Research article

Global well-posedness and large time decay for the d-dimensional tropical climate model

  • Received: 20 January 2021 Accepted: 15 March 2021 Published: 22 March 2021
  • MSC : 35Q35, 35B40, 76D03

  • This paper investigates the Cauchy problem on the d-dimensional tropical climate model with fractional hyperviscosity. We establish the small data global well-posedness of solutions to this model with supercritical dissipation. Furthermore, we study the asymptotic stability of these global solutions and obtain the optimal decay rates by using energy method and the method of bootstrapping argument.

    Citation: Zhaoxia Li, Lihua Deng, Haifeng Shang. Global well-posedness and large time decay for the d-dimensional tropical climate model[J]. AIMS Mathematics, 2021, 6(6): 5581-5595. doi: 10.3934/math.2021330

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  • This paper investigates the Cauchy problem on the d-dimensional tropical climate model with fractional hyperviscosity. We establish the small data global well-posedness of solutions to this model with supercritical dissipation. Furthermore, we study the asymptotic stability of these global solutions and obtain the optimal decay rates by using energy method and the method of bootstrapping argument.



    Consider the d-dimensional (dD) tropical climate model with fractional dissipation

    {tu+uu+νΛ2αu+p+(vv)=0,tv+uv+μΛ2βv+θ+vu=0,tθ+uθ+ηΛ2γθ+v=0,u=0,u(x,0)=u0(x),v(x,0)=v0(x),θ(x,0)=θ0(x), (1.1)

    where (x,t)Rd×R+ with d2, u=(u1(x,t),u2(x,t),,ud(x,t)) is the barotropic mode, v=(v1(x,t),v2(x,t),,vd(x,t)) is the first baroclinic mode of vector velocity, p=p(x,t) is the scalar pressure and θ=θ(x,t) is the scalar temperature, respectively. vv denotes the tensor product, namely vv=(vivj) with i,j=1,2,,d, the parameters ν0, μ0, η0, α>0, β>0, γ>0 are real numbers, and Λ=(Δ)12 denotes the Zygmund operator. The fractional operator Λr is defined via the Fourier transform as

    ^Λrf(ξ)=|ξ|rˆf(ξ), ξRd, r>0.

    The inviscid case of system (1.1), namely ν=0, μ=0 and η=0, was originally derived by Frierson, Majda and Pauluis [7] for large-scale dynamics of precipitation fronts in the tropical atmosphere. The viscous counterpart of system (1.1) with the standard Laplacian can be derived by the same argument from the viscous primitive equations (see, e.g., [12]). The model considered here, namely (1.1), is appended with fractional dissipation terms, which may be relevant in the study of viscous flows in the thinning of atmosphere. Flows in the middle atmosphere traveling upward undergo changes due to the changes of atmospheric properties. The effect of kinematic and thermal diffusion is attenuated by the thinning of atmosphere. This anomalous attenuation can be modeled by using the space fractional Laplacian (see, e.g., [3]).

    Considering the 2D tropical climate model (1.1) with fractional dissipation or partial dissipation, the global well-posedness problem has recently attracted considerable attention and significant progress has been made. When there is no thermal diffusion in (1.1), namely η=0, Li and Titi in [13] and Dong, Wang, Wu and Zhang in [6] were able to establish the global regularity for the case α=β=1 and the case α+β=2, respectively. Concerning the case ν>0, μ>0 and η>0, Ye [21] obtained the global regularity for (1.1) when α>0, β=1 and γ=1. Recently, the decay estimates were studied by Li and Xiao [11] when α=β=γ=1. For more results on the 2D tropical climate model, one can refer to [3,4,5,14,15,22] for more examples.

    Concerning to the dD tropical climate model with d3, Ye in [22] proved the global regularity of this model in the case when α12+d4, α+β1+d2 and β0. When α<12+d4, whether classical solutions to this model, even for the Navier-Stokes equations (namely system (1.1) with v=θ=0), can develop finite time singularities remains outstandingly open.

