On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.


Introduction
In the present paper, we consider the two-dimensional incompressible Boussinesq equations We write x = x 1 x 2 ∈ R 2 with x 1 , x 2 denoting the horizontal and vertical components, respectively.
Let u = u 1 u 2 : [0, ∞) × R 2 → R 2 , and let denote the symmetric deformation tensor in the second equation (1.1) 2 above. The vector field e 2 denotes the unit vector in the vertical direction: e 2 = 0 1 , and βθ e 2 stands for the buoyancy force, with the constant parameter β > 0 denoting the thermodynamic dilatation coefficient which will be assumed to be 1 in the following context for simplicity.
We consider the cases when the heat diffusion and the viscosity in the fluids are sensitive to the change of temperatures, that is, the thermal diffusivity κ and the viscosity coefficient μ may depend on the temperature function θ as follows κ = a(θ), μ = b(θ), with κ * ≤ a ≤ κ * , μ * ≤ b ≤ μ * , (1.3) where κ * ≤ κ * , μ * ≤ μ * are positive constants. We will not assume any smallness conditions on κ * − κ * or μ * − μ * , and large variations in these diffusivity coefficients are permitted. The Boussinesq system (1.1) arises from the zero order approximation to the corresponding inhomogeneous hydrodynamic systems, which are nonlinear coupling between the Navier-Stokes equations or Euler equations and the thermodynamic equations for the temperature or density functions: The Boussinesq approximation [5] ignores density differences except when they appear in the buoyancy term. They are common geophysical models describing the dynamics from large-scale atmosphere and ocean flows to solar and plasma inner convection, where density stratification is a typical feature [22,34].
The temperature or density differences in the inhomogeneous fluids may cause density gradients. When the thermodynamical coefficients such as the heat conducting coefficients and the viscosity coefficients are assumed to be constant in the Boussinesq approximation [i.e., κ, μ are constants in (1.1)], density gradients influence the motion of the flows only through the buoyancy force, which may lead to finite time singularity in the flows (the formation of the finite time singularity is sensitive to the thermal and viscous dissipation and see Sect. 1.1 below for more references on this topic).
However, the temperature variations do influence the thermal conductivity and the viscosity coefficients effectively, even for simple fluids such as pure water [32,Sect. 6]. 1 , 2 In many applications in the engineering, one also aims for effective thermal conductivities in building thermal energy storage materials [21]. Therefore in plenty of physical models, density gradients would influence the motion of the fluids not only through buoyancy force, but also through the variations of the diffusion coefficients. It is then interesting to study the well-posedness and regularity problems of the Boussinesq system (1.1)-(1.3).

