Finite energy weak solution of 2d Boussinesq equation with diffusive temperature

We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.


Introduction
The Boussinesq equation was introduced for understanding the effect of potentially large conversions between internal energy and mechanical energy in fluids, and simulates many geophysical flows, such as atmospheric fronts and ocean circulations (see, for example, [36], [41]). Moreover, it was used in recent theoretical discussion of the energetics of horizontal convection and the energetics of turbulent mixing in stratified fluids.
In this paper, we consider the following 2-dimensional Boussinesq system where α < 1 is a positive number, e 2 = (0, 1) T and T 2 is 2d torus. Here, v is the velocity vector, p is the pressure, θ denotes the temperature which is a scalar function. The global well-posedness have been established by many authors for the Cauchy problem of (1.1) in 2d for regularity data(see, for example, [9], [25]). For the 3-dimensional case, the global existence of smooth solution of (1.1) remains open. To understand the turbulence phenomena in hydrodynamics, one needs to go beyond classical solutions, and in this paper we are interested in constructing weak solutions of (1.1) with bounded kinetic energy. The triple (v, p, θ) on [0, 1] × T 2 is called a weak solution of (1.1) if they belong to L ∞ (0, 1; L 2 (T 2 )) and solve (1.1) in the following sense:ˆ1 for all ϕ ∈ C ∞ c ((0, 1) × T 2 ; R 2 ) with divϕ(t, x) = 0. (∂ t φθ + v · ∇φθ + △φθ)dxdt = 0 for all φ ∈ C ∞ c ((0, 1) × T 2 ; R) andˆT 2 v(t, x) · ∇ψ(x)dx = 0 for all ψ ∈ C ∞ (T 2 ; R) and any t ∈ [0, 1]. The study of constructing non-unique or dissipative weak solution to fluid system is very fashionable in recent years, and the construction is based on convex integration method which pioneered by De Lellis-Székelyhidi Jr in [18,21], where the author tackle the Onsager conjecture for the incompressible Euler equation. So far, there are many important work about weak solution of the incompressible Euler equation, see [10,15,19,24,26,28,40,42,43,44,46,47]. The Onsager conjecture was proved by P.Isett in [29], based on a series of process on this problem in [1,3,4,5,11,16,17,22,27], see also [6] for the construction of admissible weak solution. Moreover, the idea and method can be used to construct dissipative weak solution for other model, see [7,31,34,45,48,49,50].
Recently, Buckmaster and Vicol establish the non-uniqueness of weak solution to the 3D incompressible Navier-Stokes in [8] by introducing some new ideas. The main idea is to use "intermittent" building blocks in the convex integration scheme to control the dissipative term △v, called "intermittent Beltrami flow", which are space inhomogeneous version of the classical Beltrami flow, also see [2]. Compared with the homogenous case, the "intermittent Beltrami flow" has different scaling in different L p norms. In particular, one can ensure small L p norm for small p > 1 which is key to control the dissipative term. By choosing the parameter suitably, T.Luo and E.S.Titi in [32] construct weak solution with compact support in time for hyperviscous Navier-Stokes equation. For high dimension(d ≥ 4) stationary Navier-Stokes equation, X. Luo in [35] show the non-uniqueness by constructing the concentrated Mikado flows introduced in [37,38]. Moreover, S.Modena and Székelyhidi established the non-uniqueness for the linear transport equation (and transport-diffusion equation) with divergence-free vector in some Sobolev space, see [37,38], where they use Mikado density and Mikado fields which is highly concentrated such that the L p norm of Mikado field is small for p > 1 small. Motivated by the above earlier works, we consider the 2d Boussinesq equations (1.1) and want to know if the similar phenomena can also happen when adding the temperature effects. Following the general scheme in the construction of non-uniqueness to Navier-Stokes equation in [8], we obtain the following existence result.
and for any t ∈ [0, 1] Remark 1.2. For 0 ≤ α < 1 2 , we also can construct weak solution with prescribed energy curve in the class C([0, 1], L 2 (T 2 )) by the same method. However, in a separate paper [33], we will construct Hölder continuous solution for this case. Remark 1.3. The θ we construct in this paper is rather regular and satisfies energy equality. When θ = 0, the equation (1.1) is 2d Navier-Stokes equation with fractional diffusion, and our construction also work for this case.
The rest of the paper is organized as follows. In Section 2, we state the main proposition and give a proof of Theorem 1.1. In Section 3, we collect some technical tool which will be used frequently. In Section 4, we introduce the intermittent plane wave which is the building block in our perturbation. In Section 5 and 6, we construct velocity perturbations and temperature perturbation, respectively. After the construction, we establish the related estimates. In Section 7, we construct the Reynold-Stress and establish the related estimates. Finally, in Section 8, we give a proof of Proposition 2.2.