    This paper focuses its attention on the case when α<12+d4 with d2. To the best of authors' knowledge, compared with the magnitude of research conducted on the global well-posedness problem of the model (1.1), the large-time behavior of solutions has been studied relatively little. Here we first seek small data global solutions emanating from initial data in almost critical Sobolev space, and then study the temporal decay for these global solutions. More precisely, the first result is the global stability of solutions to (1.1) in Hs(Rd), which is stated as follows.

    Theorem 1.1. Let 12<α,β,γ<12+d4 with d2. Assume that (u0,v0,θ0)Hs(Rd) with s>1+d22min{α,β,γ} and u0=0. Then there exists a positive constant C0 such that for all 0<ϵ<C0, if

    u0Hs(Rd)+v0Hs(Rd)+θ0Hs(Rd)<ϵ, (1.2)

    then system (1.1) has a unique global solution (u,v,θ) satisfying, for any T>0,

    (u,v,θ)L(0,T;Hs(Rd)), (Λαu,Λβv,Λγθ)L2(0,T;Hs(Rd)), (1.3)

    and

    u(t)Hs(Rd)+v(t)Hs(Rd)+θ(t)Hs(Rd)<ϵ. (1.4)

    Theorem 1.1 shall be proved by using the delicate energy method and fully exploiting the special structure of this model. We remark that, mathematically, system (1.1) is more complex than the magnetohydrodynamic equations ((1.1) with θ=0 and v=0), since it involves the coupling of a divergence-free vector field u and a non-divergence-free vector field v. In particular, the results obtained in this paper also hold for the magnetohydrodynamic equations.

    The second result is to explore the long time behavior with explicit decay rates for the global solution itself and its derivative to system (1.1) when the initial data is also in negative Sobolev space ˙Hσ(Rd) or negative Besov space ˙Bσ2,(Rd), which is stated as in the following theorem.

    Theorem 1.2. Let all the assumptions in Theorem 1.1 hold. Suppose also that (u0,v0,θ0)˙Hσ(Rd) with 0σ<d2 or (u0,v0,θ0)˙Bσ2,(Rd) with 0<σd2. Then for s1+d2, the global solution (u,v,θ) established in Theorem 1.1 satisfies for all t>0,

    u(t)˙Hσ(Rd)+v(t)˙Hσ(Rd)+θ(t)˙Hσ(Rd)C, (1.5)

    or

    u(t)˙Bσ2,(Rd)+v(t)˙Bσ2,(Rd)+θ(t)˙Bσ2,(Rd)C. (1.6)

    Moreover, for any real number m with 0ms,

    Dmu(t)L2(Rd)+Dmv(t)L2(Rd)+Dmθ(t)L2(Rd)C(1+t)m+σ2max{α,β,γ}. (1.7)

    Remark 1.3. Note that for σ=dpd2, Lp(Rd)˙Hσ(Rd) when σ[0,d2) and p(1,2], and Lp(Rd)˙Bσ2,(Rd) when σ(0,d2] and p[1,2), thus Theorem 1.2 also holds for (u0,v0,θ0)Lp(Rd) with p[1,2].

    The proof of Theorem 1.2 is divided into two steps. The first uses energy method to derive the evolution of the negative Sobolev and Besov norms of solutions (u,v,θ) to the system (1.1), and the second establishes the desired results in Theorem 1.2 by the method of bootstrapping argument. We remark that the negative spaces ˙Hσ(Rd) and ˙Bσ2,(Rd) were introduced to study the decay estimates of the Boltzmann equation by Guo and Wang in [8] and Sohinger and Strain in [18], respectively. The main advantages of these two negative spaces are that the negative Sobolev and Besov norms of solutions are shown to be preserved along time evolution and enhance the decay rates.

    The rest of this paper is organized as follows. In Section 2 and Section 3, we give the proofs of Theorem 1.1 and Theorem 1.2, respectively. An appendix containing the Littlewood-Paley decomposition and the definition of Besov spaces is also given for the convenience of the readers. Throughout this manuscript, to simplify the notations, we will write f for Rdfdx, fLp for fLp(Rd), f˙Hs, fHs and f˙Bs2, for f˙Hs(Rd), fHs(Rd) and f˙Bs2,(Rd) respectively. For simplicity, we set ν=1, μ=1 and η=1 in the subsequent sections.

