# The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System

Authors: Junlei Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 1. Introduction and the Main Results

The two-dimensional Boussinesq system for the homogeneous incompressible fluids with diffusion and viscosity is given by(1)ut+(u·∇)u+∇p=νΔu+θe2,θt+(u·∇)θ=κΔθ,∇·u=0,u|t=0=u0,θ|t=0=θ0,
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),
(2)vt+(v·∇)v+∇π=ρe2,ρt+(v·∇)ρ=0,∇·v=0,v∣t=0=v0,ρ∣t=0=ρ0.
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 Hs, usually one can obtain the convergence results in Hs′ 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 Hs convergence of the solution with the initial data belonging to the same space.Now we state our main results of the paper.Theorem 1.
Lets>3, (u0ν,κ,θ0ν,κ)∈Hs(ℝ2;ℝ2)×Hs(ℝ2), and (v0,ρ0)∈Hs(ℝ2;ℝ2)×Hs(ℝ2) satisfying ∇·u0ν,κ=0, ∇·v0=0, and
(3)u0ν,κ⟶v0,θ0ν,κ⟶ρ0inHs,
as (ν,κ)→0. Assume that (uν,κ,θν,κ)∈C(ℝ+;Hs×Hs) is the classical solution of (1) with initial data (u0ν,κ,θ0ν,κ), and (v,ρ)∈C([0,T*);Hs×Hs) 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 theHs-2 norm)
(4)∥uν,κ(t)-v(t)∥Hs-2+∥θν,κ(t)-ρ(t)∥Hs-2≤C1(∥θ0ν,κ-ρ0∥Hs-2ν+κ+∥u0ν,κ-v0∥Hs-2+∥θ0ν,κ-ρ0∥Hs-2),(2)
(convergence rate in theHs1 norm with s-2<s1≤s-1)
(5)∥uν,κ(t)-v(t)∥Hs1+∥θν,κ(t)-ρ(t)∥Hs1≤C2(∥θ0ν,κ-ρ0∥Hs1ν(s-s1)/2+κ(s-s1)/2+∥u0ν,κ-v0∥Hs1+∥θ0ν,κ-ρ0∥Hs1ν(s-s_1)/2),(3)
(convergence rate in theHs2 norm with s-1<s2<s)
(6)∥uν,κ(t)-v(t)∥Hs2+∥θν,κ(t)-ρ(t)∥Hs2≤C3(ν(s-s2)/2+κ(s-s2)/2+∥u0ν,κ-v0∥Hs-1s-s2+∥θ0ν,κ-ρ0∥Hs-1s-s2),(4)
(convergence in theHs norm)
(7)∥uν,κ-v∥C([0,T0];Hs)+∥θν,κ-ρ∥C([0,T0];Hs)⟶0as(ν,κ)⟶0,where the constants Ci(i=1,2,3) in (4)–(6) are dependent of T0 and the Hs 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 withs>3 replaced by s>(n/2)+2. The important part of Theorem 1 is the convergence result (7). This result tells us that the Hs 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)∥Hs+∥ρ(t)∥Hs→∞ as t↑T*. Note that the rate of Hs 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 Hs convergence rate.Theorem 2.
Suppose that the same assumptions as Theorem1 hold. Moreover, one assumes that (v0,ρ0)∈Hs+δ×Hs+δ with δ∈(0,2]. Then, for any t∈[0,T0], there hold
(8)∥uν,κ(t)-v(t)∥Hs+∥θν,κ(t)-ρ(t)∥Hs≤C4(νδ/2+κδ/2+∥u0ν,κ-v0∥Hs+∥θ0ν,κ-ρ0∥Hs),1≤δ≤2,(9)∥uν,κ(t)-v(t)∥Hs+∥θν,κ(t)-ρ(t)∥Hs≤C5(νδ/2+κδ/2+∥u0ν,κ-v0∥Hs+∥θ0ν,κ-ρ0∥Hs)+C(∥u0ν,κ-v0∥Hs-2+∥θ0ν,κ-ρ0∥Hs-2)δ,0<δ<1,
where the constants C4 and C5 depend only on T0, ∥v0∥Hs+δ, and ∥ρ0∥Hs+δ.Finally, we end this section by setting some notations which will be used throughout the paper. Forp∈[1,∞], ∥·∥Lp denotes the norm in the Lebesgue space Lp(ℝn). We set the operator J:=(I-Δ)1/2, and for s∈ℝ, we denote by Hs,p(ℝn) the nonhomogeneous Sobolev spaces with the norm defined as
(10)∥·∥Hs,p:=∥Js·∥Lp.
