We establish a new blow-up criteria for solution of the three-dimensional Boussinesq equations in Triebel-Lizorkin spaces by using Littlewood-Paley decomposition. 1. Introduction and Main Results In this paper, we consider the regularity of the following three-dimensional incompressible Boussinesq equations: where denotes the fluid velocity vector field, is the scalar pressure, is the scalar temperature, is the constant kinematic viscosity, is the thermal diffusivity, and , while and are the given initial velocity and initial temperature, respectively, with . Boussinesq systems are widely used to model the dynamics of the ocean or the atmosphere. They arise from the density-dependent fluid equations by using the so-called Boussinesq approximation which consists in neglecting the density dependence in all the terms but the one involving the gravity. This approximation can be justified from compressible fluid equations by a simultaneous low Mach number/Froude number limit; we refer to  for a rigorous justification. It is well known that the question of global existence or finite-time blow-up of smooth solutions for the 3D incompressible Boussinesq equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the blow-up criterion of the solutions for system (1.1). Recently, Fan and Zhou  and Ishimura and Morimoto  proved the following blow-up criterion, respectively: Subsequently, Qiu et al.  obtained Serrin-type regularity condition for the three-dimensional Boussinesq equations under the incompressibility condition. Furthermore, Xu et al.  obtained the similar regularity criteria of smooth solution for the 3D Boussinesq equations in the Morrey-Campanato space. Our purpose in this paper is to establish a blow-up criteria of smooth solution for the three-dimensional Boussinesq equations under the incompressibility condition in Triebel-Lizorkin spaces. Now we state our main results as follows. Theorem 1.1. Let , be the smooth solution to the problem (1.1) with the initial data for . If the solution satisfies the following condition then the solution can be extended smoothly beyond . Corollary 1.2. Let , be the smooth solution to the problem (1.1) with the initial data for . If the solution satisfies the following condition then the solution can be extended smoothly beyond . Remark 1.3. By Corollary 1.2, we can see that our main result is an improvement of (1.2). 2. Preliminaries and Lemmas The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier
E. Feireisl and A. Novotny, “The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system,” Journal of Mathematical Fluid Mechanics, vol. 11, no. 2, pp. 274–302, 2009.