Consider the following PDEs in $(0,\infty)\times \mathbb{R}$:
\begin{align*} (1)& \quad u_t + uu_x - u_{xx} =0 \\ (2)& \quad u_t + uu_x =0 \\ (3)& \quad u_t - u_{xx} =0 \end{align*} with initial condition $u(0,x) = g(x)$, $x \in \mathbb{R}$.
What are (and what is the difference between) a priori and a posteriori estimates for the solutions of these PDEs? Can you point out some references on these kinds of estimates for the problems above and possibly more general ones?