# On the Well-Posedness of the Boussinesq Equation in the Triebel-Lizorkin-Lorentz Spaces

Authors: Zhaoyin Xiang; Wei Yan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We establish the local well-posedness and obtain a blow-up criterion of smooth solutions for the Boussinesq equations in the framework of Triebel-Lizorkin-Lorentz spaces. The main ingredients of our proofs are Littlewood-Paley decomposition and the paradifferential calculus.


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## Body

## 1. Introduction

In this paper, we consider the following inviscid Boussinesq equations:(1.1)∂tu+u⋅∇u+∇p=θed,inRd×(0,T),∂tθ+u⋅∇θ=0,inRd×(0,T),div⁡u=0,inRd×(0,T),u(x,0)=u0(x),θ(x,0)=θ0(x),inRd,
with divu0=0 and ed=(0,…,0,1). This system describes the motion of lighter or denser incompressible fluid under the influence of gravitational forces, for instance, in the atmospheric sciences, where the vector u=(u1,u2,…,ud) is the velocity, the scalar function θ denotes the temperature, and p stands for the pressure in the fluid (see [1]). The term θed takes into account the influence of the gravity and the stratification on the motion of the fluid. In case d=2, it is also used as a simplified model for the 3D axisymmetric Euler equations with swirl away from the symmetric axisr=0.The regularity or singularity questions of (1.1) are an outstanding open problem in the mathematical fluid mechanics. Over the past few years, (1.1) has been studied extensively both theoretically (see [1–6] and references therein) and numerically (see [7–9]). In particular, the local well-posedness and some blow-up criteria of (1.1) are established by Chae and Nam [3] in the Sobolev spaces, Liu et al. [5, 6] and Xiang [10] in the Besov spaces, and Cui et al. [11] in the Hölder spaces. The purpose of this paper is to establish the local well-posedness for the Boussineq equations (1.1) and to obtain a blow-up criterion of the smooth solutions in the framework of Triebel-Lizorkin-Lorentz spaces, which contain the classical Triebel-Lizorkin spaces and Lorentz spaces. Indeed, it is more natural to consider the well-posedness in the Triebel-Lizorkin-type spaces than in the Besov spaces in the sense that Fp,2s=Ws,p for 1<p<∞.Before proceeding further, we mention several local well-posedness results for other fluid equations in the Triebel-Lizorkin spaces. Recently, Chae [12] introduced a family trajectory mapping {Xj(α,t)} satisfying
(1.2)ddtXj(α,t)=Sj-2v(Xj(α,t),t),Xj(α,0)=α,
where Sj-2 is a frequency projection to the ball {ξ∈ℝ2||ξ|≲2j} and used the following equivalent relation:
(1.3)‖2js‖Δjθ(Xj(α,t),t)‖Lq(Z)‖Lp(dα)∼‖2js‖Δjθ(x,t)‖Lq(Z)‖Lp(dx)=‖θ(t)‖Ḟp,qs,
to estimate the frequency-localized solutions of the Euler equations in the Triebel-Lizorkin spaces. However, it seems difficult to give a strict proof for (1.3)-type equivalent relation due to the lack of the uniform change of the coordinates independent of j. To avoid this trouble, Chen et al. [13] introduced a particle trajectory mapping X(α,t) independent of j defined by
(1.4)ddtX(α,t)=v(X(α,t),t),X(α,0)=α,
and then they established a new commutator estimate to obtain the local well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces.In this paper, we will adapt the method of Chen et al. [13] to establish the local well-posedness and to obtain a blow-up criterion of the smooth solutions for the Boussineq equations (1.1) in the framework of Triebel-Lizorkin-Lorentz spaces. Precisely, we first define the particle trajectory mappingX(α,t) by (1.4). Then to show the well-posedness in the new framework, we have to establish the commutator estimate (Proposition 2.5) and the product estimate (Proposition 2.6). For the blow-up criterion, we also need the logarithmic inequality (Proposition 2.7). Fortunately, these preliminary estimates and inequalities have been obtained in our recent work [14].Our main results can be formulated in the following way.Theorem 1.1.
(i) Local Existence.  Let1<r≤p<∞,1<q<∞, and s>d/p+1. If (u0,θ0)∈(Fp,qs,r(ℝd))d+1, then there exists T1=T1(∥u0∥Fp,qs,r,∥θ0∥Fp,qs,r)>0 such that the Boussinesq equation (1.1) has a unique solution (u,θ)∈(C([0,T1];Fp,qs,r(ℝd)))d+1.
(ii) Blow-Up Criterion. The local-in-time solution(u,θ)∈(C([0,T1];Fp,qs,r(ℝd)))d+1 constructed in (i) blows up at finite timeT*>T1, that is,
(1.5)limsup⁡t→T*-(‖u(t)‖Fp,qs,r+‖θ(t)‖Fp,qs,r)=∞,
if and only if
(1.6)∫0T*(‖u(t)‖Ḟ∞,∞1+‖θ(t)‖Ḟ∞,∞1)dt=∞.Remark 1.2.
In caseθ0≡0, we have θ≡0 and thus get the local well-posedness and blow-up criterion for the incompressible Euler equations in the Triebel-Lizorkin-Lorentz spaces, which generalize the corresponding results in [12] because of Fp,qs,p=Fp,qs. For general θ0, we also extend the results of [3] from Sobolev spaces to Triebel-Lizorkin-Lorentz spaces.Remark 1.3.