Known results
The well-posedness and regularity problems on the two-dimensional Boussinesq equations have attracted considerable attention from the PDE community. Many interesting mathematical results have been established in the past two decades, mainly in the cases with constant thermal diffusivity coefficient κ and viscosity coefficient μ: (1.4) If κ = μ = 0, the two-dimensional inviscid Boussinesq equations (1.4) can be compared with the three-dimensional incompressible axisymmetric Euler equations with swirl, where the buoyancy force corresponds to the vortex stretching mechanism [35]. The local-in-time well-posedness as well as some 1 The absolute viscosity of the water under nominal atmospheric pressure in units of millipascal seconds is given by 1.793 (0 • ), 0.547 (50 • ), 0.282 (100 • ), respectively [32, pp. 6-186]. The thermal conductivity of the water under nominal atmospheric pressure in units of Watt per meter kelvin is given by 0.5562 (0 • ), 0.6423 (50 • ), 0.6729 (100 • ), respectively [32, Page 6-214].
2 It is common to adapt the exponential viscosity law μ(T ) = C 1 exp(C 2 /(C 3 +T )) and quasi-constant heat conductivity law κ(T ) = C 4 for the liquids, while the viscosity law μ(T ) = (μ(Tm)) T Tm Tm+C 5 T +C 6 and the thermal conductivity law κ(T ) = C 6 μ(T ) for the gases, where T denotes the absolute temperature, Tm denotes the reference temperature, and C j , 1 ≤ j ≤ 6 are positive constants [37, I].
blowup criteria have been well known for decades, see, e.g., [10,11,42]. We mention that an (improved) lower bound for the lifespan which tends to infinity as the initial temperature tends to a constant (and correspondingly, as the initial swirl tends to zero for the 3D axisymmetric Euler equations) was given in [11]. The fundamental global regularity problem for the 2D inviscid Boussinesq equations remains still open. Recently, an interesting example of finite-energy strong solutions with a finite weighted Hölder norm in a wedge-shaped domain, which become singular at the origin in finite time, has been given in [19] (see also an interesting example of solutions in Hölder-type spaces with finite-time singularity for 3D axisymmetric Euler equations in [18]).
If κ > 0 and μ > 0 are positive constants, on the contrary, the convection terms can be controlled thanks to the strong diffusion effects, and the global-in-time existence and regularity results can be established (see, e.g., [7]). Particular interests then raised if only partial dissipation is present, that is, either κ = 0 whereas μ > 0 or κ > 0 whereas μ = 0 (see, e.g., H.K. Moffatt's list of the twenty first century PDE problems [36]). The global-in-time results continue to hold, thanks to a priori estimates in the L p -framework as well as the sharp Sobolev embedding inequality in dimension two with a logarithm correction, which help the partial diffusion terms to control the demanding term ∂ x1 θ successfully (see [9,27] and see [24] for less regular cases). Further developments were made for horizontal dissipation cases (see, e.g., [13]), for vertical dissipation cases (see, e.g., [8]), and for the fractional dissipation cases (see, e.g., [25,26]). See the review notes [43] and the references therein for more interesting results and sketchy proofs.
There also have been remarkable progresses in solving the two dimensional Boussinesq equations (1.1)-(1.3) when the thermal and viscosity diffusion coefficients κ, μ are variable and depend smoothly on the unknown temperature function θ. In the variational formulation framework, the global-in-time existence of a solution of (1.1)-(1.3) has been established in [17] [see [20] for a similar formulation of (1.1)-(1.3)] for the motion of the so-called Bingham fluid (as a non-Newtonian fluid), where κ is a positive constant, β = 0 and μ depends not only on θ but also on Su/|Su|. The Boussinesq-Stefan model has been investigated in [38], where the phase transition was taken into account. The global-in-time existence and the uniqueness of the solutions for (1.1)-(1.3) have been shown in [15,23,33] under Dirichlet boundary conditions and in [37] under generalized outflow boundary conditions. We remark that the resolution of the nonhomogeneous Boussinesq system under more physical boundary conditions (e.g., with Dirichlet boundary conditions only on the inflow part of the boundary while with no prescribed assumptions on the outflow part) remains unsolved.
Lorca and Boldrini [33] (see also [15,23]) studied the initial-boundary value problem of the Boussinesq system (1.1)-(1.3) in dimension two and three under the initial condition (1.2) and Dirichlet boundary conditions, and obtained a global-in-time weak solution as well as a local-in-time unique strong solution The remarkable global-in-time existence and uniqueness results of the smooth solutions have been successfully established by Wang and Zhang [41], which affirms the propagation of high regularities (without finite time singularity) of the two dimensional Boussinesq flow in the presence of viscosity variations (see [39] for the case s = 2). We remark that the L 2 x -norm of the velocity vector field may grow in time due to the buoyancy forcing term, even provided with constant diffusion coefficients and smooth and fast decaying small initial data [6], and hence the norm with respect to the time variable in (1.5) and (1.6) is only locally in time.
It is still not clear whether there will be finite time singularity for the two-dimensional Boussinesq flow (1.1)- (1.3) in the presence of viscosity variations while no heat diffusion (i.e., κ = 0, μ = b(θ)), and we mention a recent work [2] toward this direction in the case of less heat diffusion (with div(κ∇θ) replaced by (−Δ) 1/2 ) and the small viscosity variation assumption: |μ − 1| ≤ ε. A closely related question would pertain to the global-in-time well-posedness problem of the two-dimensional inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity coefficient The global-in-time existence results of weak solutions of (1.7) (see, e.g., [3,31]) as well as the local-in-time well-posedness results (see, e.g., [28]) have been well known, while the global-in-time regularities still remain open (see, e.g., [1,16] for some interesting results under the assumption on the weak inhomogeneity).
To the best of our knowledge, there are no global-in-time regularity propagation results by the twodimensional Boussinesq flow with temperature-dependent diffusion coefficients ( ) 2 with different regularity indices s θ and s u . In this paper, we are going to investigate the existence, uniqueness as well as the regularity problems in these general Sobolev functional settings.
To conclude this subsection, let us just mention some recent interesting progresses on the stability of the stationary shear flow solutions (together with the corresponding striated temperature function) to the Boussinesq equations (1.4), with full dissipation or partial dissipation, in, e.g., [14,40,44] and references therein. It should also be interesting to investigate the stability of the stationary striated solutions of the Boussinesq equations with variable diffusion coefficients (1.1). We mention a recent work in this direction on the incompressible Navier-Stokes equations with constant density function but with variable viscosity coefficient [30].

