The Euler equations as a differential inclusion

By Camillo De Lellis, L. Sz'ekelyhidi. 2007

661 citations recorded by the source corpus.

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in R n with n 2. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ea3ccc97d0ef9a32d22d0174c078b06f71ca4890

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Citations in the corpus

The $h$-principle and the equations of fluid dynamics (2011, methodology): “In applications to evolution equations, D is a space-time domain, say D = R × (0, T ), and thus this argument yields weak solutions with compact time support, as in [27, 31, 75].”
h-Principles for the Incompressible Euler Equations (2012, methodology): “In [7,8] these existence resultswere extended to solutionswith bounded velocity and pressure, and in any space dimensions.”
Lectures on the Onsager conjecture (2010, methodology): “Very recently more regular solutions have been found by De Lellis and Székelyhidi in the class u ∈ L∞L∞ also with compact support in space-time, [15].”
Convex integration and phenomenologies in turbulence (2019, neutral): “[45, 46]).”
Relaxation of the incompressible porous media equation (2011, methodology): “The methodology of how to pass from this property to weak solutions via the Baire category theorem is well known [9, 1, 4, 5], but for the convenience of the reader we present the details in Theorem 5.”
Statistical Solutions of Hyperbolic Conservation Laws: Foundations (2016, background): “have recently been able to construct infinitely many entropy solutions for prototypical multi-dimensional systems such as the Euler equations for polytropic gas dynamics (see [16, 17] and references therein).”
Partial Differential Equations (2009, background): “[20] Mancini G and Sandeep K 2004 Cylindrical symmetry of externals of a Hardy–Sobolev inequality; Ann.”
Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, neutral): “2 in [15] the construction of Beltrami waves (see also the summary given in [2]).”
Anomalous dissipation for 1/5-Hölder Euler flows (2015, background): “In [7, 8] a new point of view was introduced, relating the issue of energy conservation to Gromov’s h-principle, see also [10].”
Beltrami Fields with a Nonconstant Proportionality Factor are Rare (2014, methodology): “For instance, de Lellis and Székelyhidi have utilized strong Beltrami fields to construct Hölder continuous weak solutions to the Euler equation in the 3-torus that dissipate energy [5,8], while in References [10,11] we constructed strong Beltrami fields in R3 having vortex lines and vortex tubes (t”