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第34回PDE実解析研究会 (北大数学COE協賛)
PDE Real Analysis Seminar
Program
- 組織委員：
- 儀我美一（東大／北大）
- 日 時：
- 2007年9月5日(水) 10:30-11:30
- 場 所：
- 東京大学大学院 数理科学研究科056号室
※会場へのアクセスは下記にてご確認下さい。
駒場アクセスマップ
http://www.u-tokyo.ac.jp/campusmap/map02_02_j.html
駒場キャンパス数理科学研究科棟
http://www.u-tokyo.ac.jp/campusmap/cam02_01_27_j.html
- 講 演 者：
- Professor Reinhard Farwig (Darmstadt University of Technology)
- 演 題：
- Regularity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion
- ABSTRACT:
- Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\Omega \subset \mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\in L^s(0,T;L^q(\Omega))$ satisfies the {\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\frac{2}{s} + \frac{3}{q}=1$, $s>2,\, q>3$. Now consider $u$ such that $$u\in L^r(0,T;L^q(\Omega))\quad \mbox{ where }\quad \frac{2}{r} + \frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\it localized energy inequality}. Finally H\"older continuity of the kinetic energy in time will imply regularity.
The proofs use local in time regularity results which are based on the {\it theory of very weak solutions} and on uniqueness arguments for weak solutions.
