论文标题
确定性和随机3D NAVIER的非唯一恐怖性 - Stokes和Euler方程
Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations
论文作者
论文摘要
我们将无限的许多固定解决方案以及沿形的固定解决方案建立在确定性和随机设置中的三维纳维尔和欧拉方程中的存在，这是由加性噪声驱动的。这些解决方案属于规则性类别$ c（\ Mathbb {r}; h^{\ vartheta}）\ cap c^{\ vartheta}（\ mathbb {r}; l^{2}）$，对于某些$ \ vartheta> 0 $，并满足分析性较差的方程。 Euler方程的解决方案是作为固定溶液对Navier -Stokes方程的消失的粘度限制而获得的。此外，无论其结构如何，满足适当的力矩界限的欧拉方程的每个固定解决方案都是固定的分析性弱解决方案的限制，用于Navier的固定定律 - 固定的粘度方程，具有消失的粘度。我们的结果基于凸集成方法的新型随机版本，该方法在上述函数空间中提供了局部稳定的力矩边界。
We establish the existence of infinitely many stationary solutions, as well as ergodic stationary solutions, to the three dimensional Navier--Stokes and Euler equations in both deterministic and stochastic settings, driven by additive noise. These solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of stationary solutions to the Navier--Stokes equations. Furthermore, regardless of their construction, every stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of stationary analytically weak solutions to Navier--Stokes equations with vanishing viscosities. Our results are based on a novel stochastic version of the convex integration method, which provides uniform moment bounds locally in the aforementioned function spaces.
