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<p>We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(C^\alpha _x\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mi>x</mi> <mi>α</mi> </msubsup> </math></EquationSource> </InlineEquation>. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.</p>

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Spatial \(C^1\), \(C^2\), and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Hongjie Dong,
  • Hyunwoo Kwon

摘要

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having \(L_2\) L 2 -Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to \(C^\alpha _x\) C x α . Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.