# Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

Authors: Jamel Benameur; Mongi Blel
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the behavior at infinity in time of any global solutionθ∈𝒞(ℝ+,Ḣ2-2α(ℝ2)) of the surface quasigeostrophic equation with subcritical exponent 2/3≤α≤1. We prove that limt→∞∥θ(t)∥Ḣ2-2α=0. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that if θ∈𝒞(ℝ+,Ḣ2-2α(ℝ2)) is a global solution, then limt→∞∥θ(t)∥H2-2α=0.


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## Body

## 1. Introduction

We consider the2D dissipative quasi-geostrophic equation with subcritical exponent 1/2<α≤1,
(Sα)∂tθ+(-Δ)αθ+(u⋅∇)θ=0inR+×R2,θ(0,x)=θ0(x)inR2,
where x∈ℝ2, t>0, θ=θ(x,t) is the unknown potential temperature, and u=(u1,u2) is the divergence free velocity which is determined by the Riesz transformation of θ in the following way:
(1.1)u1=-R2θ=-∂2(-Δ)-1/2θ,u2=R1θ=∂1(-Δ)-1/2θ.
This equation is a two-dimensional model of the 3D incompressible Euler equations, and if α=1, the equation (𝒮1) is the 2D Navier-Stokes equation. We refer the reader to [1] where the authors explain the physical origin and the signification of the parameters of this equation.The critical homogeneous Sobolev space of the system(𝒮α) is Ḣ2-2α(ℝ2), and we have
(1.2)‖λ2α-1f(λ.)‖Ḣ2-2α=‖f‖Ḣ2-2α,∀λ>0.The local well-posedness of (𝒮α) with Ḣ2-2α(ℝ2) data is established by [2] and [3] separately if α∈(0,1/2]. In [4], Dong and Du study the critical case α=1/2 in the critical space Ḣ1(ℝ2). They prove the global existence if the initial condition is in the critical space H1(ℝ2).The global existence whenα∈(1/2,1] is an open problem. We have only the local existence. In this case [5], Niche and Schonbek prove that if the initial data θ0 is in L2(ℝ2), then the L2 norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay. If θ0∈Lp(ℝ2), with 1≤p≤2, they obtain a uniform decay rate in L2. They consider also the solution in other Lq spaces. For the proof of their results, they use the kernel Pα(t,x) associated to the operator ∂t+(-Δ)α, and they use the Littlewood-Paley decomposition. Our main result is the following.Theorem 1.1.
Assume that2/3≤α≤1.(i)
Ifθ∈𝒞(ℝ+,Ḣ2-2α(ℝ2)) is a global solution of (𝒮α), then
(1.3)lim⁡t→∞⁡‖θ(t)‖Ḣ2-2α=0.(ii)
Ifθ∈𝒞(ℝ+,H2-2α(ℝ2)) is a global solution of (𝒮α), then
(1.4)lim⁡t→∞⁡‖θ(t)‖H2-2α=0.