For the MHD equations, we can also generalize the local well-posedness and blow-up criterion of [13] from the Triebel-Lizorkin spaces to the Triebel-Lizorkin-Lorentz spaces by using similar arguments.The rest of this paper is organized as follows. We state some preliminaries on functional settings in Section2 and then prove the local well-posedness and blow-up criterion in Section 3.


## 2. Preliminaries

In this section, we give some definitions and basic estimates (see [14, 15] for details). First of all, we let 𝒮(ℝd) be the Schwartz class of rapidly decreasing functions. For a fixed radially symmetric bump function χ∈𝒞0∞(B(0,4/3)) with value 1 over the ballB(0,3/4), we set φ(ξ):=χ(ξ/2)-χ(ξ) and then have the following dyadic partition of unity:
(2.1)χ(ξ)+∑j≥0φ(2-jξ)=1,ξ∈Rd,∑j∈Zφ(2-jξ)=1,ξ∈Rd∖{0}.
The frequency localization operators Δj and Sj can be defined as follows:
(2.2)Δju:=φ(2-jD)u=F-1(φ(2-j⋅)F(u))=2jd(F-1φ(2j⋅))*u,j∈Z,Sju:=∑j′≤j-1Δj′u=χ(2-jD)u=2jd(F-1χ(2j⋅))*u,j∈Z,
where ℱ is the Fourier transform and D is the Fourier multiplier with symbol |ξ|. The frequency localization operators Δj and Sj have nice almost orthogonal properties in L2:
(2.3)ΔjΔj′u≡0,if|j-j′|≥2,Δj(Sj′-1uΔj′v)≡0,if|j-j′|≥5
for any u,v∈𝒮′(ℝd).Next we recall several function spaces used in this paper. The first one isthe Triebel-Lizorkin spaces. For s∈ℝ, (p,q)∈[1,+∞)×[1,+∞], we define the homogeneous Triebel-Lizorkin spaces Ḟp,qs(ℝd) as the set of tempered distributions u∈𝒮′/𝒫(ℝd) such that
(2.4)‖u‖Ḟp,qs(Rd):=‖‖2js|Δju|‖lq(Z)‖Lp(Rd)<∞,
where 𝒫 is the polynomial space. Then for s>0, (p,q)∈[1,+∞)×[1,+∞], we can define the inhomogeneous Triebel-Lizorkin spaces Fp,qs(ℝd) as the set of tempered distributions u such that
(2.5)‖u‖Fp,qs(Rd):=‖u‖Lp(Rd)+‖u‖Ḟp,qs(Rd)<∞.
The inhomogeneous Triebel-Lizorkin spaces Fp,qs are Banach spaces with norms ∥·∥Fp,qs.The second one isthe Lorentz spaces. Let u*(τ) be the distribution function of a function u(x) and let u*(t) be its nonincreasing rearrangement, that is,
(2.6)u*(τ)=meas⁡{x||u(x)|>τ},u*(τ)=inf⁡⁡{τ>0|u*(τ)≤t}.
Then for 1<p,r<∞, the Lorentz spaces Lp,r(ℝd) are defined as the set of those measurable functions u satisfying
(2.7)‖u‖Lp,r=(rp∫0∞(t1/pu**(t))rdtt)1/r<∞,
where u**(t):=(1/t)∫0tu*(s)ds is the average rearrangement. The Lorentz spaces Lp,r are Banach spaces with norms ∥·∥Lp,r. Since u*(τ) is nonincreasing in τ, it is easy to verify that the following equivalent characterization holds:
(2.8)‖u‖Lp,r~‖2ju*1/p(2j)‖lr.
Moreover, it is well known that there exist 1≤p1andp2<∞ such that Lp,r are the interpolations spaces between Lp1 and Lp2. Thus we can deduce the following basic fact, which is standard in the usual Lp spaces.Lemma 2.1.
Assumeu∈Lp,r for 1<p,r<∞. If X:α→X(α) is a volume-preserving diffeomorphism, then
(2.9)‖u(α)‖Lp,r=‖u(X(α))‖Lp,r.The last one is calledthe Triebel-Lizorkin-Lorentz spaces and is introduced recently by Yang et al. [16] to unify the Triebel-Lizorkin spaces and the Lorentz spaces.Definition 2.2.
Fors∈ℝ and 1<p,q,r<∞, a distribution u∈𝒮′ is said to be in the homogeneous Triebel-Lizorkin-Lorentz spaces Ḟp,qs,r(ℝd) if
(2.10)‖2k(meas⁡{x∈Rd∣‖2jS|Δju|‖ljq(Z)>2k})1/p‖lkr(Z)<∞.
Indeed we can deduce that u∈Ḟp,qs,r(ℝd)  if and only if∥∥2js|Δju|∥lq(ℤ)∥Lp,r<∞ by the equivalent characterization (2.8) of Lorentz space Lp,r. Then Ḟp,qs(ℝd)⋍Ḟp,qs,p(ℝd) and Lp,r(ℝd)⋍Ḟp,20,r(ℝd) (see [16, Theorem  5]). For s>0, the inhomogeneous Triebel-Lizorkin-Lorentz spaces Fp,qs,r(ℝd) are defined by
(2.11)Fp,qs,r(Rd)=Lp,r∩Ḟp,qs,r(Rd)
with norm
(2.12)‖u‖Fp,qs,r(Rd)=‖u‖Lp,r+‖u‖Ḟp,qs,r(Rd).