    This section is devoted to the proof of Theorem 1.1. For the purpose of proving this theorem, we first present an a priori estimate stated in Proposition 2.2 below, which contains a major ingredient in proving this theorem. Then we can prove this theorem by the methods of successive approximations.

    As preparations we first give the following calculus inequality involving fractional differential operators (see, e.g., [9,10]).

    Lemma 2.1. Let s>0. Let 1<r< and 1r=1p1+1q1=1p2+1q2 with q1,p2(1,) and p1,q2[1,]. Then

    Λs(fg)LrC(ΛsfLp1gLq1+fLp2ΛsgLq2),

    where C is a positive constant depending on the indices s,r,p1,q1,p2 and q2.

    As explained above, we start with an important global an a priori estimate. More precisely, we have the following proposition.

    Proposition 2.2. Let 12<α,β,γ<12+d4. Assume that (u0,v0,θ0)Hs(Rd) with s>1+d22min{α,β,γ} and u0=0. Then any solution (u,v,θ) of the system (1.1) obeys the following differential inequality

    ddt(u2Hs+v2Hs+θ2Hs)+Λαu2Hs+Λβv2Hs+Λγθ2HsC(u2Hs+v2Hs+θ2Hs)(Λαu2Hs+Λβv2Hs+Λγθ2Hs). (2.1)

    Proof. Dotting (1.1)1, (1.1)2 and (1.1)3 by u, v and θ, respectively, we obtain

    12ddt(u2L2+v2L2+θ2L2)+Λαu2L2+Λβv2L2+Λγθ2L2=(vv)uθvvuvvθ=0, (2.2)

    where we have used the facts that

    (vv)u+vuv=0,

    and

    θv+vθ=0.

    Applying Λs to the first three equations in (1.1), dotting the resulting equations with Λsu, Λsv and Λsθ respectively, integrating in space domain and adding the results up, one obtains

    12ddt(Λsu2L2+Λsv2L2+Λsθ2L2)+Λs+αu2L2+Λs+βv2L2+Λs+γθ2L2=Λs(uu)ΛsuΛs(vv)ΛsuΛs(uv)Λsv   ΛsθΛsvΛs(vu)ΛsvΛs(uθ)Λsθ   Λs(v)Λsθ=I1+I2+I3+I4+I5+I6+I7. (2.3)

    Integration by parts implies

    I4+I7=ΛsθΛsvΛs(v)Λsθ=0. (2.4)

    Applying Hölder's inequality, Lemma 2.1 and Sobolev embedding inequality, we estimate the term I1 as

    I1=Λsα(uu)Λs+αu=Λsα(uu)Λs+αuCΛs+1α(uu)L2Λs+αuL2CuLd2α1Λs+1αuL2dd+24αΛs+αuL2CΛ1+d22αuL2Λs+αu2L2.

    Similarly, we have

    I2CΛs+1α(vv)L2Λs+αuL2CvLdα+β1Λs+1αvL2dd+22(α+β)Λs+αuL2CΛ1+d2(α+β)vL2Λs+βvL2Λs+αuL2.
    I3Λs+1β(uv)L2Λs+βvL2C(Λs+1βuL2dd+22(α+β)vLdα+β1+uLd2β1Λs+1βvL2dd+24β)Λs+βvL2C(Λ1+d2(α+β)vL2Λs+αuL2Λs+βvL2+Λ1+d22βuL2Λs+βv2L2).
    I6Λs+1γ(uθ)L2Λs+γθL2C(Λs+1γuL2dd+22(α+γ)θLdα+γ1+uLd2γ1Λs+1γθL2dd+24γ)Λs+γθL2C(Λ1+d2(α+γ)θL2Λs+αuL2Λs+γθL2+Λ1+d22γuL2Λs+γθ2L2).

