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Consider the heat equation \begin{align} \partial_t u - \Delta u &= 0 && \mbox{ in }Q=\Omega\times[0,T] \\ u &= 0 && \mbox{ on }\partial \Omega \times [0,T] \\ u(\cdot,0) &= u_0 && \mbox{ on }\Omega\times\{0\} \end{align} with $u_0 \in H^1(\Omega)$ and a smooth doamin $\Omega$. For the solution of $u$ it is known that $u$ is in $$ H^{2,1}(Q) := \{ u \in L^2(0,T,H^2(\Omega)); u_t \in L^2(Q); u=0 \text{ on } \partial \Omega \times [0,T] \} $$ and further we have the bound $\Vert u \Vert_{H^{2,1}(Q)} \leq C \Vert u_0 \Vert_{H^1(\Omega)}$ with $$ \Vert w \Vert_{H^{2,1}(Q)}^2 = \sum_{|\alpha|\leq 2} \Vert D^{\alpha} w \Vert_{L^2(Q)}^2 + \Vert \partial_t w \Vert_{L^2(Q)}^2 $$ and $D^{\alpha}$ the spatial derivatives. On the domain $Q_{\varepsilon} = \Omega \times [\varepsilon,T]$ the solution is smooth as soon as $\varepsilon > 0$.

Now may question: Can I get a bound of the form $$ \Vert u \Vert_{H^{k,k}(Q_{\varepsilon})} \leq C(\varepsilon) \Vert u_0 \Vert_{H^1(\Omega)} $$ with $k \geq 2$ and if yes how does $C(\varepsilon)$ behave for $\varepsilon \rightarrow 0$?

I don't expect that a complete answer is directly in your mind, but if you have ideas on how to approach this question please share your thoughts.

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I have no ready solution. A first approach might be looking at the decay rates of Fourier coefficients; then $C(\epsilon)$ would be related to the "worst case harmonic" with the decay rate proportional to $\mathrm{sup}_{n\in \mathbb{N}}n^k e^{-\alpha n^2\epsilon}$ for the $k$ th order spatial derivatives.

Or one might start from the derivative of the fundamental solution and work from there.

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  • $\begingroup$ Thanks for your reply. I also thought in that direction and will try this further. But apart from getting a better feeling of the situation - which is definitely worth something for me - I am not sure if it'll help to show lower bound for $C(\varepsilon)$. But first let me See where this approach will lead me to. If I get something meaningful, I will share it here. $\endgroup$ – hrothgar grumpen Jan 18 '14 at 22:57
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The solution $u$ is given by an analytic semigroup (defined e.g. by means of the spectral theorem), more precisely: $u(t,x):=e^{t\Delta}u_0(x)$. Now, this semigroup (like all analytic semigroups) satisfy an estimate $$ \sup_{t>0} \|t^k \Delta^k e^{t\Delta}\|<\infty $$ for all $k\in \mathbb N$, and again in view of the spectral theorem this is optimal. Using the fact that the norm of high-order Sobolev spaces can be expressed in terms of the graph norms of $\Delta$ and its powers, you can get your conclusion - and also see that how fast your $C(\epsilon)$ will blow up for $\epsilon\to 0$.

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