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

Abstract

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 .

1. Introduction

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., [4]). 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 [7]) 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. [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 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.

Theorem 1. Let satisfy andThen, equations (10) and (11) possess a unique solution:withwhere the constant is independent of .

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.

1.2. Notations

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

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 given by

2.1. The Representation of Solutions to Linearized Equations

We study the following linearized equations associated to equations (10) and (11):

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

Settingone has

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

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 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 [18], 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

From (43) and the fact , we see one non-vanishing principle curvature of the surface . By Lemma 3, we have

Combining (36), (42), and (44), we obtain

For small , it is trivial that

From (45) and (46), we have

Thus, we complete the proof of Lemma 2.

2.3. Strichartz Estimate

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

Lemma 4. Let be a family of operators. Suppose that for all ,

Then, the estimateshold for all with satisfying

From Lemmas 1 and 4, the fact , and the scaling in time , we obtain the following.

Lemma 5. For satisfying we have the following Strichartz estimate:

3. Blowup Criterion

This section shows a blowup criterion of equations (10) and (11).

Lemma 6. Let be a solution of equations (10) and (11) defined on a time interval containing . Then, for any , we have the bounded estimate

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