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+u⋅∇u+νΛ2αu+∇p+∇⋅(v⊗v)=0,∂tv+u⋅∇v+μΛ2βv+∇θ+v⋅∇u=0,∂tθ+u⋅∇θ+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 d≥2, 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. v⊗v denotes the tensor product, namely v⊗v=(vivj) with i,j=1,2,⋅⋅⋅,d, the parameters ν≥0, μ≥0, η≥0, 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), ξ∈Rd, 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 d≥3, Ye in [22] proved the global regularity of this model in the case when α≥12+d4, α+β≥1+d2 and β≥0. 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 d≥2. 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 d≥2. Assume that (u0,v0,θ0)∈Hs(Rd) with s>1+d2−2min{α,β,γ} and ∇⋅u0=0. Then there exists a positive constant C0 such that for all 0<ϵ<C0, if
‖u0‖Hs(Rd)+‖v0‖Hs(Rd)+‖θ0‖Hs(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 ˙H−σ(Rd) or negative Besov space ˙B−σ2,∞(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)∈˙H−σ(Rd) with 0≤σ<d2 or (u0,v0,θ0)∈˙B−σ2,∞(Rd) with 0<σ≤d2. Then for s≥1+d2, the global solution (u,v,θ) established in Theorem 1.1 satisfies for all t>0,
‖u(t)‖˙H−σ(Rd)+‖v(t)‖˙H−σ(Rd)+‖θ(t)‖˙H−σ(Rd)≤C, | (1.5) |
or
‖u(t)‖˙B−σ2,∞(Rd)+‖v(t)‖˙B−σ2,∞(Rd)+‖θ(t)‖˙B−σ2,∞(Rd)≤C. | (1.6) |
Moreover, for any real number m with 0≤m≤s,
‖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=dp−d2, Lp(Rd)↪˙H−σ(Rd) when σ∈[0,d2) and p∈(1,2], and Lp(Rd)↪˙B−σ2,∞(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 ˙H−σ(Rd) and ˙B−σ2,∞(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, ‖f‖Lp for ‖f‖Lp(Rd), ‖f‖˙Hs, ‖f‖Hs and ‖f‖˙Bs2,∞ for ‖f‖˙Hs(Rd), ‖f‖Hs(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)‖Lr≤C(‖Λsf‖Lp1‖g‖Lq1+‖f‖Lp2‖Λsg‖Lq2), |
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+d2−2min{α,β,γ} and ∇⋅u0=0. Then any solution (u,v,θ) of the system (1.1) obeys the following differential inequality
ddt(‖u‖2Hs+‖v‖2Hs+‖θ‖2Hs)+‖Λαu‖2Hs+‖Λβv‖2Hs+‖Λγθ‖2Hs≤C(‖u‖2Hs+‖v‖2Hs+‖θ‖2Hs)(‖Λαu‖2Hs+‖Λβv‖2Hs+‖Λγθ‖2Hs). | (2.1) |
Proof. Dotting (1.1)1, (1.1)2 and (1.1)3 by u, v and θ, respectively, we obtain
