Chapter 1 Incompressible Euler Equations: The Blow-up Problem and Related Results

Renormalization and universality of blowup in hydrodynamic flows

“…We know that, in compressible inviscid flows, the blowup may lead to formation of a shock wave. The question of existence of the blowup in incompressible inviscid flows (incompressible Euler equations) represents a long standing and still controversial problem, see, e.g., [5][6][7]. This paper is focused on the study of scaling properties of blowups in hydrodynamic flows.…”

Criterion for Blow-Up in the Euler Equations via Certain Physical Quantities

“…Let Ω be R 3 or a smooth bounded domain in R 3 . We consider the Euler equations of incompressible fluid in Ω:…”

“…There have been developed many criteria whether the solution blows up in a finite time. Especially, blow-up criteria in terms of vorticity, deformation tensor, and the Hessian of the pressure have been developed under various situations(See for example [2,4,5,9] and [3]). Also, localization of these blowup criteria have been developed [1,8].…”

Notes on the incompressible Euler and related equations on ℝ N

“…The questions of the finite time blow-up of the local smooth solution for (E) is a wide open problem (see e.g. [1,20,14,31] for graduate level text and survey articles on the status of the problem). In this study, there is celebrated result by Beale, Kato and Majda (BKM) in [2], which says that the blow-up of solution is controlled by the temporal accumulation of the maximum of the vorticity, ω = curl v, namely The blow-up criterion incorporating the direction of vorticity has been derived by Constantin, Fefferman and Majda (CFM) in [23], which says that smooth change of vorticity directions implies regularity of the solution.…”