Research article

Global well-posedness and optimal decay rates for the n-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion

  • Received: 18 October 2024 Revised: 01 December 2024 Accepted: 03 December 2024 Published: 13 December 2024
  • MSC : 35A05, 35Q35, 76D03

  • In this paper, n-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that (u,θ)L2(Rn)0, as t. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in Hs(Rn) with s = 1+n22α(0<α<1), the global solutions are derived. Furthermore, under the assumption that the initial data (u0, θ0) belongs to Lp(where 1p<2), using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case α = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.

    Citation: Xinli Wang, Haiyang Yu, Tianfeng Wu. Global well-posedness and optimal decay rates for the n-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion[J]. AIMS Mathematics, 2024, 9(12): 34863-34885. doi: 10.3934/math.20241660

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  • In this paper, n-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that (u,θ)L2(Rn)0, as t. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in Hs(Rn) with s = 1+n22α(0<α<1), the global solutions are derived. Furthermore, under the assumption that the initial data (u0, θ0) belongs to Lp(where 1p<2), using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case α = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.



    In this paper, we consider the generalized incompressible Boussinesq equations in Rn:

    {tU+UU+ν(Δ)αU+P=Θen,tΘ+UΘ+κ(Δ)βΘ=0,U=0,U(x,0)=U0(x),Θ(x,0)=Θ0(x), (1.1)

    where α and β are nonnegative real parameters, ν0 is the fluid kinematic viscosity coefficient, and κ0 is the thermal diffusion. The unknowns are the fluid velocity field U=U(x,t) with (x,t)Rn×R+, the temperature Θ=Θ(x,t) can be understood in a physical context as a thermal variable when the κ>0, or as a density variable when κ=0, and the fluid pressure P(x,t). We denote the vector en=(0,0,,1)t. The term Θen in the momentum equation represents the influence of buoyancy on the motion of the fluid.

    The fractional Laplacian operator Λα=(Δ)α2 is characterized by the Fourier transform, specifically,

    (Δ)αˆf=^Λ2αf(ξ)=|ξ|2αˆf(ξ).

    As follows is the definition of the Fourier transform:

    Ff(ξ)=ˆf(ξ)=Rneixξf(x)dx.

    For the higher dimensional case n3, Ye [42] showed that Eq (1.1) admits a unique global classical solution as long as α12+n4, β>0. Moreover, reader's can also find many other results related to the higher dimensional cases in [18,35] and references cited therein.

    In the following, we focus on the 2D case. According to the value range of α and β, it can be classified into three distinct categories space (see [14]): The supercritical case when α+β<1, the critical case when α+β=1, and the subcritical case when α+β>1.

    The Boussinesq system is usually referred to as (1.1) for α=β=1. The Boussinesq system is widely used in the fields of atmospheric science and oceanic turbulence, where rotation and stratification are crucial factors (see, e.g., [24]).The work of Tao and Wu [29] was able to establish the stability and the enhanced dissipation phenomenon for the linearized 2D Boussinesq equations with only vertical dissipation, Shang and Xu in [26] examined the stability and the decay of the corresponding linearized systems of 3D Boussinesq equations with horizontal viscosity and horizontal thermal diffusion. The stabilizing effect of the temperature on the buoyancy-driven fluids and the stability of the hydrostatic equilibrium were discovered for several partially dissipated 2D Boussinesq systems in [20,25]. Important progress has been made on the stability and large-time behavior in [6,19,28,31]. Therefore, the Boussinesq system has been extensively studied in the past few years; see [2,3,5,7,8,22,43] and references therein.

    In the subcritical case, Xu in [34] investigated the subcritical cases where α+β>2 with α1, and established the existence, uniqueness, and regularity of 2D fractional Boussinesq equations. Miao and Xue in [23] proved global well-posedness results for rough initial data when 664<α<1, 1α<β<min{7+26α25,α(1α)62α,22α}. Constantin and Vicol in [4] established the global regularity, when β>22+α. Yang, Jiu and Wu in [36] verified the global well-posedness when β>1α2, β2+α3, β>105α104α. Ye and Xu in [37], Ye and Xu in [38], Ye, Xu, and Xue in [39], Ye in [40], Wu, Xu, and Ye in [32], Zhou, Li, Shang, Wu, and Yuan in [44], and other literatures have made significant advancements.

    The critical case is generally more challenging than the subcritical case. Hmidi, Keraani, and Rousset in [10,11] studied two specific critical cases where α+β=1 with α=0 and β=1, or with α=1 and β=0, and they established the global regularity for the 2D Boussinesq equations with fractional diffusion in both cases. Jiu, Miao, Wu, and Zhang in [14]; verified the global regularity for the general case with α+β=1 and 0.9132α0<α<1. Stefanov and Wu in [27] advanced the previous work of [14]; they proved the global existence and smoothness of the classical solution. The conditions are as follows: 17772324=0.798103.<α<1, β>0 and α+β=1. The range of α to 0.76921013<α<1 is further extended in [33] by Wu, Xu, Xue, and Ye. Some other developments are in [9,12,13,41].