Clearly, Fp,qs,r(ℝd) is a Banach space. It has been proved that there exist 1<p1andp2<∞ such that Fp,qs,r(ℝd) are the interpolation spaces between Fp1,qs(ℝd) and Fp2,qs(ℝd) (see [16, Theorem  6]). Thus by the boundedness of Riesz transform on the Triebel-Lizorkin spaces (see Frazier et al. [17]) we have the following.Lemma 2.3.
Riesz transform is bounded from the Triebel-Lizorkin-Lorentz spaceFp,qs,r(ℝd) into itself.
Also byLp,r↪Lp for r≤p, we have the following Sobolev’s embedding theorem:
(2.13)Fp,qs,r↪Fp,qs,p=Fp,qs↪L∞,fors>dp.The following lemma is referred as Bernstein’s inequality, which describes the way derivatives act on spectrally localized functions.Lemma 2.4 (Bernstein’s inequality).
Let1<p,r<∞, and u∈Lp,r(ℝd). Then
(2.14)supp⁡F(u)⊂{ξ∈Rd:|ξ|~2j}⟹‖∂αu‖Lp,r~2j|α|‖u‖Lp,r.Here and thereafter, we useA≲B and A~B to denote A≤CB and cB≤A≤CB for some constants C>c>0, respectively. Thus Bernstein’s inequality together with the Littlewood-Paley decomposition gives that(2.15)‖∂αu‖Ḟp,qs,r~‖u‖Ḟp,qs+|α|,r,1<p,r<∞
for any u∈Ḟp,qs+|α|,r(ℝd).To conclude this section, we recall three important propositions in the Triebel-Lizorkin-Lorentz spaces, which will be frequently used in the proof of Theorem1.1. For their proofs, we refer to our recent work [14] (for the case in the Triebel-Lizorkin spaces, see also Chae [12] and Chen et al. [13]). The first one is related to the product estimates.Proposition 2.5.
Let1<p,q,r<∞. There exists a constant C>0 such that for s>0(2.16)‖uθ‖Ḟp,qs,r≤C(‖u‖L∞‖θ‖Ḟp,qs,r+‖θ‖L∞‖u‖Ḟp,qs,r),
and for s>-1(2.17)‖uθ‖Fp,qs,r≤C(‖u‖L∞‖θ‖Fp,qs,r+‖θ‖L∞‖u‖Fp,qs,r),
as long as the right-hand sides of (2.16) and (2.17) are finite.The next one concerns the commutator estimates.Proposition 2.6.
Lets>0, 1<p,q,r<∞, and let the vector field v be divergence free. There exists a constant C>0 such that
(2.18)‖‖2js([v,Δj]⋅∇θ)‖lq(Z)‖Lp,r≤C(‖∇v‖L∞‖θ‖Ḟp,qs,r+‖∇θ‖L∞‖v‖Ḟp,qs,r),(2.19)‖‖2js([v,Δj]⋅∇θ)‖lq(Z)‖Lp,r≤C(‖∇v‖L∞‖θ‖Ḟp,qs,r+‖θ‖L∞‖∇v‖Ḟp,qs,r),
as long as the right-hand sides of (2.18) and (2.19) are finite.The last one is the logarithmic Triebel-Lizorkin-Lorentz inequality with features similar to the classical logarithmic Sobolev inequality [18] and to the logarithmic Triebel-Lizorkin inequality [12].Proposition 2.7.
Let1<r≤p<∞,1<q<∞, and s>d/p. There exists a constant C>0 such that
(2.20)‖u‖L∞≤C(1+‖u‖Ḟ∞,∞0(log⁡⁡+‖u‖Fp,qs,r+1)).


## 3. Proof of Theorem1.1

In this section, we prove the local well-posedness of solutions to (1.1) by constructing the successive approximations and using the a priori estimate and establish the blow-up criterion of smooth solutions by the logarithmic Triebel-Lizorkin-Lorentz inequality. The proof is divided into five steps.Step 1 (a priori estimate).
For a given fluid velocityu(x,t), we let X(α,t) be the particle-trajectory mapping defined by the solutions of the following ordinary differential equations:
(3.1)ddtX(α,t)=u(X(α,t),t),X(α,0)=α.
Then from the Boussinesq equations (1.1) and the ODEs (3.1), we have
(3.2)ddtu(X(α,t),t)=-∇p(X(α,t),t)+θ(X(α,t),t)ed,ddtθ(X(α,t),t)=0,u(X(α,0),0)=u0(α),θ(X(α,0),0)=θ0(α),
from which we get
(3.3)u(X(α,t),t)=u0(α)-∫0t∇p(X(ατ),τ)dτ+∫0tθ(X(α,τ),τ)eddτ,θ(X(α,t),t)=θ0(α).
