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 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.