If p=2, for brevity, we write Hs(ℝn) instead of Hs,2(ℝn). Obviously, H0(ℝn)=L2(ℝn). In some places, we use the notation Hs(ℝn;ℝk) to mean that this space consists of vector-valued functions f:ℝn→ℝk with each component of f belonging to Hs(ℝn). If there is no confusion, the spaces Hs(ℝn) and Hs(ℝn;ℝk) will be simply denoted by Hs. For f,g∈L2(ℝn;ℝk), we denote by 〈f,g〉 the usual inner product of f and g; namely,
(11)〈f,g〉:=∫ℝn‍f(x)·g(x)dx.
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
(12)∥u∥Lp(0,T;B):=(∫0T‍∥u(t)∥BPdt)1/p<∞
for 1≤p<∞, and
(13)∥u∥L∞(0,T;B)=esssupt∈[0,T]∥u(t)∥B<∞.
And the space C([0,T];B) denotes the set of continuous functions u:[0,T]→B with
(14)∥u∥C([0,T];B)=maxt∈[0,T]∥u(t)∥B<∞.
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 Theorem1

In this section, we present the proof of Theorem1. To this goal, we need the following calculus inequality, the proof of which can be found in [18, 19].Lemma 3.
Assume thats>0 and p∈(1,+∞). If f,g∈𝒮(ℝn), the Schwartz class, then (15)∥Js(fg)-f(Jsg)∥Lp≤C(∥∇f∥Lp1∥g∥Hs-1,p2+∥f∥Hs,p3∥g∥Lp4),(16)∥Js(fg)∥Lp≤C(∥f∥Lp1∥g∥Hs,p2+∥f∥Hs,p3∥g∥Lp4)
with p2,p3∈(1,+∞) such that
(17)1p=1p1+1p2=1p3+1p4.Of course, Lemma3 also holds when f and g are replaced by vector-valued functions. Using (15) and (16), we have the following result.Lemma 4.
Lets>1+(n/2). Then,(1)
for anyf,g∈Hs(ℝn;ℝn)∩Hr(ℝn;ℝn) with ∇·f=0, there hold
(18)|〈Jr((f·∇)g),Jrg〉|≤C∥f∥Hs∥g∥Hr2,1<r≤s,(19)|〈Jr((f·∇)g),Jrg〉|≤C∥f∥Hs∥g∥Hr2+C∥f∥Hr∥g∥Hr∥g∥Hs,r>s;(2)
for anyf∈Hs(ℝn;ℝn), g,h∈Hr(ℝn;ℝn) with r∈[0,s-1], there holds
(20)|〈Jr((g·∇)f),Jrh〉|≤C∥f∥Hs∥g∥Hr∥h∥Hr.Proof.
Using the divergence free condition and the communicator estimate (15), one sees
(21)|〈Jr((f·∇)g),Jrg〉|=|〈Jr((f·∇)g)-(f·Jr∇)g,Jrg〉|≤C(∥∇f∥L∞∥Jr-1∇g∥L2+∥Jrf∥Lp∥∇g∥Lq)∥Jrg∥L2≤C(∥f∥Hs∥g∥Hr+∥f∥Hr,p∥g∥H1,q)∥g∥Hr,
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 Hs↪Hr,p and Hr-1↪Lq. 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,
(22)|〈Jr((g·∇)f),Jrh〉|≤C(∥Jrg∥L2∥∇f∥L∞+∥g∥Lp∥Jr∇f∥Lq)∥Jrh∥L2≤C(∥g∥Hr∥f∥Hs+∥g∥Lp∥f∥Hr+1,q)∥h∥Hr.
With the condition, we can choose p and q satisfying Hr↪Lp, Hs↪Hr+1,q, and 1/p+1/q=1/2(1<p<∞), and thus (20) follows.To prove Theorem1, 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 Theorem1, then there exist T>0 and C>0 such that for sufficiently small ν, κ>0, there holds
(23)∥uν,κ∥C([0,T];Hs)+∥θν,κ∥C([0,T];Hs)≤C
with T and C both depending only on the Hs norm of (v0,ρ0) and not depending on ν and κ.Proof.