    For the supercritical case, Jiu, Wu, and Yang in [15] consider the 2D incompressible Boussinesq equations with fractional dissipation; they verified the eventual regularity for α+β<1 and 0.91322314512<α<1. Wu, Xu, Xue, and Ye in [33] proved the final regularity of the Leray-Hopf-type weak solution of the Boussinesq equation with supercritical dissipative and 0.76921013<α<1. Some global regularity conclusions for supercritical equations can be obtained in [21].

    We assume

    u(0)=(0,0,0), θ(0)=xn, p(0)=12x2n. (1.2)

    Then the perturbation (u,θ,p) with

    u=Uu(0), θ=Θθ(0), p=Pp(0). (1.3)

    Then, (u,θ,p) satisfies

    {tu+uu+ν(Δ)αu+p=θen,tθ+uθ+κ(Δ)βθ=un,u=0,u(x,0)=u0(x),θ(x,0)=θ0(x). (1.4)

    Our first result can be stated as follows.

    Theorem 1.1. Consider (1.4) with ν>0 and κ>0. Assume the initial data (u0,θ0)L2(Rn)×L2(Rn) with u0=0 and n2. Then {(1.4)} has a global weak solution (u,θ) that satisfies the following property:

    limt(u(t)L2(Rn)+θ(t)L2(Rn))=0. (1.5)

    In the paper we focus on the global well-posedness and optimal decay of solutions of (1.4) with fractional dissipations. For the sake of simplicity, we set ν=κ=1 in (1.4). We shall consider the case with 0α=β<1. In particular, we investigate the following Cauchy problem:

    {tu+uu+(Δ)αu+p=θen,tθ+uθ+(Δ)αθ=un,u=0,u(x,0)=u0(x),θ(x,0)=θ0(x), (1.6)

    where 0α<1. When α=0, (1.6) is reduced into the n-dimensional incompressible Boussinesq system with damping

    {tu+uu+u+p=θen,tθ+uθ+θ=un,u=0,u(x,0)=u0(x),θ(x,0)=θ0(x). (1.7)

    The following is our second result.

    Theorem 1.2. Let 0<α<1 and s1+n22α. Assume that the initial data (u0,θ0)Hs(Rn), u0=0 and n2. Then there exists a constant ε>0 such that, if

    u0Hs(Rn)+θ0Hs(Rn)ε, (1.8)

    then (1.6) has a global solution

    (u,θ)L([0,);Hs(Rn))L2([0,);Hs+α(Rn)). (1.9)

    In addition, for any t>0,

    u(t)2Hs(Rn)+θ(t)2Hs(Rn)+t0Λαu(τ)2Hs(Rn)+Λαθ(τ)2Hs(Rn)dτCε2. (1.10)

    Moreover, for any 0m<s,

    limt(1+t)m2α(Λmu(t)L2(Rn)+Λmθ(t)L2(Rn))=0. (1.11)

    Furthermore, suppose that (u0,θ0)Lp(Rn) with 1p<2. Then

    Λmu(t)2L2(Rn)+Λmθ(t)2L2(Rn)C(1+t)mαnα(1p12). (1.12)

    When s1+n2α, the solution derived above is unique.

    Our third result is focused on the regularity properties of the damped Boussinesq equations (1.7) with α=0.

    Theorem 1.3. Suppose the initial data (u0,θ0)Bs2,1(Rn), sn2+1 with n2 and u0=0. Then there exists a constant ϵ>0 such that, if

    u0Bs2,1+θ0Bs2,1ϵ, (1.13)

    then (1.7) has a unique global solution

    (u,θ)L([0,);Bs2,1)L2([0,);Bs2,1). (1.14)

    In addition, for any t>0,

    u(t)2Bs2,1+θ(t)2Bs2,1+t0u(τ)2Bs2,1+θ(τ)2Bs2,1dτCϵ2. (1.15)

    Furthermore, for any 0m<s,

    Λmu(t)L2(Rn)+Λmθ(t)L2(Rn)Cϵmses+mst. (1.16)

    The purpose of this section is to introduce some basic knowledge of the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces, and some useful properties (for more details, please see [1]).

    Let B and C represent the ball {ξRn:|ξ|43} and the annulus {ξRn:34|ξ|83}, respectively. There exist radial functions χ and φ, both of which take values in the interval [0,1], and which belong to C0(B) and C0(C), respectively, such that

    ξRn{0},jZφ(2jξ)=1,
    ξRn,χ(ξ)+j0φ(2jξ)=1.

    Let uS(R) with S being the set of tempered distributions. The nonhomogeneous dyadic blocks Δj are characterized by the following definition:

    Δju=0,if j2,andΔ1u=χ(D)u=Rn˜h(y)u(xy)dy,
    Δju=φ(2jD)u:=2jnRnh(2jy)u(xy)dy,ifj>0

    where h=F1φ and ˜h=F1χ. Now we will define the low-frequency cut-off operator Sj

    Sju:=jj1Δju.

