Abstract
The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L p-condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables \({u \in L^p}\) and \({u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}\), then the blow-up does not occur, provided \({\alpha > N/2}\) or \({-1 < \alpha \leq N\,/p}\). This includes the L 3 case natural for the Navier–Stokes equations. For \({\alpha = N\,/2}\) we exclude profiles with asymptotic power bounds of the form \({ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}\). Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant.
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Communicated by V. Šverák
The work of Dongho Chae was supported partially by NRF grant 2006-0093854 and also by the Chung-Ang University Research Grants in 2012, while the work of Roman Shvydkoy was partially supported by NSF grant DMS–0907812.
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Chae, D., Shvydkoy, R. On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations. Arch Rational Mech Anal 209, 999–1017 (2013). https://doi.org/10.1007/s00205-013-0630-z
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DOI: https://doi.org/10.1007/s00205-013-0630-z