I am trying to obtain some estimates on a PDE where the unknown is:
$u(t) : \mathbb{R}^n \rightarrow \mathbb{R}^n, t \in [0,T]$.
The variational formulation is given below, where the test function is set to $\phi = u(t)$ :
$\epsilon^2 ||D(u(t))||^2_{L^2(\Omega_1)^{n^2}} + \int_0^t ||D(u(t'))||^2_{L^2(\Omega_2)^{n^2}} dt' = \int_0^t \int_\Omega F u dxdt $
$\epsilon$ is a small paramter, $D(\cdot)$ is the symmetric gradient, and $\Omega = \Omega_1 \cup \Omega_2$. $F(x,t)$ is a known function and whatever regularity is needed for it can be assumed.
It should be possible to get the following estimates on $u$ from the above (the constant $C$ depends on $F$ but not on $\epsilon$):
(1) $||D(u)||_{L^\infty(0,T; L^2(\Omega_1)^{n^2})} \leq \epsilon^{-1}C$
(2) $||u||_{L^\infty(0,T; L^2(\Omega_1)^{n})} \leq C$
(3) $||D(u)||_{L^2(0,T; L^2(\Omega_2)^{n^2})} \leq C$
(4) $||u||_{L^2(0,T; L^2(\Omega_2)^{n})} \leq C$
I am able to obtain the estimate (1) by writing the RHS as: $\int_0^t \int_{\Omega_1} F u dxdt + \int_0^t \int_{\Omega_2} F u dxdt$, and then using the Cauchy-Shwartz and Young inequalities to absorb the $\Omega_2$-term on the LHS, and then using Gronwalls inequality. However, I fail to see how the rest of the estimates should be obtained.
If anything is unclear please ask.