目：The use of defect measures for the partial regularity theory in the stationary and nonstationary Navier-Stokes equations
间：2022.07.05（星期二），09:00-11:00
点：腾讯会议：416-9346-0896    会议密码：202207
要：The nonstationary Navier-Stokes equations in space dimension 4 and the stationary Navier-Stokes equations in space dimension 6 are critical when using local energy inequalities in a sense. Here, we present different variations for the use of defect measures to tackle this critical phenomena, thus giving a partial regularity theory for these critical and vectorial cases.
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目：On stable solutions to the Euler equations in convex planar domains
间：2022.07.06（星期三），09:00-11:00
点：腾讯会议：416-9346-0896    会议密码：202207
要：On convex planar domains, given an initial vorticity with one sign, I will talk about the regularity and geometric properties of the dynamically stable solutions to the Euler equations in the coadjoint orbit of the initial vorticity. These flows have elliptic stagnation points. Under some nondegeneracy conditions on the data, we show they are Holder continuous and have convex level curves. We also give a detailed description for the set of stagnation points. If the initial vorticity has nice level set topology, these stable solutions are in the L^\infty-strong closure of the coadjoint orbit. We also demonstrate the sharpness of most assumptions we made.