    Lemma 2.1. (Bernstein inequality) Let B denote a ball and C denote an annulus. There exists a constant C such that, for any nonnegative integer k, any couple (p,q)[1,] with pq, and any function fLp(Rn), we have

    SuppˆfλBDkfLqdef=sup|α|=kαfLqCk+1λk+n(1p1q)fLp,
    SuppˆfλCCk1λkfLpsup|α|=kαfLpCk+1λkfLp.

    Next we recall the definitions of the nonhomogeneous Besov spaces.

    Definition 1. Assume sR and 1p,r, the nonhomogeneous Besov space Bsp,r consists of all tempered distributions f such that

    fBsp,r:=(2jsΔjfLp)jZr(Z)<.

    When p=r=2, we have Bs2,2(Rn)=Hs(Rn), where

    fHs(Rn)(Rn(1+|ξ|2)s|ˆf(ξ)|2dξ)12.

    Lemma 2.2. Assume sR and 1p,r, fS. Consequently, the following properties are valid:

    (1) Embedding: For 1p,ˉp and 1r,ˉr, there holds

    Bsp,r(Rn)Bsn(1p1ˉp)ˉp,ˉr(Rn),
    Bsp,1(Rn)L(Rn),
    Lp(Rn)B0p,(Rn).

    (2) Interpolation: For any s1<s2 and 0<θ<1, one has

    fBθs1+(1θ)s2p,rfθBs1p,rf1θBs2p,r,
    fBθs1+(1θ)s2p,1Cs2s1(1θ+11θ)fθBs1p,f1θBs2p,.

    Lemma 2.3. Let σR, 1r, and 1pp1. Let v be a vector field over R. Assume that

    σ>nmin{1p1,1p}orσ>1nmin{1p1,1p}ifv=0.

    Define Rj[v,Δj]f. There exists a constant C, depending continuously on p, p1, σ and d, such that

    (2jsRjLp)jrCvBnpp1,LfBσp,r,ifσ<1+np1.

    Further, if σ>0 (or σ>1, if v=0) and 1p2=1p1p1, then

    (2jsRjLp)jrC(vLfBσp,r+fLp2vBσ1p1,r),

    where

    [v,Δj]f=v(Δjf)Δj(vf).

    Lemma 2.4. ([1]) Let sR and ˙Hs(Rn) be the homogeneous Sobolev space with the definition

    u2˙Hs(Rn):=Rn|ξ|2s|ˆu(ξ)|2dξ<.

    If k[0,n2), then the space ˙Hk(Rn) is continuously embedded in L2nn2k(Rn), namely

    uL2nn2kCΛkuL2.

    Lemma 2.5. ([16,17]). Let s>0, 1<p<, fW1,p1Ws,p2, gLp2Ws,p1, then

    Λs(fg)fΛsgLpC(fLp1Λs1gLq1+gLp2ΛsgLq2),
    Λs(fg)LpC(fLp1ΛsgLq1+gLp2ΛsgLq2)

    for 1p=1p1+1q1=1p2+1q2.

    Lemma 2.6. Let s0<s<s1. Then ˙Hs0˙Hs1 is included in ˙Hs, and

    ΛsuL2Λs0uθL2Λs1u1θL2

    where θ[0,1] and s, s0, s1 satisfy

    s=θs0+(1θ)s1.

    Lemma 2.7. If fLp(Rn), 1p2, and 1p+1ˉp=1, then ˆfLˉp(Rn) and it satisfies

    ˆfLˉpCfLp.

    Firstly, dotting (1.4) with (u,θ), we have

    ddt(u(t)2L2+θ(t)2L2)+2νΛαu2L2+2κΛβθ2L2=0, (3.1)

    integrating it in time yields

    u(t)2L2+θ(t)2L2u02L2+θ02L2. (3.2)

    Furthermore, by applying the Fourier transformation to (1.4), we conclude that

    tˆu(ξ,t)+ν|ξ|2αˆu(ξ,t)=ˆθ(ξ,t)en^p(ξ,t)^(u)u(ξ,t)L1(ξ,t), (3.3)
    tˆθ(ξ,t)+κ|ξ|2βˆθ(ξ,t)=^un(ξ,t)^(u)θ(ξ,t)L2(ξ,t). (3.4)

    By calculating the above equations, we obtain

    ˆu(ξ,t)=^u0(ξ)eν|ξ|2αt+t0eν|ξ|2α(tτ)L1(ξ,t)dτ, (3.5)
    ˆθ(ξ,t)=^θ0(ξ)eκ|ξ|2βt+t0eκ|ξ|2β(tτ)L2(ξ,t)dτ. (3.6)

    Due to the simple fact that u=0, we have

    |^(u)u(ξ,t)|=|^(uu)(ξ,t)||ξ|uuL1|ξ|u2L2. (3.7)

    Similarly,

    |^(u)θ(ξ,t)||ξ|uL2θL2. (3.8)

    Applying the divergence operator to the velocity equation in (1.4), we obtained

    p=(Δ)1(θenuu)=(Δ)1(θen)(Δ)1(uu). (3.9)