From the first equation of (1), we have
(24)(Jsuν,κ)t+Js[(uν,κ·∇)uν,κ]+∇Jspν,κ=νΔJsuν,κ+Jsθν,κe2.
Multiplying this equation by Jsuν,κ and integrating the result, while noting that
(25)∫ℝ2‍∇Jspν,κ·Jsuν,κdx=-∫ℝ2‍Jspν,κ·Js(∇·uν,κ)dx=0,|∫ℝ2‍Js[(uν,κ·∇)uν,κ]·Jsuν,κdx|≤C∥Jsuν,κ∥L23,
where we have used the estimate (18) in the previous inequality, then we obtain
(26)12ddt∥Jsuν,κ∥L22+ν∥∇Jsuν,κ∥L22≤C∥Jsuν,κ∥L23+∥Jsuν,κ∥L2∥Jsθν,κ∥L2,
which gives
(27)ddt∥uν,κ∥Hs≤C∥uν,κ∥Hs2+∥θν,κ∥Hs.
Using the same argument to the second equation of (1), we can obtain
(28)12ddt∥Jsθν,κ∥L22+κ∥∇Jsθν,κ∥L22≤C∥Jsuν,κ∥L2∥Jsθν,κ∥L22.
Then, we have
(29)ddt∥θν,κ∥Hs≤C∥uν,κ∥Hs∥θν,κ∥Hs.
Hence, it concludes from the estimates (27) and (29) that
(30)ddtφν,κ(t)≤C(φν,κ(t))2,φν,κ(t):=∥uν,κ(t)∥Hs+∥θν,κ(t)∥Hs+1.
Solving this ODE gives
(31)φν,κ(t)≤φν,κ(0)1-Cφν,κ(0)t.
Since (3) holds, we may assume φν,κ(0)≤C0 for small ν and κ that here, C0 depends only on ∥v0∥Hs and ∥ρ0∥Hs. Hence, we finally arrive at
(32)φν,κ(t)≤C01-CC0t.
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 Lemma5, we also see the solution of system (2) satisfying
(33)∥v∥C([0,T];Hs)+∥ρ∥C([0,T];Hs)≤C,
where T and C are the same as (23).Remark 7.
For fixedT0∈(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 definewν,κ:=uν,κ-v and χν,κ:=θν,κ-ρ. Considering the equations, for (uν,κ,θν,κ) and (v,ρ), we see (wν,κ,χν,κ) satisfying
(34)wtν,κ+(v·∇)wν,κ+(wν,κ·∇)uν,κ+∇(pν,κ-π)=νΔwν,κ+νΔv+χν,κe2,χtν,κ+(v·∇)χν,κ+(wν,κ·∇)θν,κ=κΔχν,κ+κΔρ,∇·wν,κ=0,wν,κ|t=0=u0ν,κ-v0,χν,κ|t=0=θ0ν,κ-ρ0.
For the sake of convenience, we often omit the superscripts ν and κ in the succeeding arguments; hence, w means wν,κ, θ stands for θν,κ, and so on.Proof of Theorem1.
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 Hs-2 energy for w as
(35)12ddt∥w∥Hs-22+ν∥∇w∥Hs-22≤C(∥v∥Hs+∥u∥Hs)∥w∥Hs-22+ν∥Δv∥Hs-2∥w∥Hs-2+∥χ∥Hs-2∥w∥Hs-2≤C∥w∥Hs-22+Cν∥w∥Hs-2+∥χ∥Hs-2∥w∥Hs-2,
where we have used the uniform estimates (23) and (33) in the last step. Hence, we get
(36)ddt∥w∥Hs-2≤C∥w∥Hs-2+Cν+∥χ∥Hs-2.
Similarly, the Hs-2 energy estimate for χ can be written as
(37)12ddt∥χ∥Hs-22+κ∥∇χ∥Hs-22≤C(∥Δv∥Hs∥χ∥Hs-22+∥w∥Hs-2∥θ∥Hs∥χ∥Hs-2)+κ∥Δρ∥Hs-2∥χ∥Hs-2≤C∥χ∥Hs-22+C∥w∥Hs-2∥χ∥Hs-2+Cκ∥χ∥Hs-2
which gives
(38)ddt∥χ∥Hs-2≤C∥χ∥Hs-2+C∥w∥Hs-2+Cκ.
