Energy conservation for weak solutions to the pressureless Euler–Navier–Stokes system in three‐dimensional space

Abstract

We study the energy conservation for the weak solution of the 3D Euler–Navier–Stokes system. we provide two sufficient conditions on the regularity of weak solutions to guarantee the conservation of total energy, which also including the initial energy. Due to the couple structure between inhomogeneous Euler equation and incompressible Navier–Stokes system, we adopt a variant of the method from R. Chen and C. Yu (J. Math. Pures Appl. 131 (2019), 1–16.) and J. Lions (Rend. Semin. Mat. Univ. Padova 30 (1960), 16–23.) to deal with the density of Euler flows and velocity of Navier–Stokes fluid.