论文标题
三维随机无关原始方程
Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations
论文作者
论文摘要
我们研究了对三维无粘性原始方程（PES，也称为静液压Euler方程）的随机效应。具体而言，我们考虑了比乘法噪声更大的噪声，并且由于没有水平粘度的PES的Sobolev空间中的不良性，在分析功能空间中起作用。在适当的条件下，我们证明了Martingale解决方案和路径唯一性的局部存在。通过添加垂直粘度，即考虑到静液压Navier-Stokes方程，我们可以放松对初始条件的限制，仅在垂直变量中具有Sobolev规则性的水平变量中仅是分析性的，并允许在垂直方向上传输噪声。我们建立了Martingale解决方案和路径唯一性的局部存在，并表明该解决方案在垂直变量中立即变为$ t> 0 $，并且只要溶液的存在，垂直分析半径就会增加。
We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as $t>0$ and the vertical analytic radius increases as long as the solutions exist.
