Article

Communications in Mathematical Physics

, Volume 155, Issue 2, pp 277-294

First online:

On the blow-up of solutions of the 3-D Euler equations in a bounded domain

  • Andrew B. FerrariAffiliated withDepartment of Mathematics, Duke University

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Abstract

It is shown that if [0,\(\hat T\)) is the maximal interval of existence of a smooth solutionu of the incompressible Euler equations in a bounded, simply connected domain Ω\( \subseteq\) R 3, then\(\int_0^{\hat T} {\left| {\omega ( \cdot ,t)} \right|_{L^\infty (\Omega )} } dt = \infty\), where ω=∇×u is the vorticity. Crucial to this result is a special estimate proven in Ω of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity.