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+n2−2α(0<α<1), the global solutions are derived. Furthermore, under the assumption that the initial data (u0, θ0) belongs to Lp(where 1≤p<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+n2−2α(0<α<1), the global solutions are derived. Furthermore, under the assumption that the initial data (u0, θ0) belongs to Lp(where 1≤p<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+U⋅∇U+ν(−Δ)αU+∇P=Θen,∂tΘ+U⋅∇Θ+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, ν≥0 is the fluid kinematic viscosity coefficient, and κ≥0 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)=∫Rne−ix⋅ξf(x)dx. |
For the higher dimensional case n≥3, Ye [42] showed that Eq (1.1) admits a unique global classical solution as long as α≥12+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 α≥1, 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 6−√64<α<1, 1−α<β<min{7+2√6α−25,α(1−α)√6−2α,2−2α}. 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>1−α2, β≥2+α3, β>10−5α10−4α. 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 0.9132≈α0<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: √1777−2324=0.798103.<α<1, β>0 and α+β=1. The range of α to 0.7692≈1013<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 0.9132≈23−√14512<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 0.7692≈1013<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=U−u(0), θ=Θ−θ(0), p=P−p(0). | (1.3) |
Then, (u,θ,p) satisfies
{∂tu+u⋅∇u+ν(−Δ)αu+∇p=θen,∂tθ+u⋅∇θ+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 n≥2. 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 0≤α=β<1. In particular, we investigate the following Cauchy problem:
{∂tu+u⋅∇u+(−Δ)αu+∇p=θen,∂tθ+u⋅∇θ+(−Δ)αθ=−un,∇⋅u=0,u(x,0)=u0(x),θ(x,0)=θ0(x), | (1.6) |
where 0≤α<1. When α=0, (1.6) is reduced into the n-dimensional incompressible Boussinesq system with damping
{∂tu+u⋅∇u+u+∇p=θen,∂tθ+u⋅∇θ+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 s≥1+n2−2α. Assume that the initial data (u0,θ0)∈Hs(Rn), ∇⋅u0=0 and n≥2. Then there exists a constant ε>0 such that, if
‖u0‖Hs(Rn)+‖θ0‖Hs(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)+∫t0‖Λαu(τ)‖2Hs(Rn)+‖Λαθ(τ)‖2Hs(Rn)dτ≤Cε2. | (1.10) |
Moreover, for any 0≤m<s,
limt→∞(1+t)m2α(‖Λmu(t)‖L2(Rn)+‖Λmθ(t)‖L2(Rn))=0. | (1.11) |
Furthermore, suppose that (u0,θ0)∈Lp(Rn) with 1≤p<2. Then
‖Λmu(t)‖2L2(Rn)+‖Λmθ(t)‖2L2(Rn)≤C(1+t)−mα−nα(1p−12). | (1.12) |
When s≥1+n2−α, 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), s≥n2+1 with n≥2 and ∇⋅u0=0. Then there exists a constant ϵ>0 such that, if
‖u0‖Bs2,1+‖θ0‖Bs2,1≤ϵ, | (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+∫t0‖u(τ)‖2Bs2,1+‖θ(τ)‖2Bs2,1dτ≤Cϵ2. | (1.15) |
Furthermore, for any 0≤m<s,
‖Λmu(t)‖L2(Rn)+‖Λmθ(t)‖L2(Rn)≤Cϵmse−s+m−st. | (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 {ξ∈Rn:|ξ|≤43} and the annulus {ξ∈Rn: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 C∞0(B) and C∞0(C), respectively, such that
∀ξ∈Rn∖{0},∑j∈Zφ(2−jξ)=1, |
∀ξ∈Rn,χ(ξ)+∑j≥0φ(2−jξ)=1. |
Let u∈S′(R) with S′ being the set of tempered distributions. The nonhomogeneous dyadic blocks Δj are characterized by the following definition:
Δju=0,if j≤−2,andΔ−1u=χ(D)u=∫Rn˜h(y)u(x−y)dy, |
Δju=φ(2−jD)u:=2jn∫Rnh(2jy)u(x−y)dy,ifj>0 |
where h=F−1φ and ˜h=F−1χ. Now we will define the low-frequency cut-off operator Sj
Sju:=∑j′≤j−1Δj′u. |
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 p≤q, and any function f∈Lp(Rn), we have
Suppˆf⊂λB⇒‖Dkf‖Lqdef=sup|α|=k‖∂αf‖Lq≤Ck+1λk+n(1p−1q)‖f‖Lp, |
Suppˆf⊂λC⇒C−k−1λk‖f‖Lp≤sup|α|=k‖∂αf‖Lp≤Ck+1λk‖f‖Lp. |
Next we recall the definitions of the nonhomogeneous Besov spaces.