    We cannot bound I5 as above, since v is not divergence free. Using Hölder's inequality and Lemma 2.1, we derive that

    I5=Λsβ(vu)Λs+βvΛsβ(vu)L2Λs+βvL2C(ΛsβvL2dd2βuLdβ+vLdα+β1Λsβ+1uL2dd+22(α+β))Λs+βvL2C(ΛsvL2Λ1+d2βuL2Λs+βvL2+Λ1+d2(α+β)vL2Λs+αuL2Λs+βvL2).

    Combining these bounds and (2.4) with (2.3) together, we get

    12ddt(Λsu2L2+Λsv2L2+Λsθ2L2)+Λs+αu2L2+Λs+βv2L2+Λs+γθ2L2C(uHsΛαu2Hs+vHsΛαuHsΛβvHs+uHsΛβv2Hs   +θHsΛαuHsΛγθHs+uHsΛγθ2Hs). (2.5)

    Adding (2.2) and (2.5) up, then the Young inequality implies the desired inequality (2.1). Thus the proof of Proposition 2.2 is completed.

    With Proposition 2.2 at our disposal, we are ready to prove Theorem 1.1.

    Proof of the Theorem 1.1. We apply the method of successive approximation. It consists of constructing a successive approximation sequence (un,vn,θn) with n0 and showing its convergence to the solution (u,v,θ) of the system (1.1).

    Consider successive approximation sequences (un,vn,θn) satisfying

    {u0=0,v0=0,θ0=0,tun+1+unun+1+Λ2αun+1+pn+1+vn+1vn+vnvn+1=0,tvn+1+unvn+1+Λ2βvn+1+θn+1+vnun+1=0,tθn+1+unθn+1+Λ2γθn+1+vn+1=0,un+1=0,un+1(x,0)=u0(x),vn+1(x,0)=v0(x),θn+1(x,0)=θ0(x). (2.6)

    To show that (un,vn,θn) converges, we first prove that there exists a constant ϵ>0 independent of n, such that for any T>0,

    un(t)2Hs+vn(t)2Hs+θn(t)2Hs   +12t0(Λαun(τ)2Hs+Λβvn(τ)2Hs+Λγθn(τ)2Hs)dτϵ2, (2.7)

    for all 0<tT.

    We will prove (2.7) by mathematical induction. Obviously, (2.7) holds for n=0. Assume that (2.7) is true for n0. We start to show it for n+1. We proceed as in the proof of Proposition 2.2. Actually, after going through the steps as in proof of Proposition 2.2, we arrive at

    ddt(un+12Hs+vn+12Hs+θn+12Hs)+Λαun+12Hs+Λβvn+12Hs+Λγθn+12HsC(un2Hs+vn2Hs+θn2Hs)(Λαun+12Hs+Λβvn+12Hs+Λγθn+12Hs). (2.8)

    Integrating this in [0,t], together with (1.2) and inductive assumption, we derive that

    un+1(t)2Hs+vn+1(t)2Hs+θn+1(t)2Hs   +t0(Λαun+12Hs+Λβvn+12Hs+Λγθn+12Hs)(τ)dτu02Hs+v02Hs+θ02Hs+Ct0(un2Hs+vn2Hs+θn2Hs)(Λαun+12Hs+Λβvn+12Hs+Λγθn+12Hs)(τ)dτϵ2+Cϵ2t0(Λαun+12Hs+Λβvn+12Hs+Λγθn+12Hs)(τ)dτ.

    This implies (2.7) holds for n+1 by choosing ϵ sufficiently small such that ϵ12C. Thus (2.7) is true for all n0.

    Next we show that (un,vn,θn) is a Cauchy sequence in C([0,T];Hs). Resorting to (2.8) and (2.7), it infers that for all 0t1t2T,

    |(un(t2)2Hs+vn(t2)2Hs+θn(t2)2Hs)(un(t1)2Hs+vn(t1)2Hs+θn(t1)2Hs)|=|t2t1ddτ(un(τ)2Hs+vn(τ)2Hs+θn(τ)2Hs)dτ|Cϵ2t2t1(Λαun(τ)2Hs+Λβvn(τ)2Hs+Λγθn(τ)2Hs)dτ,

    which implies that (un,vn,θn) is absolutely continuous from [0,T] to Hs or simply (un,vn,θn)C([0,T];Hs).