Main results
We are going to show the global-in-time existence of weak solutions to the Cauchy problem for the Boussinesq system (1.1)-(1.2)-(1.3) in the whole two-dimensional space R 2 under the low-regularity initial condition (θ 0 , u 0 ) ∈ L 2 (R 2 ) × (L 2 (R 2 )) 2 . The uniqueness result holds true if the initial temperature function becomes smoother (θ 0 , u 0 ) ∈ H 1 (R 2 ) × (L 2 (R 2 )) 2 . Finally, we will establish the global-in-time regularity of the solutions in the general Sobolev setting These regularity exponent ranges are optimal for the existence, uniqueness and regularity results, respectively, by view of the formulations of the Boussinesq equations (1.1) with temperature-dependent diffusion coefficients (see Remark 1.3 below for more details).
We first define the weak solutions as follows. 3) with the given initial data (θ 0 , u 0 ) ∈ (L 2 (R 2 )) 3 if the following statements hold: • The temperature function satisfies the initial condition θ| t=0 = θ 0 , the energy equality for all positive times T > 0, and the equation satisfies the initial condition u| t=0 = u 0 , the divergence-free condition div x u = 0, the energy equality (1.10) and the equation For any fixed T > 0, p ≥ 1, q ≥ 1, s ≥ 0 and for any fixed (vector-valued) function f : Theorem 1.2. (Existence, uniqueness and global-in-time regularity) For any initial data θ 0 ∈ L 2 (R 2 ) and u 0 ∈ (L 2 (R 2 )) 2 , there exists a global-in-time weak solution have finite first and second derivatives, then the weak solution is indeed unique, and satisfies , as well as the following energy estimates for any given T > 0,
(1.16) Theorem 1.2 will be proved in Sect. 2. The proof of the existence of weak solutions is rather standard, and we are going to sketch the proof in Sect. 2.1 for the reason of completeness, as we did not find the proof in the literature. As mentioned before, some well-posedness results have already been established for smooth data in the bounded domain case (see (1.5) above in, e.g., [15,17,23,33]) or in smoother functional frameworks in the whole space case (see (1.6) above in, e.g., [41]). We are going to focus on the proofs of the uniqueness result and the global-in-time regularity result (in the low regularity regimes) in Sects. 2.2 and 2.3, respectively, where different regularity exponents for different unknowns are permitted. The commutator estimates as well as the composition estimates in Lemma 2.1 will play an important role, and the a priori estimates for a general linear parabolic equation in Lemma 2.2 will be of independent interest.
We conclude this introduction part with several remarks on the results in Theorem 1.2. If we take the difference between two different weak solutions (θ 1 , u 1 ) and (θ 2 , u 2 ), the difference of the nonlinear viscosity term div(μSu) in the u-equation will become Therefore in order to ensure the L 2 x -Estimate for the velocity difference (u 1 − u 2 ), we require the H 1 x -Estimate for the temperature difference in (μ 1 − μ 2 ). And hence the initial condition θ 0 ∈ H 1 x (i.e., s θ ≥ 1 above) is required for the proof of the uniqueness result (see Sect.

