论文标题
近似于时空白噪声强迫的三维磁流失动力学系统
Approximating three-dimensional magnetohydrodynamics system forced by space-time white noise
论文作者
论文摘要
磁水动力学系统由Navier-Stokes和Maxwell的方程组成，并通过非线性术语的倍数耦合。几十年来，物理学家已经研究了这种系统，该系统已经研究了其解决方案理论的严格证明，最近在Yamazaki（2019，Arxiv：1910.04820 [Math.ap]）中使用了paraconarlolled分布理论和一种恢复术语的技术。当方程被妥善化合时，通过用参数替换分化操作员来替换分化操作员，可以广泛认为，近似方程的解决方案应在参数接近零时收敛到原始方程的解决方案。我们证明，在时空白噪声强迫的三维磁水动力学系统的情况下，我们证明了这一点。具体而言，证明解决方案对近似系统的限制具有另外的32漂移项解决了原始系统。这32个漂移术语取决于近似值的选择，可以在重态化的过程中明确计算，并且基本上代表了它的空间版本$ \ hat {\ mathrm {o}} $ - Stratonovich校正项。特别是，证明再次依赖于耦合的重量化技术的技术，并在许多情况下利用了磁性水力动力学系统的特殊结构。
The magnetohydrodynamics system consists of the Navier-Stokes and Maxwell's equations, coupled through multiples of nonlinear terms. Such a system forced by space-time white noise has been studied by physicists for decades, and the rigorous proof of its solution theory has been recently established in Yamazaki (2019, arXiv:1910.04820 [math.AP]) using the theory of paracontrolled distributions and a technique of coupled renormalizations. When an equation is well-posed, and it is approximated by replacing the differentiation operator by reasonable discretization schemes with a parameter, it is widely believed that a solution of the approximating equation should converge to the solution of the original equation as the parameter approaches zero. We prove otherwise in the case of the three-dimensional magnetohydrodynamics system forced by space-time white noise. Specifically, it is proven that the limit of the solution to the approximating system with an additional 32 drift terms solves the original system. These 32 drift terms depend on the choice of approximations, can be calculated explicitly in the process of renormalizations, and essentially represent a spatial version of It$\hat{\mathrm{o}}$-Stratonovich correction terms. In particular, the proof relies on the technique of coupled renormalizations again, as well as taking advantage of the special structure of the magnetohydrodynamics system on many occasions.