    To prove that (un,vn,θn) is a Cauchy sequence, we consider the differences

    u(n+1)=un+1un, v(n+1)=vn+1vn, θ(n+1)=θn+1θn, p(n+1)=pn+1pn,

    which satisfy

    {tu(n+1)+unu(n+1)+u(n)un+Λ2αu(n+1)+p(n+1)+v(n+1)vn+vnv(n)+vnv(n+1)+v(n)vn=0,tv(n+1)+unv(n+1)+u(n)vn+Λ2βv(n+1)+θ(n+1)+vnu(n+1)+v(n)un=0,tθ(n+1)+unθ(n+1)+u(n)θn+Λ2γθ(n+1)+v(n+1)=0,u(n+1)=0u(n+1)(x,0)=0,v(n+1)(x,0)=0,θ(n+1)(x,0)=0. (2.9)

    After going through a similar procedure as above, we obtain

    ddt(u(n+1)2Hs+v(n+1)2Hs+θ(n+1)2Hs)+Λαu(n+1)2Hs+Λβv(n+1)2Hs+Λγθ(n+1)2HsC(u(n)2Hs+v(n)2Hs+θ(n)2Hs)(Λαun2Hs+Λβvn2Hs+Λγθn2Hs)   +C(un2Hs+vn2Hs+θn2Hs)(Λαu(n+1)2Hs+Λβv(n+1)2Hs+Λγθ(n+1)2Hs). (2.10)

    Integrating this inequality with respect to time, together with (2.7), one infers that for all 0tT,

    u(n+1)(t)2Hs+v(n+1)(t)2Hs+θ(n+1)(t)2Hs   +t0(Λαu(n+1)2Hs+Λβv(n+1)2Hs+Λγθ(n+1)2Hs)(τ)dτCϵ2sup0τt(u(n)(τ)2Hs+v(n)(τ)2Hs+θ(n)(τ)2Hs)   +Cϵ2t0(Λαu(n+1)2Hs+Λβv(n+1)2Hs+Λγθ(n+1)2Hs)(τ)dτ. (2.11)

    By choosing ϵ>0 as above, it follows from (2.11) that

    sup0tT(u(n+1)(t)2Hs+v(n+1)(t)2Hs+θ(n+1)(t)2Hs)12sup0tT(u(n)(t)2Hs+v(n)(t)2Hs+θ(n)(t)2Hs), (2.12)

    which implies that (un,vn,θn) is a Cauchy sequence in C([0,T];Hs). Therefore, the limit function (u,v,θ) satisfying system (1.1) indeed exists in C([0,T];Hs). Moreover, it obeys

    u(t)2Hs+v(t)2Hs+θ(t)2Hs   +12t0(Λαu(τ)2Hs+Λβv(τ)2Hs+Λγθ(τ)2Hs)dτϵ2, (2.13)

    for all 0<t<T.

    Finally, we prove the uniqueness. Let (u,v,θ) and (˜u,˜v,˜θ) be two solutions of system (1.1) in the regularity class (2.13). Similar process as the proof of convergence above, we derive that their difference (ˉu,ˉv,ˉθ) with

    ˉu=u˜u,ˉv=v˜v,ˉθ=θ˜θ

    satisfies

    sup0tT(ˉu(t)2Hs+ˉv(t)2Hs+ˉθ(t)2Hs)12sup0tT(ˉu(t)2Hs+ˉv(t)2Hs+ˉθ(t)2Hs). (2.14)

    This inequality implies (ˉu,ˉv,ˉθ)=0 or (u,v,θ)=(˜u,˜v,˜θ) for all 0tT. Thus we complete the proof of Theorem 1.1.

    This section proves Theorem 1.2. To this end, we first establish the global a priori estimates for the global solution (u,v,θ) of system (1.1) in the negative Sobolev norm ˙Hσ with 0σ<d2 and negative Besov norm ˙Bσ2, with 0<σd2, respectively. Then we will establish Theorem 1.2 by the method of bootstrapping argument.

    As preparations we recall the Hardy-Littlewood-Sobolev inequality for fractional integration and an inequality for homogeneous Besov norm (see [19] and [18] respectively).