## 2. Notations and Preliminary Results

### 2.1. Notations and Technical Lemmas

In this short section, we collect some notations and definitions that will be used later, and we give some technical lemmas.(i)
The Fourier transformation inℝ2 is normalized as
(2.1)F(f)(ξ)=f∧(ξ)=∫R2exp⁡(-ix⋅ξ)f(x)dx,ξ=(ξ1,ξ2)∈R2.(ii)
The inverse Fourier formula is(2.2)F-1(g)(x)=(2π)-2∫R2exp⁡(iξ⋅x)f(ξ)dξ,x=(x1,x2)∈R2.(iii)
Fors∈ℝ, Hs(ℝ2) denotes the usual nonhomogeneous Sobolev space on ℝ2 and 〈·,·〉Hs(ℝ2) its scalar product.(iv)
Fors∈ℝ, Ḣs(ℝ2) denotes the usual homogeneous Sobolev space on ℝ2 and 〈·,·〉Ḣs(ℝ2) its scalar product.(v)
Fors,s′∈ℝ and t∈[0,1],(2.3)‖f‖Hts+(1-t)s′≤‖f‖Hst‖f‖Hs′1-t,(2.4)‖f‖Ḣts+(1-t)s′≤‖f‖Ḣst‖f‖Ḣs′1-t.
These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces.(i)
For any Banach space(B,∥·∥), any real number 1≤p≤∞, and any time T>0, we denote by LTp(B) the space of measurable functions t∈[0,T]↦f(t)∈B such that (t↦∥f(t)∥)∈Lp([0,T]).(ii)
Iff=(f1,f2) and g=(g1,g2) are two vector fields, we set(2.5)f⊗g∶=(g1f,g2f),div⁡(f⊗g)∶=(div⁡(g1f),div⁡(g2f)).
We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.Lemma 2.1 (see [6]).
Lets1, s2 be two real numbers such that
(2.6)s1<1,s1+s2>0.
There exists a constant C∶=C(s1,s2), such that for all f,g∈Ḣs1(ℝ2)∩Ḣs2ℝ2),
(2.7)‖fg‖Ḣs1+s2-1(R2)≤C(‖f‖Ḣs1(R2)‖g‖Ḣs2+‖f‖Ḣs2‖g‖Ḣs1).
If s1,s2<1 and s1+s2>0, there exists a constant c=c(s1,s2) such that for all f∈Ḣs1(ℝ2) and g∈Ḣs2ℝ2),
(2.8)‖fg‖Ḣs1+s2-1(R2)≤c‖f‖Ḣs1‖g‖Ḣs2.For the proof of the main result, we need the following lemma.Lemma 2.2.
With the same conditions of Theorem1.1, for all σ≥0,
(2.9)∫R2|ξ|2σ|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣσ+α‖w‖Ḣσ+α.Remark 2.3.
(i) In the case whereσ=0, the formula (2.9) gives
(2.10)∫R2|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣα‖w‖Ḣα.
In the case whereσ=2-2α, the formula (2.9) gives
(2.11)∫R2|ξ|2(2-2α)|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣ2-α‖w‖Ḣ2-α.Proof of Lemma2.2.
From the Cauchy-Schwarz inequality, we have(2.12)∫R2|ξ|2σ|F((u⋅∇)θ)F(w)|dξ≤∫R2|ξ|σ-α|F((u⋅∇)θ)||ξ|σ+α|F(w)(ξ)|dξ≤(∫R2|ξ|2(σ-α)|F((u⋅∇)θ)|2dξ)1/2‖w‖Ḣσ+α.
Using the weak derivatives properties, the product laws (Lemma 2.1), with s1+s2=σ-α+2>0, s1=2-2α<1, and s2=σ+α, we can dominate the nonlinear part as follows:
(2.13)∫R2|ξ|2(σ-α)|F((u⋅∇)θ)|2dξ≤∫R2|ξ|2(σ-α+1)(|F(θ)|*|F(θ)|)2dξ≤C‖θ‖Ḣ2-2α2‖θ‖Ḣσ+α2.


### 2.2. Existence Theorem

In [7], Wu proves an existence and uniqueness theorem of (𝒮α) in the well-known Besov spaces Ḃp,qr. We recall this theorem in the special case, where p=q=2.Theorem 2.4.
Assume thatα∈(0,1] and θ0∈Ḣ2-2α(ℝ2), then there exists a constant cα>0 such that if
(2.14)‖θ0‖Ḣ2-2α<cα,
then the initial value problem (𝒮α) has a unique solution in 𝒞b(ℝ+,Ḣ2-2α(ℝ2))∩L2(ℝ+,Ḣ2-α(ℝ2)). Moreover,
(2.15)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤cα′,∀t≥0,
where 𝒞b(ℝ+,Ḣ2-2α(ℝ2)) is the space of continuous and bounded functions from ℝ+ to Ḣ2-2α(ℝ2).In use of the fact thatḢ2-2α(ℝ2) is a Hilbert space, one deduces the following.Corollary 2.5.
Assume thatα∈(1/2,1] and θ0∈Ḣ2-2α(ℝ2), then there exists a constant cα>0 such that if
(2.16)‖θ0‖Ḣ2-2α<cα,
then the initial value problem (𝒮α) has a unique solution in 𝒞b(ℝ+,Ḣ2-2α(ℝ2))∩L2(ℝ+,Ḣ2-α(ℝ2)). Moreover,
(2.17)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤‖θ0‖Ḣ2-2α2,∀t≥0.Proof .
Taking the scalar product inḢ2-2α(ℝ2), we get
(2.18)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤|〈(u⋅∇)θ,θ〉Ḣ2-2α|≤|〈div⁡(θu),θ〉Ḣ2-2α|≤‖div⁡(θu)‖Ḣ2-3α‖θ‖Ḣ2-α≤‖θu‖Ḣ3-3α‖θ‖Ḣ2-α.
Using Lemma2.1 with s1=2-2α<1 and s2=2-α, we obtain
(2.19)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤Cα‖θ‖Ḣ2-2α‖θ‖Ḣ2-α2,(Cα=12cα).
Then the quadratic term can be absorbed,
(2.20)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤0.
Taking the integral on the interval [0,t], we obtain
(2.21)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤‖θ0‖Ḣ2-2α2,∀t≥0.


