Inviscid symmetry breaking with non-increasing energy (Q392616)

\textit{C. Bardos} et al. [SIAM J. Math. Anal. 45, No. 3, 1871--1885 (2013; Zbl 1291.35207)] constructed weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. In the present article, a positive answer is given to the question whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. The first theorem states that for a two-dimensional vector \(v_0\in^{\mathrm{H}} (\mathbb T^2)\) (two-dimensional torus), there exist infinitely many dissipative weak solutions of the incompressible Euler equations, with the \(L^2\) norm less than \(\epsilon\). Furthermore, two other theorems are proved, the second one is a corollary of the first theorem above.