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Approximate solutions of the incompressible Euler equations with no concentrations. (English) Zbl 0965.35110
The author presents 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. This characterization is applied to greatly simplify known existence results for 2D flows in the full plane (with special emphasis on rearrangement invariant regularity spaces), and new existence results of solutions without energy concentrations in any number of spatial dimensions are also obtained. The presented 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 by removing the weak control of the vorticity at infinity and it is extended to any number of space dimensions. The authors’ approach relies on using a generalized div-curl lemma to replace the role that elliptic regularity theory has played previously in this problem.
35Q05Euler-Poisson-Darboux equation and generalizations
35Q35PDEs in connection with fluid mechanics
35A35Theoretical approximation to solutions of PDE