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Abstract <p>We study the well-posedness of the distributed order time-fractional Navier–Stokes equations with damping <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \,|{\kern 1pt} u{\kern 1pt} {{|}^{{\beta - 1}}}{\kern 1pt} u\)</EquationSource> <!--ComMat2670014O-m1--> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\alpha &gt; 0)\)</EquationSource> <!--ComMat2670014O-m2--> </InlineEquation>. Firstly, a new distributed order fractional Gronwall inequality is proposed by using some properties of the distributed order calculus and Banach contraction mapping principle. Next, we establish the existence of weak solutions to the distributed order time-fractional Navier–Stokes equations with damping by the Galerkin method for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta \geqslant 1\)</EquationSource> <!--ComMat2670014O-m3--> </InlineEquation>. Finally, using the generalized Gronwall inequality, we prove the uniqueness of the weak solution and its continuous dependence on initial conditions and external force for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\beta &gt; 3\)</EquationSource> <!--ComMat2670014O-m4--> </InlineEquation>.</p>

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Well-Posedness for a Class of Distributed Order Time-Fractional Navier–Stokes Equations with Damping

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • K. N. O,
  • H. C. Choe,
  • S. A. Pak,
  • Y. D. Ri

摘要

Abstract

We study the well-posedness of the distributed order time-fractional Navier–Stokes equations with damping \(\alpha \,|{\kern 1pt} u{\kern 1pt} {{|}^{{\beta - 1}}}{\kern 1pt} u\) \((\alpha > 0)\) . Firstly, a new distributed order fractional Gronwall inequality is proposed by using some properties of the distributed order calculus and Banach contraction mapping principle. Next, we establish the existence of weak solutions to the distributed order time-fractional Navier–Stokes equations with damping by the Galerkin method for \(\beta \geqslant 1\) . Finally, using the generalized Gronwall inequality, we prove the uniqueness of the weak solution and its continuous dependence on initial conditions and external force for \(\beta > 3\) .