In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function \(\mathbb {1}_{|{\mathbf {n}}|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}\) by an appropriate polynomial \(f({\mathbf {n}})\) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider \(f({\mathbf {n}})=\mathbb {1}_{|d|\le 1}(|{\mathbf {n}}|^2-1){\mathbf {n}}\) and if the initial condition \({\mathbf {n}}_0\) satisfies \(|{\mathbf {n}}_0|\le 1\), then the solution \({\mathbf {n}}\) also remains in the unit ball.

中文翻译：

---

由乘性噪声驱动的惩罚性向列液晶的一些结果：弱解和最大原理

在本文中，我们证明了与高度非线性随机偏微分方程（PDE）系统有关的一些数学结果。这些随机方程描述了在随机外力作用下的向列型向列液晶的动力学。首先，我们证明了整体弱解的存在（从随机分析和PDE的意义上来说）。其次，我们显示了二维域中解决方案的路径唯一性。与确定性设置中的几本作品相反，我们替换了Ginzburg–Landau函数\（\ mathbb {1} _ {| {\ mathbf {n}} | \ le 1}（| {\ mathbf {n}} | ^ 2 -1）{\ mathbf {n}} \）通过适当的多项式\（f（{\ mathbf {n}}）\），我们对多项式f给出了充分的条件保持这两个结果。我们的第三个结果是最大原理类型定理。更准确地说，如果我们考虑\（f（{\ mathbf {n}}）= \ mathbb {1} _ {| d | \ le 1}（| {\ mathbf {n}} | ^ 2-1）{\ mathbf {n}} \），如果初始条件\（{\ mathbf {n}} _ 0 \）满足\（| {\ mathbf {n}} _ 0 | \ le 1 \），则解\（{\ mathbf {n}} \）也保留在单位球中。