Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$. 

What is a simple way to prove the convergence estimate 
$$\|u^\epsilon - u\|_{L^2(\mathbb R)} \le C \sqrt{\epsilon t}
$$
(for a suitable constant $C$)? 

Is this error estimate the optimal one?