    By (3.7), we obtain

    |^p(ξ,t)||ˆθ|+|ξ|uuL1|ˆθ|+|ξ|u2L2. (3.10)

    Invoking these estimates, it leads to

    |ˆu(ξ,t)||^u0(ξ)|eν|ξ|2αt+t0eν|ξ|2α(tτ)|L1(ξ,t)|dτ,|^u0(ξ)|+t0eν|ξ|2α(tτ)C(|ξ|u(τ)2L2+|ˆθ(ξ,τ)|)dτ|^u0(ξ)|+C|ξ|t0eν|ξ|2α(tτ)(u(τ)2L2+θ(τ)2L2)dτ+t0eν|ξ|2α(tτ)|ˆθ|dτ|^u0(ξ)|+C|ξ|(u02L2+θ02L2)t0eν|ξ|2α(tτ)dτ+Ct0|ˆθ|dτ|^u0(ξ)|+Ct|ξ|+Ct0|ˆθ|dτ. (3.11)

    The other term can be similarly bounded,

    |ˆθ(ξ,t)||^θ0(ξ)|+Ct|ξ|+Ct0|ˆu|dτ. (3.12)

    By employing the Plancherel theorem in conjunction with the Fourier splitting technique, we obtain

    Λαu2L2=Rn|ξ|2α|ˆu(ξ,t)|2dξ|ξ|r1|ξ|2α|ˆu(ξ,t)|2dξr2α1|ξ|r1|ˆu(ξ,t)|2dξ=r2α1(u2L2|ξ|r1|ˆu|2dξ) (3.13)

    and

    Λαθ2L2r2β2(θ2L2|ξ|r2|ˆθ|2dξ). (3.14)

    By using Fubini's theorem, the integral terms on the right-hand side of (3.13) and (3.14) can be dominated

    |ξ|r1|ˆu|2dξ|ξ|r1(|^u0(ξ)|+Ct|ξ|+Ct0|ˆθ|dτ)2dξ|ξ|r1|^u0(ξ)|2dξ+Crn+21(1+t)2+C|ξ|r1(t0|ˆθ|1dτ)2dξ|ξ|r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ct|ξ|r1t0|ˆθ|2dτdξ|ξ|r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ctt0|ξ|r1|ˆθ|2dξdτ|ξ|r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ctt0Rn|ˆθ|2dξdτ|ξ|r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ctt0θ2L2dτ. (3.15)

    Similarly,

    |ξ|r2|ˆθ|2dξ|ξ|r2|^θ0(ξ)|2dξ+Crn+22(1+t)2+Ctt0u2L2dτ. (3.16)

    Substituting (3.13)–(3.16) into (3.9), it yields

    ddt(u(t)2L2+θ(t)2L2)+2νr2α1u2L2+2κr2β2θ2L22νr2α1(|ξ|r1|ˆu(ξ)|2dξ)+2κr2β2(|ξ|r2|ˆθ(ξ)|2dξ)2νr2α1|ξ|r1|^u0(ξ)|2dξ+2κr2β2|ξ|r2|^θ0(ξ)|2dξ+C(r2α+n+21+r2α+n+22)(1+t)2+Ctr2α1t0θ2L2dτ+Ctr2β2t0u2L2dτ2νr2α1|ξ|r1|^u0(ξ)|2dξ+2κr2β2|ξ|r2|^θ0(ξ)|2dξ+C(r2α+n+21+r2α+n+22)(1+t)2+C(r2α1+r2β2)(1+t)22νr2α1|ξ|r1|^u0(ξ)|2dξ+2κr2β2|ξ|r2|^θ0(ξ)|2dξ+C(r2α+n+21+r2α+n+22)(1+t)2. (3.17)

    Let

    r1=(k2ν(1+t)ln(1+t))12α,r2=(k2κ(1+t)ln(1+t))12β (3.18)

    for some k>1+n+2min{2α,2β}. Then, we obtain

    ddt(u(t)2L2+θ(t)2L2)+k(1+t)ln(1+t)(u2L2+θ2L2)k(1+t)ln(1+t)(|ξ|max{r1,r2}|^u0(ξ)|2+|^θ0(ξ)|2dξ)+C((k2ν(1+t)ln(1+t))1+n+22α+(k2κ(1+t)ln(1+t))1+n+22β)(1+t)2. (3.19)

    Multiplying (3.19) by lnk(1+t), we have

    ddt(lnk(1+t)(u(t)2L2+θ(t)2L2))klnk1(1+t)1+t(|ξ|max{r1,r2}|^u0(ξ)|2+|^θ0(ξ)|2dξ)+Clnk(1+t)((k2ν(1+t)ln(1+t))1+n+22α+(k2κ(1+t)ln(1+t))1+n+22β)(1+t)2J1(t)+J2(t)+J3(t). (3.20)

    Integrating (3.20) in [0,t], we obtain

    u(t)2L2+θ(t)2L2lnk(1+t)(u02L2+θ02L2)+lnk(1+t)t0J1(τ)dτ+lnk(1+t)t0J2(τ)dτ+lnk(1+t)t0J3(τ)dτ. (3.21)

