# 2D Stochastic Navier Stokes equations with Navier boundary condition

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 Prato-Debussche trick, i.e., to consider the system in a deterministic way by setting $$v=u-z$$ 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 $$\partial \Omega$$.

• Suggest explicitly defining the parameter $\alpha$, the unit normal and tangent vectors, and the rate of strain tensor appearing in the boundary condition. – Nawaf Bou-Rabee Feb 20 '19 at 11:56
• @NawafBou-Rabee Thanks. Just added more explanations. – Topoguy Feb 21 '19 at 7:40
• Still missing some detail. What is $\tau$? What is $n$? – Nawaf Bou-Rabee Feb 21 '19 at 14:56
• @NawafBou-Rabee Just added. They are unit normal and unit tangent vectors, respectively. – Topoguy Feb 21 '19 at 17:25
• Lastly, please specify what type of Weiner process $W$ is. Is it an $L^2$-Wiener process? – Nawaf Bou-Rabee Feb 21 '19 at 17:43