## 2.1. Notations and Technical Lemmas

In this short section, we collect some notations and definitions that will be used later, and we give some technical lemmas.(i)
The Fourier transformation inℝ2 is normalized as
(2.1)F(f)(ξ)=f∧(ξ)=∫R2exp⁡(-ix⋅ξ)f(x)dx,ξ=(ξ1,ξ2)∈R2.(ii)
The inverse Fourier formula is(2.2)F-1(g)(x)=(2π)-2∫R2exp⁡(iξ⋅x)f(ξ)dξ,x=(x1,x2)∈R2.(iii)
Fors∈ℝ, Hs(ℝ2) denotes the usual nonhomogeneous Sobolev space on ℝ2 and 〈·,·〉Hs(ℝ2) its scalar product.(iv)
Fors∈ℝ, Ḣs(ℝ2) denotes the usual homogeneous Sobolev space on ℝ2 and 〈·,·〉Ḣs(ℝ2) its scalar product.(v)
Fors,s′∈ℝ and t∈[0,1],(2.3)‖f‖Hts+(1-t)s′≤‖f‖Hst‖f‖Hs′1-t,(2.4)‖f‖Ḣts+(1-t)s′≤‖f‖Ḣst‖f‖Ḣs′1-t.
These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces.(i)
For any Banach space(B,∥·∥), any real number 1≤p≤∞, and any time T>0, we denote by LTp(B) the space of measurable functions t∈[0,T]↦f(t)∈B such that (t↦∥f(t)∥)∈Lp([0,T]).(ii)
Iff=(f1,f2) and g=(g1,g2) are two vector fields, we set(2.5)f⊗g∶=(g1f,g2f),div⁡(f⊗g)∶=(div⁡(g1f),div⁡(g2f)).
We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.Lemma 2.1 (see [6]).
Lets1, s2 be two real numbers such that
(2.6)s1<1,s1+s2>0.
There exists a constant C∶=C(s1,s2), such that for all f,g∈Ḣs1(ℝ2)∩Ḣs2ℝ2),
(2.7)‖fg‖Ḣs1+s2-1(R2)≤C(‖f‖Ḣs1(R2)‖g‖Ḣs2+‖f‖Ḣs2‖g‖Ḣs1).
If s1,s2<1 and s1+s2>0, there exists a constant c=c(s1,s2) such that for all f∈Ḣs1(ℝ2) and g∈Ḣs2ℝ2),
(2.8)‖fg‖Ḣs1+s2-1(R2)≤c‖f‖Ḣs1‖g‖Ḣs2.For the proof of the main result, we need the following lemma.Lemma 2.2.
With the same conditions of Theorem1.1, for all σ≥0,
(2.9)∫R2|ξ|2σ|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣσ+α‖w‖Ḣσ+α.Remark 2.3.
(i) In the case whereσ=0, the formula (2.9) gives
(2.10)∫R2|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣα‖w‖Ḣα.
In the case whereσ=2-2α, the formula (2.9) gives
(2.11)∫R2|ξ|2(2-2α)|F((u⋅∇)θ)F(w)|dξ≤C‖θ‖Ḣ2-2α‖θ‖Ḣ2-α‖w‖Ḣ2-α.Proof of Lemma2.2.
From the Cauchy-Schwarz inequality, we have(2.12)∫R2|ξ|2σ|F((u⋅∇)θ)F(w)|dξ≤∫R2|ξ|σ-α|F((u⋅∇)θ)||ξ|σ+α|F(w)(ξ)|dξ≤(∫R2|ξ|2(σ-α)|F((u⋅∇)θ)|2dξ)1/2‖w‖Ḣσ+α.
Using the weak derivatives properties, the product laws (Lemma 2.1), with s1+s2=σ-α+2>0, s1=2-2α<1, and s2=σ+α, we can dominate the nonlinear part as follows:
(2.13)∫R2|ξ|2(σ-α)|F((u⋅∇)θ)|2dξ≤∫R2|ξ|2(σ-α+1)(|F(θ)|*|F(θ)|)2dξ≤C‖θ‖Ḣ2-2α2‖θ‖Ḣσ+α2.