    Naturally,

    limtlnk(1+t)(u02L2+θ02L2)=0. (3.22)

    For the second term on the right-hand side of (3.21), we have

    t0J1(τ)dτ=(|ξ|max{r1,r2}|^u0(ξ)|2+|^θ0(ξ)|2dξ)(lnk(1+t)1). (3.23)

    When t, it is easy to show that r1 and r2 will converge towards zero. By using Theorem 1.1, we have

    limt|ξ|max{r1,r2}(|^u0(ξ)|2+|^θ0(ξ)|2)dξ=0. (3.24)

    This indicates that

    limtlnk(1+t)t0J1(τ)dτ=0. (3.25)

    For the third term on the right-hand side of (3.21),

    t0J2(τ)dτ=Ct0lnk(1+τ)(1+τ)2(k2ν(1+τ)ln(1+τ))1+n+22αdτCt0ln(1+τ)kn+22α1(1+τ)n+22α1dτCt0ln(1+τ)kn+22α11+τdτ=C(ln(1+τ)kn+22α1), (3.26)

    then

    limtlnk(1+t)t0J2(τ)dτ=0. (3.27)

    Similar to J2, one gets

    limtlnk(1+t)t0J3(τ)dτ=0. (3.28)

    Inserting (3.22), (3.25), (3.27), and (3.28) into (3.21), we finally obtain

    limtu(t)2L2+θ(t)2L2=0. (3.29)

    As a result, Theorem 1.1 is proved.

    Step 1. A prior estimation and existence. By using the basic energy estimate of (1.6),

    u(t)2L2+θ(t)2L2+2t0Λαu(τ)2L2dτ+2t0Λαθ(τ)2L2dτ=u02L2+θ02L2. (3.30)

    Applying Λs to both sides of (1.6) and then dotting the results with (Λsu,Λsθ), respectively, we obtain

    12ddt(Λsu2L2+Λsθ2L2)+(Λs+αu2L2+Λs+αθ2L2)=I1+I2, (3.31)

    where we have used the following facts:

    Rn(uΛsu)Λsudx=0,Rn(uΛsθ)Λsθdx=0,ΛsθenΛsudxΛsunΛsθdx=0, (3.32)

    and

    I1=Rn[Λs(uu)uΛsu]Λsudx, (3.33)
    I2=Rn[Λs(uθ)uΛsθ]Λsθdx. (3.34)

    Using H¨older's inequality, Lemmas 2.4 and 2.5, we obtain

    I1Λs(uu)uΛsuL2ΛsuL2ΛsuL2nn2αuLnαΛsuL2+Λs1uL2nn2αuLnαΛsuL2Λs+αu2L2ΛsuL2Λαu2HsuHs (3.35)

    with s=1+n22α.

    Similarly,

    I2(Λαu2Hs+Λαθ2Hs)θHs. (3.36)

    Inserting (3.35) and (3.36) in (3.32) leads to

    12ddt(Λsu2L2+Λsθ2L2)+(Λs+αu2L2+Λs+αθ2L2)(uHs+θHs)(Λαu2Hs+Λαθ2Hs). (3.37)

    Integrating (3.37) over [0,t] and combining it with (3.30), we have

    u(t)2Hs+θ(t)2Hs+2t0(Λαu2Hs+Λαθ2Hs)dτ(u02Hs+θ02Hs)+2t0(uHs+θHs)(Λαu2Hs+Λαθ2Hs)dτ. (3.38)

    We set

    E(t)=sup0τt(u(t)2Hs+θ(t)2Hs+t0(Λαu2Hs+Λαθ2Hs)dτ). (3.39)

    Consequently,

    E(t)C0E(0)+C1E32(t). (3.40)

    By using the bootstrapping argument ([30]), if

    E(0)=u02Hs+θ02Hs<ε2 (3.41)

    for sufficiently small ε>0, we have

    u(t)2Hs(Rn)+θ(t)2Hs(Rn)+t0(Λαu2Hs(Rn)+Λαθ2Hs(Rn))dτCε2. (3.42)

    Step 2. We demonstrate (1.10) in this step. Comparable to the derivation of (3.37), for 0<ms, we obtain

    12ddt(Λmu2L2+Λmθ2L2)+(Λm+αu2L2+Λm+αθ2L2)(uHs+θHs)(Λm+αu2L2+Λm+αθ2L2). (3.43)

    Then from (3.43)

    ddt(Λmu2L2+Λmθ2L2)+(Λm+αu2L2+Λm+αθ2L2)0. (3.44)

    By Lemma 2.6, we have

    ΛmuL2Cuαm+αL2Λm+αumm+αL2. (3.45)

    Similarly,

    ΛmθL2Cθαm+αL2Λm+αθmm+αL2. (3.46)

