标题： 分数阶随机Landau-Lifshitz Navier-Stokes方程在维度$d \geq 3$中的存在性和（非）平凡性 显示英文标题

标题： Fractional stochastic Landau-Lifshitz Navier-Stokes equations in dimension $d \geq 3$: Existence and (non-)triviality

摘要： 我们研究在$d\ge 3$中的分数随机纳维-斯托克斯方程，该方程由随机力$(-\Delta)^{\frac{\theta}{2}}\xi$驱动，正如我们所展示的，这对应于物理学文献中分数版本的朗道-利夫希茨随机力。 我们在环面$\mathbb T^d$上获得了对于$\theta > \frac{d}{2}$的鞅解的存在性和唯一性。 对于$\theta \le 1$，该方程是超临界的，我们通过引入伽辽金逼近来正则化问题，并研究截断模型在$\RR^d$上的大尺度行为。 我们证明，在$\theta < 1$时，伽辽金逼近中的非线性项在大尺度上消失，并且该模型收敛到线性化方程。 对于$\theta = 1$，非线性项对大尺度行为给出了非平凡的贡献，我们猜想大尺度行为由一个有效扩散率严格更大的线性模型给出，与简单地去掉非线性项相比。 有效扩散率明确地用模型参数表示。 显示英文摘要

摘要： We investigate fractional stochastic Navier-Stokes equations in $d\ge 3$, driven by the random force $(-\Delta)^{\frac{\theta}{2}}\xi$ which, as we show, corresponds to a fractional version of the Landau-Lifshitz random force in the physics literature. We obtain the existence and uniqueness of martingale solutions on the torus $\mathbb T^d$ for $\theta > \frac{d}{2}$. For $\theta \le 1$ the equation is supercritical and we regularize the problem by introducing a Galerkin approximation and we study the large scale behavior of the truncated model on $\RR^d$. We show that the nonlinear term in the Galerkin approximation vanishes on large scales when $\theta < 1$ and the model converges to the linearized equation. For $\theta = 1$ the nonlinear term gives a nontrivial contribution to the large scale beahvior, and we conjecture that the large scale behavior is given by a linear model with strictly larger effective diffusivity compared to simply dropping the nonlinear term. The effective diffusivity is explicitly given in terms of the model parameters.