Energy conservation for the weak solutions to the incompressible inhomogeneous Euler-Korteweg equations (Q2116241)

The author considers the system of Euler-Korteweg equations for incompressible and inhomogeneous fluid on the torus. Some sufficient Besov regularity conditions for weak solutions are presented that guarantee the local energy conservation property, i.e. solutions verify the Onsager conjecture. The result is close to that in [\textit{T. Dębiec} et al., Calc. Var. Partial Differ. Equ. 57, No. 6, Paper No. 160, 12 p. (2018; Zbl 1442.76053)] (where the homogeneous case is considered), and follows immediately from a more general theory in [\textit{P. Gwiazda} et al., Arch. Ration. Mech. Anal. 229, No. 3, 1223--1238 (2018; Zbl 1398.35168)], see also [\textit{C. Bardos} et al., J. Nonlinear Sci. 29, No. 2, 501--510 (2019; Zbl 1420.35202)] and [\textit{C. Bardos} et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 475, No. 2230, Article ID 20190289, 18 p. (2019; Zbl 1472.35285)] for problems with boundary conditions.