From Boltzmann to Navier–Stokes–Fourier: A Critical Regularity Framework

Speaker：Hailiang Li (Capital Normal University)

Time：2026-6-19 9:00

Location：Conference Room C610 at Administration Building at Haiyun Campus

Abstract:

A fundamental problem in kinetic theory is to connect kinetic descriptions with macroscopic fluid models through hydrodynamic limits. However, a key challenge lies in the mismatch between the critical regularity of the Boltzmann equation and that of the incompressible Navier–Stokes–Fourier system. In this work, we characterize the critical-regularity transition from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system. We prove the global well-posedness of solutions to the Boltzmann equation in a hybrid critical space, uniformly with respect to the Knudsen number \varepsilon. More precisely, we identify a sharp frequency threshold of order 1/\varepsilon separating the low- and high-frequency regimes and show that, as the Knudsen number tends to zero, the low-frequency part of the macroscopic component is governed by the Fujita-Kato critical regularity, whereas the higher-order spatial critical norms of the solution may blow up. In particular, our result admits low-regularity initial data with large-amplitude spatial oscillations. Moreover, we rigorously justify the hydrodynamic limit and establish global-in-time strong convergence with explicit rates for ill-prepared data. This is a joint work with L.-Y. Shou, C.-J. Xu and J. Xu.