| Abstract: |
| In this talk we will present some results obtained from the analysis of the stochastic Lagrangian Averaged Euler equations and grade two fluid filling a bounded Lipschitz domain $O$.
We will mainly talk about the existence of $H^1(O)$ weak martingale solution on any bounded Lipschitz domain $O$. Furthermore, we show that when $O$ is a convex polygon the solution $\mathbf{u}$ lives in the Sobolev space $W^{2,r}(O)$ for some $r>2$. We also prove that the \textit{vorticity} $curl(\mathbf{u}-\alpha \Delta \mathbf{u})$, where $\mathbf{u}$ is the solution, is continuous in $L^2(\mathscr{O})$ with respect to the time variable. |
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