On the optimal mass threshold for the two-dimensional Keller-Segel system coupled to the Navier-Stokes equations

Abstract

We investigate the mass threshold for the two-dimensional Keller-Segel system coupled with the incompressible Navier-Stokes equations. In the absence of fluid interaction, the classical Keller-Segel system admits a well-known $4\pi$-critical-mass dichotomy: for every suitably regular initial datum of total mass $m<4\pi$, the associated no-flux initial-boundary value problem admits a global bounded classical solution, whereas for $m>4\pi$, there exist smooth initial configurations of mass $m$ such that the associated solution blows up in finite or infinite time. For the fluid-coupled system, Winkler ({SIAM J. Math. Anal.}, 52:2041-2080, 2020) first established global generalized solvability under the condition $m<2\pi$ and posed an open question whether the fluid action may lower the critical blow-up mass to some value lying between $2\pi$ and $4\pi$. Based on numerical evidence, Bonilla and Gutiérrez-Santacreu ({J. Numer. Math.}, 32:175-212, 2024) later conjectured that the critical value is still $4\pi$. In the present work, we resolve this open problem and provide a rigorous mathematical proof to the conjecture, which indicates that the presence of the fluid does not alter the critical mass threshold, and the entire classical subcritical regime remains intact---every admissible initial datum with total mass $m<4\pi$ generates a unique global classical solution that is uniformly bounded in time. Our proof relies on the construction of a new quasi-energy inequality. The key underlying observation is that combining relative isoperimetric estimates with rearrangement techniques yields a Gagliardo-Nirenberg inequality on bounded planar domains with an explicit leading constant.