    Substituting these two bounds into (3.47), we derive

    ddt(Λmu2L2+Λmθ2L2)+C(Λmu2(m+α)mL2u2αmL2+Λmθ2(m+α)mL2θ2αmL2)0, (3.47)

    which implies

    ddt(Λmu2L2+Λmθ2L2)+Cmax{u2αmL2,θ2αmL2}(Λmu2(m+α)mL2+Λmθ2(m+α)mL2)0. (3.48)

    Using the Cauchy inequality

    (f+g)k2k1(fk+gk),k1,f,g0, (3.49)

    therefore,

    ddt(Λmu2L2+Λmθ2L2)+Cmax{u2αmL2,θ2αmL2}(Λmu2L2+Λmθ2L2)m+αm0. (3.50)

    Integrating (3.50) over [0,t], we obtain

    Λmu2L2+Λmθ2L2C(1+t01max{u(τ)2αmL2,θ(τ)2αmL2}dτ)mα. (3.51)

    Multiplying (3.51) by (1+t)mα, we obtain

    (1+t)mαΛmu2L2+Λmθ2L2C(1+t1+t01max{u(τ)2αmL2,θ(τ)2αmL2}dτ)mα. (3.52)

    We found that using the L'Hopital rule, yields

    limt(1+t1+t01max{u(τ)2αmL2,θτ2αmL2}dτ)=limtmax{u(t)2αmL2,θ(t)2αmL2}. (3.53)

    Finally, we have

    limt(1+t)m2α(ΛmuL2+ΛmθL2)=0. (3.54)

    Step 3. We demonstrate (1.11) in this step.Similar to (3.13) and (3.14), we obtain

    Λm+αu2L2=Rn|ξ|2α+2m|ˆu(ξ)|2dξ|ξ|r|ξ|2α+2m|ˆu(ξ)|2dξ=r2α(Λmu2L2|ξ|r|ξ|2m|ˆu(ξ)|2dξ) (3.55)

    and

    Λm+αθ2L2r2α(Λmθ2L2|ξ|r|ξ|2m|ˆθ(ξ)|2dξ). (3.56)

    Inserting these two bounds into (3.44), it yields

    ddt(Λmu2L2+Λmθ2L2)+r2α(Λmu2L2+Λmθ2L2)r2α(|ξ|r|ξ|2m|ˆu(ξ)|2dξ+|ξ|r|ξ|2m|ˆθ(ξ)|2dξ). (3.57)

    Utilizing u=0, due to the property of the Fourier transform, which constitutes a bounded linear operator from L1 into L, one can derive the following result:

    |^Λm(u)u(ξ,t)|=|^Λm(uu)(ξ,t)||ξ|Λm(uu)L1|ξ|uL2ΛmuL2. (3.58)

    Similarly,

    |^Λm(u)θ(ξ,t)||ξ|uL2ΛmθL2+|ξ|θL2ΛmuL2, (3.59)
    |^ΛmP(ξ,t)||ξ|uL2ΛmuL2+|^Λmθ(ξ,t)|. (3.60)

    Hence,

    |^Λmu(ξ,t)||^Λmu0(ξ)|eν|ξ|2αt+t0eν|ξ|2α(tτ)||ξ|mL1(ξ,τ)|dτ|^Λmu0(ξ)|+t0eν|ξ|2α(tτ)C|ξ|uL2ΛmuL2dτ+t0eν|ξ|2α(tτ)|^Λmθ(ξ,t)|dτ|^Λmu0(ξ)|+C|ξ|t0(u(τ)L2+θ(τ)L2)(Λmu(τ)L2+Λmθ(τ)L2)dτ+t0|^Λmθ(ξ,t)|dτ|^Λmu0(ξ)|+C|ξ|t0(u0L2+θ0L2)(Λmu(τ)L2+Λmθ(τ)L2)dτ+t0|^Λmθ(ξ,t)|dτ|^Λmu0(ξ)|+C|ξ|t0(Λmu(τ)L2+Λmθ(τ)L2)dτ+Ct0|^Λmθ(ξ,t)|dτ (3.61)

    and

    |^Λmθ(ξ,t)||^Λmθ0(ξ)|+C|ξ|t0(Λmu(τ)L2+Λmθ(τ)L2)dτ+Ct0|^Λmu(ξ,t)|dτ. (3.62)

    The right-hand side of (3.57) can be bounded in the following manner:

    |ξ|r(|ξ|2m|ˆu(ξ)|2+|ξ|2m|ˆθ(ξ)|2)dξ|ξ|r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ+C|ξ|r|ξ|2(t0(ΛmuL2+ΛmθL2)dτ)2dξ+Ct|ξ|rt0(|ξ|2m|ˆu(ξ,τ)|2+|ξ|2m|ˆθ(ξ,τ)|2)dτdξ|ξ|r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ+C|ξ|r|ξ|2[(t0dτ)12(t0(ΛmuL2+ΛmθL2)2dτ)12]2dξ+Ctt0|ξ|r(|ξ|2m|ˆu(ξ,τ)|2+|ξ|2m|ˆθ(ξ,τ)|2)dξdτ|ξ|r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ+Crn+2tt0(ΛmuL2+ΛmθL2)2dτ+Ctt0(Λmu2L2+Λmθ2L2)dτ. (3.63)

