Let $[0,T]$ be the time domain, and $I:=(-\pi,\pi)$ be the space domain.
Consider a parabolic (4th order, nonlinear) PDE $$u_t=-A(u_x,u_{xx},u_{xxx})-B(u_x,u_{xx},u_{xxx},u_{xxxx}),\quad u:[0,T]\longrightarrow W^{1,q}(I), q\in (1,+\infty).$$ Here $A$ is linear, but not $B$.
We know that:
- solutions of $$v_t= -B(u_x,u_{xx},u_{xxx},u_{xxxx})$$ are $e^T$-Lipschitz with respect to $t$.
- there exists $c$ such that for almost every time $t$ it holds $$\|A(u_x,u_{xx},u_{xxx})\| \leq c\|u_{xxx}\|,$$ where $\|\cdot\|$ denotes the operator norm of $W^{-1,q^*}(I)$, with $1/q+1/q^*=1$.
We know no a priori regularity on $u_t$. Is it possible to compare solutions of $$u_t=-A(u_x,u_{xx},u_{xxx})-B(u_x,u_{xx},u_{xxx},u_{xxxx})$$ with solutions of $v_t= -B(u_x,u_{xx},u_{xxx},u_{xxxx})$ and $w_t=-w_{xxx}$ to get some regularity for $u_t$?
Thank you.