Therefore, one has
(39)ddtϕ1(t)≤Cϕ1(t)+Cν+Cκ,ϕ1(t):=∥w(t)∥Hs-2+∥χ(t)∥Hs-2.
Then, the Gronwall inequality yields that for all t∈[0,T](⊂[0,T0]),
(40)ϕ1(t)≤eCtϕ1(0)+C(ν+κ)teCt≤C1(ϕ1(0)+ν+κ),
where C1:=eCT0(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 Hs1 energy for w as
(41)12ddt∥w∥Hs12+ν∥∇w∥Hs12≤C(∥v∥Hs+∥u∥Hs)∥w∥Hs12+ν∥∇v∥Hs-1∥∇w∥H2s1-s+1+∥χ∥Hs1∥w∥Hs1≤C∥w∥Hs12+Cν∥∇w∥H2s1-s+1+∥χ∥Hs1∥w∥Hs1.
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
(42)C∥∇w∥H2s1-s+1≤C∥∇w∥Hs1-1s-s1-1∥∇w∥Hs12-s+s1≤C∥w∥Hs12(s-s1-1)/(s-s1)+∥∇w∥Hs12.
Inserting this inequality into (41) and using Young's inequality again, we thus get
(43)12ddt∥∇w∥Hs12≤C∥∇w∥Hs12+Cν∥w∥Hs12(s-s1-1)/(s-s1)+∥χ∥Hs1∥w∥Hs1≤C∥w∥Hs12+Cνs-s1+C∥χ∥Hs12.
With similar argument as abovementioned, we can also obtain the Hs1 energy for χ as
(44)12ddt∥χ∥Hs12≤C∥χ∥Hs12+C∥w∥Hs12+Cκs-s1.
The previous two estimates give
(45)ddtϕ2(t)≤Cϕ2(t)+Cνs-s1+Cκs-s1,ϕ2(t):=∥w(t)∥Hs12+∥χ(t)∥Hs12,
and by the Gronwall inequality we get
(46)ϕ2(t)≤eCtϕ2(0)+C(νs-s1+κs-s1)teCt≤C2(ϕ2(0)+νs-s1+κs-s1),∀t∈[0,T],
which implies that the estimate (5) holds on [0,T] with C2:=eCT0(1+CT0).Step 3.
To prove (7), we need to regularize the initial data. Define ψ(x)∈C0∞(ℝ2) by
(47)ψ(x)={c0exp(-11-|x|2),|x|<1,0,|x|≥1,
where the constant c0 is selected so that ∫ℝ2‍ψ(x)dx=1. Let ψϵ(x)=ϵ-2ρ(ϵ-1x), and define the mollification 𝒥ϵf of f∈Lloc1(ℝ2) by (𝒥ϵf)(x)=(ψϵ*f)(x). By this definition, one can see 𝒥ϵf∈C∞; moreover, if f∈Hs, we have 𝒥ϵf→f in Hs as ϵ→0 and
(48)Limϵ→0∥𝒥ϵf-f∥s-k≤Cskϵk∥f∥s,∥𝒥ϵf∥s+k≤Cskϵk∥f∥s.
For the proof of these properties, see Lemma 3.5 in [20]. Now let (vϵ,ρϵ) be the solution of the inviscid system (2) with initial data (𝒥ϵv0,𝒥ϵρ0); namely, (vϵ,ρϵ) solves the equations
(49)vtϵ+(vϵ·∇)vϵ+∇πϵ=ρϵe2,ρtϵ+(vϵ·∇)ρϵ=0,∇·vϵ=0,vϵ|t=0=𝒥ϵv0,ρϵ|t=0=𝒥ϵρ0.
So, the Hs energy for vϵ and ρϵ can be written as
(50)12ddt∥vϵ∥Hs2≤C∥vϵ∥Hs3+∥vϵ∥Hs∥ρϵ∥Hs,12ddt∥ρϵ∥Hs2≤C∥vϵ∥Hs∥ρϵ∥Hs2.
With the same discussion as Lemma 5, we know that there exist T′>0 and C>0 both only depending on the Hs norm of (v0,ρ0) such that
(51)∥vϵ∥C([0,T′];Hs)+∥ρϵ∥C([0,T′];Hs)≤C.
