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 LptLqx 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 W1,p(R;Lq(Rd+))∩ Lp(R;dotc W2,q(Rd+)) regularity norm of v is controlled by the dotc F1−1/2qp,q(R; Lq(Rd−1))∩ Lp(R; dotcBq,q2−1/q(Rd−1)) norm of g. The maximal regularity estimate holds for 1≤ p,q<∞, while the caloric trace estimate holds for 1<p<∞ and 1≤ q≤∞. 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.
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BibTeX
@misc{chen2026maximal,
title = {{Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation}},
author = {Hui Chen and Su Liang and Tai-Peng Tsai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.16438},
eprint = {2606.16438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16438},
}
RIS
TY - UNPB
TI - Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation
AU - Hui Chen
AU - Su Liang
AU - Tai-Peng Tsai
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2606.16438
UR - https://arxiv.org/abs/2606.16438
AB - 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(R;L^q(R^d_+))∩ L^p(R;dotc W^2,q(R^d_+)) regularity norm of v is controlled by the dotc F^1−1/2q_p,q(R; L^q(R^d−1))∩ L^p(R; dotcB_q,q^2−1/q(R^d−1)) norm of g. The maximal regularity estimate holds for 1≤ p,q<∞, while the caloric trace estimate holds for 1<p<∞ and 1≤ q≤∞. 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.
ER -