ABSTRACT

In this paper, we consider the global regularity of 2D magnetic Bénard fluid equations with almost magnetic diffusion and thermal diffusivity and without kinematic viscosity. We focus on this goal in two ways. In one way, the magnetic diffusion and thermal diffusivity are separately given by ${\mathfrak{D}}_2$ and ${\mathfrak{D}}_3$ two Fourier multipliers whose symbols $m_2$ and $m_3$ are respectively given by $m_2\left(\xi \right)\equiv |\xi {|}^2\log (e+|\xi {|}^2{)}^{\text{β}}$ and $m_3\left(\xi \right)\equiv |\xi {|}^2\log (e+|\xi {|}^2{)}^{\gamma }$. In another way, we generalize the previous case and the magnetic diffusion and thermal diffusivity are separately given by ${\mathfrak{L}}_2$ and ${\mathfrak{L}}_3$ two Fourier multipliers whose symbols $m_2$ and $m_3$ are respectively given by $m_2=|\xi {|}^2{g}_2\left(\xi \right)$ and $m_3=|\xi {|}^2{g}_3\left(\xi \right)$, where $g_j=g_j\left(|\xi |\right)$ $(j=2,3)$ are two radial non-decreasing smooth functions.

摘要

在本文中，我们考虑了具有几乎磁扩散和热扩散率且没有运动粘度的二维磁 Bénard 流体方程的全局规律性。我们以两种方式关注这一目标。在一种方式中，磁扩散率和热扩散率分别由下式给出${\mathfrak{D}}_2$和${\mathfrak{D}}_3$两个傅立叶乘法器，其符号${米}_2$和${米}_3$分别由${米}_2\left(\xi \right)\equiv |\xi {|}^2\mathrm{日志}(e+|\xi {|}^2{)}^{\text{β}}$和${米}_3\left(\xi \right)\equiv |\xi {|}^2\mathrm{日志}(e+|\xi {|}^2{)}^{\gamma }$. 以另一种方式，我们推广前一种情况，磁扩散率和热扩散率分别由下式给出${\mathfrak{大号}}_2$和${\mathfrak{大号}}_3$两个傅立叶乘法器，其符号${米}_2$和${米}_3$分别由${米}_2=|\xi {|}^2{G}_2\left(\xi \right)$和${米}_3=|\xi {|}^2{G}_3\left(\xi \right)$， 在哪里$G_j=G_j\left(|\xi |\right)$ $(j=2,3)$是两个径向非递减平滑函数。