On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations
In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with being the buoyancy frequency. It is proved that for general initial data with , the life span of the classical solutions satisfies .
The 2D inviscid Boussinesq equations read as
The unknown functions stand for the temperature, the velocity field, and the scalar pressure, respectively. is the unit vector in the vertical direction. They can describe the natural convection in the 2D inviscid incompressible fluids such as the dynamics of the ocean or the atmosphere (see, e.g., [1–3]). 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., ). Therefore, 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 , 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 ) or under adding a damp term (see [8–11]). 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.  obtained the global well-posedness for the Boussinesq equations with non-slip boundary condition, and Lai et al.  and Littman  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  proved the long-time existence which is the life span of the associated solutions is if the initial data are of size . 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 of the formsatisfying the hydrostatic balancewhere is called the buoyancy or the Brunt–Vaisala frequency and represents the strength of stable stratification. Settingwe can reformulate (1) into
Furthermore, let us setand we getwhere and is a constant matrix given by
Since , let us introduce the extended Helmholtz projiector of the velocity onto the divergence-free vector fields which is defined bywhere denote the Riesz transforms on . Applying the operator to (7) gives the following equation:
The initial data to the above equations are given by
Now we state the main result.
Remark 1. This 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
The 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 Theorem 1.
Throughout this paper, we denote by the constants which may differ from line to line. , , and denote the Lebesgue spaces, Sobolev spaces, and the inhomogeneous Besov spaces, respectively. Let denote the Fourier transformation and inverse Fourier transformation, respectively. The Littlewood–Paley multipliers are defined bywhere such that
The low-frequency multiplier is defined by
2. Linearized Equations
2.1. The Representation of Solutions to Linearized Equations
Applying the Fourier transform to (20) yieldswhere is the multiplier matrix of the operator defined bywhich is given explicitly by
Then, a direct calculation yieldsand the eigenvalues of areand the corresponding eigenvectors are
Thus, the solution of (21) is
Since , we have . Hence, we get
2.2. Decay Estimate
We derive the following decay estimate of the operator .
Lemma 1. It holds thatfor all , where the operator is defined byin which satisfies
Actually, the result of Lemma 1 is an immediate consequence of the following lemma.
Lemma 2. There exists a positive constant independent of such that
Lemma 3. Let be a surface measure on a smooth surface in and let . Suppose that for all , at least of the principle curvatures are non-zero. Then, it holds that
2.2.1. The Proof of Lemma 2
By the theory of Fourier transform of measures supported on surfaces , we havewhere is some measure supported on the surface which is given by
By Lemma 3, the decay of is determined by the number of non-vanishing principle curvatures of the surface . Equivalently, the number of non-vanishing principle curvatures of the surface is the rank of the Hessian matrix . By some computations, we obtain
From (38), we have
(39) shows that the surface has a non-vanishing principle curvatures unless . Thus, we decomposewherewhere is a smooth function on such that on and .
For the estimate of , by Lemma 3, we have
For the estimate of , since , we see that
For small , it is trivial that
Thus, we complete the proof of Lemma 2.
2.3. Strichartz Estimate
Firstly, we recall the following result obtained by Hu et al. .
Lemma 4. Let be a family of operators. Suppose that for all ,
Then, the estimateshold for all with satisfying
Lemma 5. For satisfying we have the following Strichartz estimate:
3. Blowup Criterion
Proof. For the vector variable , we obtainWe take derivatives of the first and the second equation in (53), multiply by and , respectively, and integrate over to obtainNote thatSince , we haveThe commutator-type estimate (see, e.g., [12, 15]) provides us withThus, adding (54) and (55) giveswhich implies that by the Gronwall inequality for ,The above equality implies that Lemma 6 holds.
4. Proof of Theorem 1
Applying to (10) and using the fact gives
By the Duhamel principle, we get
Due to Lemma 5 and scaling, we find that for ,
In the following, for , 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
By (63), we have
Similar to (66), we have
Let and suppose that
We can choose sufficiently large such that
Finally, the restriction (75) implies can choose
By the bootstrap principle, we deduce from (76) that (74) actually holds. Thus, from the classical local existence (see ) and the blowup criterion Lemma 6, (10) and (11) possess unique classical solutions satisfying .
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the NSFC (grant no. 11871302), China Postdoctoral Science Foundation (grant no. 2019M652348), Natural Science Foundation of Chongqing (grant no. cstc2020jcyj-msxmX0123), and Technology Research Foundation of Chongqing Educational Committee (grant nos. KJQN201900539 and KJQN202000528).
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