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

Abstract

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

1. Introduction

We consider the 2𝐷 dissipative quasi-geostrophic equation with subcritical exponent 1/2<𝛼≤1, 𝜕_𝑡𝜃+(−Δ𝜃)(^𝛼0𝜃,+𝑥)(𝑢=⋅𝜃∇₀)𝜃=0inℝ⁺×ℝ²,(𝑥)inℝ²,(𝒮^𝛼) 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: 𝑢𝑢₁=−ℛ₂𝜃=−𝜕₂(−Δ)^−1/2𝜃,₂=ℛ₁𝜃=𝜕₁(−Δ)^−1/2𝜃.(1.1) 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 (𝒮_𝛼) is 𝐻̇^2−2𝛼(ℝ²), and we have ‖‖𝜆^2𝛼−1𝑓(𝜆.)‖‖_{̇𝐻2−2𝛼}=‖𝑓‖_̇𝐻^2−2𝛼,∀𝜆>0.(1.2)

The local well-posedness of (𝒮_𝛼) with 𝐻̇^2−2𝛼(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 when 𝛼∈(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∈𝐿^𝑝(ℝ²), with 1≤𝑝≤2, they obtain a uniform decay rate in 𝐿². They consider also the solution in other 𝐿^𝑞 spaces. For the proof of their results, they use the kernel 𝑃_𝛼(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 that 2/3≤𝛼≤1.(i)If 𝜃∈𝒞(ℝ⁺,𝐻̇^2−2𝛼(ℝ²)) is a global solution of (𝒮_𝛼), then lim_𝑡→∞‖𝜃(𝑡)‖_̇𝐻^2−2𝛼=0.(1.3)(ii)If 𝜃∈𝒞(ℝ⁺,𝐻^2−2𝛼(ℝ²)) is a global solution of (𝒮_𝛼), then lim_𝑡→∞‖𝜃(𝑡)‖_𝐻^2−2𝛼=0.(1.4)

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 in ℝ² is normalized as ℱ(𝑓)(𝜉)=𝑓^∧(𝜉)=_ℝ2exp(−𝑖𝑥⋅𝜉)𝑓(𝑥)𝑑𝑥,𝜉=𝜉₁,𝜉₂∈ℝ².(2.1)(ii)The inverse Fourier formula is ℱ⁻¹(𝑔)(𝑥)=(2𝜋)⁻²_ℝ2exp(𝑖𝜉⋅𝑥)𝑓(𝜉)𝑑𝜉,𝑥=𝑥₁,𝑥₂∈ℝ².(2.2)(iii)For 𝑠∈ℝ, 𝐻^𝑠(ℝ²) denotes the usual nonhomogeneous Sobolev space on ℝ² and ⟨⋅,⋅⟩_𝐻^𝑠_(ℝ²₎ its scalar product.(iv)For 𝑠∈ℝ, 𝐻̇^𝑠(ℝ²) denotes the usual homogeneous Sobolev space on ℝ² and ⟨⋅,⋅⟩_̇𝐻^𝑠_(ℝ²₎ its scalar product.(v)For 𝑠,𝑠^∈ℝ and 𝑡∈[0,1],‖‖𝑓𝑓‖‖_{𝐻′𝑡𝑠+(1−𝑡)𝑠̇𝐻′𝑡𝑠+(1−𝑡)𝑠}≤‖𝑓‖^𝑡𝐻_𝑠‖𝑓‖^𝐻1−𝑡_𝑠′,(2.3)≤‖𝑓‖^𝑡̇𝐻_𝑠‖𝑓‖^{̇𝐻1−𝑡}_𝑠′.(2.4) 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 𝐿^𝑝𝑇(𝐵) the space of measurable functions 𝑡∈[0,𝑇]↦𝑓(𝑡)∈𝐵 such that (𝑡↦‖𝑓(𝑡)‖)∈𝐿^𝑝([0,𝑇]).(ii)If 𝑓=(𝑓₁,𝑓₂) and 𝑔=(𝑔₁,𝑔₂) are two vector fields, we setdiv(𝑓𝑓⊗⊗𝑔𝑔)∶∶==𝑔d₁iv𝑓,𝑔𝑔₂₁𝑓,𝑓,div𝑔₂𝑓.(2.5) We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.

