The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System

and Applied Analysis 3 For any Banach space B, the space Lp(0, T;B) consists of all strongly measurable functions u : [0, T] → B equipped with the norm ‖u‖ L p (0,T;B) := (∫ T 0 ‖u (t)‖PBdt) 1/p < ∞ (12) for 1 ≤ p < ∞, and ‖u‖ L ∞ (0,T;B) = ess sup 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 ] 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), the Schwartz class, then 󵄩󵄩󵄩󵄩J s (fg) − f (J s g) 󵄩󵄩󵄩󵄩Lp ≤ C ( 󵄩󵄩󵄩󵄩∇f 󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Hs−1,p2 + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Hs,p3 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Lp4 ) , (15) 󵄩󵄩󵄩󵄩J s (fg) 󵄩󵄩󵄩󵄩Lp ≤ C ( 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Hs,p2 + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Hs,p3 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Lp4 ) (16) with p 2 , p 3 ∈ (1, +∞) such that


Introduction and the Main Results
The two-dimensional Boussinesq system for the homogeneous incompressible fluids with diffusion and viscosity is given by where the space variable x = ( 1 ,  2 ) is in R 2 , u = ( 1 (, x),  2 (, x)) is the velocity,  = (, x) denotes the scalar pressure and  = (, x) the scalar temperature, e 2 = (0, 1), and  > 0 and ] > 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), 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][4][5][6][7][8][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., ] > 0 and  = 0).Meanwhile, Chae in [5] also proved global regularity for the 2D Boussinesq system (1) both with the zero diffusivity case (] > 0 and  = 0) and the zero viscosity case (] = 0 and  > 0).However, for the case ] = 0 and  = 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 (], ) → 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][12][13][14][15][16][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   , usually one can obtain the convergence results in   with   < .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   convergence of the solution with the initial data belonging to the same space.Now we state our main results of the paper.
Of course, we can generalize the previous results to arbitrary spatial dimension case with  > 3 replaced by  > (/2) + 2. The important part of Theorem 1 is the convergence result (7).This result tells us that the   convergence can be maintained by the solution at its arbitrary existence time.We emphasize that  * is not assumed to be small; indeed, the standard energy estimate yields that the classical solution (k, ) blows up at time  * if and only if ‖k()‖ H  + ‖()‖   → ∞ as  ↑  * .Note that the rate of   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 (k 0 ,  0 ), then we can obtain the following   convergence rate.
Obviously,  0 (R  ) =  2 (R  ).In some places, we use the notation H  (R  ; R  ) to mean that this space consists of vector-valued functions f : R  → R  with each component of f belonging to   (R  ).If there is no confusion, the spaces H  (R  ) and H  (R  ; R  ) will be simply denoted by   .For f, g ∈  2 (R  ; R  ), we denote by ⟨f, g⟩ the usual inner product of f and g; namely, For any Banach space B, the space   (0, ; B) consists of all strongly measurable functions u : [0, ] → B equipped with the norm for 1 ≤  < ∞, and And the space ([0, ]; B) denotes the set of continuous functions u : [0, ] → B with In this paper, the letter  is a generic constant and its value may change at each appearance.Moreover, every  is independent of the parameters ] and .

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].
To prove Theorem 1, we first establish the uniform bounds for the solutions of (1) with the bound independent of ] and .
Proof.From the first equation of (1), we have Multiplying this equation by   u ], and integrating the result, while noting that where we have used the estimate (18) in the previous inequality, then we obtain Since (3) holds, we may assume  ], (0) ≤  0 for small ] and  that here,  0 depends only on ‖k 0 ‖   and ‖ 0 ‖   .Hence, we finally arrive at The estimate (23) follows from the previous inequality provided that we select  such that  < 1/ 0 (e.g., we can choose  = 1/2 0 ).The proof of Lemma 5 is complete.
Remark 6.From the proof of Lemma 5, we also see the solution of system (2) satisfying where  and  are the same as (23).
Remark 7.For fixed  0 ∈ (0,  * ), without loss of generality, we may assume that the time  determined by Lemma 5 satisfies  <  0 .Indeed, as will be seen in the proof of Theorem 1, no matter how small the  is, we can always use bootstrap argument to extend the interval [0, ] into our desired time interval [0,  0 ].
Proof of Theorem 1.We split the proof into several steps.
Step 2. We show that (5) holds on [0, ].Applying the estimates ( 18) and ( 20) with  =  1 , then we can easily obtain the   1 energy for w as Since  1 ∈ ( − 2,  − 1], we have By the Gagliardo-Nirenberg interpolation inequality and Young's inequality, we have Inserting this inequality into (41) and using Young's inequality again, we thus get With similar argument as abovementioned, we can also obtain the   1 energy for  as The previous two estimates give and by the Gronwall inequality we get which implies that the estimate (5) holds on [0, ] with  2 :=   0 (1 +  0 ).

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.