below).
Under the lower-regularity assumption θ 0 ∈ H s x with 0 < s < 1, the coefficients κ, μ are not expected to be continuous uniformly in time, and hence no uniqueness or H s -regularity results for θ or H s1 , s 1 > 0regularity results for u are expected. Nevertheless with constant diffusion coefficients (e.g., κ = μ = 1), the uniqueness result for the weak solutions holds true by virtue of the L 2 x -energy (in)equalities (similar as the classical global-in-time well-posedness result for the classical two dimensional incompressible Navier-Stokes equations). Furthermore, if κ = 1 is a positive constant, then the H s x , s ∈ (0, 1)-Estimate for θ holds true, provided with u ∈ L 4 loc (L 4 x (R 2 )) 2 (or with u 0 ∈ (L 2 (R 2 )) 2 ), simply by an interpolation argument between (1.8) and (1.14). Similarly if μ = 1 is a positive constant, then the H s x , s > 0-Estimate for u holds true, provided with θ ∈ L 2 loc (H s−1 x (R 2 )). Thus with constant diffusion coefficients (e.g., κ = μ = 1), the Sobolev regularities can be propagated globally in time, and the admissible regularity exponent set (1.15) extends itself indeed to the closed set consisting of all nonnegative admissible regularity exponents: In order to propagate the H s θ , s θ ≥ 2-regularity of θ, we require the transport term u · ∇θ in the θ- ) and hence the initial assumption u 0 ∈ H su with the restriction s u ≥ s θ −2 (as there is a gain of regularity of oder 1 when taking L 2 -norm in the time variable in general). Similarly, in order the propagate the H su , s u ≥ 2-regularity of u, we require the viscosity term div(μSu) in the u-equation to be at least in and hence the initial assumption θ 0 ∈ H s θ with the restriction s θ ≥ s u − 1.

Remark 1.4. (Precise H s
x -Estimates in the high regularity regime) The global-in-time regularity in the high regularity regime (1.15)-(1.16) follows immediately from the following borderline a priori estimates: and for s = 2 it holds θ 2 and for s = 2 it holds u 2 (1.20) • If θ 0 ∈ H s (R 2 ), u 0 ∈ (H s−2 (R 2 )) 2 with s > 2 and the function a ∈ C [s]+1 , then for s ∈ (2, 3) it holds θ 2 and for s ≥ 3 it holds and for s ≥ 3 it holds ).
( 1.24) We are going to prove the above borderline estimates one by one in Sect. 2.3 below.
x -type estimate here, since, e.g., only the L 2 tḢ 1 x -a priori estimate for the velocity vector field is available from the energy estimates (roughly speaking, the L 2 t -in time norm asks less spatial regularity on the coefficients). See Lemma 2.2 below for the a priori H s x , s ∈ (0, 2)-Estimates for a general linear parabolic equation with divergence-free L 2 t H 1 x -velocity vector field, which is of independent interest.
It is in general not true that θ ∈ L 1 t H s+2 x ) in the low regularity regime, although it holds straightforward in the high regularity regime. Remark 1.6. (Remarks on the smoothness assumptions on the functions a, b) It is common to assume smooth heat conductivity law and viscosity law [37, I] in fluid models.
The Lipschitz continuity assumption a, b ∈ Lip is enough for the H 1 × L 2 -Estimates (1.13)-(1.14) in Theorem 1.2. As for the uniqueness result, due to the followingḢ 1 x -Estimate for the difference of the diffusion coefficinets

Proofs
Recall the Cauchy problem for the two-dimensional Boussinesq equations (1. Recall the definition of the · L q T Xx -norm in (1.12). The Gagliardo-Nirenberg's inequality as well as the equivalence relations between the norms will be used freely in the proof.

