A priori energy estimates for Burger's equation with dissipation I've prove existence using the Galerkin method forBurger's equation with dissipation:
$u_t + uu_x - u_{xx} = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity. 
Clarification: I have proven existence for $u \in L^2([0,T];H_0^1(\Omega))$, $u_t \in L^2([0,T];H^{-1}(\Omega))$.
I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,
$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.
I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?
 A: Since $u_x(0,\cdot)$ is $L^2$, $u(0,\cdot)$ is continuous over $[0,L]$. The Burgers equation satisfies the maximum principle, thus $\|u(t)\|_{L^\infty}\le\|u_0\|_{L^\infty}$. Then the last integral $I$ can be bounded by a use of the Young inequality:
$$I\le\frac12\int_0^T\int_0^Lu_t^2dx dt+\frac{\|u_0\|_{L^\infty}^2}{2}\int_0^T\int_0^Lu_x^2dx dt.$$
The first term above is absorbed by your left-hand side. The second one is bounded because of $u\in L^2(0,T;H^1_0)$. Whence the required estimate.
I am not fond of the Galerkin method for proving the existence to the Cauchy problem for the Burgers equation and related ones. It does not give uniqueness, and it is hard to get the maximum principle that way. I prefer Picard fixed point iteration, applied to the mild formulation
$$u(t)=K^t*u_0-\int_0^tK^{t-s}*(uu_x(s))ds,$$
where $K$ is the heat kernel. You may find estimates in Chapter 6 of my book Hyperbolic conservation laws I, Cambridge University Press (1999). This method has the advantage to provide regularization for $t>0$.
