# Regularity for heat equation with Neumann boundary conditions

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.

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 $$p \geq 2$$.