Lemma 2.1 (see [6]). Let 𝑠₁, 𝑠₂ be two real numbers such that 𝑠₁<1,𝑠₁+𝑠₂>0.(2.6) There exists a constant 𝐶∶=𝐶(𝑠₁,𝑠₂), such that for all 𝑓,𝑔∈𝐻̇^𝑠1(ℝ²)∩𝐻̇^𝑠2ℝ²), ‖𝑓𝑔‖_̇𝐻^{𝑠12+𝑠−1}_(ℝ²₎≤𝐶‖𝑓‖_̇𝐻^𝑠1_(ℝ²₎‖𝑔‖_̇𝐻^𝑠2+‖𝑓‖_̇𝐻^𝑠2‖𝑔‖_̇𝐻^𝑠1.(2.7) If 𝑠₁,𝑠₂<1 and 𝑠₁+𝑠₂>0, there exists a constant 𝑐=𝑐(𝑠₁,𝑠₂) such that for all 𝑓∈𝐻̇^𝑠1(ℝ²) and 𝑔∈𝐻̇^𝑠2ℝ²), ‖𝑓𝑔‖_̇𝐻^{𝑠12+𝑠−1}_(ℝ²₎≤𝑐‖𝑓‖_̇𝐻^𝑠1‖𝑔‖_̇𝐻^𝑠2.(2.8)

For the proof of the main result, we need the following lemma.

Lemma 2.2. With the same conditions of Theorem 1.1, for all 𝜎≥0, _ℝ2||𝜉||^2𝜎||ℱ((𝑢⋅∇)𝜃)ℱ(𝑤)||𝑑𝜉≤𝐶‖𝜃‖_{̇𝐻2−2𝛼}‖𝜃‖_̇𝐻^𝜎+𝛼‖𝑤‖_̇𝐻^𝜎+𝛼.(2.9)

Remark 2.3. (i) In the case where 𝜎=0, the formula (2.9) gives _ℝ2||ℱ((𝑢⋅∇)𝜃)ℱ(𝑤)||𝑑𝜉≤𝐶‖𝜃‖_{̇𝐻2−2𝛼}‖𝜃‖_̇𝐻^𝛼‖𝑤‖_̇𝐻^𝛼.(2.10)
In the case where 𝜎=2−2𝛼, the formula (2.9) gives _ℝ2||𝜉||^2(2−2𝛼)||ℱ((𝑢⋅∇)𝜃)ℱ(𝑤)||𝑑𝜉≤𝐶‖𝜃‖_{̇𝐻2−2𝛼}‖𝜃‖_̇𝐻^2−𝛼‖𝑤‖_̇𝐻^2−𝛼.(2.11)

Proof of Lemma 2.2. From the Cauchy-Schwarz inequality, we have _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‖_̇𝐻^𝜎+𝛼.(2.12) 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: _ℝ2||𝜉||^{2(𝜎−𝛼)}||ℱ((𝑢⋅∇)𝜃)||²𝑑𝜉≤≤𝐶‖_ℝ2𝜃||‖𝜉||^{2(𝜎−𝛼+1)}_{2̇𝐻2−2𝛼}||ℱ(𝜃)||∗|ℱ(𝜃)|²𝑑𝜉‖𝜃‖^2̇𝐻_𝜎+𝛼.(2.13)

2.2. Existence Theorem

In [7], Wu proves an existence and uniqueness theorem of (𝒮_𝛼) in the well-known Besov spaces 𝐵̇^{𝑟𝑝,𝑞}. We recall this theorem in the special case, where 𝑝=𝑞=2.

