{
  "annotation_id": "c3cfab03-cf23-4f57-b256-ef67ad38577f",
  "date_created": "2026-02-17T05:53:20.074000Z",
  "date_modified": "2026-02-17T05:53:20.074000Z",
  "file_hash": "72cd4a0e7cf36608d4292bdced46ae63f578f9511eb533a230f88cf277f7698c",
  "private": false,
  "record": {
    "abstract": "The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $\\Omega$, which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution $v$ with initial data $a$, we establish \\emph{a priori} bounds in $L^\\infty(0, \\infty; H^1(\\Omega)) \\cap L^2(0, \\infty; \\dot H^2(\\Omega))$, the exponential part of which is $\\exp(C \\|a\\|_{L^2(\\Omega)}^2)$. This is in contrast to the upper bounds reported in the existing literature that are double exponential. Furthermore, the uniform-in-time estimate for the Neumann condition case, in which the Poincar\\\u0027e inequality is unavailable for $v$, seems to be new.",
    "arxiv_id": "2601.09183",
    "authors": [
      "Takahito Kashiwabara"
    ],
    "categories": [
      "math.AP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Single exponential $H^1$-upper bounds for the primitive equations",
    "url": "https://arxiv.org/abs/2601.09183",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "8511bb1f-50a3-4e75-b6e4-d3a8ef10ad82",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}