Global well-posedness for the 2D Boussinesq equations with a velocity damping term

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $(0,x_2)$. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [ACWX] (see P.3597).

Many geophysical flows such as atmospheric fronts and ocean circulations (see, e.g., [3,5,8,9]) can be modeled by the Boussinesq equations. Mathematically we regard the 2D Boussinesq equations as a lower-dimensional model of the 3D hydrodynamics equations. It is an open question whether the solutions to the inviscid case (ν = 0) exist for all time or blow-up in a finite time. Indeed, adding only a velocity damping term seems not helpful to get the global solution even under the assumption that the initial data is small enough.
When adding the temperature damping term ηΨ (η > 0) to (1.1), that is, Adhikar et al. [1] proved global well-posedness to (1.2) with the initial data satisfying whereḂ 0 ∞,1 is the Besov space (see [2] for the definition), while they proposed that global wellposedness for (1.1) is an open question (see P. 3597 in that paper). Later, Wan [13] obtained global solutions to (1.2) with large initial velocity data by exploiting a new decomposition technique which is splitting the damped Navier-Stokes equations from (1.2). In fact, in [13], the initial data satisfies for some constants C > 0 and C 0 > 0, where Ω 0 def = ∇ × u 0 and A(ν, u 0 , Ω 0 ) def = ln(e + u 0 H m ν ) exp Inspired by the studies of the MHD equations ( see, e.g., [11,15,16]), the contribution of this article is the global existence and uniqueness of solutions of (1.1) with sufficiently smooth initial data (u 0 , Ψ 0 ) close to the equilibrium state (0, x 2 ). In fact, using Ψ = θ + x 2 , we return to seek the solutions of the following system is the vorticity of the velocity u. When ν = 0, Elgindi-Widmayer [4] proved the long time existence of (1.3), that is, the lifespan of the associated solutions is ǫ − 4 3 if the initial data is of size ǫ. We point out that the initial data in W s,1 (R 2 ) is very crucial in the decay estimate of the semi-group e R 1 t . Later, Wan-Chen [12] obtained global well-posedness under the initial data near a nontrivial equilibrium (0, κx 2 ) (κ large enough), which is consistent with the corresponding work of [4] if κ = 1. Recently, by establishing some Strichartz type estimates of the semi-group e R 1 t , Wan [14] obtained the same result as [4] without assuming that the initial data is in W s,1 (R 2 ).
For some convenience, we set ν = 1, that is, we consider in the present work and use the denotations below: where the definitions of the function spaces H s (R 2 ) and H s (R 2 ) are given in section 2. Now, we state the main result.
There exists a positive constant C ′ such that if then system (1.4) has a unique global solution (ω, θ) satisfying for all T > 0.
Remark 1.2. We shall point out the solutions in Sobolev space do not grow over time, which is very different from the work [15] on the 2D MHD system with a velocity damping term, where the Sobolev norm of the solutions may grow over time, see P.2631 in that paper. Furthermore, our result gives a positive answer to the question proposed in [1] under the initial data near equilibrium state (0, x 2 ).
Let us making the following comments concerning this theorem: • By the standard energy method, we need to bound the integral T 0 ∂ 2 u 2 L ∞ dt, which is very difficult to be controlled, even if one applies the techniques in the following sections. By utilising −u 2 = ∂ t θ + u · ∇θ and integrating by parts, the estimate of this integral reduces to the estimate of K(T ) (see sections 3 and 4).
• To get the estimate of K(T ), we shall estimate the integral T 0 u 2 4 3 L ∞ dt, which strongly relies on the diagonalization process (section 5) and the energy estimate II (section 6). In addition, we shall use some unexpected techniques like (5.10).
• The unnatural condition in (1.5) is that the initial data satisfies ω 0 ∈Ḣ −2 (R 2 ) and θ 0 ∈Ḣ −1 (R 2 ). In fact, these conditions play a very important role in the energy estimate II and the estimate of The present paper is structured as follows: In section 2, we provide some definitions of spaces and several lemmas. Section 3 devotes to obtaining the estimate of E(T )+E 1 (T ). Section 4 bounds the estimate of T 0  L ∞ dt. In section 6, we give the estimate of E 2 (T ). At the last section, we prove Theorem 1.1.
Let us complete this section by describing the notations we shall use in this paper. Notations For A, B two operator, we denote by [A, B] = AB − BA the commutator between A and B. In some places of this paper, we may use L p ,Ḣ s and H s to stand for L p (R 2 ),Ḣ s (R 2 ) and H s (R 2 ), respectively. a ≈ b means C −1 b ≤ a ≤ Cb for some positive constant C. t means 1 + t. The uniform constant C may be different on different lines. We use f L p to denote the L p (R 2 ) norm of f , and use L p T (X) = L p ([0, T ]; X). We shall denote by (a|b) the L 2 inner product of a and b, and

Preliminaries
In this section, we give some necessary definitions, propositions and lemmas.

Minkowski's inequality for integrals and the embedding relation
Integrating the inequality above in time, and then using Hölder's inequality can yield the desired result.

