I am working on the couple of stokes and elastic equations and I want to find the weak formulation:
$$ ρ_1v_t +∇p−μ_1∇·(∇v+∇v^T)=ρ_1f_1\ \ \ \ \text{in}\ \ (0,T)×Ω_1,$$
$$ ∇ · v = 0 \ \ \ \text{in}\ \ \ \ (0, T ) × Ω_1,$$
$$v=0\ \ \ \text{on}\ \ \ (0,T)×Γ_1\ \ \text{and}\ \ v|_t=0 =v_0\ \ \ \text{in}\ \ \ \ Ω_1,$$
$$ρ_2u_{tt} −μ_2∇·(∇u+∇uT)−λ_2∇(∇·u)=ρ_2f_2 \ \ \ \text{in}\ \ \ (0,T)×Ω_2,$$
$$u=0\ \ \ \text{on}\ \ \ (0,T)×Γ_2,\ \ \ u|_t=u_0\ \ \ \text{in}\ \ \ Ω_2,\ \ \ u_t|_t =u_1\ \ \ \text{in}\ \ \ Ω_2,$$ Where $Ω_1$ and $Ω_2$ are the fluid and solid domains respectively.
We also have the following equalities on the interface $\Gamma_0$ : $$u_t=v$$ and $$μ_2(∇u+∇uT)·n_2 +λ_2(∇·u)n_2 =pn_1 −μ_1(∇v+∇vT)·n_1$$ where $n$ is the outward normal vector.
How can I follow from the last equation on $\Gamma_0$ to get the weak formula.
The free-divergence weak formulation I have to arrive to is:
$$\frac{d}{dt} ρ_1[v, η]_{Ω_1} + ρ_2[∂_tu, η]_{Ω_2} + a_1[v, η] + a_2[u, η] =ρ_1[f_1,η]_{Ω_1} +ρ_2[f_2,η]_{Ω_2}$$ $\forall η \in H^1_0$ with zero divergence
Where $[ .,. ]$ is $L^2(\Omega)$ inner product. $$a_1[u,v]=\frac{\mu_1}{2} \int_{(\Omega_1)}(∇u+∇uT):(∇v+∇vT)dΩ. \\a_2[u,v]=\frac{\mu_2}{2} \int_{(\Omega_2)}(∇u+∇uT):(∇v+∇vT)+\lambda_2 (∇.u ) (∇.v) dΩ$$
Hint: Let $\phi \in H^1_0(\Omega)$ such that $div(\phi)=0$ where $\Omega = \Omega_1 \cup \Omega_2$ Multiplying the first and the second equations by $\phi$ and integrating over $\Omega_1$ and $\Omega_2$ respectively, then adding the two equations we will obtain the weak formula written above(in the question) and since $\Gamma_0$ is a subset of $\Omega$ then we can incorporate the stress interface condition into the weak formulation.
This means we arrive to our result without needing derive the equation given on the interface.