Definition 1. Assume s∈R and 1≤p,r≤∞, the nonhomogeneous Besov space Bsp,r consists of all tempered distributions f such that
‖f‖Bsp,r:=‖(2js‖Δjf‖Lp)j∈Z‖ℓr(Z)<∞. |
When p=r=2, we have Bs2,2(Rn)=Hs(Rn), where
‖f‖Hs(Rn)≜(∫Rn(1+|ξ|2)s|ˆf(ξ)|2dξ)12. |
Lemma 2.2. Assume s∈R and 1≤p,r≤∞, f∈S′. Consequently, the following properties are valid:
(1) Embedding: For 1≤p,ˉp≤∞ and 1≤r,ˉr≤∞, there holds
Bsp,r(Rn)↪Bs−n(1p−1ˉp)ˉp,ˉr(Rn), |
Bsp,1(Rn)↪L∞(Rn), |
Lp(Rn)↪B0p,∞(Rn). |
(2) Interpolation: For any s1<s2 and 0<θ<1, one has
‖f‖Bθs1+(1−θ)s2p,r≤‖f‖θBs1p,r‖f‖1−θBs2p,r, |
‖f‖Bθs1+(1−θ)s2p,1≤Cs2−s1(1θ+11−θ)‖f‖θBs1p,∞‖f‖1−θBs2p,∞. |
Lemma 2.3. Let σ∈R, 1≤r≤∞, and 1≤p≤p1≤∞. Let v be a vector field over R. Assume that
σ>−nmin{1p1,1p′}orσ>−1−nmin{1p1,1p′}if∇⋅v=0. |
Define Rj≜[v⋅∇,Δj]f. There exists a constant C, depending continuously on p, p1, σ and d, such that
‖(2js‖Rj‖Lp)j‖ℓr≤C‖∇v‖Bnpp1,∞∩L∞‖f‖Bσp,r,ifσ<1+np1. |
Further, if σ>0 (or σ>−1, if ∇⋅v=0) and 1p2=1p−1p1, then
‖(2js‖Rj‖Lp)j‖ℓr≤C(‖∇v‖L∞‖f‖Bσp,r+‖∇f‖Lp2‖∇v‖Bσ−1p1,r), |
where
[v⋅∇,Δj]f=v⋅∇(Δjf)−Δj(v⋅∇f). |
Lemma 2.4. ([1]) Let s∈R and ˙Hs(Rn) be the homogeneous Sobolev space with the definition
‖u‖2˙Hs(Rn):=∫Rn|ξ|2s|ˆu(ξ)|2dξ<∞. |
If k∈[0,n2), then the space ˙Hk(Rn) is continuously embedded in L2nn−2k(Rn), namely
‖u‖L2nn−2k≤C‖Λku‖L2. |
Lemma 2.5. ([16,17]). Let s>0, 1<p<∞, f∈W1,p1∩Ws,p2, g∈Lp2∩Ws,p1, then
‖Λs(fg)−fΛsg‖Lp≤C(‖∇f‖Lp1‖Λs−1g‖Lq1+‖g‖Lp2‖Λsg‖Lq2), |
‖Λs(fg)‖Lp≤C(‖f‖Lp1‖Λsg‖Lq1+‖g‖Lp2‖Λsg‖Lq2) |
for 1p=1p1+1q1=1p2+1q2.
