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<p>In this paper, we study the existence, unconditional uniqueness, and polynomial stability of mild solutions to the Patlak–Keller–Segel–Navier–Stokes systems on the whole space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ 𝕉^{d} $</EquationSource> </InlineEquation> (where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ d\geqslant 4 $</EquationSource> </InlineEquation>).We work in the framework of weak <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ L^{p} $</EquationSource> </InlineEquation> spaces, i.e., <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ L^{p,\infty}(𝕉^{d}) $</EquationSource> </InlineEquation>.First, we use dispersive estimates together with linear and bilinear estimates to prove the existence of bounded mild solutions to the corresponding linear systems.Then, by fixed point arguments, we obtain the well-posedness of mild solutions to the semilinear systems.Moreover, we prove the unconditional uniqueness in the space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ L^{p}(𝕉^{d}) $</EquationSource> </InlineEquation>.Finally, we&#xa0;establish polynomial stability for mild solutions by using the Yamazaki estimates.</p>

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Global Existence, Unconditional Uniqueness, and Stability for the Patlak–Keller–Segel–Navier–Stokes Systems on the Whole Space

  • 期刊论文

  • 发表日期：

    2026年1月29日

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  • D. T. N. Van,
  • P. T. Xuan

摘要

In this paper, we study the existence, unconditional uniqueness, and polynomial stability of mild solutions to the Patlak–Keller–Segel–Navier–Stokes systems on the whole space $ 𝕉^{d} $ (where $ d\geqslant 4 $ ).We work in the framework of weak $ L^{p} $ spaces, i.e., $ L^{p,\infty}(𝕉^{d}) $ .First, we use dispersive estimates together with linear and bilinear estimates to prove the existence of bounded mild solutions to the corresponding linear systems.Then, by fixed point arguments, we obtain the well-posedness of mild solutions to the semilinear systems.Moreover, we prove the unconditional uniqueness in the space $ L^{p}(𝕉^{d}) $ .Finally, we establish polynomial stability for mild solutions by using the Yamazaki estimates.