Existence of weak solutions if
We will follow the standard procedure to show the existence of the weak solutions under the initial condition Step 1 We construct a sequence of approximate solutions, which satisfy the energy estimates uniformly.
Step 2 We show the convergence of this approximate solution sequence to a weak solution and study the property of the weak solution. We are going to sketch the proof and pay attention to the low-regularity assumptions.

Step 1: Construction of approximate solutions with uniform bounds
We use the Friedrich's method to construct a sequence of approximate solutions. We consider the following system of (θ n , u n ) ⎧ ⎪ ⎨ ⎪ ⎩ ∂ t θ n + P n (u n · ∇θ n ) − P n div(κ n ∇θ n ) = 0, ∂ t u n + P n P(u n · ∇u n ) − P n P div(μ n Su n ) = P(θ n e 2 ), where κ n = a(θ n ) and μ n = b(θ n ). The operator P n , n ∈ N, is the low-frequency cutoff operator which is defined as follows where B n ⊂ R 2 is the disk with center at 0 and radius n, and F, F −1 are the standard Fourier and inverse Fourier transformations. The operator P in (2.4) denotes the Leray-Helmholtz projector on R 2 , which decomposes the tempered distributions v ∈ S (R 2 ; R 2 ) into div-free and curl-free parts as follows where Notice that P maps L p (R 2 ; R 2 ) into itself for any p ∈ (1, ∞) and it is commutative with the projection operator P n . We define the Banach spaces L 2 n and L 2,σ n as following The system (2.4) turns out to be an ordinary differential equation system in L 2 n (R 2 ) × L 2,σ n (R 2 ). Indeed, the following estimates hold Hence, for any n ∈ N, there exists T n > 0 such that the system (2.4) has a solution (θ n , u n ) ∈ C([0, T n ]; L 2 n (R 2 )) × C([0, T n ]; L 2,σ n (R 2 )). We take the L 2 (R 2 )-inner product of Eq. (2.4) 1 and θ n to derive 1 2 d dt Then, the following uniform estimate for (θ n ) holds Similarly, we take the L 2 (R 2 )-inner product of Eq. (2.4) 2 and u n to derive for all positive times T > 0, and thus by Gronwall's inequality we arrive at the following uniform estimate for (u n ) (noticing Su n Thus, the approximate solutions (θ n , u n ) exist for all positive times.

Energy estimates and uniqueness of the weak solutions if
We first introduce a scalar function η, which is given in terms of the temperature function as follows As A (θ) = a(θ) ≥ κ * > 0, the function A is invertible and we can write (2.12) 3 We can easily compute ∇η = a(θ)∇θ, ∇θ = 1 a(A −1 (η)) ∇η, That is,

be a weak solution of the Cauchy problem (2.1) in the sense of Definition 1.1 with
) is an algebra (in the sense that the product of any two elements in Y still belongs to Y ), we can multiply the above θ-equation by κ = a(θ) (with a(θ)−a(0) ∈ Y ), to arrive at the parabolic equation We are going to derive the H 1 -Estimate for η (and hence for θ 4 ) as well as the L 2 -Estimate for u first. Then, we will show the uniqueness result of the weak solutions by considering the difference of two possible weak solutions. The procedure is standard (see, e.g., Sect. 2 [29]) and we are going to sketch the proof.
By virtue of the energy equalities (1.8) and (1.10) and the derivation of the uniform estimates (2.6) and (2.7), we have the L 2 -Estimate We assume a priori that the function η is smooth and decay sufficiently fast at infinity. We test the η-equation (2.15) by Δη to derive by integration by parts that By Gagliardo-Nirenberg's inequality (2.2), the equivalence Δη L 2 x (R 2 ) ∼ ∇ 2 η L 2 x (R 2 ) and Young's inequality we arrive at Gronwall's inequality gives x for any positive time T > 0. Thus by the η-equation By virtue of the equivalence relation (2.12): The introduction of the η-function makes the derivation of the H 1 -Estimate for θ straightforward (and possible). and (2.16)-(2.17), we have the a priori H 1 -Estimate (1.14) for θ: (2.18) Therefore, both the parabolic equations (2.14) and (2.15) for θ and η hold in L 2 loc ([0, ∞); L 2 (R 2 )). A standard density argument ensures the H 1 -Estimate (1.14) for θ, and hence θ ∈ C([0, ∞); H 1 x (R 2 )).

