标题： 随机幂律流的遍历勒瓦-霍普夫解的存在性与非唯一性 显示英文标题

标题： Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows

摘要： 我们研究在$d \geq 3$维空间中由迹类加性随机强迫驱动的剪切变稀流体流动的长时间行为，其幂律指数范围从$1$到$ \frac{2d}{d+2}$。 我们特别关注勒雷-霍普夫解，即满足能量不等式的解析弱解。 将一种新的与能量相关的泛函引入凸积分技术中，使得可以构造出无限多个对于特定初始值为概率强解的此类解。 此外，我们提供了全局时间估计，从而得出存在无限多个平稳甚至遍历的勒雷-霍普夫解。 这些结果代表了在该幂律指数范围内的随机剪切变稀流体框架中勒雷-霍普夫解的首次构造。 显示英文摘要

摘要： We study long time behavior of shear-thinning fluid flows in $d \geq 3$ dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from $1$ to $ \frac{2d}{d+2}$. We particularly focus on Leray-Hopf solutions, i.e. on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global i time estimates which lead to the existence of infinitely many stationary and even ergodic Leray--Hopf solutions. These results represent the first construction of Leray-Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices.