Contents

• TOP
• Organizers and Board Members
• Purpose of the Symposium
• History of the Symposium
  (Under Construction)
• Archives of the Symposium
  

Outline

Organizers :

Y.Giga (University of Tokyo/Hokkaido University)

Period :

September 5, 2007 (Wednesday) 10:30-11:30

Place :

Graduate School of Mathematical Sciences the University of Tokyo, Room #056

Programme :

Professor Reinhard Farwig (Darmstadt University of Technology)

TITLE:

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.