Energy estimate I
In this section, we prove some a priori estimates, which are given by the following lemma: Remark 3.2. ω 0 ∈Ḣ −1 is a natural condition, which is equal to u 0 ∈ L 2 . Thanks to this condition, together with θ 0 ∈ L 2 , we can get the global kinetic energy.
Proof. Using the cancelation relations Using (u · ∇Λ s ω|Λ s ω) = 0 and (2.1), we have For the estimate of I, using (u · ∇∂ s+1 θ|∂ s+1 θ) = 0, we obtain Depending on the derivatives ∂ s+1 , we will split the estimate into two cases: (1) ∂ s+1 including at least one derivative on x 1 and (2) ∂ s+1 = ∂ s+1 2 . For the case (1), it is easy to get For the case (2), thanks to where we have used ∂ 2 u 2 = −∂ 1 u 1 and integration by parts two times, then where 3) The estimate of I 1 will be given in section 4. So (3.2) reduces to Next, we will find the dissipation of θ. As a matter of fact, we have where we have used (u · ∇ω|∂ 1 θ) + (u · ∇∂ 1 θ|ω) = 0. It is easy to get Using the cancelation property Multiplying (3.4) by 2C 0 , and adding the resulting inequality to (3.6), we can get where I 1 is defined by (3.3). Using H s , and integrating (3.7) in time can lead to the desired estimate (3.1).

The estimate of
In order to bounding T 0 I 1 (t)dt, we suffice to show the estimate of which is given by the lemma below.
then we shall bound T 0 ∂ 2 u 2 L ∞ dt, which seems difficult, see section 5. In fact, following the techniques in section 5, we can only bound the integral T 0 ∂ 2 u 2 4 3 L ∞ dt. In the following proof, we shall repeatedly use −u 2 = ∂ t θ + u · ∇θ Proof. We shall find a new way to bound this integral. In fact, using u 2 = ∂ t θ + u · ∇θ, we have • The estimate of J 1 Using ∂ t θ = −u · ∇θ − u 2 , we can obtain For J 11 , using ∂ 2 u 2 = −∂ 1 u 1 and integrating by parts two times leads to For J 12 , we have By using ∂ 2 u 2 = −∂ 1 u 1 and integration by parts, we have By using the previous approach, one can easily get the estimate as follows: As for the estimate of K 3 , using the equation of θ two times and ∂ 2 u 2 = −∂ 1 u 1 , we have Integrating by parts two times, one can get Integrating by parts and using Lemma 2.2, we have . Similarly, using Lemma 2.2 again, we can get So we can get the estimate of K 3 , and combining with the estimates of K 1 and K 2 yields For the estimate of J 122 , we have Hence, • The estimate of J 2 We have It is easy to get while the second term can be bounded as the estimate of K 3 . So we can get We can get the desired result by combining with the estimates of J 1 and J 2 .
One can deduce from Lemma 3.1 and Lemma 4.1 that It is easy to see that M (T ) is a good term. Next, we suffice to show the estimate of K(T ). Thanks to L ∞ dt, we shall first investigate the spectrum properties to the following system: then we can get from (5.1) that One can get the eigenvalues of the matrix A as follows: and where the matrixes P and P −1 are given by Thanks to (5.2) and (5.4), denote So we can get ω(ξ, t) =M 1 (t) ω 0 (ξ) + M 2 (t) θ 0 (ξ) and , when |ξ| ≥ 2|ξ 1 |, In this case, one can get from (5.3), (5.6) and (5.7) that Consequently, using (5.5), we can obtain Using (5.9), we have which yields where 0 < η ≪ 1, and by Young's inequality, (2.1) and H 1+η (R 2 ) ֒→ L ∞ (R 2 ), then we have For the last term L 4 , we shall first give some analysis on u · ∇θ. Applying u · ∇ = u 1 ∂ 1 + u 2 ∂ 2 , we can get where L 41 can be bounded like L 3 , indeed, denote similarly, we also have However, this strategy can not be used to the estimate of L 42 , due to the fact that can not be bounded by E 1 (T ). In order to avoiding this bad term, we shall seek a different approach. We begin with the analysis of u 2 ∂ 2 θ. Using u 2 = −∂ 1 Λ −2 ω, one has with (5.7) leads to So we have (5.11) Denote g 3 (s) = Λ −2 ω H 1+η ∂ 1 ∂ 2 θ H 1+η χ (0,T ) , by Young's inequality and (2.1), we have . Denote g 4 (s) = Λ −2 ω∂ 2 θ H 2+η χ (0,T ) , by Young's inequality, (2.1) and Thus we have with the estimate of L 41 leads to Therefore, we can get  In this case, we have and then It means M i (t) (i = 1, 2) admits a faster decay then the Case 1. So one can obtain by following the previous procedure that We have Similarly, we also have Then one can get by following the previous procedure line by line that To closing this estimate, we need the estimate of E 2 (T ), which is the main part in the following section.

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1. Local well-posedness can be proved by using standard method. Here we only show the global a priori bound. Combining with (5.13) and (6.4), we can get A(T ) + A 1 (T ) ≤C 2 A(0) + C 2 A(T ){A(T ) + A 1 (T ) + A(T )    where T ⋆ > 0 is the maximal existence time of the local solution. AssumeT < T ⋆ . Thanks to (1.5), we can get from (7.1) that which yields a contradiction withT < T ⋆ by the continuous arguments. Thus we can get T = T ⋆ , which leads to the desired result.