On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations
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 Lp-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 L3 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.
- Publication:
-
Archive for Rational Mechanics and Analysis
- Pub Date:
- September 2013
- DOI:
- 10.1007/s00205-013-0630-z
- arXiv:
- arXiv:1201.6009
- Bibcode:
- 2013ArRMA.209..999C
- Keywords:
-
- Vortex;
- Vorticity;
- Euler Equation;
- Exterior Domain;
- Incompressible Euler Equation;
- Mathematics - Analysis of PDEs;
- Mathematics - Numerical Analysis
- E-Print:
- A revised version with improved notation, proofs, etc. 19 pages