Local Hadamard well-posedness of the 2D primitive equations without thermal diffusion under strict stratification
Abstract
We study the two-dimensional non-rotating primitive equations with anisotropic viscosity and no thermal diffusion. For doubly periodic perturbations of a hydrostatic equilibrium with an affine temperature profile, we establish local Hadamard well-posedness in Sobolev spaces under strict stratification. The initial horizontal velocity and temperature have sufficient regularity and vanishing vertical means, but need not be small. The estimates close at the same Sobolev order for these variables despite the horizontal derivative loss in the diagnostic vertical velocity. Stratification-weighted energy estimates and a horizontal compensation argument avoid an additional temperature derivative while retaining the full vertical dependence and the vertically homogeneous velocity component determined by periodic compatibility. A one-sided comparison places higher-order norms of the more regular solution only in terms involving lower-order differences. This structure allows the Bona--Smith method to yield continuous dependence in the original Sobolev topology. We obtain a unique maximal strong solution in the strictly stratified class, together with continuous dependence of the diagnostic vertical velocity and pressure in their natural Sobolev spaces and lower semicontinuity of the maximal lifespan.
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BibTeX
@misc{luo2026local,
title = {{Local Hadamard well-posedness of the 2D primitive equations without thermal diffusion under strict stratification}},
author = {Fengjie Luo and Quan Wang and Zhengdong Du},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17995},
eprint = {2610.17995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17995},
}
RIS
TY - UNPB
TI - Local Hadamard well-posedness of the 2D primitive equations without thermal diffusion under strict stratification
AU - Fengjie Luo
AU - Quan Wang
AU - Zhengdong Du
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17995
UR - https://arxiv.org/abs/2610.17995
AB - We study the two-dimensional non-rotating primitive equations with anisotropic viscosity and no thermal diffusion. For doubly periodic perturbations of a hydrostatic equilibrium with an affine temperature profile, we establish local Hadamard well-posedness in Sobolev spaces under strict stratification. The initial horizontal velocity and temperature have sufficient regularity and vanishing vertical means, but need not be small. The estimates close at the same Sobolev order for these variables despite the horizontal derivative loss in the diagnostic vertical velocity. Stratification-weighted energy estimates and a horizontal compensation argument avoid an additional temperature derivative while retaining the full vertical dependence and the vertically homogeneous velocity component determined by periodic compatibility. A one-sided comparison places higher-order norms of the more regular solution only in terms involving lower-order differences. This structure allows the Bona--Smith method to yield continuous dependence in the original Sobolev topology. We obtain a unique maximal strong solution in the strictly stratified class, together with continuous dependence of the diagnostic vertical velocity and pressure in their natural Sobolev spaces and lower semicontinuity of the maximal lifespan.
ER -