Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation
Abstract
We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range 1<α≤2. Together with the known ill-posedness results for 0≤α<1, this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework. The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the L∞ norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.
Related
- Global Well-Posedness and Persistence of H2 Regularity for the Three-Dimensional Primitive Equations with Horizontal and Topography-Adapted Thermal Diffusion
- Global well-posedness of the primitive equations with stochastic wind-driven boundary conditions
- Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
- Well-posedness and large deviations for 3D primitive equations with jumps
- Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
BibTeX
@misc{lin2026global,
title = {{Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation}},
author = {Quyuan Lin and Changhui Tan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04840},
eprint = {2610.04840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04840},
}
RIS
TY - UNPB
TI - Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation
AU - Quyuan Lin
AU - Changhui Tan
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04840
UR - https://arxiv.org/abs/2610.04840
AB - We establish global well-posedness of classical solutions to the two-dimensional primitive equations with fractional horizontal dissipation for arbitrarily large initial data in the full subcritical range 1<α≤2. Together with the known ill-posedness results for 0≤α<1, this establishes the sharp dissipation threshold for large-data global well-posedness in the corresponding solution framework. The key ingredient is a hydrostatic energy estimate obtained by splitting the nonlinear energy into symmetric and antisymmetric parts and exploiting the commutator structure of the latter. Using anisotropy and incompressibility, we bound the nonlinear energy by the L^∞ norm of the hydrostatic vorticity times a quadratic velocity norm with only one-half additional horizontal derivative. The vorticity maximum principle then yields enhanced velocity bounds, which close the vorticity estimates and verify the continuation criterion throughout the subcritical regime.
ER -