Theorem 2.4. Assume that 𝛼∈(0,1] and 𝜃⁰∈𝐻̇^2−2𝛼(ℝ²), then there exists a constant 𝑐_𝛼>0 such that if ‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}<𝑐_𝛼,(2.14) then the initial value problem (𝒮_𝛼) has a unique solution in 𝒞_𝑏(ℝ⁺,𝐻̇^2−2𝛼(ℝ²))∩𝐿²(ℝ⁺,𝐻̇^2−𝛼(ℝ²)). Moreover, ‖𝜃(𝑡)‖^2̇𝐻_2−2𝛼+^𝑡0‖𝜃(𝜏)‖^2̇𝐻_2−𝛼𝑑𝜏≤𝑐^𝛼,∀𝑡≥0,(2.15) where 𝒞_𝑏(ℝ⁺,𝐻̇^2−2𝛼(ℝ²)) is the space of continuous and bounded functions from ℝ⁺ to 𝐻̇^2−2𝛼(ℝ²).

In use of the fact that 𝐻̇^2−2𝛼(ℝ²) is a Hilbert space, one deduces the following.

Corollary 2.5. Assume that 𝛼∈(1/2,1] and 𝜃⁰∈𝐻̇^2−2𝛼(ℝ²), then there exists a constant 𝑐_𝛼>0 such that if ‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}<𝑐_𝛼,(2.16) then the initial value problem (𝒮_𝛼) has a unique solution in 𝒞_𝑏(ℝ⁺,𝐻̇^2−2𝛼(ℝ²))∩𝐿²(ℝ⁺,𝐻̇^2−𝛼(ℝ²)). Moreover, ‖𝜃(𝑡)‖^2̇𝐻_2−2𝛼+^𝑡0‖𝜃(𝜏)‖^2̇𝐻_2−𝛼𝑑𝜏≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼,∀𝑡≥0.(2.17)

Proof . Taking the scalar product in 𝐻̇^2−2𝛼(ℝ²), we get 12𝜕^𝑡‖𝜃‖^2̇𝐻_2−2𝛼+‖𝜃‖^2̇𝐻_2−𝛼≤≤||||⟨⟨(d𝑢iv⋅∇(𝜃)𝑢𝜃),,𝜃𝜃⟩⟩_{̇𝐻2−2𝛼̇𝐻2−2𝛼}||≤‖div(𝜃𝑢)‖_{̇𝐻2−3𝛼}||‖𝜃‖_̇𝐻^2−𝛼≤‖𝜃𝑢‖_̇𝐻^3−3𝛼‖𝜃‖_̇𝐻^2−𝛼.(2.18)
Using Lemma 2.1 with 𝑠₁=2−2𝛼<1 and 𝑠₂=2−𝛼, we obtain 12𝜕^𝑡‖𝜃‖^2̇𝐻_2−2𝛼+‖𝜃‖^2̇𝐻_2−𝛼≤𝐶_𝛼‖𝜃‖_̇𝐻^2−2𝛼‖𝜃‖^2̇𝐻_2−𝛼,𝐶_𝛼=12𝑐_𝛼.(2.19) Then the quadratic term can be absorbed, 12𝜕^𝑡‖𝜃‖^2̇𝐻_2−2𝛼+‖𝜃‖^2̇𝐻_2−𝛼≤0.(2.20) Taking the integral on the interval [0,𝑡], we obtain ‖𝜃(𝑡)‖^2̇𝐻_2−2𝛼+^𝑡0‖𝜃(𝜏)‖^2̇𝐻_2−𝛼𝑑𝜏≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼,∀𝑡≥0.(2.21)

