Time: 8:00-9:30, Friday, September 6 2024

Venue: E4-201, Yungu Campus

Zoom ID: 942 8621 5408

Passcode: 712615

Speaker: Jeaheang Bang, Purdue University

Title: Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions

Abstract: Solutions with scaling-invariant bounds, such as self-similar solutions, play an important role in understanding the regularity and asymptotic structures of solutions to the Navier-Stokes equations. On the other hand, the higher dimensional steady Navier-Stokes equations have attracted attention because the 5D steady case shares the same scaling dimension as the 3D evolutionary case. We recently proved that any steady solution $u$ must be trivial if  $|u(x)|\leq C/|x|$ (scaling-invariant bound) in the entire Euclidean space for some constant $C>0$ when the dimension is at least 4, without assuming any smallness, self-similarity or axisymmetry. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates. Our proof is elementary and short. As applications, we also proved a removable singularity in a punctured domain and identified a leading term describing the large-distance behavior of a general solution in an exterior domain. This is a collaboration with Changfeng Gui, Chunjing Xie, Yun Wang and Hao Liu.