    Lemma 3.1. Let 0σ<d2 and 1<p2 with 12+σd=1p. Then

    ΛσfL2(Rd)CfLp(Rd). (3.1)

    Lemma 3.2. Let 0<σd2 and 1p<2 with 12+σd=1p. Then

    f˙Bσ2,(Rd)CfLp(Rd). (3.2)

    Now we show the global a priori estimates for the global solution (u,v,θ) established in Theorem 1.1 in ˙Hσ with 0σ<d2. More precisely, we have the following lemma.

    Lemma 3.3. Let the assumptions stated in Theorem 1.2 hold. Then for s>d2, (u,v,θ) obeys

    ddt(u2˙Hσ+v2˙Hσ+θ2˙Hσ)C(u4s+2σd22sL2+v4s+2σd22sL2+θ4s+2σd22sL2)(ud+22σ2s˙Hs+vd+22σ2s˙Hs+θd+22σ2s˙Hs)   ×(u˙Hσ+v˙Hσ+θ˙Hσ). (3.3)

    Proof. Applying Λσ to (1.1)1(1.1)3, and taking the L2-inner products with Λσu, Λσv and Λσθ respectively, we obtain

    12ddt(Λσu2L2+Λσv2L2+Λσθ2L2)+(Λασu2L2+Λβσv2L2+Λγσθ2L2)=Λσ(uu)ΛσuΛσ(vv)ΛσuΛσ(uv)Λσv   Λσ(vu)ΛσvΛσ(uθ)Λσθ:=K1+K2+K3+K4+K5, (3.4)

    where we have used the fact

    ΛσθΛσv+Λσ(v)Λσθ=0.

    Using Hölder's inequality, Lemma 3.1 and the Gagliardo-Nirenberg inequality, we derive that

    K1=Λσ(uu)ΛσuΛσ(uu)L2ΛσuL2CuuL2dd+2σΛσuL2CuLdσuL2ΛσuL2Cu4s+2σd22sL2Λsud+22σ2sL2ΛσuL2.

    Similarly, we have

    K2Cv4s+2σd22sL2Λsvd+22σ2sL2ΛσuL2.

    Again applying Hölder's inequality, Lemma 3.1 and the Gagliardo-Nirenberg inequality, we derive that

    K3=Λσ(uv)ΛσvΛσ(uv)L2ΛσvL2CuvL2dd+2σΛσvL2CuLdσvL2ΛσvL2Cu2s+2σd2sL2Λsud2σ2sL2vs1sL2Λsv1sL2ΛσvL2.

    Similarly, we obtain

    K4Cv2s+2σd2sL2Λsvd2σ2sL2us1sL2Λsu1sL2ΛσvL2.
    K5Cu2s+2σd2sL2Λsud2σ2sL2θs1sL2Λsθ1sL2ΛσθL2.

    Inserting the above bounds into (3.4), together with the Young inequality, leads to (3.3). Thus the proof of Lemma 3.3 is completed.

    Next we establish the global a priori estimates for the global solution (u,v,θ) established in Theorem 1.1 in ˙Bσ2, with 0<σd2, as stated in the following lemma.

    Lemma 3.4. Let the assumptions stated in Theorem 1.2 hold. Then for s>d2, (u,v,θ) obeys

    ddt(u2˙Bσ2,+v2˙Bσ2,+θ2˙Bσ2,)C(u4s+2σd22sL2+v4s+2σd22sL2+θ4s+2σd22sL2)(ud+22σ2s˙Hs+vd+22σ2s˙Hs+θd+22σ2s˙Hs)   ×(u˙Bσ2,+v˙Bσ2,+θ˙Bσ2,). (3.5)

    Proof. We remark that the argument is similar to the proof of Lemma 3.3, here we give the details for reader's convenience. Applying ˙Δj, which definition is in the appendix, to (1.1)1(1.1)3, taking the L2-inner products with ˙Δju, ˙Δjv and ˙Δjθ respectively, multiplying the results by 22σj, and taking the supremum over jZ, we conclude that

    12ddt(u2˙Bσ2,+v2˙Bσ2,+θ2˙Bσ2,)supjZ22σj|˙Δj(uu)˙Δju|+supjZ22σj|˙Δj(vv)˙Δju|   +supjZ22σj|˙Δj(uv)˙Δjv|+supjZ22σj|˙Δj(vu)˙Δjv|   +supjZ22σj|˙Δj(uθ)˙Δjθ|:=M1+M2+M3+M4+M5, (3.6)

    where we used the fact that

    ˙Δjθ˙Δjv+˙Δj(v)˙Δjθ=0.

