Global regularity of generalized MHD-Boussinesq equations(广义磁Boussinesq方程的整体正则性)

The purpose of this paper is to study the global well-posedness of generalized MHD-Boussinesq equations with only velocity dissipation in whole space Rn (n≥2). Firstly, by exploiting the structure of this system, we obtain the uniform L2-bound of the global solutions. Then, the uniform H1-bound of the global solutions is proved by making use of logarithmic type interpolation inequality and the improved Gronwall inequality. Finally, by using delicate energy estimates, we overcome the difficulty of lack of dissipation and establish the a priori uniform Hss>1+n/2 estimate which proves the global existence and uniqueness of the classical solutions to this system.(在全空间Rn（n≥2）中研究仅具有速度场耗散的广义磁Boussinesq方程的整体适定性。首先，利用方程的结构得到整体解的一致L2界；然后，利用对数型的插值不等式和改进的Gronwall 不等式证明了整体解的一致H1界；最后，利用精细的能量估计，克服方程耗散缺失带来的困难，建立了解的整体一致Hss>1+n/2先验估计，证明了该方程经典解的整体存在唯一性。)