Lemma 2.6. Let s0<s<s1. Then ˙Hs0∩˙Hs1 is included in ˙Hs, and
‖Λsu‖L2≤‖Λs0u‖θL2‖Λs1u‖1−θL2 |
where θ∈[0,1] and s, s0, s1 satisfy
s=θs0+(1−θ)s1. |
Lemma 2.7. If f∈Lp(Rn), 1≤p≤2, and 1p+1ˉp=1, then ˆf∈Lˉp(Rn) and it satisfies
‖ˆf‖Lˉp≤C‖f‖Lp. |
Firstly, dotting (1.4) with (u,θ), we have
ddt(‖u(t)‖2L2+‖θ(t)‖2L2)+2ν‖Λαu‖2L2+2κ‖Λβθ‖2L2=0, | (3.1) |
integrating it in time yields
‖u(t)‖2L2+‖θ(t)‖2L2≤‖u0‖2L2+‖θ0‖2L2. | (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)e−ν|ξ|2αt+∫t0e−ν|ξ|2α(t−τ)L1(ξ,t)dτ, | (3.5) |
ˆθ(ξ,t)=^θ0(ξ)e−κ|ξ|2βt+∫t0e−κ|ξ|2β(t−τ)L2(ξ,t)dτ. | (3.6) |
Due to the simple fact that ∇⋅u=0, we have
|^(u⋅∇)u(ξ,t)|=|^∇⋅(u⊗u)(ξ,t)|≤|ξ|‖u⊗u‖L1≤|ξ|‖u‖2L2. | (3.7) |
Similarly,
|^(u⋅∇)θ(ξ,t)|≤|ξ|‖u‖L2‖θ‖L2. | (3.8) |
Applying the divergence operator to the velocity equation in (1.4), we obtained
∇p=(−Δ)−1∇∇⋅(θen−u⋅∇u)=(−Δ)−1∇∇⋅(θen)−(−Δ)−1∇∇⋅∇⋅(u⊗u). | (3.9) |
By (3.7), we obtain
|^∇p(ξ,t)|≤|ˆθ|+|ξ|‖u⊗u‖L1≤|ˆθ|+|ξ|‖u‖2L2. | (3.10) |
Invoking these estimates, it leads to
|ˆu(ξ,t)|≤|^u0(ξ)|e−ν|ξ|2αt+∫t0e−ν|ξ|2α(t−τ)|L1(ξ,t)|dτ,≤|^u0(ξ)|+∫t0e−ν|ξ|2α(t−τ)C(|ξ|‖u(τ)‖2L2+|ˆθ(ξ,τ)|)dτ≤|^u0(ξ)|+C|ξ|∫t0e−ν|ξ|2α(t−τ)(‖u(τ)‖2L2+‖θ(τ)‖2L2)dτ+∫t0e−ν|ξ|2α(t−τ)|ˆθ|dτ≤|^u0(ξ)|+C|ξ|(‖u0‖2L2+‖θ0‖2L2)∫t0e−ν|ξ|2α(t−τ)dτ+C∫t0|ˆθ|dτ≤|^u0(ξ)|+Ct|ξ|+C∫t0|ˆθ|dτ. | (3.11) |
The other term can be similarly bounded,
|ˆθ(ξ,t)|≤|^θ0(ξ)|+Ct|ξ|+C∫t0|ˆu|dτ. | (3.12) |
By employing the Plancherel theorem in conjunction with the Fourier splitting technique, we obtain
‖Λαu‖2L2=∫Rn|ξ|2α|ˆu(ξ,t)|2dξ≥∫|ξ|≥r1|ξ|2α|ˆu(ξ,t)|2dξ≥r2α1∫|ξ|≥r1|ˆu(ξ,t)|2dξ=r2α1(‖u‖2L2−∫|ξ|≤r1|ˆu|2dξ) | (3.13) |
and
‖Λαθ‖2L2≥r2β2(‖θ‖2L2−∫|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|+C∫t0|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|≤r1∫t0|ˆθ|2dτdξ≤∫|ξ|≤r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ct∫t0∫|ξ|≤r1|ˆθ|2dξdτ≤∫|ξ|≤r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ct∫t0∫Rn|ˆθ|2dξdτ≤∫|ξ|≤r1|^u0(ξ)|2dξ+Crn+21(1+t)2+Ct∫t0‖θ‖2L2dτ. | (3.15) |
Similarly,