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 ‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}<𝑐_𝛼,(3.1) with 𝑐_𝛼 a sufficient small number. Then from Corollary 2.5, 𝜃∈‖𝜃𝒞‖_𝑏ℝ⁺_{2̇𝐻2−2𝛼},𝐻̇^2−2𝛼ℝ²∩𝐿²ℝ⁺,𝐻̇^2−𝛼ℝ²,(3.2)+^𝑡0‖𝜃‖^2̇𝐻_2−𝛼≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼,∀𝑡≥0.(3.3)
For a strictly positive real number 𝛿 and a given distribution 𝑓, we define the operators 𝐴_𝛿(𝐷) and 𝐵_𝛿(𝐷), respectively, by the following: 𝐵𝐴_𝛿(𝐷)𝑓∶=𝜒_𝐵(0,𝛿)||𝐷||𝑓=ℱ⁻¹𝜒_𝐵(0,𝛿)ℱ(𝑓)_𝛿,(𝐷)𝑓∶=1−𝐴_𝛿(𝐷)𝑓=ℱ⁻¹1−𝜒_𝐵(0,𝛿)ℱ(𝑓).(3.4) We define 𝑤_𝛿=𝐴_𝛿(𝐷)𝜃 and 𝑣_𝛿=𝐵_𝛿(𝐷)𝜃; ℱ(𝜃)=ℱ(𝑤_𝛿)+ℱ(𝑣_𝛿). Then, 𝜕𝜕_𝑡𝑤_𝛿+(−Δ)^𝛼𝑤_𝛿+𝐴_𝛿(𝐷)(𝑢⋅∇𝜃)=0,_𝑡‖‖𝑤_𝛿‖‖^2̇𝐻_2−2𝛼+2‖‖𝑤_𝛿‖‖^2̇𝐻_2−𝛼≤𝐶‖𝜃‖_̇𝐻^2−2𝛼⋅‖𝜃‖_̇𝐻^2−𝛼⋅‖‖𝑤_𝛿‖‖_{̇𝐻2−𝛼}.(3.5) We deduce that ‖‖𝑤_𝛿‖‖^2̇𝐻_2−2𝛼≤‖‖𝑤_𝛿(0)‖‖^2̇𝐻_2−2𝛼+𝐶‖𝜃(0)‖_̇𝐻^2−2𝛼^∞0‖𝜃‖_̇𝐻^2−𝛼‖‖𝑤_𝛿‖‖_{̇𝐻2−𝛼}𝑑𝜏.(3.6) Since ‖𝑤_𝛿‖_{̇𝐻2−𝛼}≤‖𝜃‖_̇𝐻^2−𝛼, then from the dominate convergence theorem and (3.3), we have lim_𝛿→0Sup_𝑡≥0‖‖𝑤_𝛿‖‖_{̇𝐻2−2𝛼}=0.(3.7) The function 𝑣_𝛿 satisfies 𝜕𝜕_𝑡𝑣_𝛿+(−Δ)^𝛼𝑣_𝛿+𝐵_𝛿(𝐷)(𝑢⋅∇𝜃)=0,_𝑡||ℱ𝑣_𝛿||²+2||𝜉||^2𝛼||ℱ𝑣_𝛿||²≤||ℱ(𝑢⋅∇𝜃)ℱ𝑣_𝛿||.(3.8) Multiplying this equation by |𝜉|^2(2−2𝛼)𝑒^{2𝑡|𝜉|2𝛼}, we deduce that ‖‖𝑣_𝛿‖‖^2̇𝐻_2−2𝛼≤_|𝜉|>𝛿||𝜉||^2(2−2𝛼)𝑒^{−2𝑡|𝜉|2𝛼}||ℱ𝑣^0𝛿||²+^𝑡0_|𝜉|>𝛿≤𝑒_{−2𝑡𝛿2𝛼}||𝜉||^2(2−2𝛼)𝑒^{−2(𝑡−𝜏)|𝜉|2𝛼}||ℱ(𝑢⋅∇𝜃)ℱ𝑣_𝛿||𝑑𝜉𝑑𝜏‖‖𝑣^0𝛿‖‖^2̇𝐻_2−2𝛼+𝐶^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}_𝜉||𝜉||^2(2−2𝛼)||ℱ(𝑢⋅∇𝜃)ℱ𝑣_𝛿||𝑑𝜉𝑑𝜏.