12ddt(‖u‖2L2+‖v‖2L2+‖θ‖2L2)+‖Λαu‖2L2+‖Λβv‖2L2+‖Λγθ‖2L2=−∫∇⋅(v⊗v)⋅u−∫∇θ⋅v−∫v⋅∇u⋅v−∫∇⋅vθ=0, | (2.2) |
where we have used the facts that
∫∇⋅(v⊗v)⋅u+∫v⋅∇u⋅v=0, |
and
∫∇θ⋅v+∫∇⋅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(‖Λsu‖2L2+‖Λsv‖2L2+‖Λsθ‖2L2)+‖Λs+αu‖2L2+‖Λs+βv‖2L2+‖Λs+γθ‖2L2=−∫Λs(u⋅∇u)⋅Λsu−∫Λs∇⋅(v⊗v)⋅Λsu−∫Λs(u⋅∇v)⋅Λsv −∫Λs∇θ⋅Λsv−∫Λs(v⋅∇u)⋅Λsv−∫Λs(u⋅∇θ)⋅Λsθ −∫Λs(∇⋅v)⋅Λsθ=I1+I2+I3+I4+I5+I6+I7. | (2.3) |
Integration by parts implies
I4+I7=−∫Λs∇θ⋅Λsv−∫Λs(∇⋅v)⋅Λsθ=0. | (2.4) |
Applying Hölder's inequality, Lemma 2.1 and Sobolev embedding inequality, we estimate the term I1 as
I1=−∫Λs−α(u⋅∇u)⋅Λs+αu=−∫Λs−α∇⋅(u⊗u)⋅Λs+αu≤C‖Λs+1−α(u⊗u)‖L2‖Λs+αu‖L2≤C‖u‖Ld2α−1‖Λs+1−αu‖L2dd+2−4α‖Λs+αu‖L2≤C‖Λ1+d2−2αu‖L2‖Λs+αu‖2L2. |
Similarly, we have
I2≤C‖Λs+1−α(v⊗v)‖L2‖Λs+αu‖L2≤C‖v‖Ldα+β−1‖Λs+1−αv‖L2dd+2−2(α+β)‖Λs+αu‖L2≤C‖Λ1+d2−(α+β)v‖L2‖Λs+βv‖L2‖Λs+αu‖L2. |
I3≤‖Λs+1−β(u⊗v)‖L2‖Λs+βv‖L2≤C(‖Λs+1−βu‖L2dd+2−2(α+β)‖v‖Ldα+β−1+‖u‖Ld2β−1‖Λs+1−βv‖L2dd+2−4β)‖Λs+βv‖L2≤C(‖Λ1+d2−(α+β)v‖L2‖Λs+αu‖L2‖Λs+βv‖L2+‖Λ1+d2−2βu‖L2‖Λs+βv‖2L2). |
I6≤‖Λs+1−γ(u⊗θ)‖L2‖Λs+γθ‖L2≤C(‖Λs+1−γu‖L2dd+2−2(α+γ)‖θ‖Ldα+γ−1+‖u‖Ld2γ−1‖Λs+1−γθ‖L2dd+2−4γ)‖Λs+γθ‖L2≤C(‖Λ1+d2−(α+γ)θ‖L2‖Λs+αu‖L2‖Λs+γθ‖L2+‖Λ1+d2−2γu‖L2‖Λs+γθ‖2L2). |
We cannot bound I5 as above, since v is not divergence free. Using Hölder's inequality and Lemma 2.1, we derive that
I5=−∫Λs−β(v⋅∇u)⋅Λs+βv≤‖Λs−β(v⋅∇u)‖L2‖Λs+βv‖L2≤C(‖Λs−βv‖L2dd−2β‖∇u‖Ldβ+‖v‖Ldα+β−1‖Λs−β+1u‖L2dd+2−2(α+β))‖Λs+βv‖L2≤C(‖Λsv‖L2‖Λ1+d2−βu‖L2‖Λs+βv‖L2+‖Λ1+d2−(α+β)v‖L2‖Λs+αu‖L2‖Λs+βv‖L2). |
Combining these bounds and (2.4) with (2.3) together, we get
12ddt(‖Λsu‖2L2+‖Λsv‖2L2+‖Λsθ‖2L2)+‖Λs+αu‖2L2+‖Λs+βv‖2L2+‖Λs+γθ‖2L2≤C(‖u‖Hs‖Λαu‖2Hs+‖v‖Hs‖Λαu‖Hs‖Λβv‖Hs+‖u‖Hs‖Λβv‖2Hs +‖θ‖Hs‖Λαu‖Hs‖Λγθ‖Hs+‖u‖Hs‖Λγθ‖2Hs). | (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 n≥0 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+un⋅∇un+1+Λ2αun+1+∇pn+1+∇⋅vn+1vn+vn⋅∇vn+1=0,∂tvn+1+un⋅∇vn+1+Λ2βvn+1+∇θn+1+vn⋅∇un+1=0,∂tθn+1+un⋅∇θn+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 +12∫t0(‖Λαun(τ)‖2Hs+‖Λβvn(τ)‖2Hs+‖Λγθn(τ)‖2Hs)dτ≤ϵ2, | (2.7) |
for all 0<t≤T.
