高斯波动：在二维及以上维度的随机Landau-Lifshitz Navier-Stokes方程的高斯波动
Gaussian Fluctuations for the Stochastic Landau-Lifshitz Navier-Stokes Equation in Dimension $D\geq2$
AI总结：
本文通过分析随机Landau-Lifshitz Navier-Stokes方程的高斯波动，利用特定方法得到随机热方程的收敛性，并修正了先前猜想中的错误。
摘要：
我们重新审视随机Landau-Lifshitz Navier-Stokes方程（LLNS）在临界点及以上规模的高斯波动，使用\cite{CGT24}中的方法。在$d\geq3$中采用经典扩散尺度，在$d=2$中采用弱耦合尺度，我们得到正则化LLNS收敛到具有非平凡重新规范化的系数的随机热方程。此外，我们得到了当$d\geq3$时有效系数的渐近展开，并证明了\cite[猜想6.5]{JP24}中的系数是错误的。我们证明的新成分是通过跟踪向量在Leray投影作用下的演化进行逐个情况分析，结合使用生成元的反对称部分和旋转坐标变换，从原始耦合系统推导出所需的解耦随机热方程。
英文摘要：
We revisit the large-scale Gaussian fluctuations for the stochastic Landau-Lifshitz Navier-Stokes equation (LLNS) at and above criticality, using the method in \cite{CGT24}. With the classical diffusive scaling in $d\geq 3$ and weak coupling scaling in $d=2$, we obtain the convergence of the regularised LLNS to a stochastic heat equation with a non-trivially renormalized coefficient. Moreover, we obtain an asymptotic expansion of the effective coefficient when $d\geq3$, and show that the one in \cite[Conjecture 6.5]{JP24} is incorrect. The new ingredient in our proof is a case-by-case analysis to track the evolution of the vector under the action of the Leray projection, combined with the use of the anti-symmetric part of the generator and a rotational change of coordinates to derive the desired decoupled stochastic heat equation from the original coupled system.
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