On the Euler equations of incompressible fluids

Abstract

Abstract. Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$ 2 / 3 de-aliasing method

“…In the absence of any information re:the smoothness of the underlying Euler solutions (-as loss of smoothness for the 3D Euler equations is still a challenging open problem), energy-preserving numerical method need not shed light on the question of global regularity vs. finite-time blow-up. Recall that L 2 -energy conservation was conjectured by Onsager [ON49] and verified in [Ey94,CET94,BT10] under the assumption of minimal smoothness of u, but otherwise is not supported by the energy decreasing solutions of Euler equation, [Co07,DeLS12,Buck13].…”

“…Although there is no known energy dissipation-based selection principle to identify a unique solution of Euler equations within the class of "rough" data (similar to the entropy dissipation selection principle for Burgers' equations), nevertheless we argue that the L 2 -energy conservation of the (pseudo-)spectral approximations may be responsible to their unstable behavior. While L 2 -energy conservation holds for weak solutions with a minimal degree of 1/3-order of smoothness (Onsager's conjecture proved in [Ey94,CET94,BT10]), there are experimental and numerical evidence for the other part of Onsager's conjecture that anomalous dissipation of energy shows up for "physical-turbulent" L 2 -solutions of Euler equations [Co07]. Whether this observed anomalous dissipation of energy should be due to spontaneous appearance of singularities in smooth solutions of the Euler equation or to the fact that physical initial data may be rough is a completely open problem.…”

Blow up of incompressible Euler solutions

“…In this note we address the analogous problem for the inviscid incompressible Euler equations, which for some reason is not explicitely a Millennium Problem, although mentioned briefly in [7] and in [6] referred to as "a major open problem in PDE theory, of far greater physical importance than the blowup problem for Navier-Stokes equations, which of course is known to the nonspecialists because it is a Clay Millenium Problem". In fact, since the viscosity in the Navier-Stokes equations is allowed to be arbitarily small and solutions of the Euler equations are defined as viscosity solutions of the Navier-Stokes equations under vanishing viscosity, the Euler equations effectively are included in the Millenium Problem.…”

“…In the work cited in [5,6,10,11,12], blowup is identified by the development of an infinite velocity gradient of a specific initially smooth exact Euler solution as time approaches a blowup time. This form of blowup detection, which we refer to as local blowup, requires pointwise accurate information (analytically or computationally) of the blowup to infinity, which (so far) has shown to be impossible to obtain.…”

“…A corner stone in the approach of local blowup presented in [5,6] is a result from 1984 by Beale-Kato-Majda [1] stating that "the maximum norm of the vorticity alone controls the breakdown of smooth solutions for the 3d Euler equations": An initially smooth Euler solution does not blow up on [0, T ] if and only if…”