Energy conservation of weak solutions for the incompressible Euler equations via vorticity

Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the Lp control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion & omega; = curl v & ISIN; L3 (0, T; L n+23n (2)) (n = 2, 3), that generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatiotemporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and backward difference & RADIC;& rho; belongs to L & INFIN;(0, T ; Ln(Tn)) (n = 2, 3), which gives an affirmative answer to a problem proposed by Chen and Yu in [5]. & COPY; 2023 Elsevier Inc. All rights reserved.