Proof of the uniqueness
Let (θ 1 , u 1 , Π 1 ) and (θ 2 , u 2 , Π 2 ) be two weak solutions of the Cauchy problem (2.1) with the same initial data (θ 0 , u 0 ) ∈ H 1 (R 2 ) × (L 2 (R 2 )) 2 , which satisfy the energy estimates (1.13)-(1.14). Recall (2.10) for the definition of the function A, and we set We consider the difference which lies in Similarly as in (2.12) we have the following equivalence relationships 20) and correspondingly we have (2.21) We are going to sketch the derivation of the H 1 × L 2 -Estimate for (η,u). (i) L 2 estimate ofη. We take the L 2 (R 2 )-inner product between (2.19) 1 andη to derive The right hand side can be bounded by He and X. Liao ZAMP We take the L 2 inner product of the equation (2.19) 2 andu to derive 1 2 d dt The right hand side can be bounded by We take the L 2 inner product of Eq. (2.19) 1 andΔη to derive 1 2 d dt x . To conclude, by virtue of the above estimates and (2.21), we have the following inequality d dt η 2 ∞)). Gronwall's inequality implies thenη = 0 andu = 0. The uniqueness of the weak solutions follows.

Propagation of the general H s -regularities
After the derivation of the a priori H s x , s ∈ (0, 2)-Estimate for a general linear parabolic equation in Sect. For readers' convenience, we recall here briefly the Littlewood-Paley dyadic decomposition and the definition of the H s (R n )-norms (see, e.g., Chapter 2 in the book [4] for more details). We fix a nonincreasing radial function χ ∈ C ∞ c (B 4 3 ) with χ(x) = 1 for x ∈ B 1 , where B r ⊂ R n denotes the ball centered at 0 with radius r. We define the function ϕ(ξ) = χ( ξ 2 ) − χ(ξ) and ϕ j (ξ) = ϕ(2 −j ξ) with j ≥ 0. We do the Littlewood-Paley decomposition in the following way where and F denotes the Fourier transform. We have the following Bernstein's inequalities for some universal constant C (depending only on n) Let s ≥ 0 and p, r ≥ 1. We define the nonhomogeneous Besov spaces B s p,r (R n ) as the spaces consisting of all tempered distributions g ∈ S (R n ) satisfying where the H s (R n )-norm reads in terms of Littlewood-Paley decomposition as follows It is straightforward to derive the following interpolation inequality We are going to use the following known estimates to control the nonlinear terms in the Boussinesq system (1.1).

Lemma 2.1.
We have the following commutator, product, and composition estimates.

28)
where C is a constant depending only on s, ν.
(2) [4, Lemma 2.100] For any s > 0, the following commutator estimate holds true with k = [s] ∈ N, then the above estimate can be improved in the spatial dimension two as follows (2.32) The commutator estimate (2.28) will present its power in the low regularity regime (see Sects. 2.3.1-2.3.3 below), and the classical commutator estimate (2.29) will help in the high regularity regime (see Sect. 2.3.4 below).
The composition estimate (2.32) will help to bound the diffusion coefficients κ, μ in terms of θ in the low regularity regime, where only H 1 (R 2 )-norm (instead of L ∞ x -norm) of θ is available.

