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?
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2$\begingroup$ I think the energy method is proving something slightly different, namely that $F \in L^2_t H^1_x$ implies $u \in L^\infty_t H^2_x$. Note that the energy method can certainly get the right fixed-time $H^s_x \to H^2_x$ bounds for the homogeneous heat propagator $e^{t\Delta}$, at which point Duhamel finishes the job; but it appears to be crucial that the energy method is applied to a homogeneous component of the solution rather than the solution itself. $\endgroup$– Terry TaoCommented Jul 31, 2012 at 6:26
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$\begingroup$ @Terry: Thanks a lot! It was very helpful. $\endgroup$– timurCommented Jul 31, 2012 at 6:44
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