12 - Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions

Summary

In Chaps. 10 and 11 we introduced the notion of an approximate-solution sequence to the 2D Euler equations. The theory of such sequences is important in understanding the kinds of small-scale structures that form in the zero-viscosity limiting process and also for modeling the complex phenomena associated with jets and wakes. One important result of Chap. 11 was the use of the techniques developed in this book to prove the existence of solutions to the 2D Euler equation with vortex-sheet initial data when the vorticity has a fixed sign.

To understand the kinds of phenomena that can occur when vorticity has mixed sign and is in three dimensions, we address three important topics in this chapter. First, we analyze more closely the case of concentration by devising an effective way to measure the set on which concentration takes place. In Chap. 11 we showed that a kind of “concentration–cancellation” occurs for solution sequences that approximate a vortex sheet when the vorticity has distinguished sign. This cancellation property yielded the now-famous existence result for vortex sheets of distinguished sign (see Section 11.4). In this chapter we show that for steady approximate-solution sequences with L¹ vorticity control, concentration–cancellation occurs even in the case of mixed-sign vorticity (DiPerna and Majda, 1988).

We go on to discuss what kinds of phenomena can occur when L¹ vorticity control is not known. This topic is especially relevant to the case of 3D Euler solutions in which no a priori estimate for L¹ vorticity control is known.