(3.9) Using Remark 2.3 and (3.3), we get ‖‖𝑣_𝛿‖‖^2̇𝐻_2−2𝛼≤𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2̇𝐻_2−2𝛼+𝐶‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_2−𝛼𝑑𝜏.(3.10) We set 𝐹_𝛿(𝑡)=𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2̇𝐻_2−2𝛼+𝐶‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_2−𝛼𝑑𝜏,_0+∞𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2̇𝐻_2−2𝛼𝑑𝑡=‖‖𝑣^0𝛿‖‖^2̇𝐻_2−2𝛼2𝛿^2𝛼≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼2𝛿^2𝛼_0+∞,^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_2−𝛼𝑑𝜏𝑑𝑡=^0+∞^𝜏+∞=1𝑒^{−2(𝑡−𝜏)𝛿2𝛼}𝑑𝑡‖𝜃‖^2̇𝐻_2−𝛼𝑑𝜏2𝛿^2𝛼^0+∞‖𝜃‖^2̇𝐻_2−𝛼𝑑𝑡≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼4𝛿^2𝛼.(3.11) Then, ^0+∞𝐹_𝛿(𝑡)𝑑𝑡≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼𝛿^2𝛼.(3.12) Let 𝜀>0, from (3.7), there exists 𝛿₀>0 such that ‖‖𝑤_𝛿0‖‖_{̇𝐻2−2𝛼}≤𝜀2,∀𝑡≥0.(3.13) Let 𝐸_𝛿0={𝑡≥0;‖𝑣_𝛿0‖_̇𝐻^2−2𝛼>𝜀/2}, then ^0+∞‖‖𝑣_𝛿0‖‖^2̇𝐻_2−2𝛼𝑑𝑡≥_𝐸𝛿0‖‖𝑣_𝛿0‖‖^2̇𝐻_2−2𝛼𝑑𝑡≥𝜀2²𝜆₁𝐸_𝛿0,(3.14) where 𝜆₁(𝐸_𝛿0) is the Lebesgue measure of 𝐸_𝛿0. If 𝑇_𝜀=2𝜀²^0+∞‖‖𝑣_𝛿0‖‖^2̇𝐻_2−2𝛼𝑑𝑡,(3.15) then 𝜆₁(𝐸_𝛿0)≤𝑇_𝜀. For 𝜂>0, there exists 𝑡₀∈[0,𝑇_𝜀+𝜂] such that 𝑡₀∉𝐸_𝛿0, and it results that ‖‖𝑣_𝛿0𝑡₀‖‖_̇𝐻^2−2𝛼≤𝜀2.(3.16) Equation (3.13) and (3.16) give that ‖‖𝜃𝑡₀‖‖_̇𝐻^2−2𝛼≤𝜀.(3.17) Thus, lim_𝑡→+∞‖𝜃(𝑡)‖_̇𝐻^2−2𝛼=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 some 𝑡₀≥0 such that ‖‖𝜃𝑡₀‖‖_̇𝐻^2−2𝛼<𝑐_𝛼.(3.18) Let 𝜃⁰=𝑎⁰+𝑟⁰, with 𝑎⁰∶=ℱ𝑟⁻¹𝟏_{{1/𝑁<|𝜉|<𝑁}}ℱ𝜃⁰₀,‖‖𝑟₀∶=𝜃⁰−𝑎⁰,‖‖_{̇𝐻2−2𝛼}<𝑐_𝛼.(3.19)