We will prove (2.7) by mathematical induction. Obviously, (2.7) holds for n=0. Assume that (2.7) is true for n≥0. 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+1‖2Hs+‖vn+1‖2Hs+‖θn+1‖2Hs)+‖Λαun+1‖2Hs+‖Λβvn+1‖2Hs+‖Λγθn+1‖2Hs≤C(‖un‖2Hs+‖vn‖2Hs+‖θn‖2Hs)(‖Λαun+1‖2Hs+‖Λβvn+1‖2Hs+‖Λγθn+1‖2Hs). | (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+1‖2Hs+‖Λβvn+1‖2Hs+‖Λγθn+1‖2Hs)(τ)dτ≤‖u0‖2Hs+‖v0‖2Hs+‖θ0‖2Hs+C∫t0(‖un‖2Hs+‖vn‖2Hs+‖θn‖2Hs)(‖Λαun+1‖2Hs+‖Λβvn+1‖2Hs+‖Λγθn+1‖2Hs)(τ)dτ≤ϵ2+Cϵ2∫t0(‖Λαun+1‖2Hs+‖Λβvn+1‖2Hs+‖Λγθn+1‖2Hs)(τ)dτ. |
This implies (2.7) holds for n+1 by choosing ϵ sufficiently small such that ϵ≤1√2C. Thus (2.7) is true for all n≥0.
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 0≤t1≤t2≤T,
|(‖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|≤Cϵ2∫t2t1(‖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+1−un, v(n+1)=vn+1−vn, θ(n+1)=θn+1−θn, p(n+1)=pn+1−pn, |
which satisfy
{∂tu(n+1)+un⋅∇u(n+1)+u(n)⋅∇un+Λ2αu(n+1)+∇p(n+1)+∇⋅v(n+1)vn+∇⋅vnv(n)+vn⋅∇v(n+1)+v(n)⋅∇vn=0,∂tv(n+1)+un⋅∇v(n+1)+u(n)⋅∇vn+Λ2βv(n+1)+∇θ(n+1)+vn⋅∇u(n+1)+v(n)⋅∇un=0,∂tθ(n+1)+un⋅∇θ(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)‖2Hs≤C(‖u(n)‖2Hs+‖v(n)‖2Hs+‖θ(n)‖2Hs)(‖Λαun‖2Hs+‖Λβvn‖2Hs+‖Λγθn‖2Hs) +C(‖un‖2Hs+‖vn‖2Hs+‖θn‖2Hs)(‖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 0≤t≤T,
‖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)dτ≤Cϵ2sup0≤τ≤t(‖u(n)(τ)‖2Hs+‖v(n)(τ)‖2Hs+‖θ(n)(τ)‖2Hs) +Cϵ2∫t0(‖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
sup0≤t≤T(‖u(n+1)(t)‖2Hs+‖v(n+1)(t)‖2Hs+‖θ(n+1)(t)‖2Hs)≤12sup0≤t≤T(‖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 +12∫t0(‖Λαu(τ)‖2Hs+‖Λβv(τ)‖2Hs+‖Λγθ(τ)‖2Hs)dτ≤ϵ2, | (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
sup0≤t≤T(‖ˉu(t)‖2Hs+‖ˉv(t)‖2Hs+‖ˉθ(t)‖2Hs)≤12sup0≤t≤T(‖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 0≤t≤T. 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 ˙H−σ with 0≤σ<d2 and negative Besov norm ˙B−σ2,∞ with 0<σ≤d2, 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 0≤σ<d2 and 1<p≤2 with 12+σd=1p. Then
