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?