Now, consider the following system: 𝜕_𝑡𝑟+(−Δ)^𝛼𝑟𝑟+(0()𝑅=⋅𝑟∇₀)𝑟=0inℝ⁺×ℝ²,𝑅=∇_⟂inℝ²,Δ^−1/2𝑟.(3.20) By Corollary 2.5, there is a unique solution 𝑟∈𝒞_𝑏(ℝ⁺,𝐻̇^2−2𝛼(ℝ²))∩𝐿²(ℝ⁺,𝐻̇^2−𝛼(ℝ²)) such that ‖𝑟(𝑡)‖^2̇𝐻_2−2𝛼+^𝑡0‖𝑟(𝜏)‖^2̇𝐻_2−𝛼𝑑𝜏≤‖‖𝑟⁰‖‖^2̇𝐻_2−2𝛼.(3.21) Let 𝑎∶=𝜃−𝑟∈𝒞(ℝ⁺,𝐻̇^2−2𝛼(ℝ²)), then 𝑎 is a solution of the following system: 𝜕_𝑡𝑎+(−Δ)^𝛼𝑎+(𝐴⋅∇)𝑎𝑎+(0()𝐴=⋅𝑎∇₀)𝑟+(𝑅⋅∇)𝑎=0inℝ⁺×ℝ²,𝐴=∇_⟂inℝ²,Δ^−1/2𝑎.(S¹) Taking a scalar product in 𝐿²(ℝ²), we obtain 𝜕_𝑡‖𝑎(𝑡)‖^2𝐿₂+2‖𝑎(𝑡)‖^2̇𝐻_𝛼≤≤22||||||||_ℝ2ℝ2(𝐴⋅∇)𝑟𝑎||||≤2‖𝑟𝐴‖div_{̇𝐻1−𝛼}(𝑟𝐴)𝑎||||‖𝑎‖_̇𝐻^𝛼.(3.22) Using the product law in Lemma 2.1, with 𝑠₁=2−2𝛼<1 and 𝑠₂=𝛼<1, ||⟨(𝐴⋅∇)𝑟,𝑎⟩_𝐿2(ℝ2)||≤𝐶(𝛼)‖𝑟‖_{̇𝐻2−2𝛼}‖𝐴‖_̇𝐻^𝛼‖𝑎‖_̇𝐻^𝛼≤𝐶(𝛼)‖𝑟‖_̇𝐻^2−2𝛼‖𝑎‖^2̇𝐻_𝛼≤‖𝑎‖^2̇𝐻_𝛼,(3.23) then, for all 𝑡≥0, ‖𝑎(𝑡𝜕)_𝑡‖‖𝑎_2𝐿2(𝑡)‖^2𝐿₂+‖𝑎(𝑡)‖^2̇𝐻_𝛼≤0,+^𝑡0‖𝑎(𝜏)‖^2̇𝐻_𝛼𝑑𝜏≤‖‖𝑎⁰‖‖^2𝐿₂,(3.24) then 2−2𝛼=𝜆×0+(1−𝜆)𝛼, with 𝜆∶=3−(2/𝛼)∈[0,1], ‖𝑎(𝑡)‖_̇𝐻^2−2𝛼≤‖𝑎(𝑡)‖^{𝐿3−2/𝛼}₂‖𝑎(𝑡)‖^{̇𝐻2/𝛼−2}_𝛼≤‖‖𝑎⁰‖‖^{𝐿3−2/𝛼}₂‖𝑎(𝑡)‖^{̇𝐻2/𝛼−2}_𝛼.(3.25) Then, ^∞0‖𝑎(𝑡)‖^{̇𝐻𝛼/(1−𝛼)}_2−2𝛼𝑑𝑡≤‖‖𝑎⁰‖‖^{𝐿1/(1−𝛼)}₂.(3.26) Now define the set 𝑆_𝜀∶=𝑡≥0;‖𝑎(𝑡)‖_̇𝐻^2−2𝛼>𝜀(3.27) as a measurable with respect to the Lebesgue measure. We have 𝜀^{𝛼/(1−𝛼)}𝜆₁𝑆_𝜀≤_𝑆𝜀‖𝑎(𝑡)‖^{̇𝐻𝛼/(1−𝛼)}_2−2𝛼𝑑𝑡≤‖‖𝑎⁰‖‖^{𝐿1/(1−𝛼)}₂.(3.28) So 𝜆₁(𝑆_𝜀)<∞ and 𝜆₁(𝑆_𝜀)≤𝜀^{𝛼/(1−𝛼)}‖𝑎⁰‖^{𝐿1/(1−𝛼)}₂, then there is 𝑡₀∈0,𝜆₁𝑆_𝜀+1⧵𝑆_𝜀.(3.29) Then, ‖‖𝑎𝑡₀‖‖_̇𝐻^2−2𝛼<𝜀,(3.30) and then ‖‖𝜃𝑡₀‖‖_̇𝐻^2−2𝛼≤‖‖𝑟𝑡₀‖‖_̇𝐻^2−2𝛼+‖‖𝑎𝑡₀‖‖_̇𝐻^2−2𝛼<𝜀2+2𝜀=𝜀.(3.31)