Moreover, taking the Hs+k energy for vϵ and ρϵ and using (19), then for k∈ℕ+,
(52)12ddt∥vϵ∥Hs+k2≤C∥vϵ∥Hs∥vϵ∥Hs+k2+∥vϵ∥Hs+k∥ρϵ∥Hs+k≤C∥vϵ∥Hs+k2+C∥ρϵ∥Hs+k2,(53)12ddt∥ρϵ∥Hs+k2≤C∥vϵ∥Hs∥ρϵ∥Hs+k2+C∥vϵ∥Hs+k∥ρϵ∥Hs+k∥ρϵ∥Hs≤C∥vϵ∥Hs+k2+C∥ρϵ∥Hs+k2.
Using (48), we deduce from the Previous energy estimate that
(54)∥vϵ∥C([0,T′];Hs+k)+∥ρϵ∥C([0,T′];Hs+k)≤CeCT0ϵk,k∈ℕ+
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.
Setv~ϵ=vϵ-v and ρ~ϵ=ρϵ-ρ; then, from (2) and (49), we know that (v~ϵ,ρ~ϵ) satisfies
(55)v~tϵ+(v·∇)v~ϵ+(v~ϵ·∇)vϵ+∇(πϵ-π)=ρ~ϵe2,ρ~tϵ+(v·∇)ρ~ϵ+(v~ϵ·∇)ρϵ=0.
Using (18) with r=s, we have
(56)|〈Js((vϵ·∇)v~ϵ),Jsv~ϵ〉|≤C∥vϵ∥Hs∥v~ϵ∥Hs2.
By the Cauchy-Schwarz inequality and (16), one has
(57)|〈Js((v~ϵ·∇)vϵ),Jsv~ϵ〉|≤C(∥Jsv~ϵ∥L2∥∇vϵ∥L∞+∥v~ϵ∥L∞∥Js∇vϵ∥L2)∥Jsv~ϵ∥L2≤C(∥v~ϵ∥Hs∥vϵ∥Hs+∥v~ϵ∥L∞∥vϵ∥Hs+1)∥v~ϵ∥Hs.
With these two estimates, it is easy to obtain the following Hs energy estimate for v~ϵ:
(58)12ddt∥v~ϵ∥Hs2≤C(∥vϵ∥Hs+∥v∥Hs)∥v~ϵ∥Hs2+C∥v~ϵ∥L∞∥vϵ∥Hs+1∥v~ϵ∥Hs+∥ρ~ϵ∥Hs∥v~ϵ∥Hs≤C∥v~ϵ∥Hs2+CeCT0ϵ-1∥v~ϵ∥L∞∥v~ϵ∥Hs+∥ρ~ϵ∥Hs∥v~ϵ∥Hs,
where we have used (54) and the uniform estimate (51) in the last step. Similarly, we can obtain the following Hs energy estimate for ρ~ϵ:
(59)12ddt∥ρ~ϵ∥Hs2≤C(∥v∥Hs+∥ρ~ϵ∥Hs)∥ρ~ϵ∥Hs2+C∥v~ϵ∥L∞∥ρϵ∥Hs+1∥ρ~ϵ∥Hs≤C∥ρ~ϵ∥Hs2+CeCT0ϵ-1∥v~ϵ∥L∞∥ρ~ϵ∥Hs.
Now, we have to estimate ∥v~ϵ∥L∞. From (55), we take the Hs-2 energy for (v~ϵ,ρ~ϵ) and obtain (using (18) and (20))
(60)12ddt∥v~ϵ∥Hs-22≤C∥v~ϵ∥Hs-22+C∥ρ~ϵ∥Hs-2∥v~ϵ∥Hs-2,(61)12ddt∥ρ~ϵ∥Hs-22≤C∥ρ~ϵ∥Hs-22+C∥ρ~ϵ∥Hs-2∥v~ϵ∥Hs-2;
Then, the Gronwall inequality and (48) yield
(62)∥v~ϵ(t)∥Hs-2+∥ρ~ϵ(t)∥Hs-2≤eCT0(∥v~ϵ(0)∥Hs-2+∥ρ~ϵ(0)∥Hs-2)≤CeCT0ϵ2,
for all t∈[0,T], which in turn by Sobolev embedding theorem gives ∥v~ϵ(t)∥L∞+∥ρ~ϵ(t)∥L∞≤CeCT0ϵ2 since s>3. Inserting this estimate into (58) and (59), we can see
(63)ddtϕ3≤CeCT0ϕ3+CeCT0ϵ,ϕ3:=∥v~ϵ∥Hs+∥ρ~ϵ∥Hs,
where we have used the relation eCT0·eCT0=e2CT0=:eCT0 in the last step since the value of C at each appearance may be different. Hence, the Gronwall inequality gives
(64)ϕ3(t)≤eCT0eCT0(1+CT0eCT0)(ϕ3(0)+ϵ),∀t∈[0,T].Step 5.