It follows from divu=0 that the mapping α→X(α,t) is a volume-preserving diffeomorphism for any given t>0. Thus by Lemma 2.1 we have
(3.4)‖u(t)‖Lp,r+‖θ(t)‖Lp,r=‖u(X(α,t),t)‖Lp,r+‖θ(X(α,t),t)‖Lp,r≤‖u0‖Lp,r+‖θ0‖Lp,r+∫0t‖∇p(X(α,τ),τ)‖Lp,rdτ+∫0t‖θ(X(α,τ),τ)‖Lp,rdτ=‖u0‖Lp,r+‖θ0‖Lp,r+∫0t‖∇p(τ)‖Lp,rdτ+∫0t‖θ(τ)‖Lp,rdτ.
To deal with the pressure term, we take the divergence on both sides of (1.1)1 and then obtain the representation of the pressure:
(3.5)p=(-Δ)-1∂iuj∂jui-(-Δ)-1∂dθ=(-Δ)-1∂i∂j(uiuj)-(-Δ)-1∂dθ,
where we used the Einstein convention on the summation over repeated indices and the fact that divu=0. Thus by the boundedness of Riesz transform on Lp,r, we have
(3.6)‖∇p(τ)‖Lp,r≤‖∇(-Δ)-1∂i∂j(uiuj)(τ)‖Lp,r+‖∇(-Δ)-1∂dθ(τ)‖Lp,r≲‖u⋅∇u(τ)‖Lp,r+‖θ(τ)‖Lp,r≤‖u(τ)‖Lp,r‖∇u(τ)‖∞+‖θ(τ)‖Lp,r.
Substituting the last inequality into (3.4), we obtain
(3.7)‖u(t)‖Lp,r+‖θ(t)‖Lp,r≲‖u0‖Lp,r+‖θ0‖Lp,r+∫0t‖u(τ)‖Lp,r‖∇u(τ)‖∞dτ+∫0t‖θ(τ)‖Lp,rdτ.
On the other hand, we apply the frequency projectionΔj to both sides of (1.1) to obtain
(3.8)∂tΔju+u⋅∇Δju=[u,Δj]⋅∇u-∇Δjp+Δjθed,∂tΔjθ+u⋅∇Δjθ=[u,Δj]⋅∇θ.
Then by (3.1), we have
(3.9)ddtΔju(X(α,t),t)=[u,Δj]⋅∇u(X(α,t),t)-∇Δjp(X(α,t),t)+Δjθ(X(α,t),t)ed,ddtΔjθ(X(α,t),t)=[u,Δj]⋅∇θ(X(α,t),t).
Integrating the previous two equations from 0 to t gives
(3.10)Δju(X(α,t),t)+Δjθ(X(α,t),t)=Δju0(α)+Δjθ0(α)+∫0t([u,Δj]⋅∇u(X(α,τ),τ)+[u,Δj]⋅∇θ(X(α,τ),τ))dτ-∫0t∇Δjp(X(α,τ),τ)dτ+∫0tθ(X(α,τ),τ)eddτ.
We first multiply both sides by 2js, take the lq(ℤ) norm, and then use Minkowski’s inequality to obtain
(3.11)‖2js|Δju(X(α,t),t)|‖lq(Z)+‖2js|Δjθ(X(α,t),t)|‖lq(Z)≤‖2js|Δju0(α)|‖lq(Z)+‖2js|Δjθ0(α)|‖lq(Z)+∫0t(‖2js|[u,Δj]⋅∇u(X(α,τ),τ)|‖lq(Z)+‖2js|[u,Δj]⋅∇θ(X(α,τ),τ)|‖lq(Z))dτ+∫0t‖2js|∇Δjp(X(α,τ),τ)|‖lq(Z)dτ+∫0t‖2js|θ(X(α,τ),τ)|‖lq(Z)dτ,
and then we take the Lp,r norm with respect to α and use Minkowski’s inequality to get
(3.12)‖‖2js|Δju(X(α,t),t)|‖lq(Z)‖Lp,r+‖‖2js|Δjθ(X(α,t),t)|‖lq(Z)‖Lp,r≤‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r+∫0t‖‖2js|[u,Δj]⋅∇u(X(α,τ),τ)|‖lq(Z)‖Lp,rdτ+∫0t‖‖2js|[u,Δj]⋅∇θ(X(α,τ),τ)|‖lq(Z)‖Lp,rdτ+∫0t‖‖2js|∇Δjp(X(α,τ),τ)|‖lq(Z)‖Lp,rdτ+∫0t‖‖2js|θ(X(α,τ),τ)|‖lq(Z)‖Lp,rdτ.
Recalling the fact that α→X(α,t) is a volume-preserving diffeomorphism for any given t>0 and using Lemma 2.1, we have
(3.13)‖u(t)‖Ḟp,qs,r+‖θ(t)‖Ḟp,qs,r≲‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r+∫0t‖‖2js|[u,Δj]⋅∇u(τ)|‖lq(Z)‖Lp,r+‖‖2js|[u,Δj]⋅∇θ(τ)|‖lq(Z)‖Lp,rdτ+∫0t‖∇p(τ)‖Ḟp,qs,rdτ+∫0t‖θ(τ)‖Ḟp,qs,rdτ.
