Energy conservation for 3D Euler and Navier-Stokes equations in a bounded domain. Applications to Beltrami flows
Abstract: In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier-Stokes equations. Subsequently, we utilise this finding, which is based on a proper approximation of the velocity (and doesn't require estimates or additional assumptions on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier-Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field.
- K. Abe. Existence of vortex rings in Beltrami flows. Comm. Math. Phys., 391(2):873–899, 2022.
- C. Bardos and E.S. Titi. Onsager’s conjecture for the incompressible Euler equations in bounded domains. Arch. Ration. Mech. Anal., 228(1):197–207, 2018.
- Remarks on the breakdown of smooth solutions for the 3333-D Euler equations. Comm. Math. Phys., 94(1):61–66, 1984.
- H. Beirão da Veiga and J. Yang. On the energy equality for solutions to Newtonian and non-Newtonian fluids. Nonlinear Anal., 185:388–402, 2019.
- E. Beltrami. Sui principii fondamentali dell’idrodinamica razionale. Mem. dell’Accad. Scienze Bologna, page 394, 1873.
- L. C. Berselli and S. Georgiadis. Three results on the energy conservation for the 3D Euler equations. NoDEA Nonlinear Differential Equations Appl., 31:33, 2024.
- L. C. Berselli and R. Sannipoli. Velocity-vorticity geometric constraints for the energy conservation of 3D ideal incompressible fluids, Technical report, arXiv:2405.08461, 2024.
- Onsager’s conjecture for admissible weak solutions. Comm. Pure Appl. Math., 72(2):229–274, 2019.
- Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys., 165(1):207–209, 1994.
- C. De Lellis and Jr. L. Székelyhidi. The Euler equations as a differential inclusion. Ann. of Math. (2), 170(3):1417–1436, 2009.
- L. De Rosa. On the helicity conservation for the incompressible Euler equations. Proc. Amer. Math. Soc., 148(7):2969–2979, 2020.
- A. Enciso and D. Peralta-Salas. Beltrami fields with a nonconstant proportionality factor are rare. Arch. Ration. Mech. Anal., 220(1):243–260, 2016.
- U. Frisch. Turbulence, The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge, 1995.
- On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond. SMAI J. Comput. Math., 5:89–129, 2019.
- P. Isett. A proof of Onsager’s conjecture. Ann. of Math. (2), 188(3):871–963, 2018.
- Energy conservation of weak solutions for the incompressible Euler equations via vorticity. J. Differential Equations, 372:254–279, 2023.
- Energy equalities for compressible Navier-Stokes equations. Nonlinearity, 32(11):4206–4231, 2019.
- L. Onsager. Statistical hydrodynamics. Nuovo Cimento (9), 6(Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)):279–287, 1949.
- On the energy and helicity conservation of the incompressible Euler equations. Technical Report arXiv:2307.08322, 2023.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
