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<p>This paper focuses on the two-dimensional hydrostatic Navier-Stokes equations in the strip <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{R}\times \mathbb{T}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> </math></EquationSource> </InlineEquation>. We first prove the long-time well-posedness for the hydrostatic Navier-Stokes equations in Sobolev space <i>H</i><sup><i>m</i></sup> when the initial data is a small perturbation of some convex function. Then we justify the limit from the anisotropic Navier-Stokes equations to the hydrostatic Navier-Stokes equations and get the optimal convergence rate in <i>H</i><sup><i>m</i></sup>.</p>

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On the Navier-Stokes equations in a thin strip: long time well-posedness and hydrostatic limit in Sobolev space

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Zhan Xu,
  • Tianyuan Yu

摘要

This paper focuses on the two-dimensional hydrostatic Navier-Stokes equations in the strip \(\mathbb{R}\times \mathbb{T}\) R × T . We first prove the long-time well-posedness for the hydrostatic Navier-Stokes equations in Sobolev space Hm when the initial data is a small perturbation of some convex function. Then we justify the limit from the anisotropic Navier-Stokes equations to the hydrostatic Navier-Stokes equations and get the optimal convergence rate in Hm.