
BlowUp Criteria for ThreeDimensional Boussinesq Equations in TriebelLizorkin SpacesDOI: 10.1155/2012/539278 Abstract: We establish a new blowup criteria for solution of the threedimensional Boussinesq equations in TriebelLizorkin spaces by using LittlewoodPaley decomposition. 1. Introduction and Main Results In this paper, we consider the regularity of the following threedimensional 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 densitydependent fluid equations by using the socalled 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 [1] for a rigorous justification. It is well known that the question of global existence or finitetime blowup of smooth solutions for the 3D incompressible Boussinesq equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the blowup criterion of the solutions for system (1.1). Recently, Fan and Zhou [2] and Ishimura and Morimoto [3] proved the following blowup criterion, respectively: Subsequently, Qiu et al. [4] obtained Serrintype regularity condition for the threedimensional Boussinesq equations under the incompressibility condition. Furthermore, Xu et al. [5] obtained the similar regularity criteria of smooth solution for the 3D Boussinesq equations in the MorreyCampanato space. Our purpose in this paper is to establish a blowup criteria of smooth solution for the threedimensional Boussinesq equations under the incompressibility condition in TriebelLizorkin 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
