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Reduced Hausdorff dimension and concentration-cancellation for two- dimensional incompressible flow. (English) Zbl 0707.76026

We study the detailed limiting behavior of approximate solution sequences for 2-D Euler with vortex sheet initial data. A sequence of smooth velocity fields \(\nu^{\epsilon}(x,t)\) is an approximate solution sequence for 2-D Euler equation provided that \(\nu\) is incompressible, i.e. div \(\nu\) \(=0\), and satisfies the following properties:
(1) The velocity fields \(\nu^{\epsilon}\) have uniformly bounded local kinetic energy,
(2) The corresponding vorticity, \(\omega^{\epsilon}=curl \nu^{\epsilon}\), is uniformly bounded in \(L^ 1,\)
(3) The velocity field \(\nu^{\epsilon}\) is weakly consistent with 2-D Euler.
This paper as well as the author’s companion paper [Commun. Pure Appl. Math. 60, 301-345 (1987)] are concerned with the following basic questions: If \(\nu_ 0\) is vortex sheet initial data, is there a weak solution of 2-D Euler on \(R^ 2\times (0,\infty)\) with initial data \(\nu_ 0?\) If \(\nu^{\epsilon}\) is an approximate solution sequence for 2-D Euler, does \(\nu^{\epsilon}\) converge to a weak solution of 2-D Euler as \(\epsilon\downarrow 0?\) Do new phenomena occur in the limiting process?
To the authors’ knowledge, this paper as well as the companion paper (loc. cit.) are the first to address these questions in the mathematical literature. We introduce a new tool to measure weak convergence, the reduced defect measure.
Our main results are the following. We prove that the reduced defect measure associated with an approximation sequence of velocity fields for 2-D Euler always concentrates on a space-time set of Hausdorff dimension less than or equal to one. This concentration phenomena can occur despite the fact that the standard weak-star defect measure introduced by P. L. Lions [e.g.: Ann. Inst. Henri Poincare, Anal. Non Lineaire 1, 109-145 (1984; Zbl 0541.49009); and ibid. 1, 223-283 (1984)] may attach positive mass to space-time sets of positive three-dimensional Lebesgue measure. Examples given by the authors [loc. cit., and Commun. Math. Phys. 108, 667-689 (1987; Zbl 0626.35059)] prove that this result is sharp. Furthermore, under the assumption that the reduced defect measure concentrates on a space-time set with Hausdorff dimension strictly less than one, we prove that there is concentration-cancellation and \(\nu\) is a weak solution of the 2-D Euler equations.

MSC:

76B99 Incompressible inviscid fluids
35D99 Generalized solutions to partial differential equations
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