Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation

Abstract

We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L^p_tL^q_x$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin--Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural \[\dotc W^{1,p}\big(\R;L^q(\R^d_+)\big)\cap L^p\big(\R;\dotc W^{2,q}(\R^d_+)\big)\] regularity norm of $v$ is controlled by the \[\dotc {F}^{1-\frac{1}{2q}}_{p,q}\big(\R;\,L^{q}(\R^{d-1})\big)\cap L^{p}\big(\R;\,\dotc{B}_{q,q}^{2-\frac{1}{q}}(\R^{d-1})\big)\] norm of $g$. The maximal regularity estimate holds for $1\leq p,q<\infty$, while the caloric trace estimate holds for $1<p<\infty$ and $1\leq q\leq\infty$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier--Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier--Stokes flows exhibiting strong boundary singularities.