𝐯^𝜖_𝐻𝑠+2𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠𝐔^𝜖_𝐻𝑠≤𝐶𝐔^𝜖2_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜈𝜖^𝛿−2𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠𝐔^𝜖_𝐻𝑠, (88) where we have used (83) in the last step. Simultaneously, the 𝐻^𝑠 energy estimate for Θ^𝜖 is estimated as 12𝑑Θ𝑑𝑡^{𝜖2𝐻𝑠}≤𝐶Θ^𝜖2_𝐻𝑠+𝐶Θ^𝜖_𝐻𝑠𝐔^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜅𝜖^𝛿−2Θ^𝜖_𝐻𝑠. (89) Then Gronwall inequality yields that 𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠≤𝑒^{𝐶𝑇0𝑒0𝐶𝑇}1+𝐶𝑇₀𝑒^𝐶𝑇0×𝐔^𝜖(0)_𝐻𝑠+Θ^𝜖(0)_𝐻𝑠+(𝜅+𝜈)𝜖^𝛿−2. (90) Combining (87) and (90), we can arrive at ‖𝐰‖_𝐻^𝑠+𝜒_𝐻𝑠≤𝐶₄‖𝐰(0)‖_𝐻^𝑠+𝜒(0)_𝐻𝑠+(𝜅+𝜈)𝜖^𝛿−2+𝜖^𝛿 (91) for some 𝐶₄>0, from which we know that (8) holds by choosing 𝜖^𝛿=(𝜅+𝜈)𝜖^𝛿−2.
Case 2 (0<𝛿<1). In this case, we have 𝑣^𝜖_{𝐶([0,𝑇0];𝐻𝑠+1)}+𝜌^𝜖_{𝐶([0,𝑇0];𝐻𝑠+1)}≤𝐶𝑒^𝐶𝑇0𝜖^𝛿−1. (92) And also one obtains from (48) and the first inequality of (62) 𝐯^𝜖_𝐿∞+𝜌^𝜖_𝐿∞≤𝑒^𝐶𝑇0𝑣^𝜖(0)_𝐻𝑠−2+𝜌^𝜖(0)_𝐻𝑠−2≤𝐶𝑒^𝐶𝑇0𝜖^2+𝛿. (93) Applying the previous two estimates into (58) and (59), we get 𝑑𝐯𝑑𝑡^𝜖𝐻𝑠≤𝐶𝐯^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖^1+2𝛿+𝜌^𝜖_𝐻𝑠,𝑑𝜌𝑑𝑡^𝜖𝐻𝑠≤𝐶𝜌^𝜖_𝐻𝑠+𝐶𝑒^𝐶𝑇0𝜖^1+2𝛿, (94) which yields that 𝐯^𝜖_𝐻𝑠+𝜌^𝜖_𝐻𝑠≤𝑒^{𝐶𝑇0𝑒0𝐶𝑇}1+𝐶𝑇₀𝑒^𝐶𝑇0×𝐯^𝜖(0)_𝐻𝑠+𝜌^𝜖(0)_𝐻𝑠+𝜖^1+2𝛿≤𝐶𝑒^{𝐶𝑇0𝑒0𝐶𝑇}1+𝑇₀𝑒^𝐶𝑇0𝜖^𝛿. (95) On the other hand, the estimates (71) and (93) imply 𝐔^𝜖_𝐿∞≤𝐶^′𝜙₁(0)+𝜈+𝜅+𝜖^2+𝛿,𝐶^′∶=max𝐶^∗,𝐶𝑒^𝐶𝑇0. (96) Then, we insert this inequality and (83) and (92) into (66) and obtain 𝑑𝐔𝑑𝑡^𝜖𝐻𝑠≤𝐶𝐔^𝜖_𝐻𝑠+𝐶𝐶^′𝑒^𝐶𝑇0𝜖^𝛿−1𝜙₁(0)+𝜈+𝜅+𝜖^2+𝛿+𝐶𝑒^𝐶𝑇0𝜈𝜖^𝛿−2+Θ^𝜖_𝐻𝑠. (97) In the same way, the estimate (68) is replaced by 𝑑Θ𝑑𝑡^𝜖𝐻𝑠≤𝐶Θ^𝜖_𝐻𝑠+𝐶𝐔^𝜖_𝐻𝑠+𝐶𝐶^′𝑒^𝐶𝑇0𝜖^𝛿−1𝜙₁(0)+𝜈+𝜅+𝜖^2+𝛿+𝐶𝑒^𝐶𝑇0𝜅𝜖^𝛿−2. (98) Hence, by the Gronwall inequality, we have 𝐔^𝜖_𝐻𝑠+Θ^𝜖_𝐻𝑠≤𝐶^′′𝐔^𝜖(0)_𝐻𝑠+Θ^𝜖(0)_𝐻𝑠+𝜖^𝛿−1𝜙₁(0)+(𝜈+𝜅)𝜖^𝛿−2+𝜖^1+2𝛿 (99) for some 𝐶^′′>0. This inequality, together with (95), gives ‖𝐰‖_𝐻^𝑠+𝜒_𝐻𝑠≤𝐶₅‖𝐰(0)‖_𝐻^𝑠+𝜒(0)_𝐻𝑠+𝜖^𝛿−1𝜙₁(0)+(𝜈+𝜅)𝜖^𝛿−2+𝜖^𝛿 (100) for some 𝐶₅>0. So, (9) follows from (100) provided that we choose 𝜖 satisfying 𝜖=𝜙₁(0)+(𝜅+𝜈)^(1/2).

Acknowledgments

The author would like to thank the referee's valuable suggestions which improved the presentation considerably. This work is supported by the National Science Foundation (Grant no. 11271334), Zhejiang Provincial Natural Science Foundation of China (Grant no. LY12A01019) and the Science Research Foundation of Jiaxing University.