‖Λ−σf‖L2(Rd)≤C‖f‖Lp(Rd). | (3.1) |
Lemma 3.2. Let 0<σ≤d2 and 1≤p<2 with 12+σd=1p. Then
‖f‖˙B−σ2,∞(Rd)≤C‖f‖Lp(Rd). | (3.2) |
Now we show the global a priori estimates for the global solution (u,v,θ) established in Theorem 1.1 in ˙H−σ with 0≤σ<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(‖u‖2˙H−σ+‖v‖2˙H−σ+‖θ‖2˙H−σ)≤C(‖u‖4s+2σ−d−22sL2+‖v‖4s+2σ−d−22sL2+‖θ‖4s+2σ−d−22sL2)(‖u‖d+2−2σ2s˙Hs+‖v‖d+2−2σ2s˙Hs+‖θ‖d+2−2σ2s˙Hs) ×(‖u‖˙H−σ+‖v‖˙H−σ+‖θ‖˙H−σ). | (3.3) |
Proof. Applying Λ−σ to (1.1)1−(1.1)3, and taking the L2-inner products with Λ−σu, Λ−σv and Λ−σθ respectively, we obtain
12ddt(‖Λ−σu‖2L2+‖Λ−σv‖2L2+‖Λ−σθ‖2L2)+(‖Λα−σu‖2L2+‖Λβ−σv‖2L2+‖Λγ−σθ‖2L2)=−∫Λ−σ(u⋅∇u)⋅Λ−σu−∫Λ−σ∇⋅(v⊗v)⋅Λ−σu−∫Λ−σ(u⋅∇v)⋅Λ−σv −∫Λ−σ(v⋅∇u)⋅Λ−σv−∫Λ−σ(u⋅∇θ)⋅Λ−σθ:=K1+K2+K3+K4+K5, | (3.4) |
where we have used the fact
∫Λ−σ∇θ⋅Λ−σv+∫Λ−σ(∇⋅v)⋅Λ−σθ=0. |
Using Hölder's inequality, Lemma 3.1 and the Gagliardo-Nirenberg inequality, we derive that
K1=−∫Λ−σ(u⋅∇u)⋅Λ−σu≤‖Λ−σ(u⋅∇u)‖L2‖Λ−σu‖L2≤C‖u⋅∇u‖L2dd+2σ‖Λ−σu‖L2≤C‖u‖Ldσ‖∇u‖L2‖Λ−σu‖L2≤C‖u‖4s+2σ−d−22sL2‖Λsu‖d+2−2σ2sL2‖Λ−σu‖L2. |
Similarly, we have
K2≤C‖v‖4s+2σ−d−22sL2‖Λsv‖d+2−2σ2sL2‖Λ−σu‖L2. |
Again applying Hölder's inequality, Lemma 3.1 and the Gagliardo-Nirenberg inequality, we derive that
K3=−∫Λ−σ(u⋅∇v)⋅Λ−σv≤‖Λ−σ(u⋅∇v)‖L2‖Λ−σv‖L2≤C‖u⋅∇v‖L2dd+2σ‖Λ−σv‖L2≤C‖u‖Ldσ‖∇v‖L2‖Λ−σv‖L2≤C‖u‖2s+2σ−d2sL2‖Λsu‖d−2σ2sL2‖v‖s−1sL2‖Λsv‖1sL2‖Λ−σv‖L2. |
Similarly, we obtain
K4≤C‖v‖2s+2σ−d2sL2‖Λsv‖d−2σ2sL2‖u‖s−1sL2‖Λsu‖1sL2‖Λ−σv‖L2. |
K5≤C‖u‖2s+2σ−d2sL2‖Λsu‖d−2σ2sL2‖θ‖s−1sL2‖Λsθ‖1sL2‖Λ−σθ‖L2. |
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 ˙B−σ2,∞ with 0<σ≤d2, 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(‖u‖2˙B−σ2,∞+‖v‖2˙B−σ2,∞+‖θ‖2˙B−σ2,∞)≤C(‖u‖4s+2σ−d−22sL2+‖v‖4s+2σ−d−22sL2+‖θ‖4s+2σ−d−22sL2)(‖u‖d+2−2σ2s˙Hs+‖v‖d+2−2σ2s˙Hs+‖θ‖d+2−2σ2s˙Hs) ×(‖u‖˙B−σ2,∞+‖v‖˙B−σ2,∞+‖θ‖˙B−σ2,∞). | (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 2−2σj, and taking the supremum over j∈Z, we conclude that