## 2.2. Existence Theorem

In [7], Wu proves an existence and uniqueness theorem of (𝒮α) in the well-known Besov spaces Ḃp,qr. We recall this theorem in the special case, where p=q=2.Theorem 2.4.
Assume thatα∈(0,1] and θ0∈Ḣ2-2α(ℝ2), then there exists a constant cα>0 such that if
(2.14)‖θ0‖Ḣ2-2α<cα,
then the initial value problem (𝒮α) has a unique solution in 𝒞b(ℝ+,Ḣ2-2α(ℝ2))∩L2(ℝ+,Ḣ2-α(ℝ2)). Moreover,
(2.15)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤cα′,∀t≥0,
where 𝒞b(ℝ+,Ḣ2-2α(ℝ2)) is the space of continuous and bounded functions from ℝ+ to Ḣ2-2α(ℝ2).In use of the fact thatḢ2-2α(ℝ2) is a Hilbert space, one deduces the following.Corollary 2.5.
Assume thatα∈(1/2,1] and θ0∈Ḣ2-2α(ℝ2), then there exists a constant cα>0 such that if
(2.16)‖θ0‖Ḣ2-2α<cα,
then the initial value problem (𝒮α) has a unique solution in 𝒞b(ℝ+,Ḣ2-2α(ℝ2))∩L2(ℝ+,Ḣ2-α(ℝ2)). Moreover,
(2.17)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤‖θ0‖Ḣ2-2α2,∀t≥0.Proof .
Taking the scalar product inḢ2-2α(ℝ2), we get
(2.18)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤|〈(u⋅∇)θ,θ〉Ḣ2-2α|≤|〈div⁡(θu),θ〉Ḣ2-2α|≤‖div⁡(θu)‖Ḣ2-3α‖θ‖Ḣ2-α≤‖θu‖Ḣ3-3α‖θ‖Ḣ2-α.
Using Lemma2.1 with s1=2-2α<1 and s2=2-α, we obtain
(2.19)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤Cα‖θ‖Ḣ2-2α‖θ‖Ḣ2-α2,(Cα=12cα).
Then the quadratic term can be absorbed,
(2.20)12∂t‖θ‖Ḣ2-2α2+‖θ‖Ḣ2-α2≤0.
Taking the integral on the interval [0,t], we obtain
(2.21)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤‖θ0‖Ḣ2-2α2,∀t≥0.


