Consider the heat equation $$ u_t = u_{xx} + f, $$ on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ for some $s>0$, and if the initial condition is smooth. I am wondering if one can get this result by energy method. A naive application gives estimates of the form $$ \frac{\mathrm{d}}{\mathrm{d}t}\\partial^2 u\^2 + \\partial^3u\^2 \lesssim \langle \partial^2f,\partial^2u\rangle\leq \\partial f\\\partial^3 u\, $$ where the norms are the $L^2$norms, and $\partial$ denotes a generic derivative. This seems to require that $f\in H^1$ rather than $f\in H^s$ with some $s>0$ (which can be smaller than $1$). Is there a trick to get around this in the energy method?
