标题： 关于与二维和三维随机Navier-Stokes方程相关的鞅问题 显示英文标题

标题： On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations

摘要： 我们考虑与二维或三维纳维-斯托克斯方程相关的鞅问题。 存在性是已知的，并且最近已经证明可以构造与这些方程相关的马尔可夫转移半群。 我们研究与这些方程相关的科尔莫戈罗夫算子。 它可以在无限维希尔伯特空间上形式地定义为一个微分算子。 也可以以抽象的方式定义为转移半群的无穷小生成元。 我们明确这些抽象算子的核，并将它们与这些核上的具体微分算子相匹配。 在二维情况下，核是明确的，我们可以使用经典论证来证明鞅问题的唯一性。 在三维情况下，我们只能展示一个抽象定义的核，该核无法证明鞅问题的唯一性。 相反，我们展示了一个修改后的科尔莫戈罗夫算子的核，该核使我们能够证明在解保持正则的时间内鞅问题的唯一性。 显示英文摘要

摘要： We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.