On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity

The goal of this paper is to study the two-dimensional inviscid Boussinesq equations with temperature-dependent thermal diffusivity. Firstly we establish the global existence theory and regularity estimates for this system with Yudovich's type initial data. Then we investigate the vortex patch problem, and proving that the patch remains in Holder class \begin{document}$ C^{1+s}\; (0 for all the time.

The Boussinesq system describe the influence of convection phenomenon in the dynamics of the ocean or of the atmosphere (see e.g. [33]). Mathematically, the global well-posedness for system (1) in the case ν, κ are positive constant has been solved in [4,20]. But it the case ν = κ = 0, it is still an unsolved problem that whether we can construct global unique solutions for some non-trivial θ 0 . So 5712 MARIUS PAICU AND NING ZHU this system has been extensively studied in the last few years due to the physical background and mathematical challenging.
For the constant viscosity case ν(θ) = ν > 0, κ = 0, Chae in [5] and Hou, Li in [26] obtained the global well-posedness result for regular initial data. Later, Abidi and Hmidi studied this system in the Besov space in [1]. For lower regularity initial data, the global weak solution with finite energy has been constructed in [23] and has been proved to be unique later in [13]. On the other hand, for the constant thermal diffusive case κ(θ) = κ > 0, ν = 0, the global well-posed for regular initial data has been obtained by Chae in [5]. Later, Hmidi and Keraani extended this result to rough initial data in some Besov space in [24]. Danchin and the first author studied this system in [14] with Yudovich's type data.
For the temperature dependent viscosity, system (1) has been studied in [36], and they obtained the global well-posedness result for smooth data with De Giorgi method. Later, Li and Xu studied the case ν = 0, κ(θ) > 0 in [28] with smooth initial data. In contrast, the case ν(θ) > 0, κ(θ) = 0 still unsolved even for smooth initial data. Other interesting results corresponding to this model can be found in [27,29,35,37].
In this paper, we investigate the 2D Boussinesq equations with only temperaturedependent thermal diffusion under the Yudovich's type initial data, the system reads: x ∈ R 2 , t > 0, ∂ t θ + u · ∇θ − ∇ · (κ(θ)∇θ) = 0, ∇ · u = 0, u(0, x) = u 0 (x), θ(0, x) = θ 0 (x). (2) Through out this paper, we assume that κ(θ) satisfies Here we want to introduce an important quantity ω ∂ 1 u 2 −∂ 2 u 1 called vorticity which measures how fast the fluid rotates and its control plays an important role in the literature we mentioned above. The Yudovich's type initial data is (u 0 , θ 0 ) in L 2 with bounded vorticity ω 0 ∂ 1 u 2 0 − ∂ 2 u 1 0 . Taking curl operator to the first equation of (2) we can obtain the corresponding vorticity equation Let us denote by ψ(t, ·) the flow associated with the vector field u, that is The classical vortex patch problem is concerned about the following 2D incompressible Euler equations   The associate vorticity ω satisfies the following transport equation ON THE 2D BOUSSINESQ SYSTEM 5713 If the initial vorticity where D 0 is a connected bounded domain, χ D0 is the standard characteristic function of D 0 . Then according to the properties of the flow ψ, we have ω(t) = χ Dt with D t = χ(D 0 , t). A natural problem is that whether the regularity of the boundary ∂D t preserving through the evolution of the flow. It has been proved by Chemin (see e.g. [6,7]) that the regularity of the boundary can be persisted for all the time. Later, Gamblin and Saint-Raymond studied the vortex patch problem for 3D Euler equations in [18]. As for Boussinesq system, the vortex patch problem for inviscid Boussinesq equations has been discussed by Hassainia and Hmidi in [21]. Then Danchin and Zhang in [15], Gancedo and García-Juárez in [19] considered the temperature patch problem associate to the Boussinesq system with full Laplacian dissipation in velocity and no diffusion in temperature. Then for the stratified Euler system, which is system (2) with constant temperature diffusion, Hmidi and Zerguine studied the vortex patch problem in [25]. Many similar studies have been subsequently implemented by numerous authors for homogeneous (inhomogeneous) Navier-Stokes and other viscous (inviscid) flows, see for instance [3,8,9,10,11,12,16,17,22,30,31,32] and the references therein. In order to understand the striated regularity clearly, we need first to introduce some notations and definitions which will be used to describe the boundary regularity. Let X 0 be a vector field defined on D 0 , X is the evolution of X 0 along the flow ψ defining as follows, where ∂ X0 f X 0 · ∇f denoting the standard directional derivative. Taking time derivative of (8), one can obtain X satisfies the following transport equation, It is not hard to check that ∂ X satisfies, where [A, B] AB − BA represents the standard commutator. Applying div operator to (9) and combining with the divergence-free condition of u, we obtain in addition Therefore, the divergence-free property can be preserved through the evolution.
The following definition of I(x) is needed in order to estimate the striated regularity and state our result, which can be found in [2,7]. Definition 1.1. A family (X λ ) λ∈Λ of vector fields over R 2 is said to be nondegenerate whenever Let r ∈ (0, 1) and (X λ ) λ∈Λ be a non-degenerate family of C r vector fields over R 2 .
A bounded function f is said to be in the function space C r X if it satisfies Then we give the definition about how to describe a boundary curve in C s class.
Definition 1.2. Let 0 < s < 1 and Ω be a bounded domain in R 2 . We say that Ω is of class C 1+s if there exists a compactly supported function f ∈ C 1+s (R 2 ) and a neighborhood V of ∂Ω such that The main result of our paper can be stated as follows.
Theorem 1.3. Assume u 0 ∈ L 2 be a divergence-free vector field, the corresponding vorticity system (2) has a global solution (u, θ) satisfies for any T > 0 and some σ > 1. If p and r satisfy 1 p + 1 r ≤ 1, then the solution is unique.

