For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u  u\cdot \nabla u  \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the Da PratoDebussche trick, i.e., to consider the system in a deterministic way by setting $v=uz$ where $z$ is the solution of the corresponding stochastic heat equation. Then, the system becomes $$v_t  \Delta v + (v+z)\cdot \nabla (v+z) + \nabla p=0$$ My question is: what will be the boundary condition for $v$, assuming the Navier boundary condition for $u$ is: $$u\cdot n=0$$ and $$2D(u)n\cdot \tau + \alpha u\cdot \tau=0$$ on $\partial \Omega$ where $\Omega \subseteq \mathbb R^2$ is a bounded domain. $\alpha>0$ is a constant and $$D(u)=\frac{1}{2}(\nabla u + (\nabla u)^{T})$$ The stress tensor $T=(T_{ij})$ and $$T_{ij}=\delta_{ij}p + 2D_{ij}u$$ $n$ and $\tau$ are unit normal and tangent vectors to the boundary $\partial \Omega$.
Asked
Viewed
64 times
1
$\begingroup$
$\endgroup$

1$\begingroup$ Suggest explicitly defining the parameter $\alpha$, the unit normal and tangent vectors, and the rate of strain tensor appearing in the boundary condition. $\endgroup$ – Nawaf BouRabee Feb 20 at 11:56

$\begingroup$ @NawafBouRabee Thanks. Just added more explanations. $\endgroup$ – Topoguy Feb 21 at 7:40

1$\begingroup$ Still missing some detail. What is $\tau$? What is $n$? $\endgroup$ – Nawaf BouRabee Feb 21 at 14:56

$\begingroup$ @NawafBouRabee Just added. They are unit normal and unit tangent vectors, respectively. $\endgroup$ – Topoguy Feb 21 at 17:25

1$\begingroup$ Lastly, please specify what type of Weiner process $W$ is. Is it an $L^2$Wiener process? $\endgroup$ – Nawaf BouRabee Feb 21 at 17:43