Applying the conclusion of Theorem 1.1 for (𝒮_𝛼) system starting at 𝜃(𝑡₀), we can deduce the desired result.

In the nonhomogeneous case, we suppose that 𝜃∈𝒞(ℝ⁺,𝐻^2−2𝛼), then lim_𝑡→∞‖𝜃(𝑡)‖_̇𝐻^2−2𝛼=0.(3.32)

We can suppose that ‖𝜃‖_̇𝐻^2−2𝛼<𝑐_𝛼, and for all 𝑡≥0, ‖𝜃(𝑡)‖^2̇𝐻_2−2𝛼+^𝑡0‖𝜃(𝜏)‖^2̇𝐻_2−𝛼𝑑𝜏≤‖‖𝜃⁰‖‖^2̇𝐻_2−2𝛼.(3.33)

Thus, it suffices to prove that lim_𝑡→∞‖𝜃(𝑡)‖_𝐿²=0.(3.34)

Let 𝛿>0, then we recall the operators 𝐵𝐴_𝛿(𝐷)𝜃=ℱ⁻¹𝜒_𝐵(0,𝛿)ℱ(𝜃)_𝛿,(𝐷)𝜃=ℱ⁻¹1−𝜒_𝐵(0,𝛿)ℱ(𝜃).(3.35) We define 𝑤_𝛿=𝐴_𝛿(𝐷)(𝜃) and 𝑣_𝛿=𝐵_𝛿(𝐷)(𝜃). Then, 𝜕_𝑡𝑤_𝛿+(−Δ)^𝛼𝑤_𝛿+𝐴_𝛿(𝐷)𝑢_𝜃⋅∇𝜃=0,(3.36) and from Lemma 2.2, 𝜕_𝑡‖‖𝑤_𝛿‖‖^2𝐿₂+2‖‖𝑤_𝛿‖‖^2̇𝐻_𝛼≤𝐶‖𝜃‖_̇𝐻^2−2𝛼.‖𝜃‖_̇𝐻^𝛼⋅‖‖𝑤_𝛿‖‖_̇𝐻𝛼.(3.37) We deduce that ‖‖𝑤_𝛿‖‖^2𝐿₂≤‖‖𝑤_𝛿(0)‖‖^2𝐿₂+𝐶‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}^0+∞‖𝜃‖_̇𝐻^𝛼‖‖𝑤_𝛿‖‖_̇𝐻𝛼𝑑𝜏.(3.38) Then from the dominate convergence theorem and the following 𝐿²energy estimate ‖𝜃‖^2𝐿₂+2^𝑡0‖𝜃‖^2̇𝐻_𝛼𝑑𝜏≤‖‖𝜃⁰‖‖^2𝐿₂,(3.39) we deduce that 𝜕lim_𝛿→0Sup_𝑡≥0‖‖𝑤_𝛿‖‖_𝐿2=0,_𝑡𝜕𝑣_𝛿+(−Δ)^𝛼𝑣_𝛿+𝐵_𝛿(𝐷)𝑢_𝜃⋅∇𝜃=0,(3.40)_𝑡||ℱ𝑣_𝛿||²+2||𝜉||^2𝛼||ℱ𝑣_𝛿||²≤||ℱ(𝑢⋅∇𝜃)ℱ𝑣_𝛿||.(3.41)

