Approximate solutions of the incompressible Euler equations with no concentrations

✍ Scribed by Milton C Lopes Filho; Helena J Nussenzveig Lopes; Eitan Tadmor

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 281 KB
• Volume: 17
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(00)00113-x

✦ Synopsis

We present a sharp local condition for the lack of concentrations in (and hence the L 2 convergence of) sequences of approximate solutions to the incompressible Euler equations. We apply this characterization to greatly simplify known existence results for 2D flows in the full plane (with special emphasis on rearrangement invariant regularity spaces), and obtain new existence results of solutions without energy concentrations in any number of spatial dimensions.

Our results identify the 'critical' regularity which prevents concentrations, regularity which is quantified in terms of Lebesgue, Lorentz, Orlicz and Morrey spaces. Thus, for example, the strong convergence criterion cast in terms of circulation logarithmic decay rates due to DiPerna and Majda is simplified (removing the weak control of the vorticity at infinity) and extended (to any number of space dimensions).

Our approach relies on using a generalized div-curl lemma (interesting for its own sake) to replace the role that elliptic regularity theory has