Thanks to the commutator estimates (2.18), the integrand in the first integral is dominated by
(3.14)‖∇u(τ)‖L∞‖u(τ)‖Ḟp,qs,r+‖∇u(τ)‖L∞‖θ(τ)‖Ḟp,qs,r+‖∇θ(τ)‖L∞‖u(τ)‖Ḟp,qs,r,
and thus
(3.15)‖u(t)‖Ḟp,qs,r+‖θ(t)‖Ḟp,qs,r≲‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r+∫0t‖∇p(τ)‖Ḟp,qs,rdτ+∫0t(1+‖∇u(τ)‖L∞+‖∇θ(τ)‖L∞)(‖u(τ)‖Ḟp,qs,r+‖θ(τ)‖Ḟp,qs,r)dτ.
For the term related to the pressure, we use (3.5) to obtain
(3.16)∂k∂lp=∂k∂l(-Δ)-1∂iuj∂jui-∂k∂l(-Δ)-1∂dθ:=RkRl(∂iuj∂jui)-RkRl(∂dθ),
where RkandRl are Riesz transform, and then
(3.17)‖∇p‖Ḟp,qs,r≤‖∂k∂lp‖Ḟp,qs-1,r≤‖∂iuj∂jui‖Ḟp,qs-1,r+‖∂dθ‖Ḟp,qs-1,r≤‖u‖Ḟp,qs,r‖∇u‖+‖θ‖Ḟp,qs,r,
where we used the boundedness of Riesz transform on Ḟp,qs,r and the product estimate (2.16). Substituting the last inequality into (3.15), we have
(3.18)‖u(t)‖Ḟp,qs,r+‖θ(t)‖Ḟp,qs,r≲‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r+∫0t(1+‖∇u(τ)‖L∞+‖∇θ(τ)‖L∞‖θ(τ)‖Ḟp,qs,r)(‖u(τ)‖Ḟp,qs,r+‖θ(τ)‖Ḟp,qs,r)dτ.
This together with Gronwall’s inequality gives the a priori estimates in the homogeneous Triebel-Lizorkin-Lorentz spaces:
(3.19)‖u(t)‖Ḟp,qs,r+‖θ(t)‖Ḟp,qs,r≤C(‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r)exp⁡(C∫0t(1+‖∇v(τ)‖L∞+‖∇θ(τ)‖L∞)dτ).
Similarly, by (3.7) and (3.18), we get
(3.20)‖u(t)‖Fp,qs,r+‖θ(t)‖Fp,qs,r≲‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r+∫0t(1+‖∇u(τ)‖L∞+‖∇θ(τ)‖L∞‖θ(τ)‖Fp,qs,r)(‖u(τ)‖Fp,qs,r+‖θ(τ)‖Fp,qs,r)dτ,
and thus Gronwall’s inequality yields the a priori estimates in the inhomogeneous spaces:
(3.21)‖u(t)‖Fp,qs,r+‖θ(t)‖Fp,qs,r≤C(‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r)exp⁡⁡(C∫0t(1+‖∇u(τ)‖L∞+‖∇θ(τ)‖L∞)dτ).Step 2 (approximate solutions and uniform estimates).
In order to construct the approximate solutions of (1.1), we first set (u(0),θ(0))=(0,0) and then define {(u(n),θ(n))}n∈ℕ as the solutions of the linear equations:
(3.22)∂tu(n+1)+u(n)⋅∇u(n+1)+∇p(n+1)=θ(n+1)ed,inRd×(0,∞),∂tθ(n+1)+u(n)⋅∇θ(n+1)=0,inRd×(0,∞),div⁡u(n)=0,inRd×(0,∞),
with initial datau(n+1)(x,0)=u0(n+1)(x):=Sn+2u0(x),θ(n+1)(x,0)=θ0(n+1)(x):=Sn+2θ0(x).
For eachn∈ℕ, similar to Step 1, we let the particle-trajectory mapping X(n)(α,t) be the solution of the following ordinary differential equation:
(3.23)ddtX(n)(α,t)=u(n)(X(n)(α,t),t),X(n)(α,0)=α.
Then following the same procedure of estimate leading to (3.20), we obtain
(3.24)‖u(n+1)(t)‖Fp,qs,r+‖θ(n+1)(t)‖Fp,qs,r≲‖Sn+2u0‖Fp,qs,r+‖Sn+2θ0‖Fp,qs,r+∫0t(1+‖∇u(n)(τ)‖L∞)×(‖u(n+1)(τ)‖Fp,qs,r+‖θ(n+1)(τ)‖Fp,qs,r)dτdτ.+∫0t(‖∇u(n+1)(τ)‖L∞+‖∇θ(n+1)(τ)‖L∞)‖u(n)(τ)‖Fp,qs,r.
Note that
(3.25)‖Sn+2u0‖Fp,qs,r≤‖u0‖Fp,qs,r,‖Sn+2θ0‖Fp,qs,r≤‖θ0‖Fp,qs,r.
This together with Sobolev embedding theorem Fp,qs-1,r↪L∞ for s-1>d/p yields that
(3.26)‖u(n+1)(t)‖Fp,qs,r+‖θ(n+1)(t)‖Fp,qs,r≲‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r+∫0t(1+‖u(n)(τ)‖Fp,qs,r)(‖u(n+1)(τ)‖Fp,qs,r+‖θ(n+1)(τ)‖Fp,qs,r)dτ.