    By the following inequality

    |ξ|r|^u0(ξ)|2dξ(|ξ|r|^u0(ξ)|pp1dξ)2p2p(|ξ|rdξ)2p2Crn(2p1)u02Lp, (3.64)

    and

    |ξ|r|^θ0(ξ)|2dξ(|ξ|r|^θ0(ξ)|pp1dξ)2p2p(|ξ|rdξ)2p2Crn(2p1)θ02Lp. (3.65)

    Thus,

    ddt(Λmu2L2+Λmθ2L2)+r2α(Λmu2L2+Λmθ2L2)r2α(|ξ|r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ)+Cr2α+2+ntt0(ΛmuL2+ΛmθL2)2dτ+Cr2αtt0(Λmu2L2+Λmθ2L2)dτr2m+2α+n(2p1)(u02Lp+θ02Lp)+Cr2α+2+n(1+t)t0(Λmu2L2+Λmθ2L2)dτ. (3.66)

    Now we set r=(k1+t)12α with k>nα(1p12)+mα, and insert it into (3.63), we obtain

    ddt(1+t)k(Λmu2L2+Λmθ2L2)C(1+t)kmα1nα(1p12)+C(1+t)k2+n2αt0(Λmu2L2+Λmθ2L2)dτ. (3.67)

    Then integrating (3.67) from 0 to t, one has

    (Λmu(t)2L2+Λmθ(t)2L2)C(1+t)k(Λmu02L2+Λmθ02L2)+C(1+t)kt0(1+τ)kmα1nα(1p12)dτ+C(1+t)kt0(1+τ)k2+n2ατ0(Λmu2L2+Λmθ2L2)dzdτC(1+t)k(Λmu02L2+Λmθ02L2)+C(1+t)mαnα(1p12)+C(1+t)12+n2αt0(Λmu2L2+Λmθ2L2)dτ. (3.68)

    By the following basic inequality

    (1+t)mα+nα(1p12)t0(1+τ)mαnα(1p12)dτ{(1+t)ln(1+t),nα(1p12)+mα=1,C(1+t),nα(1p12)+mα1,

    we have

    (1+t)mα+nα(1p12)(Λmu(t)2L2+Λmθ(t)2L2)C(1+t)mα+nα(1p12)k(Λmu02L2+Λmθ02L2)+C+C(1+t)mα+nα(1p12)+12+n2α×t0C(1+t)mαnα(1p12)dτsup0τt(1+τ)mα+nα(1p12)(Λmu(τ)2L2+Λmθ(τ)2L2)C+C(1+t)22+n2αmax{1,ln(1+t)}×sup0τt((1+τ)mα+nα(1p12)(Λmu(τ)2L2+Λmθ(τ)2L2)). (3.69)

    Since 0<α<1<2+n4, we obtain,

    (Λmu(t)2L2+Λmθ(t)2L2)C(1+t)mαnα(1p12). (3.70)

    Step 4. Uniqueness. Assume that

    (U,Θ),(u,θ)L([0,T];Hs)L2([0,T];Hs+α). (3.71)

    are two solutions of (1.6). Consider the difference (˜u,˜θ) with

    ˜u=uU,˜θ=θΘ. (3.72)

    Then it follows that

    {t˜u+u˜u+˜uU+(Δ)α˜u+˜p=˜θen,t˜θ+u˜θ+˜uΘ+(Δ)α˜θ=~un,˜u=0,˜u(x,0)=0,˜θ(x,0)=0, (3.73)

    where ˜p=pP. Dotting (3.73) with (˜u,˜θ), we obtain

    12ddt(˜u2L2+˜θ2L2)+(Λα˜u2L2+Λα˜θ2L2)=Rn˜uU˜udxRn˜uΘ˜θdx=K1+K2. (3.74)

    By Young's inequality and Lemma 2.4,

    |K1|=|Rn˜uU˜udx|˜uL2˜uL2nn2αULnα˜uL2Λα˜uL2nn2αΛn2+1αUL214Λα˜u2L2+C˜u2L2ΛsU2L2. (3.75)

    Similarly,

    |K2|=|Rn˜uΘ˜θdx|14Λα˜θ2L2+C˜u2L2ΛsΘ2L2. (3.76)

    Invoking the estimates in (3.75) and (3.76), it infers that

    ddt(˜u2L2+˜θ2L2)+(Λα˜u2L2+Λα˜θ2L2)C˜u2L2(ΛsU2L2+ΛsΘ2L2)(˜u2L2+˜θ2L2)(ΛsU2L2+ΛsΘ2L2). (3.77)

    The application of Gronwall's inequality consequently establishes the required uniqueness. Then this concludes the proof of Theorem 1.2.