Estimates for the general parabolic equations.
We derive in this paragraph a priori H s , s ∈ (0, 2)-Estimates for a general linear parabolic equation, which should be of independent interest.

(2.34)
Proof. It is straightforward to derive the following L 2 x -Estimate by simply taking the L 2 (R 2 ) inner product of the equation (2.33) and ψ itself We next consider the a priori estimates for the H s (R 2 )-norm. By virtue of the description (2.26) of the H s (R 2 )-norm, we consider the dyadic piece of ψ: where the operator Δ j is defined in (2.24). We apply Δ j to the linear ψ-equation to derive the equation for ψ j : We take the L 2 inner product of the equation (2.37) and ψ j and make use of divu = 0 and κ ≥ κ * to derive By use of Bernstein's inequality (2.25), we have We make use of the commutator estimate (2.28) in Lemma 2.1 to estimate the commutators [u, Δ j ] · ∇ψ L 2 x and 2 j [κ, Δ j ]∇ψ j L 2 x in the above inequality in the following way. Let (l j ) j≥0 be a normalized sequence in 1 (N) such that l j ≥ 0 and j≥0 l j = 1. Then, we have for ν ∈ (−1, 1), s ∈ (−1, ν + 1).
(2.39) Therefore, we have We use Duhamel's Principle to derive We multiply the inequality (2.40) by 2 js to derive We take L ∞ ([0, T ])-norm in t of (2.41) and the L 2 ([0, T ])-norm in t of 2 j ·(2.41), to derive by use of Young's inequality that We take square of (2.42) and sum them up for j ∈ N to derive that is, by virtue of the L 2 -Estimate (2.35), We next consider the norm ∇ψ H s−ν x . By the interpolation inequality (2.27), we have To conclude, by taking ε small enough and Gronwall's inequality, we derive the H s -Estimate (2.34). (θ 0 , u 0 ) ∈ H s (R 2 ) × (L 2 (R 2 )) 2 , s ∈ (1, 2]. In this subsection, we are going to prove the H s -Estimates (1.17) for the unique solution (θ, u) of the Boussinesq equations (1.1) with the initial data (θ 0 , u 0 ) ∈ H s (R 2 ) × (L 2 (R 2 )) 2 , s ∈ (1, 2), following exactly the procedure in Sect. 2.3.1. We will pay more attention on the "nonlinearities" in the equations such as κ = a(θ), u · ∇u when using the commutator estimates and will sketch the proof. The endpoint estimate (1.18) for (θ 0 , u 0 ) ∈ H 2 (R 2 ) × (L 2 (R 2 )) 2 follows similarly as in the proof for the H 1 -Estimate for θ in Sect. 2.2 and we will sketch the proof.
We test the above η-equation (2.46) by Δ 2 η, to arrive at By integration by parts, divu = 0 and the embedding L 1 (R 2 ) → H −1 (R 2 ), we derive x . To conclude, we have the following a prioriḢ 2 x -Estimate for η and any positive time T > 0 by Gronwall's inequality By view of the equivalence relation (2.12) as well as 5  We deal with the endpoint case (θ 0 , u 0 ) ∈ H 1 (R 2 ) × (H 2 (R 2 )) 2 similarly as for the endpoint case above.

Case
Recall (2.5) for the definition of the Leray-Helmholtz projector P such that We apply P to the velocity equation (1.1) 2 to arrive at ∂ t u + P(u · ∇u) − P div(μSu) = P(θ e 2 ).
By use of the embedding L 1 (R 2 ) → H −1 (R 2 ) again, the right hand side can be bounded by Thus, we have the following a prioriḢ 2 x -Estimate for u and any positive time T > 0 by Young's inequality and Gronwall's inequality which gives (1.20). Case (θ 0 , u 0 ) ∈ H s (R 2 ) × (H s−2 (R 2 )) 2 , s > 2 We can view the transport term u · ∇θ simply as a source term of the θ-equation: