Tag: Compatibility

Consider the following coupled Stokes-Darcy systems in a simplified domain $Omega=(-1,1)times(-1,1)$, WLOG, we may assume the horizontal variable to be periodic. The Stokes flow occuplied the upper part $Omega_{c}=(-1,1)times(0,1)$, and the lower part is Darcy flow $Omega_{m}=(-1,1)times(-1,0)$, clearly $Omega=Omega_{c}cupOmega_{m}$. The two fluid contact on the interface $Gamma=Omega_{c}capOmega_{m}=(-1,1)times{0}$. We name the top and bottom of $Omega$ to be $Gamma_{c}$ and $Gamma_{m}$.

begin{equation}
leftlbrace
begin{aligned}
partial_{t}u_{c}-nablacdotmathbb{T}(u_{c},p_{c}) & =f, & text{in} Omega_{c}, \
nablacdot u_{c} & =0, & text{in} Omega_{c}, \
u_{m} & =-nabla p_{m}, & text{in} Omega_{m}, \
nablacdot u_{m} & =0, & text{in} Omega_{m}, \
u_{c}cdot n & =u_{m}cdot n, & text{on} Gamma, \
taucdotmathbb{T}(u_{c},p_{c})n & =-alpha u_{c}cdottau, & text{on} Gamma, \
ncdotmathbb{T}(u_{c},p_{c})n & =-p_{m}, & text{on} Gamma, \
u_{c} & =0, & text{on} Gamma_{c}, \
partial_{x_{2}}p_{m} & =0, & text{on} Gamma_{m},
end{aligned}
right.
end{equation}
where $mathbb{T}(u,p)=frac{1}{text{Re}}(nabla u+nabla u^{T})-pmathbb{I}$, $alpha>0$ is a constant.

Question: What are the compatibility conditions satisfying by the initial Stokes velocity $u_{c}|_{t=0}=u_{0}$ to ensure the global existence of classical solution to this system?