Consider the following advection-diffusion equation $$ \begin{cases} u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\ u^\epsilon(0,\cdot) = u_0, \end{cases} $$
How can one prove an estimate on the difference $$\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $$ where $u^\epsilon$ solves the advection-diffusion equation $$ \begin{cases} u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\ u^\epsilon(0,\cdot) = u_0, \end{cases} $$ and $u^\eta$ solves $$ \begin{cases} u^\eta_t + f(u^\eta)_x = \eta \Delta u^\eta\\ u^\eta(0,\cdot) = u_0, \end{cases} $$ for the same initial data $u_0$?