## 3. Proof of the Main Theorem

The proof of the first part will be in two steps.First Step (Small Initial Data)
In this case, we suppose that(3.1)‖θ0‖Ḣ2-2α<cα,
with cα a sufficient small number. Then from Corollary 2.5,
(3.2)θ∈Cb(R+,Ḣ2-2α(R2))∩L2(R+,Ḣ2-α(R2)),(3.3)‖θ‖Ḣ2-2α2+∫0t‖θ‖Ḣ2-α2≤‖θ0‖Ḣ2-2α2,∀t≥0.
For a strictly positive real numberδ and a given distribution f, we define the operators Aδ(D) and Bδ(D), respectively, by the following:
(3.4)Aδ(D)f∶=χB(0,δ)(|D|)f=F-1(χB(0,δ)F(f)),Bδ(D)f∶=(1-Aδ(D))f=F-1((1-χB(0,δ))F(f)).
We define wδ=Aδ(D)θ and vδ=Bδ(D)θ; ℱ(θ)=ℱ(wδ)+ℱ(vδ). Then,
(3.5)∂twδ+(-Δ)αwδ+Aδ(D)(u⋅∇θ)=0,∂t‖wδ‖Ḣ2-2α2+2‖wδ‖Ḣ2-α2≤C‖θ‖Ḣ2-2α⋅‖θ‖Ḣ2-α⋅‖wδ‖Ḣ2-α.
We deduce that
(3.6)‖wδ‖Ḣ2-2α2≤‖wδ(0)‖Ḣ2-2α2+C‖θ(0)‖Ḣ2-2α∫0∞‖θ‖Ḣ2-α‖wδ‖Ḣ2-αdτ.
Since ∥wδ∥Ḣ2-α≤∥θ∥Ḣ2-α, then from the dominate convergence theorem and (3.3), we have
(3.7)lim⁡δ→0Sup⁡t≥0⁡‖wδ‖Ḣ2-2α=0.
The function vδ satisfies
(3.8)∂tvδ+(-Δ)αvδ+Bδ(D)(u⋅∇θ)=0,∂t|F(vδ)|2+2|ξ|2α|F(vδ)|2≤|F(u⋅∇θ)F(vδ)|.
Multiplying this equation by |ξ|2(2-2α)e2t|ξ|2α, we deduce that
(3.9)‖vδ‖Ḣ2-2α2≤∫|ξ|>δ|ξ|2(2-2α)e-2t|ξ|2α|F(vδ0)|2+∫0t∫|ξ|>δ|ξ|2(2-2α)e-2(t-τ)|ξ|2α|F(u⋅∇θ)F(vδ)|dξdτ≤e-2tδ2α‖vδ0‖Ḣ2-2α2+C∫0te-2(t-τ)δ2α∫ξ|ξ|2(2-2α)|F(u⋅∇θ)F(vδ)|dξdτ.
Using Remark 2.3 and (3.3), we get
(3.10)‖vδ‖Ḣ2-2α2≤e-2tδ2α‖vδ0‖Ḣ2-2α2+C‖θ0‖Ḣ2-2α∫0te-2(t-τ)δ2α‖θ‖Ḣ2-α2dτ.
We set
(3.11)Fδ(t)=e-2tδ2α‖vδ0‖Ḣ2-2α2+C‖θ0‖Ḣ2-2α∫0te-2(t-τ)δ2α‖θ‖Ḣ2-α2dτ,∫0+∞e-2tδ2α‖vδ0‖Ḣ2-2α2dt=‖vδ0‖Ḣ2-2α22δ2α≤‖θ0‖Ḣ2-2α22δ2α,∫0+∞∫0te-2(t-τ)δ2α‖θ‖Ḣ2-α2dτdt=∫0+∞(∫τ+∞e-2(t-τ)δ2αdt)‖θ‖Ḣ2-α2dτ=12δ2α∫0+∞‖θ‖Ḣ2-α2dt≤‖θ0‖Ḣ2-2α24δ2α.
Then,
(3.12)∫0+∞Fδ(t)dt≤‖θ0‖Ḣ2-2α2δ2α.
Let ɛ>0, from (3.7), there exists δ0>0 such that
(3.13)‖wδ0‖Ḣ2-2α≤ɛ2,∀t≥0.
Let Eδ0={t≥0;∥vδ0∥Ḣ2-2α>ɛ/2}, then
(3.14)∫0+∞‖vδ0‖Ḣ2-2α2dt≥∫Eδ0‖vδ0‖Ḣ2-2α2dt≥(ɛ2)2λ1(Eδ0),
where λ1(Eδ0) is the Lebesgue measure of Eδ0. If