∫|ξ|≤r2|ˆθ|2dξ≤∫|ξ|≤r2|^θ0(ξ)|2dξ+Crn+22(1+t)2+Ct∫t0‖u‖2L2dτ. | (3.16) |
Substituting (3.13)–(3.16) into (3.9), it yields
ddt(‖u(t)‖2L2+‖θ(t)‖2L2)+2νr2α1‖u‖2L2+2κr2β2‖θ‖2L2≤2νr2α1(∫|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+Ctr2α1∫t0‖θ‖2L2dτ+Ctr2β2∫t0‖u‖2L2dτ≤2νr2α1∫|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)2≤2νr2α1∫|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)(‖u‖2L2+‖θ‖2L2)≤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))≤klnk−1(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)2≜J1(t)+J2(t)+J3(t). | (3.20) |
Integrating (3.20) in [0,t], we obtain
‖u(t)‖2L2+‖θ(t)‖2L2≤ln−k(1+t)(‖u0‖2L2+‖θ0‖2L2)+ln−k(1+t)∫t0J1(τ)dτ+ln−k(1+t)∫t0J2(τ)dτ+ln−k(1+t)∫t0J3(τ)dτ. | (3.21) |
Naturally,
limt→∞ln−k(1+t)(‖u0‖2L2+‖θ0‖2L2)=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
limt→∞ln−k(1+t)∫t0J1(τ)dτ=0. | (3.25) |
For the third term on the right-hand side of (3.21),
∫t0J2(τ)dτ=C∫t0lnk(1+τ)(1+τ)2(k2ν(1+τ)ln(1+τ))1+n+22αdτ≤C∫t0ln(1+τ)k−n+22α−1(1+τ)n+22α−1dτ≤C∫t0ln(1+τ)k−n+22α−11+τdτ=C(ln(1+τ)k−n+22α−1), | (3.26) |
then
limt→∞ln−k(1+t)∫t0J2(τ)dτ=0. | (3.27) |
Similar to J2, one gets
limt→∞ln−k(1+t)∫t0J3(τ)dτ=0. | (3.28) |
Inserting (3.22), (3.25), (3.27), and (3.28) into (3.21), we finally obtain
limt→∞‖u(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+2∫t0‖Λαu(τ)‖2L2dτ+2∫t0‖Λαθ(τ)‖2L2dτ=‖u0‖2L2+‖θ0‖2L2. | (3.30) |
Applying Λs to both sides of (1.6) and then dotting the results with (Λsu,Λsθ), respectively, we obtain
12ddt(‖Λsu‖2L2+‖Λsθ‖2L2)+(‖Λs+αu‖2L2+‖Λs+αθ‖2L2)=I1+I2, | (3.31) |
where we have used the following facts:
∫Rn(u⋅∇Λsu)⋅Λsudx=0,∫Rn(u⋅∇Λsθ)⋅Λsθdx=0,∫Λsθen⋅Λsudx−∫Λsun⋅Λsθdx=0, | (3.32) |
and
I1=−∫Rn[Λs(u⋅∇u)−u⋅∇Λsu]⋅Λsudx, | (3.33) |
I2=−∫Rn[Λs(u⋅∇θ)−u⋅∇Λsθ]⋅Λsθdx. | (3.34) |
Using H¨older's inequality, Lemmas 2.4 and 2.5, we obtain
I1≤‖Λs(u⋅∇u)−u⋅∇Λsu‖L2‖Λsu‖L2≲‖Λsu‖L2nn−2α‖∇u‖Lnα‖Λsu‖L2+‖∇Λs−1u‖L2nn−2α‖∇u‖Lnα‖Λsu‖L2≲‖Λs+αu‖2L2‖Λsu‖L2≲‖Λαu‖2Hs‖u‖Hs | (3.35) |
with s=1+n2−2α.
