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Abstract <p>In this paper, the Cauchy problem for the stationary and instationary nonlocal imcompressible abstract Stokes equation is considered. The equation involve a convolution term and an abstract operator in a Banach space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(E\)</EquationSource> <!--RusMath2570062Shakhmurov-m1--> </InlineEquation>. The existence, uniqueness and coercive estimates are derived in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({{L}^{p}}\)</EquationSource> <!--RusMath2570062Shakhmurov-m2--> </InlineEquation> spaces. We can obtain different classes of Stokes equations by choosing the space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(E\)</EquationSource> <!--RusMath2570062Shakhmurov-m3--> </InlineEquation> and the linear operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(A\)</EquationSource> <!--RusMath2570062Shakhmurov-m4--> </InlineEquation> which occur in a wide variety of physical systems. In application the existence, uniqueness and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({{L}^{p}}\)</EquationSource> <!--RusMath2570062Shakhmurov-m5--> </InlineEquation>-maximal regularity properties to solution of initial value problems for nonlocal degenerate Stokes equations and nonlocal Stokes equations with discontinuous coefficients are established.</p>

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Nonlocal Abstract Stokes Operators and Applications

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • V. B. Shakhmurov

摘要

Abstract

In this paper, the Cauchy problem for the stationary and instationary nonlocal imcompressible abstract Stokes equation is considered. The equation involve a convolution term and an abstract operator in a Banach space \(E\) . The existence, uniqueness and coercive estimates are derived in \({{L}^{p}}\) spaces. We can obtain different classes of Stokes equations by choosing the space \(E\) and the linear operator \(A\) which occur in a wide variety of physical systems. In application the existence, uniqueness and \({{L}^{p}}\) -maximal regularity properties to solution of initial value problems for nonlocal degenerate Stokes equations and nonlocal Stokes equations with discontinuous coefficients are established.