Main proposition and proof of main theorem
In this section, we state our main iterative proposition and give a proof of theorem 1.1 by the help of main proposition. (2.2) Here and throughout the paper, S 2×2 0 is the set of trace-free symmetric 2 × 2 matrices. We now state our main proposition, and Theorem 1.1 is a corollary.
Proposition 2.2. Let e(t), α be as in Theorem 1.1 and ε 0 be a universal constant from the Geometric Lemma 4.2. Then there exist universal constant M andM such that the following hold.

5)
and We will prove the above proposition in the next several sections. Here we first give a proof of Theorem 1.1.
Passing into the limit in (2.2), we conclude that (v, θ) solve (1.1) in the sense of distribution. Moreover, by (2.7), Moreover, by (2.12), we deduce that the temperature θ satisfies the energy equality: for every t ∈ [0, 1] This complete the proof of Theorem 1.1. The rest of this paper will be dedicated to prove Proposition 2.2. First, we add perturbations to v 0 and get new functions v 1 as following: 1 , w c 1 , w t 1 are smooth functions given by explicit formulas. We introduce some parameters λ 1 , µ 1 , r 1 , σ 1 satisfying the relation (5.24). After the construction of new velocity v 1 , we construction new temperature θ 1 by solving the following transport-diffusion equation: there exists where θ 0 is the function appeared in Proposition 2.2. After construction of v 1 , θ 1 , we mainly focus on finding functionsR 1 , p 1 with the desired estimates and solving system (2.2).

Technical tool
In this section, we collect some technical tools which will be frequently used in the following.
3.1. Properties of fast oscillatory. In this subsection, we discuss some properties of fast oscillatory and the proof can be found in [37], [38], which was inspired by [8]. More precisely, we give an improved Hölder inequality which concern the L p norm of the product of a slow oscillating function with a fast oscillating function, and a mean value estimate which concern the mean value of the product of a slow oscillating function with a fast oscillating function .
For a given function f : Lemma 3.1. Let f, g : T 2 → R be smooth functions, λ ∈ N. Then for every p ∈ [1, +∞], we have Lemma 3.2. Let f, g : T 2 → R be smooth function with ffl T 2 g(x)dx = 0, and λ ∈ N. Then there hold