    Applying Hölder's inequality, Lemma 3.2 and the Gagliardo-Nirenberg inequality, one infers that

    M1uu˙Bσ2,u˙Bσ2,uuL2dd+2σu˙Bσ2,CuLdσuL2u˙Bσ2,Cu4s+2σd22sL2Λsud+22σ2sL2u˙Bσ2,.

    Similarly, we have

    M2Cv4s+2σd22sL2Λsvd+22σ2sL2u˙Bσ2,.
    M3Cu2s+2σd2sL2Λsud2σ2sL2vs1sL2Λsv1sL2v˙Bσ2,.
    M4Cv2s+2σd2sL2Λsvd2σ2sL2us1sL2Λsu1sL2v˙Bσ2,.
    M5Cu2s+2σd2sL2Λsud2σ2sL2θs1sL2Λsθ1sL2θ˙Bσ2,.

    Then (3.5) eventually follows from the above bounds, (3.6) and the Young inequality. This completes the proof of Lemma 3.4.

    With Lemma 3.3 and Lemma 3.4 at our disposal, we are ready to prove Theorem 1.2 by the method of bootstrapping argument.

    Proof of the Theorem 1.2. We will just focus on the case (u0,v0,θ0)˙Hσ. The case (u0,v0,θ0)˙Bσ2, can be treated similarly. Assume that

    u02˙Hσ+v02˙Hσ+θ02˙Hσ=C0. (3.7)

    Suppose that for all t[0,T],

    u(t)2˙Hσ+v(t)2˙Hσ+θ(t)2˙Hσ2C0. (3.8)

    If we can derive that for all t[0,T],

    u(t)2˙Hσ+v(t)2˙Hσ+θ(t)2˙Hσ3C02, (3.9)

    then an application of the bootstrapping argument would imply that the solution (u,v,θ) of system (1.1) satisfies (3.9) for all t[0,T], which implies (1.5).

    With (3.7) and (3.8) at our disposal, we shall show that (3.9) holds. At the same time, the decay estimates (1.5) will be established in this process. Similar as the proof of (2.5), one can show that for 0ms,

    12ddt(Λmu2L2+Λmv2L2+Λmθ2L2)+Λm+αu2L2+Λm+βv2L2+Λm+γθ2L2C(uHsΛαu2Hm+vHsΛαuHmΛβvHm+uHsΛβv2Hm   +θHsΛαuHmΛγθHm+uHsΛγθ2Hm). (3.10)

    Then this inequality together with the Young inequality implies

    ddt(Λmu2L2+Λmv2L2+Λmθ2L2)+Λm+αu2L2+Λm+βv2L2+Λm+γθ2L2C(u2Hs+v2Hs+θ2Hs)(Λm+αu2L2+Λm+βv2L2+Λm+γθ2L2). (3.11)

    Using (1.4) with ϵ<12C, it follows from (3.11) that

    ddt(Λmu2L2+Λmv2L2+Λmθ2L2)+12(Λm+αu2L2+Λm+βv2L2+Λm+γθ2L2)0. (3.12)

    Applying the Gagliardo-Nirenberg inequality, together with (3.8), we obtain

    ΛmuL2Cuαm+α+σ˙HσΛm+αum+σm+α+σL2CΛm+αum+σm+α+σL2. (3.13)

    Similarly, we have

    ΛmvL2CΛm+βvm+σm+β+σL2. (3.14)
    ΛmθL2CΛm+γθm+σm+γ+σL2. (3.15)

    Inserting (3.13)–(3.15) into (3.12), there exists a positive constant C1>0 such that

    ddt(Λmu2L2+Λmv2L2+Λmθ2L2)+C1(Λmu2L2+Λmv2L2+Λmθ2L2)m+a+σm+σ0

    with a0=max{α,β,γ}. Integrating this inequality with respect to time, we derive that

    Λmu2L2+Λmv2L2+Λmθ2L2C(1+t)m+σa0, (3.16)

    which implies (1.7).

