School of Mathematical Sciences Academic Report [2025] 054  

(High-Level University Construction Series Report No. 1076)  

Lecture Title: Stability and decay rates for the 3D compressible Navier-Stokes equations with eddy diffusion  

Speaker: Assoc. Prof. Guangyi Hong (South China University of Technology)  

Date & Time: 10:30 AM – 11:30 AM, June 22, 2025  

Venue: Classroom No. 4, Convergence Stars Building  

Abstract: We address the Cauchy problem for 3D compressible Navier-Stokes equations with eddy diffusion—a model widely used in geophysical flows (cf. Jabin-Bresch, Ann. of Math. 2018). The model’s hallmark is the absence of vertical velocity dissipation. Nonlinear asymptotic stability of the constant equilibrium state (strictly positive constant density, vanishing velocity) is established under small initial perturbations in regular Sobolev spaces. Specifically, we prove the convergence of density and velocity towards equilibrium in H^2(\mathbb{R}^3) with almost optimal decay rates. The proof leverages Green’s function analysis, time-weighted energy estimates, and the intrinsic coupling structure of \mathop{\mathrm{div}}\nolimits \mathbf{u} and \nabla \rho.  

Biography: Guangyi Hong, Associate Professor at South China University of Technology, specializes in theoretical analysis of nonlinear PDEs and their applications. His research has been published in Math. Ann., Indiana Univ. Math. J., J. Math. Pures Appl., SIAM J. Math. Anal., J. Math. Anal. Appl., M3AS, Proc. London Math. Soc, J. Differential Equations, and Nonlinearity.  

All faculty and students are welcome to attend!  

Invited by: School of Mathematical Sciences  

School of Mathematical Sciences  

June 22, 2025