Intermittent plane waves
In this section, we describe in detail the construction of the intermittent plane waves which will form the building blocks of the convex integration scheme.
We first recall the following stationary solution for the 2d Euler equation. Our building block in this paper is the inhomogeneous version of it.
Lemma 4.2 (Geometric Lemma). There exists ε 0 > 0, and smooth positive functions γ ξ : Remark 4.3. By rotational symmetry, Geometric Lemma 4.2 also holds for ξ ∈ Λ + 1 . It is convenient to introduce a small geometric constant c 0 ∈ (0, 1) such that Thus, taking trace in both side, we obtain (4.14) 4.2. Intermittent plane flow. The Dirichlet kernelD r is defined as , and it obeys the estimates: for any p > 1, where the implicit constant only depend only on p. Define a 2d square and normalizing to unit size in L 2 , we obtain a kernel which has the property: where the implicit constant only depend on p. This computation is very easy due to the fact: We first fixed a large parameter λ ∈ N, then introduce a parameter σ such that λσ ∈ N which parameterizes the spacing between frequencies. We assume that where c 0 is the constant in Remark 4.3, and N is a fixed integer(for example, we can take N = 5). Furthermore, we introduce a parameter µ ∈ (0, λ), which describes temporal oscillation in the building blocks. As in [8], for ξ ∈ Λ + j , we define a directed and rescaled ( 2π where N is a integer(we can set N = 5 duo to our construction of Λ 0 , Λ 1 ). Observe that η (ξ) (x, t) satisfies the following important identity: for all 1 < p ≤ ∞, for any t.
Let W (ξ) (x) be the stationary wave at frequency λ, namely We define the intermittent plane waves W (ξ) as Remark 4.5. The explicit representation of W (ξ) (x, t) is as following: Some facts about the frequency support of η (ξ) and W (ξ) : Lemma 4.6. We have the following frequency support property: For ξ + ξ ′ = 0, by the definition of c 0 , we have These facts can be obtained directly from the definition. In fact, the frequency support of η (ξ) is obvious. Then, using the fact that 2λσrN ≤ c 0 λ 5 , it's easy to obtain the frequency support of W (ξ) . Finally, a direct computation gives that we obtain the frequency support of From these frequency support properties (from which we can use the Berstein inequality) and the estimates for Dirichlet kernel, we have the following estimates.
Proposition 4.7. Let W (ξ) be defined as above. Then These estimates are direct, and we omit the proof here.

The velocity perturbation: Construction and Estimates
In this section, we construct the perturbation of velocity and give some estimates for it.

Construction of the velocity perturbation.
In this subsection, we give the detailed construction of velocity perturbation. for any y > 0, whereχ j (y) =χ(4 −j y). We then define for all j ≥ 0. Here and throughout the paper we use the notation A = (1 + |A| 2 ) 1 2 where |A| denotes the standard norm of the matrix A. By the definition of the cutoff functions, we have In fact, Thus, for fixed t ∈ [0, 1], we have Hence, for fixed t ∈ [0, 1], we haveˆT Thus, the assumption (2.3) tells us that for every t ∈ [0, 1], where ρ j , j ≥ 1, are defined by and ρ 0 is defined later. We first claim that a (ξ,j) (t, x) is well-defined for j ≥ 1. We only need to show that Id −R 0 ρ j ∈ B ε 0 (Id). In fact, when χ j = 0, there holds Thus, thus a (ξ,j) (t, x) is well-defined for j ≥ 1. We define ρ 0 (t) as following: Due to (5.19), we deduce that ρ 0 (t) is well-defined. Next, we show that In fact, using the assumption (2.4), we know that On the support of χ 0 , there holds Thus, on the support of χ 0 , for any t ∈ [0, 1], there holds Thus, a (ξ,0) is well-defined.

5.1.2.
Construction of velocity perturbation. Let us fix λ 1 , σ 1 , r 1 , µ 1 such that λ 1 σ 1 ∈ N and the integer r 1 , the parameter σ 1 and µ 1 are defined by The principle part of perturbation w p 1 will be defined as where 0 ≤ j ≤ j max . Here and throughout the paper, η (ξ) , W (ξ) denote, respectively, where ξ ∈ Λ (j) . Then we define an incompressibility corrector Here and throughout the paper, we denote ∇ ⊥ as ∇ ⊥ = (−∂ 2 , ∂ 1 ). Thus, we have and div(w p 1 + w c 1 ) = 0. As in paper [8], in addition to the incompressibility corrector w c 1 , we introduce a temporal corrector w t 1 , which is defined by Here Finally, we define the velocity increment w 1 by After the construction of w 1 , we define the new velocity field v 1 as v 1 := v 0 + w 1 .

