Blow-up criterion for the density dependent inviscid Boussinesq equations

In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in RN(N≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{N}\ (N\geq 2)$\end{document}. By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.


Introduction
This paper is devoted to investigating the initial value problem associated to the following density-dependent inviscid incompressible Boussinesq equations in (x, t) ∈ R N × (0, +∞) with N ≥ 2: (ρ, v, θ )| t=0 = (ρ 0 , v 0 , θ 0 ), (1.1) where e N denotes the vertical unit vector (0, . . . , 0, 1), and ρ, v, θ , and P denote the fluid density, velocity field, temperature, and pressure, respectively, while ρ 0 , v 0 , and θ 0 are the given corresponding initial data with ∇ · v 0 = 0. When θ ≡ 0, system (1.1) reduces to the initial value problem associated to the incompressible density-dependent Euler equations. Chae and Lee [4] showed the local wellposedness of the incompressible density-dependent Euler equations in the L 2 -type critical Besov space. Zhou et al. [18] generalized the result of [4] to the L p -type critical Besov space and obtained the following blow-up criterion: for 1 < p < ∞. Very recently, Bae et al. [1] derived a refined blow-up criterion When ρ is constant, system (1.1) becomes the initial value problem associated to the homogeneous inviscid Boussinesq equations. The local well-posedness and regularity criteria are well-established; see, for example, [2,3,5,7,9,12,16]. In particular, by using Littlewood-Paley method, the authors in [2] and [7] derived the blow-up criterion (1.3) in Besov-Morrey spaces (see Remark 1.3 in [2]) and Hölder spaces [7], respectively. Let us mention that the global regularity question of the inviscid Boussinesq system (1.1) is a rather challenging problem.
Compared with the homogeneous flow, fewer works are concerned with the nonhomogeneous system (1.1). Regarding the local existence and blow-up criteria results, one can refer to [14,17]. Precisely, Qiu and Yao [14] developed the methods of [4] and [18] and got the blow-up criterion (1.2) in the Besov framework. Xu [17] obtained the blow-up criterion (1.3) for smooth solutions to the 2-dimensional compressible Boussinesq equations. In this paper, we are going to establish the local existence and blow-up criterion (1.3) for the N -dimensional (N ≥ 2) system (1.1) by applying the standard energy method. We suppose that where ρ and ρ are positive constants and assume ρ 0 → ρ as |x| → ∞. Different from the homogeneous case, the classical energy method cannot be applied directly to the equation To obtain the H s estimate of v, we need the elaborate estimates of P. To this end, as in [1], we introduce the following two variables to deal with the term 1 ρ ∇P: As a consequence, we use the usual energy method to deal with P, which satisfies (1.5) By virtue of (1.1) 1 , we see that a and b satisfy with the initial data respectively.
The main result of this paper is stated as follows.

Proof of the main result
The proof of Theorem 1.1 is divided into two parts, i.e., the local existence and the blow-up criterion.
Proof (Local existence). We first recall some basic lemmas that will be applied to the proof of the local existence.
Then for any X 0 ∈ O, there exists a time T such that the ODE Let us first briefly explain the idea of the proof of the local well-posedness, see [13,Chap. 3], or [5] for details. As in [5], we regularize system (1.1) and then due to Lemmas 2.1 and 2.2, for any > 0, we obtain the global solution (a , b , v , θ ) of the regularized Boussinesq equations in Let us mention that, for the proof of the above global existence of regularized solutions, one can refer to Theorem 3.2 in [13]. Next, noting that 4 ) for any s < s. As a consequence, we can find a solution Then, we can prove which is unique.
Moreover, there exist a maximal time of existence T * (possibly infinite) and unique solution Through Sobolev imbedding, we have which means that (a, b, v, θ ) is a classical solution of system (1.1).
Based on the above arguments, here we only present the key part, that is, the solution (a , b , v , θ ) of the regularized Boussinesq equations is uniformly bounded in L ∞ ([0, T]; (H s ) 4 ) with respect to . The remaining parts such as the approximation to system (1.1), the process of taking limits, and that the solution is continuous in time in the highest norm H s are omitted, which can be referred to [13] and [5] for details. To simplify the presentation, we also omit the superscript and denote def = √ -throughout the paper.
Step 1. H s estimate of (a, b, v, θ ). Since divv = 0, it is easy to deduce (see [11, Applying the operator s to the first equation in (1.6) and taking the L 2 inner product with itself, we have as divv = 0, the last term is zero. One gets that Here and in what follows, we will frequently use the following two estimates for s > 0 (see [10]): Similarly, for b and θ , we have Next, we deal with v. Multiplying (1.1) 2 by v and (1.1) 3 by θ , respectively, integrating in R N and combining the resulting equations together, we have which, together with Gronwall's inequality and the bound of ρ, yields we have which yields Step 2. H s estimate of ∇P. We first give the L 2 bound of ∇P. Since 1/ρ ≥ 1/ρ > 0, the classical L 2 theory used to (1.5) which, together with (2.4), gives (2.7) Thanks to (1.5) again, one infers Taking the L 2 inner product with s P in (2.8) yields that Based on that 1/ρ ≥ 1/ρ > 0, we derive That is, which, combined with (2.7), implies Step 3. L ∞ estimate of ∇P. Firstly, by interpolation inequality, we have for N < p < ∞ that (2.12) In order to estimate P L p , we have from (1.5) that Then, by the interpolation inequality and Young's inequality again, one deduces This, together with (2.12) and (2.7), gives Step 4. A priori estimates. Combining (2.6), (2.11), and (2.14) together, we end up with By Sobolev embedding H s → W 1,p ∩ W 1,∞ for s > 1 + N 2 and N < p < ∞, we have This completes the proof of local well-posedness for system (1.1) in H s . Next, we present the proof of the second part in Theorem 1.1, namely, the blow-up criterion.
(Blow-up criterion). We first show the " ⇒ part in (1.7). From the equations of a, b, and θ , we obtain To deal with ∇v L p , we define the vorticity as w Then we turn to consider the following equations: (2.17) where ∇ ⊥ = (-∂ 2 , ∂ 1 ) and ∧ represents the wedge product. Next we only estimate the case N = 3 since the other two cases could be handled similarly.