Remark 1.
Noticing that with Yudovich's type data, ∂ X ω is defined as ∂ X ω div (ωX) − ωdiv X, in the sense of distribution. For the sake of simplicity, we assume div X 0 = 0 in Theorem 1.3, which can be preserved for all the time. As for the general case (div X 0 = 0), we just need to propagate the regularity of div X, which is easy to get because div X satisfies a transport type equation (11). We will explain the details in Remark 3.

Remark 2.
In the process of proving ∇u in L 1 ([0, T ]; L ∞ ), we only need ∂ X0 ω 0 ∈ C s−1 , which leads to the estimate of ∂ X ω in C s−1 . This is enough to obtain the Lipschitz bound of the velocity u. The reason we choose L p space here is because L p → C s−1 and the L p norm of ∂ X ω will be used in the proof of X ∈ L ∞ ([0, T ];Ẇ 1,p ). Theorem 1.3 can be used to deal with the following vortex patch problem directly.
Let ω 0 defined as (7), the solution ω(t, x) = ω 1 (t, x) + ω 2 (t, x) where ω 1 is the solution of the system ∂ t ω 1 + u · ∇ω 1 = 0, and ω 2 is the solution of the system Then the main result can be stated as follows.
Corollary 1. Assume u 0 be a divergence free vector field with vorticity ω 0 defined as in (7) and D 0 be a connected bounded domain with its boundary ∂D 0 in Hölder class C 1+s (0 < s < 1), θ 0 defined as Theorem 1.3. Then system (2) exists a unique global solution satisfies the properties shows in Theorem 1.3. Moreover, the solution of systems (13) and (14) satisfying with D t ψ(D 0 , t) and the boundary of the domain remains in the class C 1+s .
The rest of this paper is divided into four sections and an appendix. Some tool lemmas will be given in section 2. In the third section, we prove the well-posedness result for system (2) which shows the proof of results of the first part in Theorem 1.3. Section 4 is devoted to give some estimates of the striated regularity which complete the proof the Theorem 1.3. The last section presents the proof of Corollary 1, which solves the corresponding vortex patch problem for system (2). In the appendix, we provide the description of the Littlewood-Paley decomposition, the Besov space and some related facts used in the previous sections.

2.
Preparations. In this section, we will give some lemmas which will be used in the next several sections. Throughout this paper, C stands for some real positive constant which may be different in each occurrence. C(t) is also a constant which depending on t and the initial data.
Noticing that if u is a divergence-free vector field in R 2 , then there exists a stream function ψ such that u = ∇ ⊥ ψ. Then we can obtain that the velocity u can be recovered from the corresponding vorticity ω by means of the following Biot-Savart law Combining the classical Calderón-Zygmund estimate with (15), it can lead to the following lemma (see [7] for details).
Lemma 2.1. For any smooth divergence-free vector field u with its vorticity ω ∈ L p and p ∈ (1, ∞), there exists a constant C such that Then we present the maximum regularity estimate for heat semi-group, which play an important role in the regularity estimate for θ of system (2). (see e.g. Lem. 7.3 in [34] for the proof.) and A be an operator satisfies