5.2.
Estimate of the perturbation. In this subsection, we establish some estimates for the velocity perturbation.
Firstly, we collect some estimates concerning the cutoffs function χ j (t, x).
Lemma 5.1. There exists a j max = j max (R 0 , δ) such that χ j (t, x) = 0, for all j > j max .
Moreover, for all 0 ≤ j ≤ j max , there holds Proof. For j ≥ 1, Thus, χ j (t, x) = 0 implies Thus, there exists j max = j max (R 0 , δ) such that More precisely, we have where L is integer, and the constant C also depend on ε 0 , but ε 0 is a universal constant and we omit it.
Proof. Direct computation gives that By the inequality we know that χ j C L t,x ≤ C(ε 0 ,R 0 , δ, L).

Lemma 5.3 (Estimate on the amplitude).
For 0 ≤ j ≤ j max , we have Proof. Recall that By Lemma 5.2, the estimate on a (ξ,j) L ∞ and a (ξ,j) C L t,x is obvious. In fact, noticing (5.28), we have In particular, we have Proof.
Step 1: L 2 estimate. Recall the definition (5.25) of w p 1 , using the support property (5.18) of cutoff function χ i , Proposition 4.7, Lemma 5.3 and Lemma 3.1, we have Moreover, for any t ∈ [0, 1], there hold Thus, we obtain Finally, due to the definition of ρ 0 and estimate (5.23), we deduce that where C 0 is a universal constant. Taking M to be a universal constant, we obtain From the definition (5.26) of w c 1 , Proposition 4.7 and lemma 5.3, and noticing the fact (5.28)(we use this fact frequently below), we deduce that

Recalling (5.26), there holds
Thus, by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we get Recalling (5.27), we have Thus, by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we get The same argument gives that is similar, and we omit the detail here.
Step 4: Time derivative estimate. A direct computation gives that thus by Lemma 3.1, Proposition 4.7 and Lemma 5.3, we deduce that Corollary 5.4. For all 1 < p < 2, by taking λ 1 large enough, we have Proof. From the definition of w 1 , by the L 2 estimate in Proposition 5.1, we deduce that Using the relationship of parameter (5.24) and taking λ 1 large enough, we obtain .
Similarly, we get Then, the parameter relationship gives

Construction and Estimate on temperature perturbation
After the construction of new velocity v 1 , we construct new temperature θ 1 as following. Consider the transport-diffusion equation: where θ 0 (x) is the function in Proposition 2.2. From the standard theory, we know that there exists a unique solution θ 1 ∈ C ∞ ([0, 1] × T 2 ) and it obeys the following estimates: Direct energy estimate gives that

Reynold Stress: Construction and Estimate
7.1. Anti-divergence operator. We first recall the anti-divergence operator: There exists an operator R satisfying the following property: Proof. Let u ∈ C ∞ 0 (T 2 ) be a solution to Then R satisfies the above property.

7.2.
Construction of new errorR 1 . In this subsection, we define the new errorR 1 . We first compute the interaction w p 1 ⊗ w p 1 of principle perturbation. Recalling the definition (5.25) of w p 1 , we have However, by Geometric Lemma 4.2, Thus, there holdsR Furthermore, by (4.15) and using the identity divf = P =0 divf , there hold div j ξ∈Λ + (j) (ξ,j) Using (5.18), we have It follows from Proposition 4.1 that Thus, we obtain Finally, by combining the definition (5.27) of w t 1 , we obtain where we define the new pressure p 1 such that and oscillatory term (7.34) From (7.33), we know that ffl T 2 T 1,osc (t, x)dx = 0. After the computation of interaction of principle perturbation, we defineR 1 as follows: Rtem .

Proof of main Proposition
In this section, we give a proof of Proposition 2.2 by combining the above construction and estimate.
Noticing that 3α − 1 − 2α p < 1 for 1 < p < 2α 3α−1 , thus we first take λ 1 to be a integer, large enough such that Thus, there holds To complete the proof of Proposition 2.2, we only need to estimate the energy difference between e(t) and´T 2 |v 1 (t, x)| 2 dx.
Finally, collecting estimate (1)-(5), noticing the parameter relationship (5.24), taking p sufficiently close to 1 and parameter λ 1 sufficiently large, we arrive at This completes the proof.