    Applying the inhomogeneous blocks Δj operator to Eq (1.7) yields

    {tΔju+Δj(uu)+Δju+Δjp=Δjθen,tΔjθ+Δj(uθ)+Δjθ=Δjun. (3.78)

    Taking the inner product of (3.78) with Δju and Δjθ respectively, we have

    12ddt(Δju2L2+Δjθ2L2)+(Δju2L2+Δjθ2L2)=Δj(uu),ΔjuΔj(uθ),Δjθ=Z1+Z2, (3.79)

    where

    Z1=Rn[Δj(uu)uΔju]Δjudx, (3.80)
    Z2=Rn[Δj(uθ)uΔjθ]Δjθdx. (3.81)

    By Lemma 2.2, it yields

    fLfBn22,1fBs12,1fBs2,1, (3.82)

    where sn2+1. By Lemma 2.3 and (3.82),

    |Z1||Rn[Δj(uu)uΔju]Δjudx|ΔjuL2[u,Δj]uL2Ccj,12jsu2Bs2,1ΔiuL2. (3.83)
    |Z2||Rn[Δj(uθ)uΔjθ]Δjθdx|ΔjθL2[u,Δj]θL2Ccj,12jsuBs2,1θBs2,1ΔiθL2. (3.84)

    Here cj,1l1=1, where l1 stands for

    cj,1l1=jZ|cj,1|. (3.85)

    Using these estimates, it leads to

    12ddt(Δju2L2+Δjθ2L2)+(Δju2L2+Δjθ2L2)cj,12js(ΔjuL2+ΔjθL2)(u2Bs2,1+θ2Bs2,1)cj,12js(Δju2L2+Δjθ2L2)12(u2Bs2,1+θ2Bs2,1)12(Δju2L2+Δjθ2L2)+Ccj,12js(u2Bs2,1+θ2Bs2,1)2. (3.86)

    According to the Cauchy inequality

    2(f+g)2(f2+g2)12f+g(f,g0) (3.87)

    and Lemma 2.1, we obtain

    ddt(Δju2L2+Δjθ2L2)12+12(ΔjuL2+ΔjθL2)ddt(Δju2L2+Δjθ2L2)12+(Δju2L2+Δjθ2L2)12cj,12js(u2Bs2,1+θ2Bs2,1). (3.88)

    Integrating (3.88) in time from 0 to t yields

    2(Δju(t)L2+Δjθ(t)L2)+2t0(ΔjuL2+ΔjθL2)dτ2(Δju0L2+Δjθ0L2)+Ct0cj,12js(u2Bs2,1+θ2Bs2,1)dτ. (3.89)

    Multiplying it by 22js and subsequently performing the summation with respect to j results in the following transformation

    2(uBs2,1+θBs2,1)+2t0(ΔjuBs2,1+ΔjθBs2,1)dτ2(u0Bs2,1+θ0Bs2,1)+Ct0cj,12js(u2Bs2,1+θ2Bs2,1)dτ. (3.90)

    Set

    E(t)=sup0τt(uBs2,1+θBs2,1)+t0(uBs2,1+θBs2,1)dτ. (3.91)

    Consequently, (3.90) implies that

    E(t)C0E(0)+C1E2(t). (3.92)

    According to the bootstrapping argument, if

    E(0)=u2Bs2,1+θ2Bs2,1<ε2 (3.93)

    for sufficiently small ε>0, then

    u2Bs2,1+θ2Bs2,1+t0(u(τ)2Bs2,1+θ(τ)2Bs2,1)dτCε2 (3.94)

    for any t>0.

    Multiplying (1.7) with et, it yields that

    {ettu+u(etu)+etu+etp=etθen,ettθ+u(etθ)+etθ=etun,u=0,u(x,0)=u0(x),θ(x,0)=θ0(x). (3.95)

    Dotting (3.95) with (etu,etθ), it yields that

    12ddt(etu2L2+etθ2L2)=0. (3.96)

    Integrating it in [0,t], we have

    e2t(u2L2+θ2L2)=etu2L2+etθ2L2=u02L2+θ02L2. (3.97)

    Therefore,

    u(t)2L2+θ(t)2L2Cet. (3.98)

    By utilizing the interpolation inequality, we have

    Λmu(t)L2+Λmθ(t)L2CΛs(u,θ)msL2(u,θ)smsL2C(u,θ)msB2,1(u,θ)smsL2Cεmses+mst. (3.99)

    Hence, Theorem 1.3 has been proved.

    In summary, our study provides a comprehensive analysis of the n-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion, including conditions for global existence, convergence, decay, uniqueness, and regularity of solutions, depending on the size and nature of the initial data and the presence of fractional dissipation.

    Xinli Wang developed the concept with her supervisor, designed the manuscript and provided key information; Haiyang Yu and Tianfeng Wu helped revise the manuscript and provided the intellectual support. All of authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are grateful to the anonymous referees for their carefully reading of the manuscript and the numerous very helpful suggestions which have helped to improve the exposition of this paper greatly.

    The research of the authors was partially supported by the Natural Science Foundation of Sichuan Province of China (No. 2022NSFSC1799) and Opening Fund of Geomathematics Key Laboratory of Sichuan Province (No. scsxdz2023-2).

    The authors declare no conflict of interest.



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