Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations

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dc.creatorChae, Dongho
dc.date2006-04-11
dc.date2006-04-27
dc.date.accessioned2026-07-07T07:10:44Z
dc.date.available2026-07-07T07:10:44Z
dc.descriptionIn this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/math/0604234
dc.identifierhttp://arxiv.org/abs/math/0604234
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/111516
dc.subjectAnalysis of PDEs
dc.titleNonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations
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