It follows from Gronwall’s inequality that
(3.27)‖u(n+1)(t)‖Fp,qs,r+‖θ(n+1)(t)‖Fp,qs,r≤C(‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r)exp⁡⁡(C∫0t(1+‖u(n)(τ)‖Fp,qs,r)dτ)
for some C>0 independent of n. Choosing T0:=T0(∥u0∥Fp,qs,r,∥θ0∥Fp,qs,r)>0 such that
(3.28)C(1+2C‖u0‖Fp,qs,r+2C‖θ0‖Fp,qs,r)T0≤log⁡⁡2,
and using the standard induction arguments, we obtain for any n∈ℕ0(3.29)sup⁡⁡0≤t≤T0(‖u(n+1)(t)‖Fp,qs,r+‖θ(n+1)(t)‖Fp,qs,r)≤2C(‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r).Step 3 (existence).
To prove the local existence, we show that there existsT1∈(0,T0] independent of n such that (u(n),θ(n)) is a Cauchy sequence in (C([0,T1];Fp,qs-1,r(ℝd)))d+1. For this purpose, we denote
(3.30)δu(n+1):=u(n+1)-u(n),δθ(n+1):=θ(n+1)-θ(n),δp(n+1):=p(n+1)-p(n).
It follows that (δu(n+1),δθ(n+1),δp(n+1)) satisfies the following:
(3.31)∂tδu(n+1)+u(n)⋅∇δu(n+1)+∇δp(n+1)=-δu(n)⋅∇u(n)+δθ(n+1)ed,inRd×(0,∞),∂tδθ(n+1)+u(n)⋅∇δθ(n+1)=-δu(n)⋅∇θ(n),inRd×(0,∞),div⁡u(n)=0,inRd×(0,∞),δu(n+1)(x,0)=Δn+1u0(x),δθ(n+1)(x,0)=Δn+1θ0(x),inRd.
Since X(n)(α,t) is a solution of the ordinary differential equations (3.23), we have
(3.32)ddtδu(n+1)(X(n)(α,t),t)=-δu(n)⋅∇u(n)(X(n)(α,t),t)-∇δp(n+1)(X(n)(α,t),t)+δθ(n+1)(X(n)(α,t),t)ed,ddtδθ(n+1)(X(n)(α,t),t)=-δu(n)⋅∇θ(n)(X(n)(α,t),t).
Integrating the previous equations on [0,t], we have
(3.33)δu(n+1)(X(n)(α,t),t)+δθ(n+1)(X(n)(α,t),t)=Δn+1u0(α)+Δn+1θ0(α)-∫0tδu(n)⋅∇(u(n)+θ(n))(X(n)(α,τ),τ)dτ-∫0t∇δp(n+1)(X(n)(α,τ),τ)dτ+∫0tδθ(n+1)(X(n)(α,τ),τ)eddτ.
Similar to Step 1, we take the Lp,r norm on both sides, use Minkowski’s inequality and Hölder’s inequality in Lorentz spaces, and then obtain
(3.34)‖δu(n+1)(t)‖Lp,r+‖δθ(n+1)(t)‖Lp,r≲‖Δn+1u0‖Lp,r+‖Δn+1θ0‖Lp,r+∫0t‖δu(n)⋅∇(u(n)+θ(n))(τ)‖Lp,rdτ+∫0t‖∇δp(n+1)‖Lp,rdτ+∫0t‖δθ(n+1)(τ)‖Lp,rdτ≲‖Δn+1u0‖Lp,r+‖Δn+1θ0‖Lp,r+∫0t‖δu(n)‖Lp,r(‖∇u(n)(τ)‖∞+‖∇θ(n)(τ)‖L∞)dτ+∫0t(1+‖∇u(n)(τ)‖L∞)(‖δu(n+1)(τ)‖Lp,r+‖δθ(n+1)(τ)‖Lp,r)dτ≲‖Δn+1u0‖Lp,r+‖Δn+1θ0‖Lp,r+∫0t‖δu(n)‖Lp,r(‖u(n)(τ)‖Fp,qs,r+‖θ(n)(τ)‖Fp,qs,r)dτ+∫0t(1+‖u(n)(τ)‖Fp,qs,r)(‖δu(n+1)(τ)‖Lp,r+‖δθ(n+1)(τ)‖Lp,r)dτ,
where we used Fp,qs-1,r↪L∞ for s-1>d/p in the last inequality.
Next, we applyΔj to the first equation of (3.31) to obtain
(3.35)∂tΔjδu(n+1)+u(n)⋅∇Δjδu(n+1)=[u(n),Δj]⋅∇δu(n+1)-Δj(δu(n)⋅∇u(n))-∇Δjδp(n+1)+Δjδθ(n+1)ed,∂tΔjδθ(n+1)+u(n)⋅∇Δjδθ(n+1)=[u(n),Δj]⋅∇δθ(n+1)-Δj(δu(n)⋅∇θ(n)).
It follows from the definition of X(n)(α,t) that
(3.36)ddtΔjδu(n+1)(X(n)(α,t),t)=[u(n),Δj]⋅∇δu(n+1)(X(n)(α,t),t)-Δj(δu(n)⋅∇u(n))(X(n)(α,t),t)-∇Δjδp(n+1)(X(n)(α,t),t)+Δjδθ(n+1)(X(n)(α,t),t)ed,ddtΔjδθ(n+1)(X(n)(α,t),t)=[u(n),Δj]⋅∇δθ(n+1)(X(n)(α,t),t)-Δj(δu(n)⋅∇θ(n))(X(n)(α,t),t).