12ddt(‖u‖2˙B−σ2,∞+‖v‖2˙B−σ2,∞+‖θ‖2˙B−σ2,∞)≤supj∈Z2−2σj|∫˙Δj(u⋅∇u)⋅˙Δju|+supj∈Z2−2σj|∫˙Δj∇⋅(v⊗v)⋅˙Δju| +supj∈Z2−2σj|∫˙Δj(u⋅∇v)⋅˙Δjv|+supj∈Z2−2σj|∫˙Δj(v⋅∇u)⋅˙Δjv| +supj∈Z2−2σj|∫˙Δj(u⋅∇θ)⋅˙Δjθ|:=M1+M2+M3+M4+M5, | (3.6) |
where we used the fact that
∫˙Δj∇θ⋅˙Δjv+∫˙Δj(∇⋅v)⋅˙Δjθ=0. |
Applying Hölder's inequality, Lemma 3.2 and the Gagliardo-Nirenberg inequality, one infers that
M1≤‖u⋅∇u‖˙B−σ2,∞‖u‖˙B−σ2,∞≤‖u⋅∇u‖L2dd+2σ‖u‖˙B−σ2,∞≤C‖u‖Ldσ‖∇u‖L2‖u‖˙B−σ2,∞≤C‖u‖4s+2σ−d−22sL2‖Λsu‖d+2−2σ2sL2‖u‖˙B−σ2,∞. |
Similarly, we have
M2≤C‖v‖4s+2σ−d−22sL2‖Λsv‖d+2−2σ2sL2‖u‖˙B−σ2,∞. |
M3≤C‖u‖2s+2σ−d2sL2‖Λsu‖d−2σ2sL2‖v‖s−1sL2‖Λsv‖1sL2‖v‖˙B−σ2,∞. |
M4≤C‖v‖2s+2σ−d2sL2‖Λsv‖d−2σ2sL2‖u‖s−1sL2‖Λsu‖1sL2‖v‖˙B−σ2,∞. |
M5≤C‖u‖2s+2σ−d2sL2‖Λsu‖d−2σ2sL2‖θ‖s−1sL2‖Λsθ‖1sL2‖θ‖˙B−σ2,∞. |
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)∈˙H−σ. The case (u0,v0,θ0)∈˙B−σ2,∞ can be treated similarly. Assume that
‖u0‖2˙H−σ+‖v0‖2˙H−σ+‖θ0‖2˙H−σ=C0. | (3.7) |
Suppose that for all t∈[0,T],
‖u(t)‖2˙H−σ+‖v(t)‖2˙H−σ+‖θ(t)‖2˙H−σ≤2C0. | (3.8) |
If we can derive that for all t∈[0,T],
‖u(t)‖2˙H−σ+‖v(t)‖2˙H−σ+‖θ(t)‖2˙H−σ≤3C02, | (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 0≤m≤s,
12ddt(‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2)+‖Λm+αu‖2L2+‖Λm+βv‖2L2+‖Λm+γθ‖2L2≤C(‖u‖Hs‖Λαu‖2Hm+‖v‖Hs‖Λαu‖Hm‖Λβv‖Hm+‖u‖Hs‖Λβv‖2Hm +‖θ‖Hs‖Λαu‖Hm‖Λγθ‖Hm+‖u‖Hs‖Λγθ‖2Hm). | (3.10) |
Then this inequality together with the Young inequality implies
ddt(‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2)+‖Λm+αu‖2L2+‖Λm+βv‖2L2+‖Λm+γθ‖2L2≤C(‖u‖2Hs+‖v‖2Hs+‖θ‖2Hs)(‖Λm+αu‖2L2+‖Λm+βv‖2L2+‖Λm+γθ‖2L2). | (3.11) |
Using (1.4) with ϵ<1√2C, it follows from (3.11) that
ddt(‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2)+12(‖Λm+αu‖2L2+‖Λm+βv‖2L2+‖Λm+γθ‖2L2)≤0. | (3.12) |
Applying the Gagliardo-Nirenberg inequality, together with (3.8), we obtain
‖Λmu‖L2≤C‖u‖αm+α+σ˙H−σ‖Λm+αu‖m+σm+α+σL2≤C‖Λm+αu‖m+σm+α+σL2. | (3.13) |
Similarly, we have
‖Λmv‖L2≤C‖Λm+βv‖m+σm+β+σL2. | (3.14) |
‖Λmθ‖L2≤C‖Λm+γθ‖m+σm+γ+σL2. | (3.15) |