Similarly,
I2≲(‖Λαu‖2Hs+‖Λαθ‖2Hs)‖θ‖Hs. | (3.36) |
Inserting (3.35) and (3.36) in (3.32) leads to
12ddt(‖Λsu‖2L2+‖Λsθ‖2L2)+(‖Λs+αu‖2L2+‖Λs+αθ‖2L2)≲(‖u‖Hs+‖θ‖Hs)(‖Λαu‖2Hs+‖Λαθ‖2Hs). | (3.37) |
Integrating (3.37) over [0,t] and combining it with (3.30), we have
‖u(t)‖2Hs+‖θ(t)‖2Hs+2∫t0(‖Λαu‖2Hs+‖Λαθ‖2Hs)dτ≲(‖u0‖2Hs+‖θ0‖2Hs)+2∫t0(‖u‖Hs+‖θ‖Hs)(‖Λαu‖2Hs+‖Λαθ‖2Hs)dτ. | (3.38) |
We set
E(t)=sup0≤τ≤t(‖u(t)‖2Hs+‖θ(t)‖2Hs+∫t0(‖Λαu‖2Hs+‖Λαθ‖2Hs)dτ). | (3.39) |
Consequently,
E(t)≤C0E(0)+C1E32(t). | (3.40) |
By using the bootstrapping argument ([30]), if
E(0)=‖u0‖2Hs+‖θ0‖2Hs<ε2 | (3.41) |
for sufficiently small ε>0, we have
‖u(t)‖2Hs(Rn)+‖θ(t)‖2Hs(Rn)+∫t0(‖Λαu‖2Hs(Rn)+‖Λαθ‖2Hs(Rn))dτ≤Cε2. | (3.42) |
Step 2. We demonstrate (1.10) in this step. Comparable to the derivation of (3.37), for 0<m≤s, we obtain
12ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+(‖Λm+αu‖2L2+‖Λm+αθ‖2L2)≲(‖u‖Hs+‖θ‖Hs)(‖Λm+αu‖2L2+‖Λm+αθ‖2L2). | (3.43) |
Then from (3.43)
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+(‖Λm+αu‖2L2+‖Λm+αθ‖2L2)≤0. | (3.44) |
By Lemma 2.6, we have
‖Λmu‖L2≤C‖u‖αm+αL2‖Λm+αu‖mm+αL2. | (3.45) |
Similarly,
‖Λmθ‖L2≤C‖θ‖αm+αL2‖Λm+αθ‖mm+αL2. | (3.46) |
Substituting these two bounds into (3.47), we derive
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+C(‖Λmu‖2(m+α)mL2‖u‖2αmL2+‖Λmθ‖2(m+α)mL2‖θ‖2αmL2)≤0, | (3.47) |
which implies
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+Cmax{‖u‖2αmL2,‖θ‖2αmL2}(‖Λmu‖2(m+α)mL2+‖Λmθ‖2(m+α)mL2)≤0. | (3.48) |
Using the Cauchy inequality
(f+g)k≤2k−1(fk+gk),k≥1,f,g≥0, | (3.49) |
therefore,
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+Cmax{‖u‖2αmL2,‖θ‖2αmL2}(‖Λmu‖2L2+‖Λmθ‖2L2)m+αm≤0. | (3.50) |
Integrating (3.50) over [0,t], we obtain
‖Λmu‖2L2+‖Λmθ‖2L2≤C(1+∫t01max{‖u(τ)‖2αmL2,‖θ(τ)‖2αmL2}dτ)−mα. | (3.51) |
Multiplying (3.51) by (1+t)mα, we obtain
(1+t)mα‖Λmu‖2L2+‖Λmθ‖2L2≤C(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,‖θτ‖2αmL2}dτ)=limt→∞max{‖u(t)‖2αmL2,‖θ(t)‖2αmL2}. | (3.53) |
Finally, we have
limt→∞(1+t)m2α(‖Λmu‖L2+‖Λmθ‖L2)=0. | (3.54) |
