On the Euler equations of incompressible fluids

By Peter Constantin. 2007

266 citations recorded by the source corpus.

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 RayleighTaylor 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.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/04768c8e582656d520824560d51231348f6deac5

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An Lp Theory (2010, background): “Concerning the Navier–Stokes equations, the first thought goes to the classical non-slip boundary condition u = 0 on . However, it is well known that in this case “strong” convergence does not hold up to the boundary, since boundary layers appear; see the thorough analysis of [ 15 ], in particular S”
All speed scheme for the low Mach number limit of the isentropic Euler equations (2009, background): “, when the fluid velocity is small compared with the speed of sound [3], the solution of (1.”
Asymptotic Models for Internal Waves (2007, background): “Indeed, these can be subject to Kelvin–Helmholtz type instabilities (see for instance [4] and [21]), although one might expect a “stability of the instability” result even in the face of such instabilities (see [16]).”
Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces (2013, unknown): “We shall only mention a few and refer to Majda-Bertozzi [23] and Constantin [8] for more extensive references.”
Convex integration and phenomenologies in turbulence (2019, background): “As for their viscous counterpart, the global in time well-posedness for the three-dimensional Euler equations remains an outstanding open problem, arguably of greater physical significance [34].”
On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations (2019, background): “The fundamental question regarding the global regularity of the 3D Euler and Navier-Stokes equations for general smooth initial data with finite energy remains open, and it is generally viewed as one of the most important open questions in mathematical fluid mechanics; see the surveys [10, 18, 19, 2”
Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, background): “See, for instance, the survey articles [2, 9, 10], the books [7, 28], and references therein for a review of the subject.”