(3.15)Tɛ=(2ɛ)2∫0+∞‖vδ0‖Ḣ2-2α2dt,
then λ1(Eδ0)≤Tɛ. For η>0, there exists t0∈[0,Tɛ+η] such that t0∉Eδ0, and it results that
(3.16)‖vδ0(t0)‖Ḣ2-2α≤ɛ2.
Equation (3.13) and (3.16) give that
(3.17)‖θ(t0)‖Ḣ2-2α≤ɛ.
Thus, limt→+∞∥θ(t)∥Ḣ2-2α=0, and this finishes the proof in this case.Second Step (Large Initial Data)
To prove the result for any initial data, it suffices to prove the existence of somet0≥0 such that
(3.18)‖θ(t0)‖Ḣ2-2α<cα.
Let θ0=a0+r0, with
(3.19)a0∶=F-1(1{1/N<|ξ|<N}F(θ0)),r0∶=θ0-a0,‖r0‖Ḣ2-2α<cα.
Now, consider the following system:(3.20)∂tr+(-Δ)αr+(R⋅∇)r=0inR+×R2,r(0)=r0inR2,R=∇⊥Δ-1/2r.
By Corollary 2.5, there is a unique solution r∈𝒞b(ℝ+,Ḣ2-2α(ℝ2))∩L2(ℝ+,Ḣ2-α(ℝ2)) such that
(3.21)‖r(t)‖Ḣ2-2α2+∫0t‖r(τ)‖Ḣ2-α2dτ≤‖r0‖Ḣ2-2α2.
Let a∶=θ-r∈𝒞(ℝ+,Ḣ2-2α(ℝ2)), then a is a solution of the following system:
(S1)∂ta+(-Δ)αa+(A⋅∇)a+(A⋅∇)r+(R⋅∇)a=0inR+×R2,a(0)=a0inR2,A=∇⊥Δ-1/2a.
Taking a scalar product in L2(ℝ2), we obtain
(3.22)∂t‖a(t)‖L22+2‖a(t)‖Ḣα2≤2|∫R2(A⋅∇)ra|≤2|∫R2div⁡(rA)a|≤2‖rA‖Ḣ1-α‖a‖Ḣα.
Using the product law in Lemma 2.1, with s1=2-2α<1 and s2=α<1,
(3.23)|〈(A⋅∇)r,a〉L2(R2)|≤C(α)‖r‖Ḣ2-2α‖A‖Ḣα‖a‖Ḣα≤C(α)‖r‖Ḣ2-2α‖a‖Ḣα2≤‖a‖Ḣα2,
then, for all t≥0,
(3.24)∂t‖a(t)‖L22+‖a(t)‖Ḣα2≤0,‖a(t)‖L22+∫0t‖a(τ)‖Ḣα2dτ≤‖a0‖L22,
then 2-2α=λ×0+(1-λ)α, with λ∶=3-(2/α)∈[0,1],
(3.25)‖a(t)‖Ḣ2-2α≤‖a(t)‖L23-2/α‖a(t)‖Ḣα2/α-2≤‖a0‖L23-2/α‖a(t)‖Ḣα2/α-2.
Then,
(3.26)∫0∞‖a(t)‖Ḣ2-2αα/(1-α)dt≤‖a0‖L21/(1-α).
Now define the set
(3.27)Sε∶={t≥0;‖a(t)‖Ḣ2-2α>ε}
as a measurable with respect to the Lebesgue measure. We have
(3.28)ɛα/(1-α)λ1(Sε)≤∫Sε‖a(t)‖Ḣ2-2αα/(1-α)dt≤‖a0‖L21/(1-α).
So λ1(Sɛ)<∞ and λ1(Sɛ)≤ɛα/(1-α)∥a0∥L21/(1-α), then there is
(3.29)t0∈[0,λ1(Sε)+1]∖Sε.
Then,
(3.30)‖a(t0)‖Ḣ2-2α<ε,
and then