Step 3. We demonstrate (1.11) in this step.Similar to (3.13) and (3.14), we obtain
‖Λm+αu‖2L2=∫Rn|ξ|2α+2m|ˆu(ξ)|2dξ≥∫|ξ|≥r|ξ|2α+2m|ˆu(ξ)|2dξ=r2α(‖Λmu‖2L2−∫|ξ|≤r|ξ|2m|ˆu(ξ)|2dξ) | (3.55) |
and
‖Λm+αθ‖2L2≥r2α(‖Λmθ‖2L2−∫|ξ|≤r|ξ|2m|ˆθ(ξ)|2dξ). | (3.56) |
Inserting these two bounds into (3.44), it yields
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+r2α(‖Λmu‖2L2+‖Λmθ‖2L2)≤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(u⊗u)(ξ,t)|≤|ξ|‖Λm(u⊗u)‖L1≤|ξ|‖u‖L2‖Λmu‖L2. | (3.58) |
Similarly,
|^Λm(u⋅∇)θ(ξ,t)|≤|ξ|‖u‖L2‖Λmθ‖L2+|ξ|‖θ‖L2‖Λmu‖L2, | (3.59) |
|^Λm∇P(ξ,t)|≤|ξ|‖u‖L2‖Λmu‖L2+|^Λmθ(ξ,t)|. | (3.60) |
Hence,
|^Λmu(ξ,t)|≤|^Λmu0(ξ)|e−ν|ξ|2αt+∫t0e−ν|ξ|2α(t−τ)||ξ|mL1(ξ,τ)|dτ≤|^Λmu0(ξ)|+∫t0e−ν|ξ|2α(t−τ)C|ξ|‖u‖L2‖Λmu‖L2dτ+∫t0e−ν|ξ|2α(t−τ)|^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(‖u0‖L2+‖θ0‖L2)(‖Λmu(τ)‖L2+‖Λmθ(τ)‖L2)dτ+∫t0|^Λmθ(ξ,t)|dτ≤|^Λmu0(ξ)|+C|ξ|∫t0(‖Λmu(τ)‖L2+‖Λmθ(τ)‖L2)dτ+C∫t0|^Λmθ(ξ,t)|dτ | (3.61) |
and
|^Λmθ(ξ,t)|≤|^Λmθ0(ξ)|+C|ξ|∫t0(‖Λmu(τ)‖L2+‖Λmθ(τ)‖L2)dτ+C∫t0|^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(‖Λmu‖L2+‖Λmθ‖L2)dτ)2dξ+Ct∫|ξ|≤r∫t0(|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(‖Λmu‖L2+‖Λmθ‖L2)2dτ)12]2dξ+Ct∫t0∫|ξ|≤r(|ξ|2m|ˆu(ξ,τ)|2+|ξ|2m|ˆθ(ξ,τ)|2)dξdτ≤∫|ξ|≤r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ+Crn+2t∫t0(‖Λmu‖L2+‖Λmθ‖L2)2dτ+Ct∫t0(‖Λmu‖2L2+‖Λmθ‖2L2)dτ. | (3.63) |
By the following inequality
∫|ξ|≤r|^u0(ξ)|2dξ≤(∫|ξ|≤r|^u0(ξ)|pp−1dξ)2p−2p(∫|ξ|≤rdξ)2−p2≤Crn(2p−1)‖u0‖2Lp, | (3.64) |
and
∫|ξ|≤r|^θ0(ξ)|2dξ≤(∫|ξ|≤r|^θ0(ξ)|pp−1dξ)2p−2p(∫|ξ|≤rdξ)2−p2≤Crn(2p−1)‖θ0‖2Lp. | (3.65) |
Thus,
ddt(‖Λmu‖2L2+‖Λmθ‖2L2)+r2α(‖Λmu‖2L2+‖Λmθ‖2L2)≤r2α(∫|ξ|≤r|^Λmu0(ξ)|2+|^Λmθ0(ξ)|2dξ)+Cr2α+2+nt∫t0(‖Λmu‖L2+‖Λmθ‖L2)2dτ+Cr2αt∫t0(‖Λmu‖2L2+‖Λmθ‖2L2)dτ≤r2m+2α+n(2p−1)(‖u0‖2Lp+‖θ0‖2Lp)+Cr2α+2+n(1+t)∫t0(‖Λmu‖2L2+‖Λmθ‖2L2)dτ. | (3.66) |
Now we set r=(k1+t)12α with k>nα(1p−12)+mα, and insert it into (3.63), we obtain
ddt(1+t)k(‖Λmu‖2L2+‖Λmθ‖2L2)≤C(1+t)k−mα−1−nα(1p−12)+C(1+t)k−2+n2α∫t0(‖Λmu‖2L2+‖Λmθ‖2L2)dτ. | (3.67) |
