The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem (Q1989852)

The Euler equations for an incompressible fluid is considered on the torus \(\mathbb T^d\). The initial value problem is investigated by a concave maximization method following \textit{V. Scheffer} [J. Geom. Anal. 3, No. 4, 343--401 (1993; Zbl 0836.76017)], \textit{A. Shnirelman} [Commun. Pure Appl. Math. 50, No. 12, 1261--1286 (1997; Zbl 0909.35109)], \textit{C. De Lellis} and \textit{L. Székelyhidi jun.} [Ann. Math. (2) 170, No. 3, 1417--1436 (2009; Zbl 1350.35146)]. It is proved that any local smooth solution of the Euler equations can be recovered from the maximization problem for a sufficiently short time. A connection between the maximization problem and the theory of sub-solutions to the Euler equations is established. The results are extended to a class of compressible fluids.