Inserting (3.13)–(3.15) into (3.12), there exists a positive constant C1>0 such that
ddt(‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2)+C1(‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2)m+a+σm+σ≤0 |
with a0=max{α,β,γ}. Integrating this inequality with respect to time, we derive that
‖Λmu‖2L2+‖Λmv‖2L2+‖Λmθ‖2L2≤C(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<t≤T, together with (3.16), (3.7) and (1.4), one infers that
‖u(t)‖2˙H−σ+‖v(t)‖2˙H−σ+‖θ(t)‖2˙H−σ≤‖u0‖2˙H−σ+‖v0‖2˙H−σ+‖θ0‖2˙H−σ+Csup0≤τ≤t(‖u(τ)‖˙H−σ+‖v(τ)‖˙H−σ+‖θ(τ)‖˙H−σ) ×∫t0(‖u(τ)‖4s+2σ−d−22sL2+‖v(τ)‖4s+2σ−d−22sL2+‖θ(τ)‖4s+2σ−d−22sL2) ×(‖u(τ)‖d+2−2σ2s˙Hs+‖v(τ)‖d+2−2σ2s˙Hs+‖θ(τ)‖d+2−2σ2s˙Hs)dτ≤C0+Csup0≤τ≤t(‖u(τ)‖˙H−σ+‖v(τ)‖˙H−σ+‖θ(τ)‖˙H−σ) ×ϵϵ0∫t0(1+τ)−(σ2a0(4s+2σ−d−22s−ϵ0)+(s+σ)(d+2−2σ)4a0s)dτ≤C0+Cϵϵ0sup0≤τ≤t(‖u(τ)‖˙H−σ+‖v(τ)‖˙H−σ+‖θ(τ)‖˙H−σ), | (3.17) |
where ϵ0>0 is chosen small enough such that σ2a0(4s+2σ−d−22s−ϵ0)+(s+σ)(d+2−2σ)4a0s>1, which is meaningful since assumptions 12<a0<d+24 and 0≤σ<d2 implies that σ(4s+2σ−d−2)+(s+σ)(d+2−2σ)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 j∈Z
Aj={ξ∈Rd:2j−1≤|ξ|<2j+1}. |
The Littlewood-Paley decomposition asserts the existence of a sequence of functions {Φj}j∈Z∈S such that
suppˆΦj⊂Aj,ˆΦj(ξ)=ˆΦ0(2−jξ)orΦj(x)=2jdΦ0(2jx), |
and
∞∑j=−∞ˆΦj(ξ)={1,ifξ∈Rd∖{0},0,ifξ=0. |
Therefore, for a general function ψ∈S, we have
∞∑j=−∞ˆΦj(ξ)ˆψ(ξ)=ˆψ(ξ)for ξ∈Rd∖{0}. |
We now choose Ψ∈S such that
ˆΨ(ξ)=1−∞∑j=0ˆΦj(ξ),ξ∈Rd. |
Then, for any ψ∈S,
Ψ∗ψ+∞∑j=0Φj∗ψ=ψ |
and hence
Ψ∗f+∞∑j=0Φj∗f=f | (4.1) |
in S′ for any f∈S′. To define the inhomogeneous Besov space, we set
Δjf={0,ifj≤−2,Ψ∗f,ifj=−1,Φj∗f,ifj=0,1,2,⋯. | (4.2) |
To define the homogeneous Besov space, we set
˙Δjf=Φj∗f,ifj=0,±1,±2,⋯. | (4.3) |
Definition 4.1. The inhomogeneous and homogeneous Besov spaces Bsp,q and ˙Bsp,q with s∈R and p,q∈[1,∞] consists of f∈S′ satisfying
‖f‖Bsp,q≡‖2js‖Δjf‖Lp‖lqj<∞, |
and
‖f‖˙Bsp,q≡‖2js‖˙Δjf‖Lp‖lqj<∞, |
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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