Integrating the previous equations on [0,t] yields
(3.37)Δjδu(n+1)(X(n)(α,t),t)+Δjδθ(n+1)(X(n)(α,t),t)=ΔjΔn+1u0(α)+ΔjΔn+1θ0(α)+∫0t[u(n),Δj]⋅∇δu(n+1)(X(n)(α,τ),τ)dτ+∫0t[u(n),Δj]⋅∇δθ(n+1)(X(n)(α,τ),τ)dτ-∫0tΔj(δu(n)⋅∇u(n))(X(n)(α,τ),τ)dτ-∫0tΔj(δu(n)⋅∇θ(n))(X(n)(α,τ),τ)dτ-∫0t∇Δjδp(n+1)(X(n)(α,τ),τ)dτ+∫0tΔjδθ(n+1)(X(n)(α,τ),τ)eddτ.
Then following the similar procedure of estimate leading to (3.20), we get
(3.38)‖δu(n+1)(t)‖Ḟp,qs-1,r+‖δθ(n+1)(t)‖Ḟp,qs-1,r≲‖Δn+1u0‖Ḟp,qs-1,r+‖Δn+1θ0‖Ḟp,qs-1,r+∫0t‖‖2j(s-1)|[u(n),Δj]⋅∇δu(n+1)(τ)|‖lq(Z)‖Lp,rdτ+∫0t‖‖2j(s-1)|[u(n),Δj]⋅∇δθ(n+1)(τ)|‖lq(Z)‖Lp,rdτ+∫0t‖δu(n)⋅∇u(n)‖Ḟp,qs-1,rdτ+∫0t‖δu(n)⋅∇θ(n)‖Ḟp,qs-1,rdτ+∫0t‖∇δp(n+1)‖Ḟp,qs-1,rdτ+∫0t‖δθ(n+1)‖Ḟp,qs-1,rdτ≲‖Δn+1u0‖Ḟp,qs-1,r+‖Δn+1θ0‖Ḟp,qs-1,r+∫0t(1+‖∇u(n)(τ)‖∞)(‖δu(n+1)(τ)‖Ḟp,qs-1,r+‖δθ(n+1)(τ)‖Ḟp,qs-1,r)dτ+∫0t(‖δu(n+1)(τ)‖∞+‖δθ(n+1)(τ)‖∞)‖∇u(n)(τ)‖Ḟp,qs-1,rdτ+∫0t‖δu(n)(τ)‖∞(‖∇u(n)(τ)‖Ḟp,qs-1,r+‖∇θ(n)(τ)‖Ḟp,qs-1,r)dτ+∫0t‖δu(n)(τ)‖Ḟp,qs-1,r(‖∇u(n)(τ)‖∞+‖∇θ(n)(τ)‖∞)dτ,
where we used the commutator estimates (2.19) and the product estimates (2.16) in last inequality. This together with (3.34), Sobolev embedding theorem, and the boundedness of Riesz transform gives that
(3.39)‖δu(n+1)(t)‖Fp,qs-1,r+‖δθ(n+1)(t)‖Fp,qs-1,r≲‖Δn+1u0‖Fp,qs-1,r+‖Δn+1θ0‖Fp,qs-1,r+∫0t(1+‖u(n)(τ)‖Fp,qs,r)(‖δu(n+1)(τ)‖Ḟp,qs-1,r+‖δθ(n+1)(τ)‖Ḟp,qs-1,r)dτ+∫0t‖δu(n)(τ)‖Fp,qs-1,r(‖u(n)(τ)‖Fp,qs,r+‖θ(n)(τ)‖Fp,qs,r)dτ≲2-(n+1)(‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r)+T1(1+sup⁡⁡t∈[0,T1]‖u(n)(t)‖Fp,qs,r)sup⁡t∈[0,T1](‖δu(n+1)(t)‖Ḟp,qs-1,r+‖δθ(n+1)(t)‖Ḟp,qs-1,r)+T1sup⁡t∈[0,T1](‖u(n)(τ)‖Fp,qs,r+‖θ(n)(τ)‖Fp,qs,r)sup⁡⁡t∈[0,T1]‖δu(n)(t)‖Fp,qs-1,r.
It follows from the uniform bounds (3.29) that
(3.40)sup⁡⁡t∈[0,T1](‖δu(n+1)(t)‖Fp,qs-1,r+‖δθ(n+1)(t)‖Fp,qs-1,r)≤C2-(n+1)(‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r)+CT1sup⁡⁡t∈[0,T1](‖δu(n+1)(t)‖Fp,qs-1,r+‖δθ(n+1)(t)‖Fp,qs-1,r)+CT1sup⁡⁡t∈[0,T1](‖δu(n)(t)‖Fp,qs-1,r+‖δθ(n)(t)‖Fp,qs-1,r).
If we choose T1 small enough such that CT1≤1/2, then
(3.41)sup⁡⁡t∈[0,T1](‖δu(n+1)(t)‖Fp,qs-1,r+‖δθ(n+1)(t)‖Fp,qs-1,r)≤C2-(n+1)(‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r)+2CT1sup⁡⁡t∈[0,T1](‖δu(n)(t)‖Fp,qs-1,r+‖δθ(n)(t)‖Fp,qs-1,r).