LetUϵ=u-vϵ, Θϵ=θ-ρϵ, and recall that here u=uν,κ and θ=θν,κ; so, one can deduce from (2) and (55) that (Uϵ,Θϵ) solves
(65)Utϵ+(u·∇)Uϵ+(Uϵ·∇)vϵ+∇(p-πϵ)=νΔUϵ+νΔvϵ+Θϵe2,Θtϵ+(u·∇)Θϵ+(Uϵ·∇)ρϵ=κΔΘϵ+κΔρϵ.
Using the same reasonings that lead to (58), we have the Hs energy estimate for Uϵ as
(66)12ddt∥Uϵ∥Hs2+ν∥∇Uϵ∥Hs2≤C(∥u∥Hs+∥vϵ∥Hs)∥Uϵ∥Hs2+C∥vϵ∥Hs+1∥Uϵ∥L∞∥Uϵ∥Hs+ν∥vϵ∥Hs+2∥Uϵ∥Hs+∥Θϵ∥Hs∥Uϵ∥Hs≤C∥Uϵ∥Hs2+CeCT0ϵ-1∥Uϵ∥L∞∥Uϵ∥Hs+CeCT0νϵ-2∥Uϵ∥Hs+∥Θϵ∥Hs∥Uϵ∥Hs,
which yields
(67)ddt∥Uϵ∥Hs≤C∥Uϵ∥Hs+CeCT0ϵ-1∥Uϵ∥L∞+CeCT0νϵ-2+∥Θϵ∥Hs.
Similarly, we can obtain the Hs energy for Θϵ as
(68)ddt∥Θϵ∥Hs≤C∥Θϵ∥Hs+C∥Uϵ∥Hs+CeCT0ϵ-1∥Uϵ∥L∞+CeCT0κϵ-2.
Now, we should estimate ∥Uϵ∥L∞. Note that Uϵ=w-v~ϵ; so,
(69)∥Uϵ∥L∞≤∥w∥L∞+∥v~ϵ∥L∞.
By (40), (62), and Sobolev embedding theorem, we have
(70)∥w∥L∞(0,T;L∞)≤C∥w∥L∞(0,T;Hs-2)≤C1(ϕ1(0)+ν+κ),∥v~ϵ∥L∞(0,T;L∞)≤C∥v~ϵ∥L∞(0,T;Hs-2)≤CeCT0ϵ2,
which gives
(71)∥Uϵ∥L∞≤C*(ϕ1(0)+ν+κ+ϵ2),C*:=max{C1,CeCT0}.
Inserting this estimate into (67) and (68), one has
(72)ddtϕ4≤Cϕ4+CeCT0ϵ-1∥Uϵ∥L∞+CeCT0(ν+κ)ϵ-2≤Cϕ4+CeCT0ϵ-1C*(ϕ1(0)+ν+κ+ϵ2)+CeCT0(ν+κ)ϵ-2≤C**ϕ4+C**ϵ+C**ϵ-1ϕ1(0)+C**ϵ-2(ν+κ),
where ϕ4:=∥Uϵ∥Hs+∥Θϵ∥Hs and C**:=max{C,CeCT0C*,CeCT0}. By the Gronwall inequality, one obtains
(73)ϕ4(t)≤eC**T0(1+C**T0)×(ϕ4(0)+ϵ+ϵ-1ϕ1(0)+ϵ-2(ν+κ)).