Then we have
The next lemma shows the Hölder estimate for transport equation, which is useful in the estimate of the striated regularity. The proof can be found in [7]. Lemma 2.3. Let u be a smooth divergence-free vector field, r ∈ (−1, 1). Consider two functions f ∈ L ∞ loc (R; C r ) and g ∈ L 1 loc (R; C r ) satisfy the transport equation Then we have and the constant C depends only on r.
Next we give lemma which showing a logarithmic inequality which can be found in [2,7]. This inequality plays an important role in the proof of the vortex patch problem for Euler equations.
Lemma 2.4. Let r ∈ (0, 1) and (X λ ) λ∈Λ be a non-degenerate family of C r vector fields over R 2 . Let u be a divergence-free vector field over R 2 with vorticity ω ∈ C r X . Assume in addition that u ∈ L q for some q ∈ [1, +∞] or that ∇u ∈ L p for some finite p. Then there exists a constant C depending on p and r such that Finally we give a commutator estimate for tangential derivatives and Riesz transform , which will be used in the regularity estimate for X. The proof can be found in the Appendix of [32].
3. Well-posedness and global regularity estimate for System (2). In this section, we will give some a priori estimates for (u, θ), then obtain the global existence and uniqueness results for system (2). Firstly, basic L 2 energy estimate for θ shows that Then using the estimate (18) and the divergence free condition of u, one can obtain the L 2 estimate for u and ω that, and Similarly, L q estimate of ω can be obtained by the same way that, So we need to do the L q estimate for ∇θ firstly. The following proposition alerting the W 2,p estimate for θ. Proposition 1. Let (u 0 , θ 0 ) satisfies the assumptions in Theorem 1.3, then for some σ ≥ 1, we have Proof. Multiplying the second equation of system (2) by |θ| q−2 θ (2 < q < ∞) and integrating over R 2 with respect to x, according to the divergence free condition of u, we have 1 q Then integrating for time variable, we obtain In order to obtain the first and second order derivative estimates for θ, we need first rewrite the second equation of system (2) as Then θ can be represented by where (e t∆ ) t>0 stands for the heat semi-group. Applying ∇ to (24) and taking L 2σ For F 0 , according to Lemma A.2, we have For the case 1 p + 1 r ≤ 1, by taking σ = r and making use of Proposition 4, we obtain θ 0 For the case 1 p + 1 r > 1. Taking 1 σ = 1 p + 2 r − 1 (which implies σ > 1) and by Proposition 4, θ 0 Then we estimate F 1 . Making use of Lemma 2.2 and Hölder inequality, we can bound F 1 by

MARIUS PAICU AND NING ZHU
For u L ∞ t (L ∞ x ) , by interpolation and Lemma 15, for any 2 < p < ∞.
Combining with (19) and (21), we obtain Inserting (28) into (27), we deduce that Because θ 0 ∈ B 2− 2 r p,r , by Proposition 4 in the Appendix, it is not hard to see θ 0 ∈ L 2p . Combining with the estimate (22), we have θ L ∞ t (L 2p x ) is bounded. Then making use of Young's inequality, we get F 1 is bounded by For F 2 , also by Lemma 2.2, According to (12), we obtain Inserting the estimates (26), (30) and (32) into (25), one can deduce Then we give the estimate of ∇ 2 θ. According to (23), Then we have According to Lemma A.2 in the Appendix, In the case 1 p + 1 r ≤ 1, we take σ = r, so we have ON THE 2D BOUSSINESQ SYSTEM 5719 For 1 p + 1 r > 1. Noticing that 1 σ = 1 p + 2 r − 1, that is 1 < σ < r. Then by Proposition 4, Next we estimate G 1 , by Lemma 2.2 and Hölder inequality and estimate By interpolation , and combining with the estimate (19), (20), (33) and (37), we can deduce Similarly, Inserting the estimates (36)-(40) to (35), we deduce that , which complete the proof of this proposition.
With these regularity estimates of (u, θ), it is enough to derive the existence and uniqueness results for system (2). For the proof of the existence, we make use of the Friedrichs method. First we define the spectral cut-off as follows: N ) is the characteristic function on B(0, N ). We define Let P denote the Leray projector over divergence free vector fields. Now we consider the following approximate system with smooth initial data From the Cauchy-Lipschitz Theorem, we can get a unique smooth solution (u N , θ N ) in C 1 ([0, T * ); L 2 N ). Due to P 2 = P, J 2 N = J N and PJ N = J N P, we can discover that (Pu N , θ N ) and (J N u N , J N θ N ) are also solutions to the approximate system (42) with the same initial condition. Thanks to the uniqueness, we deduce Thus the approximate system (42) reduces to According to the previous a priori estimates, we have for any T > 0, Because L 2 is (locally) compactly embedded in H −1 . Then by the standard Aubin-Lions theorem and Cantor diagonal process, we can prove there exists a subsequence of (u N , θ N ) N ∈N strong convergence to its limit (u, θ) Then with these results, it is enough to pass the limit in (43) to obtain the existence result for (2).
(44) Standard L 2 estimate combined with Hölder inequality yields for all q ∈ [2, ∞), Then by interpolation, this inequality can be written as: with Noticing that in (20) and (41) Next we deal with δθ. Also from L 2 estimate, Then by interpolation and Young's inequality, we get 1 2 Let ε be a small parameter (tend to 0). Denote

Performing a time integration yields
Having ε tend to 0, we end up with for all t > 0. Because ∇u 1 (t) L is locally bounded. Hence one may find a positive time T * such that Taking q tend to infinity in (47) entails that (δθ, δu) ≡ 0 on [0, T * ]. Then by standard connectivity argument, we can conclude the uniqueness for [0, T ] with all T > 0.

