caa a22 4500 463170997 CHVBK 20180406164813.0 cr unu---uuuuu 170326e20070801xx s 000 0 eng 10.1007/s00021-005-0205-3 doi (NATIONALLICENCE)springer-10.1007/s00021-005-0205-3 He Xinyu Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK aut An Example of Finite-time Singularities in the 3d Euler Equations [Elektronische Daten] [Xinyu He] Abstract.: Let $$\Omega = {\user2{\mathbb{R}}}^{3} \backslash \overline{B} _{1} (0)$$ be the exterior of the closed unit ball. Consider the self-similar Euler system $$\alpha u + \beta y \cdot \nabla u + u \cdot \nabla u + \nabla p = 0, \quad \hbox{div}\, u = 0 \quad {\hbox{in}} \quad\Omega.$$ Setting α = β = 1/2 gives the limiting case of Leray's self-similar Navier-Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution $$(u,p) \in \user1{\mathcal{C}}^1 (\Omega ;\user2{\mathbb{R}}^3 \times\user2{\mathbb{R}}) $$ , vanishing at infinity, precisely $$u(y) \downarrow 0\quad {\hbox{as}}\quad |y| \uparrow \infty ,\quad {\hbox{with}}\quad u = \user1{\mathcal{O}}(|y|^{ - 1} ),\quad \nabla u = \user1{\mathcal{O}}(|y|^{ - 2} ). $$ The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v. Birkhäuser Verlag, Basel, 2006 Euler equations nationallicence singular self-similar solutions nationallicence Journal of Mathematical Fluid Mechanics Birkhäuser-Verlag; www.birkhäuser.ch 9/3(2007-08-01), 398-410 1422-6928 9:3<398 2007 9 21 https://doi.org/10.1007/s00021-005-0205-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00021-005-0205-3 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- He Xinyu Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK aut NATIONALLICENCE 773 0- Journal of Mathematical Fluid Mechanics Birkhäuser-Verlag; www.birkhäuser.ch 9/3(2007-08-01), 398-410 1422-6928 9:3<398 2007 9 21 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer