On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

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                        <mi>N</mi>
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                           <mrow>
                              <mi>s</mi>
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                  </jats:inline-formula> with <jats:inline-formula>
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                        <mi>s</mi>
                        <mo>></mo>
                        <mn>3</mn>
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                  </jats:inline-formula>, the life span of the classical solutions satisfies <jats:inline-formula>
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                        <mi>T</mi>
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                        <mi>C</mi>
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                                 <mn>3</mn>
                                 <mo>/</mo>
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                  </jats:inline-formula>.</jats:p>


Introduction
e 2D inviscid Boussinesq equations read as z t η +(u · ∇)η � 0, (x, t) ∈ R 2 × R + , e unknown functions η � η(x, t), u � (u 1 (x, t), u 2 (x, t)), q � q(x, t) stand for the temperature, the velocity field, and the scalar pressure, respectively. e → 2 � (0, 1) is the unit vector in the vertical direction. ey can describe the natural convection in the 2D inviscid incompressible fluids such as the dynamics of the ocean or the atmosphere (see, e.g., [1][2][3]). Besides the physical significance, there is also a strong mathematical motivation for studying these equations. In fact, the 2D Boussinesq equation can be used as a model for the 3D axisymmetric Euler equations (see, e.g., [4]). erefore, the study of equation (1) can provide us with the useful information to understand the Euler equations.
Due to the physical and strong mathematical significance of the 2D Boussinesq equations, many researchers have paid much attention on its study. For the Cauchy problem around the stationary solution (0, 0, 0), global regularity of solutions is known when the classical dissipation is present in at least one of the equations (1) (see [5,6]) or under a variety of more general conditions on dissipation (see [7]) or under adding a damp term (see [8][9][10][11]). In contrast, the global regularity problem on the inviscid 2D Boussinesq equations (1) now is still open. Some attempts have been made for a step forward in this direction, which appears to be out of reach in spite of the progress on the local well-posedness and regularity criteria (see [12,13]). On the other hand, for the initial boundary problem, Hu et al. [14] obtained the global well-posedness for the Boussinesq equations with non-slip boundary condition, and Lai et al. [2] and Littman [4] obtained the local well-posedness for the Boussinesq equations with slip boundary condition. Recently, some researchers started to consider the global well-posedness around the stationary solution of the strong stratification [11,15,16]. In particular, Elgindi and Widmayer [15] proved the long-time existence which is the life span of the associated solutions is ε − (4/3) if the initial data are of size ε. In this paper, we will present a new life span which is suitable for general initial data by the method of Strichartz estimate combining with a blowup criterion.
To state our result more precisely, we firstly consider the solutions of (1) around a stratified solution and rewrite equation (1). It is well known that equation (1) admits an explicit stationary solution (u s , q s , η s ) of the form u s , q s , η s � 0, satisfying the hydrostatic balance where N is called the buoyancy or the Brunt-Vaisala frequency and represents the strength of stable stratification. Setting we can reformulate (1) into Furthermore, let us set and we get where ∇ � (∇, 0) T and J is a constant matrix given by Since ∇ · U � 0, let us introduce the extended Helmholtz projiector of the velocity u onto the divergence-free vector fields which is defined by where R j 1≤j≤2 denote the Riesz transforms on R 2 . Applying the operator P to (7) gives the following equation: e initial data to the above equations are given by Now we state the main result.
Theorem 1. Let s ∈ N satisfy s > 3 and en, equations (10) and (11) possess a unique solution: with where the constant C is independent of N.

Remark 1.
is result implies that the existence time will be larger as the buoyancy increases and the life span of the classical solutions satisfies 1.1. Plan of the Article. e paper is organized as follows. In Section 2, we first derive the explicit formula of solutions to the linearized equations of (10) and (11), and then we establish the decay estimate and Strichartz estimate of a linear propagator. In Section 3, we establish the blowup criterion of equations (10) and (11). In Section 4, we present the proof of eorem 1.

Notations.
roughout this paper, we denote by C the constants which may differ from line to line.
, and B s p,q , s ∈ R, 1 ≤ p, q ≤ ∞ denote the Lebesgue spaces, Sobolev spaces, and the inhomogeneous Besov spaces, respectively. Let F, F − 1 denote the Fourier transformation and inverse Fourier transformation, respectively. e Littlewood-Paley multipliers Δ j∈Z are defined by where e low-frequency multiplier χ(ξ) is defined by

Linearized Equations
In this section, we derive the representation of solutions to the linearized equations of (10) and (11) and establish the decay estimate and Strichartz estimate of the linear propagator e ±iNtw(D) given by

e Representation of Solutions to Linearized Equations.
We study the following linearized equations associated to equations (10) and (11): Applying the Fourier transform to (20) yields where P(ξ) is the multiplier matrix of the operator P defined by which is given explicitly by en, a direct calculation yields and the eigenvalues of P(ξ)JP(ξ) are and the corresponding eigenvectors are us, the solution of (21) is Since ∇ · U � 0, we have 〈U 0 (ξ), e 0 〉 � 0. Hence, we get Setting one has

Decay Estimate.
We derive the following decay estimate of the operator e ±itNw(D) .

Lemma 1. It holds that
in which Φ(ξ) satisfies Actually, the result of Lemma 1 is an immediate consequence of the following lemma.

Lemma 2.
ere exists a positive constant C independent of (t, x) ∈ R 1+2 such that Next, we give the details of the proof of Lemma 2. Firstly, we recall an important lemma (see Keel and Tao [17] and Majda [18]).

Lemma 3.
Let dσ be a surface measure on a smooth surface S in R n and let ϕ ∈ C ∞ 0 (R n ). Suppose that for all x ∈ S, at least k of the principle curvatures are non-zero. en, it holds that where μ is some measure supported on the surface S which is given by By Lemma 3, the decay of dμ(η) is determined by the number of non-vanishing principle curvatures of the surface S. Equivalently, the number of non-vanishing principle curvatures of the surface S is the rank of the Hessian matrix Hρ � (z 2 ρ/zξ j zξ k ) 1≤j,k≤2 . By some computations, we obtain

Journal of Mathematics
From (38), we have (39) shows that the surface S has a non-vanishing principle curvatures unless ξ 2 � 0. us, we decompose where where φ is a smooth function on R such that φ � 1 on For the estimate of ‖I 2 ‖ L ∞, since ξ 2 � 0, we see that From (43) and the fact |ξ| ∼ 1 ((1/4) ≤ |ξ| ≤ 4), we see one non-vanishing principle curvature of the surface S. By Lemma 3, we have Combining (36), (42), and (44), we obtain For small |t|, it is trivial that From (45) and (46), we have us, we complete the proof of Lemma 2.

Strichartz Estimate.
Firstly, we recall the following result obtained by Hu et al. [14].

Lemma 4. Let S(t), t ∈ R be a family of operators. Suppose that for all t, s ∈ R,
en, the estimates hold for all 2 ≤ q, r ≤ ∞ with (q, r, σ) ≠ (2, ∞, 1) satisfying From Lemmas 1 and 4, the fact Φ(D)Δ 0 � Δ 0 , and the scaling in time t↦Nt, we obtain the following.

Blowup Criterion
is section shows a blowup criterion of equations (10) and (11).

Lemma 6. Let U(t) be a solution of equations (10) and (11) defined on a time interval containing [0, T]. en, for any s > 0, we have the bounded estimate
Proof. For the vector variable U(t) � (u(t), u 3 ) T , we obtain We take s derivatives of the first and the second equation in (53), multiply by ∇ s u 3 and ∇ s u, respectively, and integrate over R 2 to obtain Note that Since ∇ · u � 0, we have (57) e commutator-type estimate (see, e.g., [12,15]) provides us with us, adding (54) and (55) gives which implies that by the Gronwall inequality for t ∈ [0, T], e above equality implies that Lemma 6 holds.
By the Duhamel principle, we get Due to Lemma 5 and scaling, we find that for q ≥ 4, In the following, for α > 0, we are going to derive the estimates of By (63) and the Bernstein inequality, we have On the other hand, we have from (61) that (65) and (66) give By (63), we have Similar to (66), we have (68) and (69) imply Combining (67) and (70) yields Define We get from (71) and Lemma 6 that Let q � 4 and suppose that We can choose N sufficiently large such that Combining (73) with (74) gives Finally, the restriction (75) implies T can choose By the bootstrap principle, we deduce from (76) that (74) actually holds. us, from the classical local existence (see [12]) and the blowup criterion Lemma 6, (10) and (11) possess unique classical solutions satisfying u ∈ C([0, T]; H s ).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.