Then integrating (3.67) from 0 to t, one has
(‖Λmu(t)‖2L2+‖Λmθ(t)‖2L2)≤C(1+t)−k(‖Λmu0‖2L2+‖Λmθ0‖2L2)+C(1+t)−k∫t0(1+τ)k−mα−1−nα(1p−12)dτ+C(1+t)−k∫t0(1+τ)k−2+n2α∫τ0(‖Λmu‖2L2+‖Λmθ‖2L2)dzdτ≤C(1+t)−k(‖Λmu0‖2L2+‖Λmθ0‖2L2)+C(1+t)−mα−nα(1p−12)+C(1+t)1−2+n2α∫t0(‖Λmu‖2L2+‖Λmθ‖2L2)dτ. | (3.68) |
By the following basic inequality
(1+t)mα+nα(1p−12)∫t0(1+τ)−mα−nα(1p−12)dτ≤{(1+t)ln(1+t),nα(1p−12)+mα=1,C(1+t),nα(1p−12)+mα≠1, |
we have
(1+t)mα+nα(1p−12)(‖Λmu(t)‖2L2+‖Λmθ(t)‖2L2)≤C(1+t)mα+nα(1p−12)−k(‖Λmu0‖2L2+‖Λmθ0‖2L2)+C+C(1+t)mα+nα(1p−12)+1−2+n2α×∫t0C(1+t)−mα−nα(1p−12)dτsup0≤τ≤t(1+τ)mα+nα(1p−12)(‖Λmu(τ)‖2L2+‖Λmθ(τ)‖2L2)≤C+C(1+t)2−2+n2αmax{1,ln(1+t)}×sup0≤τ≤t((1+τ)mα+nα(1p−12)(‖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)−mα−nα(1p−12). | (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=u−U,˜θ=θ−Θ. | (3.72) |
Then it follows that
{∂t˜u+u⋅∇˜u+˜u⋅∇U+(−Δ)α˜u+∇˜p=˜θen,∂t˜θ+u⋅∇˜θ+˜u⋅∇Θ+(−Δ)α˜θ=−~un,∇⋅˜u=0,˜u(x,0)=0,˜θ(x,0)=0, | (3.73) |
where ˜p=p−P. Dotting (3.73) with (˜u,˜θ), we obtain
12ddt(‖˜u‖2L2+‖˜θ‖2L2)+(‖Λα˜u‖2L2+‖Λα˜θ‖2L2)=−∫Rn˜u⋅∇U⋅˜udx–∫Rn˜u⋅∇Θ⋅˜θdx=K1+K2. | (3.74) |
By Young's inequality and Lemma 2.4,
|K1|=|∫Rn˜u⋅∇U˜udx|≤‖˜u‖L2‖˜u‖L2nn−2α‖∇U‖Lnα≤‖˜u‖L2‖Λα˜u‖L2nn−2α‖Λn2+1−αU‖L2≤14‖Λα˜u‖2L2+C‖˜u‖2L2‖ΛsU‖2L2. | (3.75) |
Similarly,
|K2|=|∫Rn˜u⋅∇Θ˜θdx|≤14‖Λα˜θ‖2L2+C‖˜u‖2L2‖ΛsΘ‖2L2. | (3.76) |
Invoking the estimates in (3.75) and (3.76), it infers that
ddt(‖˜u‖2L2+‖˜θ‖2L2)+(‖Λα˜u‖2L2+‖Λα˜θ‖2L2)≤C‖˜u‖2L2(‖ΛsU‖2L2+‖ΛsΘ‖2L2)≤(‖˜u‖2L2+‖˜θ‖2L2)(‖ΛsU‖2L2+‖ΛsΘ‖2L2). | (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(u⋅∇u)+Δju+∇Δjp=Δjθen,∂tΔjθ+Δj(u⋅∇θ)+Δjθ=−Δjun. | (3.78) |
Taking the inner product of (3.78) with Δju and Δjθ respectively, we have
12ddt(‖Δju‖2L2+‖Δjθ‖2L2)+(‖Δju‖2L2+‖Δjθ‖2L2)=−⟨Δj(u⋅∇u),Δju⟩−⟨Δj(u⋅∇θ),Δjθ⟩=Z1+Z2, | (3.79) |
where
Z1=−∫Rn[Δj(u⋅∇u)−u⋅Δj∇u]⋅Δjudx, | (3.80) |
Z2=−∫Rn[Δj(u⋅∇θ)−u⋅Δj∇θ]⋅Δjθdx. | (3.81) |
By Lemma 2.2, it yields
‖∇f‖L∞≲‖∇f‖Bn22,1≲‖∇f‖Bs−12,1≤‖∇f‖Bs2,1, | (3.82) |
where s≥n2+1. By Lemma 2.3 and (3.82),
|Z1|≤|−∫Rn[Δj(u⋅∇u)−u⋅Δj∇u]⋅Δjudx|≤‖Δju‖L2‖[u⋅∇,Δj]u‖L2≤Ccj,12−js‖u‖2Bs2,1‖Δiu‖L2. | (3.83) |