By the standard induction arguments, we obtain
(3.42)sup⁡⁡t∈[0,T1](‖δu(n+1)(t)‖Fp,qs-1,r+‖δθ(n+1)(t)‖Fp,qs-1,r)≤2C2-n(‖u0‖Ḟp,qs,r+‖θ0‖Ḟp,qs,r),
which implies that (u(n)(x,t),θ(n)(x,t)) is a Cauchy sequence in (C([0,T1];Fp,qs-1,r))d+1 with T1≤min{T0,1/2C}. Thus u(n)(x,t)→u(x,t) and θ(n)(x,t)→θ(x,t) in (C([0,T1];Fp,qs,r))d+1 as n→∞. From (3.22) and the uniform estimates (3.29), we have actually (u(x,t),θ(x,t))∈(C([0,T1];Fp,qs,r))d+1 solving the Boussinesq equation (1.1).Step 4 (uniqueness).
To get the uniqueness we closely follow the arguments in last step. Assume that(u,θ) and (ũ,θ̃) are two solutions of the Boussinesq equations (1.1) with the same initial data (u0,θ0) in the class L∞([0,T1];Fp,qs,r(ℝd)). Denote
(3.43)δu:=ũ-u,δθ:=θ̃-θ,δp:=p̃-p.
Then it follows that δθ satisfies the following:
(3.44)∂tδu+ũ⋅∇δu+∇δp=-δu⋅∇u+δθed,inRd×(0,T1),∂tδθ+ũ⋅∇δθ=-δu⋅∇θ,inRd×(0,T1),div⁡ũ=0,inRd×(0,T1),δu(x,0)=0,δθ(x,0)=0,inRd.
We follow the strategy used to derive (3.40) to obtain
(3.45)sup⁡⁡t∈[0,T](‖δu(t)‖Fp,qs-1,r+‖δθ(t)‖Fp,qs-1,r)≤CTsup⁡⁡t∈[0,T](‖δu(t)‖Fp,qs-1,r+‖δθ(t)‖Fp,qs-1,r)
for any T≤T1. Whenever T is small enough such that CT<1, we have δu(x,t)≡δθ(x,t)≡0 for any t≤T, that is, ũ(x,t)≡u(x,t) and θ̃(x,t)≡θ(x,t). By using the standard continuity argument, we see ũ(x,t)≡u(x,t) and θ̃(x,t)≡θ(x,t) for any t≤T1.Step 5 (blow-up criterion).
By thea priori estimate (3.21), we only need to dominate ∥∇u∥∞ and ∥∇θ∥∞. Indeed, it follows from the logarithmic Triebel-Lizorkin-Lorentz inequality (2.20) with s-1>d/p that
(3.46)‖∇u‖L∞≲1+‖∇u‖Ḟ∞,∞0(log⁡+‖∇u‖Fp,qs-1,r+1)≲1+‖u‖Ḟ∞,∞1(log⁡+‖u‖Fp,qs,r+1),‖∇θ‖L∞≲1+‖∇θ‖Ḟ∞,∞0(log⁡+‖∇θ‖Fp,qs-1,r+1)≲1+‖θ‖Ḟ∞,∞1(log⁡+‖θ‖Fp,qs,r+1).
Thus, the a priori estimate (3.21) gives that
(3.47)‖u(t)‖Fp,qs,r+‖θ(t)‖Fp,qs,r≤C(‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r)exp⁡⁡(C∫0t(‖u(τ)‖Ḟ∞,∞1+‖θ(τ)‖Ḟ∞,∞1)×(log⁡+(‖u(τ)‖Fp,qs,r+‖θ(τ)‖Fp,qs,r)+1)dτ∫0t).
By using Gronwall’s inequality, we have
(3.48)‖u(t)‖Fp,qs,r+‖θ(t)‖Fp,qs,r≤C(‖u0‖Fp,qs,r+‖θ0‖Fp,qs,r)exp⁡⁡(Cexp⁡⁡(C∫0t(‖u(τ)‖Ḟ∞,∞1+‖θ(τ)‖Ḟ∞,∞1)dτ)).
Thus if limsupt→T*-(∥u(t)∥Fp,qs,r+∥θ(t)∥Fp,qs,r)=∞, then ∫0T*(∥u(t)∥Ḟ∞,∞1+∥θ(t)∥Ḟ∞,∞1)dt=∞.
On the other hand, it follows from Sobolev embeddingFp,qs,r↪W1,∞↪Ḟ∞,∞1 for s-1>d/p that
(3.49)∫0T*(‖u(t)‖Ḟ∞,∞1+‖θ(t)‖Ḟ∞,∞1)dt≤T*sup⁡⁡0≤τ≤T*(‖u(τ)‖Ḟ∞,∞1+‖θ(τ)‖Ḟ∞,∞1)≲T*sup⁡⁡0≤τ≤T*(‖∇u(τ)‖L∞+‖∇θ(τ)‖L∞)≲T*sup⁡⁡0≤τ≤T*(‖u(τ)‖Fp,qs,r+‖θ(τ)‖Fp,qs,r).
Thus ∫0T*(∥u(t)∥Ḟ∞,∞1+∥θ(t)∥Ḟ∞,∞1)dt=∞ implies limsupt→T*-(∥u(t)∥Fp,qs,r+∥θ(t)∥Fp,qs,r)=∞.



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*Source: 573087-2012-05-29.xml*