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Consider the following inhomogeneous heat equation for $f=f(x,t)$ with Neumann boundary conditions on a bounded domain $\Omega$ and initial value $f(x,0)=f_0(x)$: \begin{align*} \partial_t f - \Delta f &= g \quad \text{on } \Omega \times [0,T], \\ \nabla f \cdot \hat n &= 0 \quad \text{on } \partial \Omega \times [0,T], \\ f(0) & = f_0 \quad \text{on } \Omega. \end{align*} I am looking for statements of the following form: If $g, f_0$ and $\Omega$ are smooth enough, then any weak solution $f$ is also a strong solution and has higher regularity.

More precisely, let $k \in \mathbb N_0$. Then under what conditions for $g, f_0$ and $\Omega$ do we obtain that $$f \in L^2(0,T;H^{2k-2s}( \Omega)) , \; \frac{d^s}{dt^s} f \in L^2(0,T;H^{2k+2-2s}(\Omega)) , \quad s= 0, \ldots,m+1? $$

One could assume it is like

  • $\Omega$ has $C^k$-boundary
  • $\frac{d^s}{dt^s} g \in L^2(0,T; H^{2k-2s}(\Omega))$ for all $s=0,\ldots,k$
  • $f_0 \in H^{2k+1}(\Omega)$
  • $g,f_0$ satisfy some compatibilty condition.

I cannot find any regularity results to this problem in the literatur; only for the case of Dirichlet boundary condition, see Evans(1997) Thm 6 in Section 7.1 on page 365.

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1 Answer 1

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I found an answer to parts of the problem in a paper of Jan Prüss, 2019

MAXIMAL REGULARITY FOR ABSTRACT PARABOLIC PROBLEMS WITH INHOMOGENEOUS BOUNDARY DATA IN $L_p$-SPACES

In a nutshell, we obtain that a right-hand side $g$ in $L^p( \Omega \times (0,T))$ implies that weak solutions belong to the following Bessel potential spaces $$ H^1_p(0,T;L^p(\Omega)) \cap L^p(0,T;H^2_p(\Omega)). $$ For the readers not familiar with the space $H^s_p$ for $s \in \mathbb R$, it has - more or less - equivalent embeddings as the Sobolev space $W^{s,p}$ if $s \in \mathbb N$ and $p \geq 2$.

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