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<p>In this paper, we study the initial-boundary value problem for the Stokes system in the three-dimensional infinite layer domain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega = (0,\frac{\pi }{2}) \times \mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, subject to upper stress-free and lower slip-type boundary conditions. Using the Fourier method, we derive an integral formulation for regular solutions. Based on this formula, we demonstrate the solvability of the problem in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-based Sobolev space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(H^2(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Solution Formula and \(H^2\)-Solvability for the Stokes System in An Infinite Layer with Upper Stress-Free and Lower Slip-Type Boundary Conditions

  • 期刊论文

  • 发表日期：

    2026年4月8日

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  • Daisuke Hirata

摘要

In this paper, we study the initial-boundary value problem for the Stokes system in the three-dimensional infinite layer domain \(\Omega = (0,\frac{\pi }{2}) \times \mathbb {R}^2\) Ω = ( 0 , π 2 ) × R 2 , subject to upper stress-free and lower slip-type boundary conditions. Using the Fourier method, we derive an integral formulation for regular solutions. Based on this formula, we demonstrate the solvability of the problem in the \(L^2\) L 2 -based Sobolev space \(H^2(\Omega )\) H 2 ( Ω ) .