永利app官网登录入口使用教程 - 永利app官网登录入口

In this talk, we will introduce our results on the Cauchy problem of the 3D compressible Navier-Stokes equations with eddy diffusion which is commonly used in the study of geophysical flows (cf. Jabin-Bresch, Ann. of Math. 2018). The prominent character of the model is the lack of vertical dissipation of the velocity field. The nonlinear asymptotic stability of the constant equilibrium state with strictly positive constant density and vanishing velocity is established under suitable small initial perturbation in regular Sobolev spaces. Specifically, we show the convergence of the density and velocity towards the corresponding equilibrium state in H²(R³) with almost optimal decay rates. The proof is based on the detailed analysis of the Green function, the time-weighted energy estimates, and the intrinsic coupling structure of the system. As an application, we will also introduce our recent results on the 3D compressible MHD equations with eddy diffusion for the velocity field and horizontal dissipation for the magnetic field.