Recall that
(74)w=Uϵ+v~ϵ,χ=Θϵ+ρ~ϵ;
therefore, it follows from (64) and (73) that
(75)∥w(t)∥Hs+∥χ(t)∥Hs≤ϕ3(t)+ϕ4(t)≤C~(ϕ3(0)+ϕ4(0)+ϵ+ϵ-1ϕ1(0)+ϵ-2(ν+κ))≤2C~(ϵ-2(ν+κ)ϕ3(0)+∥w(0)∥Hs+∥χ(0)∥Hs+ϵ+ϵ-1ϕ1(0)+ϵ-2(ν+κ))
for all t∈[0,T], where we use ϕ4(0)≤ϕ1(0)+ϕ3(0) in the last step and
(76)C~:=max{eC**T0(1+C**T0),eCT0eCT0(1+CT0eCT0)}.
Note that (3) gives
(77)∥w(0)∥Hs+∥χ(0)∥Hs=∥u0ν,κ-v0∥Hs+∥θ0ν,κ-ρ0∥Hs⟶0
as (ν,κ)→0, and the property of the operator 𝒥ϵ yields
(78)ϕ3(0)=∥v~ϵ(0)∥Hs+∥ρ~ϵ(0)∥Hs=∥vϵ(0)-v(0)∥Hs+∥ρϵ(0)-ρ(0)∥Hs=∥𝒥ϵv0-v0∥Hs+∥𝒥ϵρ0-ρ0∥Hs⟶0
as ϵ→0. Now, we choose ϵ=ϵ(ν,κ)>0 satisfying the following properties: (1)
lim(ν,κ)→0ϵ=0,(2)
lim(ν,κ)→0ϵ-1ϕ1(0)=lim(ν,κ)→0ϵ-1(∥u0ν,κ-v0∥Hs-2+∥θ0ν,κ-ρ0∥Hs-2)=0,(3)
lim(ν,κ)→0ϵ-2(ν+κ)=0.Hence, combining the previous convergence results, it is easy to obtain from (75) that
(79)lim(ν,κ)→0(∥wν,κ∥C([0,T];Hs)+∥χν,κ∥C([0,T];Hs))=0.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 (uν,κ(T1),θν,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 ∥uν,κ(T1)∥Hs, ∥θν,κ(T1)∥Hs, ∥v(T1)∥Hs, and ∥ρ(T1)∥Hs depend only on the Hs norm of the initial data (v0,ρ0). Then, we repeat the previous argument and find a positive sequence {Tk}k=1∞ such that (4), (5), and (7) hold on [0,T1+⋯+Tk]. We assert that ∑k=1∞‍Tk=T*. Indeed, if ∑k=1∞‍Tk=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)∥Hs+∥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 ∥uν,κ-v∥C([0,T0];Hs)+∥θν,κ-ρ∥C([0,T0];Hs)≤C; then, the convergence result (6) follows from (5) and the following interpolation inequality:
(80)∥f∥Hs′≤C∥f∥Hs-1s-s′∥f∥Hs1-(s-s′),s′∈[s-1,s].
Therefore, we finish the proof of Theorem 1.


## 3. TheHs Convergence Rate with Some Loss of Derivatives

In this section, we will prove Theorem2, and we still use the same notations that are used in the proof of Theorem 1.Proof of Theorem2.
Since (7) holds, without loss of generality, we may assume that
(81)∥u∥C([0,T0];Hs)+∥θ∥C([0,T0];Hs)≤C
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
(82)∥vϵ∥C([0,T0];Hs+δ)+∥ρϵ∥C([0,T0];Hs+δ)≤C,(83)∥vϵ∥C([0,T0];Hs+2)+∥ρϵ∥C([0,T0];Hs+2)≤CeCT0ϵδ-2.
Now the proof is divided into two cases.
Case   1 (0≤δ≤2). In this case, the estimate (82) implies
(84)∥vϵ∥C([0,T0];Hs+1)+∥ρϵ∥C([0,T0];Hs+1)≤C.
Using the estimates ∥v~ϵ∥L∞≤C∥v~ϵ∥Hs and (84), we deduce from the first inequality of (58) that
(85)12ddt∥v~ϵ∥Hs2≤C∥v~ϵ∥Hs2+∥ρ~ϵ∥Hs∥v~ϵ∥Hs.
Similarly, one can infer from the first inequality of (59) that
(86)12ddt∥ρ~ϵ∥Hs2≤C∥ρ~ϵ∥Hs2+C∥ρ~ϵ∥Hs∥v~ϵ∥Hs.
