Inviscid symmetry breaking with non-increasing energy

In a recent article, C. Bardos et. al. constructed weak solutions of the three-dimensional incompressible Euler equations which emerge from two-dimensional initial data yet become fully three-dimensional at positive times. They asked whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. In this note, we give a positive answer to this question and show that such a construction is possible for a large class of initial data. We use convex integration techniques as developed by De Lellis-Székelyhidi.

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