Abstract
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
$\mathds{1}_{\lvert \d\rvert \le 1}(\lvert \d\rvert^2-1)\d$ by an appropriate polynomial $f(\d)$ 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 $f(\d)=\mathds{1}_{\lvert d\rvert \le 1}(\lvert \d\rvert^2-1)\d$ and if the initial condition $\d_0$ satisfies $\lvert \d_0\rvert\le 1$, then the solution $\d$ also remains in the unit ball.
$\mathds{1}_{\lvert \d\rvert \le 1}(\lvert \d\rvert^2-1)\d$ by an appropriate polynomial $f(\d)$ 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 $f(\d)=\mathds{1}_{\lvert d\rvert \le 1}(\lvert \d\rvert^2-1)\d$ and if the initial condition $\d_0$ satisfies $\lvert \d_0\rvert\le 1$, then the solution $\d$ also remains in the unit ball.
| Original language | English |
|---|---|
| Number of pages | 59 |
| Journal | Stochastic Partial Differential Equations: Analysis and Computations |
| Early online date | 24 Jan 2019 |
| Publication status | E-pub ahead of print - 24 Jan 2019 |