|Z2|≤|−∫Rn[Δj(u⋅∇θ)−u⋅Δj∇θ]⋅Δjθdx|≤‖Δjθ‖L2‖[u⋅∇,Δj]θ‖L2≤Ccj,12−js‖u‖Bs2,1‖θ‖Bs2,1‖Δiθ‖L2. | (3.84) |
Here ‖cj,1‖l1=1, where l1 stands for
‖cj,1‖l1=∑j∈Z|cj,1|. | (3.85) |
Using these estimates, it leads to
12ddt(‖Δju‖2L2+‖Δjθ‖2L2)+(‖Δju‖2L2+‖Δjθ‖2L2)≲cj,12−js(‖Δju‖L2+‖Δjθ‖L2)(‖u‖2Bs2,1+‖θ‖2Bs2,1)≲cj,12−js(‖Δju‖2L2+‖Δjθ‖2L2)12(‖u‖2Bs2,1+‖θ‖2Bs2,1)≲12(‖Δju‖2L2+‖Δjθ‖2L2)+Ccj,12−js(‖u‖2Bs2,1+‖θ‖2Bs2,1)2. | (3.86) |
According to the Cauchy inequality
√2(f+g)≥√2(f2+g2)12≥f+g(f,g≥0) | (3.87) |
and Lemma 2.1, we obtain
ddt(‖Δju‖2L2+‖Δjθ‖2L2)12+1√2(‖Δju‖L2+‖Δjθ‖L2)≤ddt(‖Δju‖2L2+‖Δjθ‖2L2)12+(‖Δju‖2L2+‖Δjθ‖2L2)12≲cj,12−js(‖u‖2Bs2,1+‖θ‖2Bs2,1). | (3.88) |
Integrating (3.88) in time from 0 to t yields
√2(‖Δju(t)‖L2+‖Δjθ(t)‖L2)+√2∫t0(‖Δju‖L2+‖Δjθ‖L2)dτ≤2(‖Δju0‖L2+‖Δjθ0‖L2)+C∫t0cj,12−js(‖u‖2Bs2,1+‖θ‖2Bs2,1)dτ. | (3.89) |
Multiplying it by 22js and subsequently performing the summation with respect to j results in the following transformation
√2(‖u‖Bs2,1+‖θ‖Bs2,1)+√2∫t0(‖Δju‖Bs2,1+‖Δjθ‖Bs2,1)dτ≤2(‖u0‖Bs2,1+‖θ0‖Bs2,1)+C∫t0cj,12−js(‖u‖2Bs2,1+‖θ‖2Bs2,1)dτ. | (3.90) |
Set
E(t)=sup0≤τ≤t(‖u‖Bs2,1+‖θ‖Bs2,1)+∫t0(‖u‖Bs2,1+‖θ‖Bs2,1)dτ. | (3.91) |
Consequently, (3.90) implies that
E(t)≤C0E(0)+C1E2(t). | (3.92) |
According to the bootstrapping argument, if
E(0)=‖u‖2Bs2,1+‖θ‖2Bs2,1<ε2 | (3.93) |
for sufficiently small ε>0, then
‖u‖2Bs2,1+‖θ‖2Bs2,1+∫t0(‖u(τ)‖2Bs2,1+‖θ(τ)‖2Bs2,1)dτ≤Cε2 | (3.94) |
for any t>0.
Multiplying (1.7) with et, it yields that
{et∂tu+u⋅∇(etu)+etu+et∇p=etθen,et∂tθ+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(‖etu‖2L2+‖etθ‖2L2)=0. | (3.96) |
Integrating it in [0,t], we have
e2t(‖u‖2L2+‖θ‖2L2)=‖etu‖2L2+‖etθ‖2L2=‖u0‖2L2+‖θ0‖2L2. | (3.97) |
Therefore,
‖u(t)‖2L2+‖θ(t)‖2L2≤Ce−t. | (3.98) |
By utilizing the interpolation inequality, we have
‖Λmu(t)‖L2+‖Λmθ(t)‖L2≤C‖Λs(u,θ)‖msL2‖(u,θ)‖s−msL2≤C‖(u,θ)‖msB2,1‖(u,θ)‖s−msL2≤Cεmse−s+m−st. | (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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