an:06813725
Zbl 1379.35255
Wu, Jiahong; Xu, Xiaojing; Ye, Zhuan
Global regularity for several incompressible fluid models with partial dissipation
EN
J. Math. Fluid Mech. 19, No. 3, 423-444 (2017).
00369544
2017
j
35Q35 35B45 35B65 76D03 76D09 35Q31 86A05 26A33
Euler equation; surface quasi-geostrophic equation; Boussinesq equations; partial dissipation; global regularity
Summary: This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity
for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.