论文标题
具有奇异潜力的随机Allen-Cahn-Navier-Stokes系统
A stochastic Allen-Cahn-Navier-Stokes system with singular potential
论文作者
论文摘要
我们在平滑的两维域中使用随机初始数据研究了Allen-Cahn-Navier-Stokes系统的随机版本。该系统由Navier-Stokes方程和对流allen-cahn方程，以及两个独立的随机性来源，由一般乘法型Wiener噪声给出。特别是，艾伦-CAHN方程的特征是对数类型的奇异潜力，如模型的经典热力学推导所规定的。该问题具有（体积平均）速度字段的无滑动边界条件，以及该顺序参数的均匀的Neumann条件。我们首先证明了在两个和三个空间维度中存在分析性弱的Martingale解决方案。然后，在两个维度上，我们还可以避开路线唯一性以及独特的概率较长的解决方案。最终，还显示了通过将经典的DE Rham定理概括为随机过程的适当概括，也显示了压力的存在和独特性。
We investigate a stochastic version of the Allen-Cahn-Navier-Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier-Stokes equation coupled with a convective Allen-Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen-Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also estabilish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown.
