Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces

Min Fu,Chao Cai

doi:10.1155/2017/6754780

Abstract

This study is focused on the pressure blow-up criterion for a smooth solution of three-dimensional zero-diffusion Boussinesq equations. With the aid of Littlewood-Paley decomposition together with the energy methods, it is proved that if the pressure satisfies the following condition on margin Besov spaces,π(x,t)∈L2/(2+r)(0,T;B˙∞,∞r)forr=±1,then the smooth solution can be continually extended to the interval(0,T⁎)for someT⁎>T. The findings extend largely the previous results.

Highlights

Introduction and Main ResultsIt is well known that mathematical models in fluid dynamics have attracted more and more attention in the past ten years [1]
We consider the dynamical models of the ocean or the atmosphere which arise from the density dependent incompressible Navier-Stokes equations by using the so-called Boussinesq approximation [2]
Jia et al [13] recently studied the blow-up criterion for local smooth solutions of zero-diffusive Boussinesq equations (1) in the large critical Besov space

Summary

Introduction and Main Results

It is well known that mathematical models in fluid dynamics have attracted more and more attention in the past ten years [1]. Jia et al [13] recently studied the blow-up criterion for local smooth solutions of zero-diffusive Boussinesq equations (1) in the large critical Besov space. Wang [14] proved the blow-up criterion for the zero-diffusive Boussinesq equation when the velocity components satisfy. The aim of the present paper is to improve the pressure blow-up criterion for smooth solution of three-dimensional zero-diffusion Boussinesq equations in the margin Besov spaces r = ±1 in (9); more precisely, we will prove the following result. Compared with the many previous results on the pressure regularity criterion for full viscous fluid dynamical models such as Navier-Stokes equations and MHD equations (see [18]), the zero-diffusion Boussinesq equations (1) do not have the important inequality. Since it is interesting and important to consider this issue in some working space such as Morrey Space (see [17, 21, 22]), we will focus on this problem in the forthcoming paper

Preliminary

Proof of Theorem 1