The previous two inequalities together with (48) give
(87)∥v~ϵ(t)∥Hs+∥ρ~ϵ(t)∥Hs≤CeCT0ϵδ,∀t∈[0,T0].
On the other hand, from (81) and the first inequality of (66), we can get
(88)12ddt∥Uϵ∥Hs2≤C∥Uϵ∥Hs2+ν∥vϵ∥Hs+2∥Uϵ∥Hs+∥Θϵ∥Hs∥Uϵ∥Hs≤C∥Uϵ∥Hs2+CeCT0νϵδ-2∥Uϵ∥Hs+∥Θϵ∥Hs∥Uϵ∥Hs,
where we have used (83) in the last step. Simultaneously, the Hs energy estimate for Θϵ is estimated as
(89)12ddt∥Θϵ∥Hs2≤C∥Θϵ∥Hs2+C∥Θϵ∥Hs∥Uϵ∥Hs+CeCT0κϵδ-2∥Θϵ∥Hs.
Then Gronwall inequality yields that
(90)∥Uϵ∥Hs+∥Θϵ∥Hs≤eCT0eCT0(1+CT0eCT0)×(∥Uϵ(0)∥Hs+∥Θϵ(0)∥Hs+(κ+ν)ϵδ-2).
Combining (87) and (90), we can arrive at
(91)∥w∥Hs+∥χ∥Hs≤C4(∥w(0)∥Hs+∥χ(0)∥Hs+(κ+ν)ϵδ-2+ϵδ)
for some C4>0, from which we know that (8) holds by choosing ϵδ=(κ+ν)ϵδ-2.
Case   2 (0<δ<1). In this case, we have
(92)∥vϵ∥C([0,T0];Hs+1)+∥ρϵ∥C([0,T0];Hs+1)≤CeCT0ϵδ-1.
And also one obtains from (48) and the first inequality of (62)
(93)∥v~ϵ∥L∞+∥ρ~ϵ∥L∞≤eCT0(∥v~ϵ(0)∥Hs-2+∥ρ~ϵ(0)∥Hs-2)≤CeCT0ϵ2+δ.
Applying the previous two estimates into (58) and (59), we get
(94)ddt∥v~ϵ∥Hs≤C∥v~ϵ∥Hs+CeCT0ϵ1+2δ+∥ρ~ϵ∥Hs,ddt∥ρ~ϵ∥Hs≤C∥ρ~ϵ∥Hs+CeCT0ϵ1+2δ,
which yields that
(95)∥v~ϵ∥Hs+∥ρ~ϵ∥Hs≤eCT0eCT0(1+CT0eCT0)×(∥v~ϵ(0)∥Hs+∥ρ~ϵ(0)∥Hs+ϵ1+2δ)≤CeCT0eCT0(1+T0eCT0)ϵδ.
On the other hand, the estimates (71) and (93) imply
(96)∥Uϵ∥L∞≤C′(ϕ1(0)+ν+κ+ϵ2+δ),C′≔max{C*,CeCT0}.
Then, we insert this inequality and (83) and (92) into (66) and obtain
(97)ddt∥Uϵ∥Hs≤C∥Uϵ∥Hs+CC′eCT0ϵδ-1(ϕ1(0)+ν+κ+ϵ2+δ)+CeCT0νϵδ-2+∥Θϵ∥Hs.
In the same way, the estimate (68) is replaced by
(98)ddt∥Θϵ∥Hs≤C∥Θϵ∥Hs+C∥Uϵ∥Hs+CC′eCT0ϵδ-1(ϕ1(0)+ν+κ+ϵ2+δ)+CeCT0κϵδ-2.
Hence, by the Gronwall inequality, we have
(99)∥Uϵ∥Hs+∥Θϵ∥Hs≤C′′(ϵ1+2δ∥Uϵ(0)∥Hs+∥Θϵ(0)∥Hs+ϵδ-1ϕ1(0)+(ν+κ)ϵδ-2+ϵ1+2δ)
for some C′′>0. This inequality, together with (95), gives
(100)∥w∥Hs+∥χ∥Hs≤C5(ϵδ∥w(0)∥Hs+∥χ(0)∥Hs+ϵδ-1ϕ1(0)+(ν+κ)ϵδ-2+ϵδ)
for some C5>0. So, (9) follows from (100) provided that we choose ϵ satisfying ϵ=ϕ1(0)+(κ+ν)(1/2).



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