Multiplying this equation by 𝑒^{2𝑡|𝜉|2𝛼}, we have‖‖𝑣_𝛿‖‖^2𝐿₂≤𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2𝐿₂+𝐶^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}||||𝑢⋅∇𝜃𝑣_𝛿||||^2𝐿_2ℝ2≤𝑒_{−2𝑡𝛿2𝛼}𝑑𝜏‖‖𝑣^0𝛿‖‖^2𝐿₂+𝐶‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_𝛼𝑑𝜏.(3.42) We set 𝐹_𝛿(𝑡)=𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2𝐿₂+𝐶‖‖𝜃⁰‖‖_{̇𝐻2−2𝛼}^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_𝛼𝑑𝜏,_0+∞𝑒^{−2𝑡𝛿2𝛼}‖‖𝑣^0𝛿‖‖^2𝐿₂𝑑𝑡=‖‖𝜃⁰‖‖^2𝐿₂2𝛿^2𝛼_0+∞,^𝑡0𝑒^{−2(𝑡−𝜏)𝛿2𝛼}‖𝜃‖^2̇𝐻_𝛼𝑑𝜏𝑑𝑡=^0+∞^𝜏+∞=1𝑒^{−2(𝑡−𝜏)𝛿2𝛼}𝑑𝑡‖𝜃‖^2̇𝐻_𝛼𝑑𝜏2𝛿^2𝛼^0+∞‖𝜃‖^2̇𝐻_𝛼𝑑𝑡≤‖‖𝜃⁰‖‖^2𝐿₂2𝛿^2𝛼.(3.43) Then,^0+∞𝐹_𝛿(𝑡)𝑑𝑡≤‖‖𝜃⁰‖‖^2𝐿₂𝛿^2𝛼.(3.44) Let 𝜀>0, from (3.40), then there exists 𝛿₀>0 such that ‖‖𝑤_𝛿0‖‖_𝐿2≤𝜀2,∀𝑡≥0.(3.45) Let 𝐸_𝛿0={𝑡≥0;‖𝑣_𝛿0‖_𝐿²>𝜀/2}, then ^0+∞‖‖𝑣_𝛿0‖‖^2𝐿₂𝑑𝑡≥_𝐸𝛿0‖‖𝑣_𝛿0‖‖^2𝐿₂𝑑𝑡≥𝜀2²𝜆₁𝐸_𝛿0,(3.46) where 𝜆₁(𝐸_𝛿0) is the Lebesgue measure of 𝐸_𝛿0. If 𝑇_𝜀=2𝜀²^0+∞‖‖𝑣_𝛿0‖‖^2𝐿₂𝑑𝑡,(3.47) then 𝜆₁(𝐸_𝛿0)≤𝑇_𝜀. For 𝜂>0, there exists 𝑡₀∈[0,𝑇_𝜀+𝜂] such that 𝑡₀∉𝐸_𝛿0, then ‖‖𝑣_𝛿0𝑡₀‖‖_𝐿²≤𝜀2.(3.48) The equations (3.45) and (3.48) give that ‖‖𝜃𝑡₀‖‖_𝐿²<𝜀.(3.49) Thus, lim_𝑡→+∞‖𝜃(𝑡)‖_𝐿²=0, and this finishes the proof.

Acknowledgments

This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02. The authors thank the referee for his/her careful reading of the paper and corrections.