How can I find a-priori estimates for u, $u_x$ and $u_{xx}$ which do not depend on time? The two independent PDEs that I would like to find these estimates for are provided below:
$u_t + u^2u_x - \epsilon u_{xx} = 0$
$u_t = (u^m)_{xx} + u^n$
For both PDEs, assume that $u(x,0) = g(x)$ and $g(x) = g(x+1)$ (periodic domain). Prove uniqueness of classical solution.