论文标题
部分可观测时空混沌系统的无模型预测
Large time behavior of the solutions to 3D incompressible MHD system with horizontal dissipation or horizontal magnetic diffusion
论文作者
论文摘要
在本文中，我们考虑了全球解决方案对3D各向异性不可压缩的MHD系统的渐近行为。对于具有水平耗散和充分磁性扩散的3D MHD系统，这表明$ \ uh（t）$以$ o（t^{ - （1- \ f {1} {p} {p} {p} {p} {p}}）}）$，$ u_3（t）$衰变$ o（t^{ - \ f {3} {2}（1- \ f {1} {p}）}）$和$ b（t）$以$ o（t^{ - \ f { - \ f {3} {2} {2} {2} {2} {1- \ f {1} {1} {p} {p} {p} {p {p} {1}} $此外，我们给出了解决方案的渐近扩展。我们证明，$ \ uh（t）$的前项是线性解和两个非线性耦合效果的两个积分的组合，而对于$ u_3（t）$，仅由线性解决方案给出，而没有磁场的影响。尽管速度的耗散较弱，但我们表明，完整的磁扩散足以保持磁场的渐近膨胀基本上是预期的。但是，事实证明，磁场会影响$ u_3（t）$的高阶渐近扩展。获得的结果揭示了磁场在各向异性不可压缩MHD方程的渐近分析中所起的至关重要的作用，并有望从数学角度增强对不可压缩MHD方程的理解。
In this paper, we consider the asymptotic behavior of global solutions to 3D anisotropic incompressible MHD systems. For the 3D MHD system with horizontal dissipation and full magnetic diffusion, it is shown that $\uh(t)$ decays at the rate of $O(t^{-(1-\f{1}{p})})$, $u_3(t)$ decays at the rate of $O(t^{-\f{3}{2}(1-\f{1}{p})})$ and $B(t)$ decays at the rate of $O(t^{-\f{3}{2}(1-\f{1}{p})-\f{1}{2}})$. Furthermore, we give the asymptotic expansion of solutions. We prove that the leading term of $\uh(t)$ is a combination of linear solution and two integrals from nonlinear coupling effects, while for $u_3(t)$ the leading term is given by only the linear solution without the influence of magnetic field. Though the dissipation of velocity is weak, we show that the full magnetic diffusion is robust enough to keep the asymptotic expansion of magnetic field basically expected. However, the magnetic field turns out to affect the higher order asymptotic expansions of $u_3(t)$. The obtained results reveal the crucial role played by the magnetic field in the asymptotic analysis of anisotropic incompressible MHD equations and are expected to enhance the understanding of incompressible MHD equations from the viewpoint of mathematics.