(3.31)‖θ(t0)‖Ḣ2-2α≤‖r(t0)‖Ḣ2-2α+‖a(t0)‖Ḣ2-2α<ε2+ε2=ε.Applying the conclusion of Theorem1.1 for (𝒮α) system starting at θ(t0), we can deduce the desired result.In the nonhomogeneous case, we suppose thatθ∈𝒞(ℝ+,H2-2α), then
(3.32)lim⁡t→∞‖θ(t)‖Ḣ2-2α=0.We can suppose that∥θ∥Ḣ2-2α<cα, and for all t≥0,
(3.33)‖θ(t)‖Ḣ2-2α2+∫0t‖θ(τ)‖Ḣ2-α2dτ≤‖θ0‖Ḣ2-2α2.Thus, it suffices to prove that(3.34)lim⁡t→∞‖θ(t)‖L2=0.Letδ>0, then we recall the operators
(3.35)Aδ(D)θ=F-1(χB(0,δ)F(θ)),Bδ(D)θ=F-1((1-χB(0,δ))F(θ)).
We define wδ=Aδ(D)(θ) and vδ=Bδ(D)(θ). Then,
(3.36)∂twδ+(-Δ)αwδ+Aδ(D)(uθ⋅∇θ)=0,
and from Lemma 2.2,
(3.37)∂t‖wδ‖L22+2‖wδ‖Ḣα2≤C‖θ‖Ḣ2-2α.‖θ‖Ḣα⋅‖wδ‖Ḣα.
We deduce that
(3.38)‖wδ‖L22≤‖wδ(0)‖L22+C‖θ0‖Ḣ2-2α∫0+∞‖θ‖Ḣα‖wδ‖Ḣαdτ.
Then from the dominate convergence theorem and the following L2energy estimate
(3.39)‖θ‖L22+2∫0t‖θ‖Ḣα2dτ≤‖θ0‖L22,
we deduce that
(3.40)lim⁡δ→0⁡Sup⁡t≥0‖wδ‖L2=0,∂tvδ+(-Δ)αvδ+Bδ(D)(uθ⋅∇θ)=0,(3.41)∂t|F(vδ)|2+2|ξ|2α|F(vδ)|2≤|F(u⋅∇θ)F(vδ)|.Multiplying this equation bye2t|ξ|2α, we have(3.42)‖vδ‖L22≤e-2tδ2α‖vδ0‖L22+C∫0te-2(t-τ)δ2α|〈u⋅∇θvδ〉|L2(R2)2dτ≤e-2tδ2α‖vδ0‖L22+C‖θ0‖Ḣ2-2α∫0te-2(t-τ)δ2α‖θ‖Ḣα2dτ.
We set
(3.43)Fδ(t)=e-2tδ2α‖vδ0‖L22+C‖θ0‖Ḣ2-2α∫0te-2(t-τ)δ2α‖θ‖Ḣα2dτ,∫0+∞e-2tδ2α‖vδ0‖L22dt=‖θ0‖L222δ2α,∫0+∞∫0te-2(t-τ)δ2α‖θ‖Ḣα2dτdt=∫0+∞(∫τ+∞e-2(t-τ)δ2αdt)‖θ‖Ḣα2dτ=12δ2α∫0+∞‖θ‖Ḣα2dt≤‖θ0‖L222δ2α.
Then,(3.44)∫0+∞Fδ(t)dt≤‖θ0‖L22δ2α.
Let ɛ>0, from (3.40), then there exists δ0>0 such that
(3.45)‖wδ0‖L2≤ɛ2,∀t≥0.
Let Eδ0={t≥0;∥vδ0∥L2>ɛ/2}, then
(3.46)∫0+∞‖vδ0‖L22dt≥∫Eδ0‖vδ0‖L22dt≥(ε2)2λ1(Eδ0),
where λ1(Eδ0) is the Lebesgue measure of Eδ0. If
(3.47)Tɛ=(2ε)2∫0+∞‖vδ0‖L22dt,
then λ1(Eδ0)≤Tɛ. For η>0, there exists t0∈[0,Tɛ+η] such that t0∉Eδ0, then
(3.48)‖vδ0(t0)‖L2≤ε2.
The equations (3.45) and (3.48) give that
(3.49)‖θ(t0)‖L2<ε.
Thus, limt→+∞∥θ(t)∥L2=0, and this finishes the proof.



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*Source: 627813-2012-05-09.xml*