中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

综合报告

报告人： 吴昪 博士（Institute for advanced study）

题  目：On the regularity of solutions to the Naiver-Stokes equations satisfying local energy inequalities

时  间：2022.07.01（星期五），09:00-10:00

地  点：腾讯会议：355-277-371   会议密码：202207

摘  要：The regularity problems on the PDEs in fluid dynamics (in particular, the Navier Stokes equations and the Euler equations) become harder when the space dimension gets higher. I will present three results in different dimensions. In space dimension 4, I will show that there exist global-in-time partially regular weak solutions to the nonstationary Navier-Stokes equations whose singular sets have finite 2-dimensional parabolic Hausdorff measure. These solutions satisfy local energy inequalities. In space dimension 3, I will show the singularity formation in a linear toy model of the axi-symmetric Navier-Stokes equations. As a by-product, I construct sharp time-independent supercritical drifts such that the Harnack inequality and the Holder continuity fail in both elliptic and parabolic equations associated to these drifts. These singular solutions satisfy local energy inequalities even at the blow-up time. In space dimension 2, the global regularity of the Navier Stokes equations and the Euler equations is well understood. I will present a recent result on the Holder regularity and the geometric properties of stable solutions to the Euler equations in convex domains, which connects the (formal) Hamiltonian structure of 2D Euler equations and the study of semilinear elliptic equations.