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<p>While finite-time blowup solutions have been studied in depth for the Keller–Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller–Segel–Navier–Stokes system with buoyancy in 3<i>D</i>. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3<i>D</i> Keller–Segel equation with the abstract semigroup tool from Merle et al. (Invent Math 227:247–413, 2022), which partially generalizes the radial stability result (Glogić and Schörkhuber in Arch Ration Mech Anal 248:4, 2024) to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.</p>

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Finite-Time Blowup for Keller–Segel–Navier–Stokes System in Three Dimensions

  • 期刊论文

  • 发表日期：

    2025年7月2日

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  • Zexing Li,
  • Tao Zhou

摘要

While finite-time blowup solutions have been studied in depth for the Keller–Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller–Segel–Navier–Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller–Segel equation with the abstract semigroup tool from Merle et al. (Invent Math 227:247–413, 2022), which partially generalizes the radial stability result (Glogić and Schörkhuber in Arch Ration Mech Anal 248:4, 2024) to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.