    Now we start to show (3.9). Integrating (3.3) in [0,t] with 0<tT, together with (3.16), (3.7) and (1.4), one infers that

    u(t)2˙Hσ+v(t)2˙Hσ+θ(t)2˙Hσu02˙Hσ+v02˙Hσ+θ02˙Hσ+Csup0τt(u(τ)˙Hσ+v(τ)˙Hσ+θ(τ)˙Hσ)   ×t0(u(τ)4s+2σd22sL2+v(τ)4s+2σd22sL2+θ(τ)4s+2σd22sL2)      ×(u(τ)d+22σ2s˙Hs+v(τ)d+22σ2s˙Hs+θ(τ)d+22σ2s˙Hs)dτC0+Csup0τt(u(τ)˙Hσ+v(τ)˙Hσ+θ(τ)˙Hσ)   ×ϵϵ0t0(1+τ)(σ2a0(4s+2σd22sϵ0)+(s+σ)(d+22σ)4a0s)dτC0+Cϵϵ0sup0τt(u(τ)˙Hσ+v(τ)˙Hσ+θ(τ)˙Hσ), (3.17)

    where ϵ0>0 is chosen small enough such that σ2a0(4s+2σd22sϵ0)+(s+σ)(d+22σ)4a0s>1, which is meaningful since assumptions 12<a0<d+24 and 0σ<d2 implies that σ(4s+2σd2)+(s+σ)(d+22σ)4a0s>1. By choosing ϵ sufficiently small, then (3.17) together with the Young inequality yields (3.9) for all t[0,T], which closes the proof. Thus we complete the proof of Theorem 1.2.

    This appendix provides the definition of the Littlewood-Paley decomposition and the definition of Besov spaces. Some related facts used in the previous sections are also included. Materials presented in this appendix can be found in several books and many papers (see, e.g., [1,2,16,17,20]).

    We start with several notation. S denotes the usual Schwarz class and S its dual, the space of tempered distributions. To introduce the Littlewood-Paley decomposition, we write for each jZ

    Aj={ξRd:2j1|ξ|<2j+1}.

    The Littlewood-Paley decomposition asserts the existence of a sequence of functions {Φj}jZS such that

    suppˆΦjAj,ˆΦj(ξ)=ˆΦ0(2jξ)orΦj(x)=2jdΦ0(2jx),

    and

    j=ˆΦj(ξ)={1,ifξRd{0},0,ifξ=0.

    Therefore, for a general function ψS, we have

    j=ˆΦj(ξ)ˆψ(ξ)=ˆψ(ξ)for ξRd{0}.

    We now choose ΨS such that

    ˆΨ(ξ)=1j=0ˆΦj(ξ),ξRd.

    Then, for any ψS,

    Ψψ+j=0Φjψ=ψ

    and hence

    Ψf+j=0Φjf=f (4.1)

    in S for any fS. To define the inhomogeneous Besov space, we set

    Δjf={0,ifj2,Ψf,ifj=1,Φjf,ifj=0,1,2,. (4.2)

    To define the homogeneous Besov space, we set

    ˙Δjf=Φjf,ifj=0,±1,±2,. (4.3)

    Definition 4.1. The inhomogeneous and homogeneous Besov spaces Bsp,q and ˙Bsp,q with sR and p,q[1,] consists of fS satisfying

    fBsp,q2jsΔjfLplqj<,

    and

    f˙Bsp,q2js˙ΔjfLplqj<,

    respectively.

    The authors would like to thank the anonymous referees for their useful suggestions, which improved the presentation of the manuscript. This work was partially supported by NSFC (No.11201124) and a Foundation for University Key Teacher by the Henan Province (No. 2015GGJS-070).

    The authors declare that they have no conflict of interest.



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