4.
A priori estimates for the striated regularity. In this subsection, we will give the estimates of tangential derivatives of ω and regularity estimates of X. The first lemma gives L p (p ∈ [1, ∞]) estimate of X.
Then the solution X of equation (9) satisfies for any t > 0.
Proof. Multiplying (9) by |X| p−2 X (1 ≤ r < ∞) and integrating over R 2 with respect to x, by Hölder inequality, we obtain Then by Grönwall's Lemma, we deduce the second inequality of (48). The first equality can be obtained by the time reversibility. And taking r → ∞ to obtain the result for the case r = ∞.
Then we give a proposition which alerting the C s (0 < s < 1) estimate for X. As a by-product, we can also obtain the Lipschitz information for the velocity u. Moreover, Before we prove this proposition, we need firstly give the L p estimate of ∂ X ω. Applying ∂ X to the vorticity equation (3), according to (10), we get ∂ X ω satisfies the following equation Multiplying the equation (51) by |∂ X ω| p−2 ∂ X ω (2 ≤ p < ∞), and integrating over R 2 with respect to x, because u satisfies the divergence-free condition, by Hölder inequality, Then integrating in time and combining with the results of Proposition 1 and Lemma 4.1, Then we give the proof of Proposition 2.
Proof of Proposition 2. Firstly, making use of Lemma 2.3 to (9), we can get the Hölder norm of X satisfies that where we can chooseC > 2. In order to estimate Hölder norm of ∂ X u, we need the following estimate which proof can be found in [7,2], By Sobolev embedding L p → C s−1 (1 − s = 2 p ) and estimate (52), we obtain Inserting (54) and (55) into (53), one can deduce that Denoting Then according to the above estimates, we obtain By Grönwall's Lemma, According to the definition of F (t), we obtain the Hölder estimate of X that, Recalling the logarithmic inequality in Lemma 2.4 that where ω C s X is defined in Definition 1.1. Noticing that ω L 2 ∩L ∞ has been proven to be boundedness in section 3. Then inserting the estimates (55), (56) into (57), it follows that Making use of Grönwall's Lemma, one can deduce

MARIUS PAICU AND NING ZHU
Combining the estimates (56) and (58), we can obtain the desired Hölder norm of X, Then inserting the estimate (58) into (52), we can conclude ∂ X ω L ∞ t (L p x ) is bounded, which completes the proof of this proposition.
So compared with (51), we need to estimate divX additionally. Because divX satisfies a transport equation (11), using Lemma 2.3, we have which share a similar structure with (52), so the other estimates are the same as Proposition 2.
Then we give a proposition about theẆ 1,p estimate for X. Proof. Applying ∂ i (i = 1, 2) to (9), we can obtain ∂ i X satisfies the following equation, Multiplying by |∂ i X| p−2 ∂ i X and integrating over R 2 with respect to x, according to the divergence-free condition of u and Hölder inequality, Noticing that by equation (15), Then we can write For M 1 , by Hölder inequality,

ON THE 2D BOUSSINESQ SYSTEM 5725
Thus we have M 1 can be bounded by For M 2 , by Hölder inequality and the boundedness of Riesz transform in L p (1 < p < ∞), Inserting estimates (63) and (64) into (62), According to the estimate of (41), Proposition 2 and then making use of Grönwall's Lemma, we obtain which completes the proof of this proposition.
A. Appendix. This appendix provides the definitions of Besov space and some related facts are used in the previous sections. Firstly we present the classical Littlewood-Paley theory in R d which plays an important role in the proof of our result. Let χ be a smooth function support on the ball B {ξ ∈ R d : |ξ| ≤ 4 3 } and ϕ be a smooth function support on the ring C {ξ ∈ R d : 3 4 ≤ ξ ≤ 8 3 } such that χ(ξ) + q≥0 ϕ(2 −q ξ) = 1, for all ξ ∈ R d , q∈Z ϕ(2 −q ξ) = 1, for all ξ ∈ R d \ {0}.
Next we state the definition of homogeneous and non-homogeneous Besov spaces through the dyadic decomposition, more details can be found in [2]. 2 qs ∆ q f L p for r = ∞.
The following proposition lists some useful equivalence and embedding relations. (R d ).
The above properties also valid for homogeneous Besov space. For Besov space with negative index, we have the following equivalent definition.