题目：Optimal Initial Values and Regularity Conditions for Weak Solutions to the Navier-Stokes System

报告人🥫：Prof. Dr. Reinhard Farwig （TU Darmstadt）

地点：宁静楼108室

时间：2017年11月15日 下午2点半到3点半

摘要

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded smooth domain $/Omega$. It is well known that any solenoidal initial value $u_0$ in $L^2(/Omega)$ with a vanishing normal component on the boundary admits a global in time weak solution. Moreover, if $u_0 /in H^1$ or even only $u_0 /in /mathcal{D}(A^{1/4})/subset L^3$, where $A =-P/Delta$ denotes the Stokes operator, then $u_0$ admits a unique local in time regular (strong) solution in Serrin’s class $L^s(0,T;L^q (/Omega))$ where $2/s + 3/q = 1$ for some $T = T(u_0)/leq/infty$.

The optimal class of initial values $u_0 /in L^2$ with this property was determined by H. Sohr, W. Varnhorn and myself in 2009 and is given by a certain Besov space with negative order of differentiability. This Besov space condition is used at (almost) all $t > 0$ along a given weak solution to find various new conditions on regularity and uniqueness of weak solutions.

个人简历

Reinhard Farwig 教授1982年博士毕业于波恩大学，1995年起成为德国达姆施塔特工业大学数学系教授✌🏼。Farwig 教授发表了100多篇文章☀️👨🏻‍🏭，其中引用率**次。他的主要贡献在于Navier-Stokes方程, Euler方程，调和分析☆，线性和非线性泛函分析以及发展方程等🏌🏻‍♀️。

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