On Admissibility Criteria for Weak Solutions of the Euler Equations

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

573 citations recorded by the source corpus.

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper, we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct, in more than one space dimension, we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b6a381da58e760d297c329331b5d94c345d502f8

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Citations in the corpus

Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations (2013, methodology): “ though taking advantage of its main new ideas, and showed the existence of C0,θ solutions whose total kinetic energy is dissipated for any θ<1 5. The aim of this paper is to extend the results of [DLS10] showing that the set of initial data for which one can get inﬁnitely many Ho¨lder solutions ”
On Nonperiodic Euler Flows with Hölder Regularity (2014, methodology): “Other results that help draw attention to this point of view are the works of [11,16,21].”
Convex integration and phenomenologies in turbulence (2019, background): “The work [48], has since been extended and adapted by various authors to various problems arising in mathematical physics [46, 41, 181, 201, 27], see the reviews [47, 188, 50, 51] and references therein.”
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].”
Dissipative Euler Flows and Onsager's Conjecture (2012, background): “The ones of [29, ?] are just square summable functions of time and space, whereas the example of [31] was the first to be in the energy space and the constructions of [12, 13] gave bounded solutions.”
Dissipative Euler Flows with Onsager‐Critical Spatial Regularity (2014, background): “Furthermore, in [9, 10] the failure of energy conservation (as well as the nonuniqueness) were identified as a weak hprinciple—we refer to the survey [11] for more details.”
Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, background): “It follows from [8, 23] that no additional constraints on R̄ exist for L∞ weak solutions, but for Cθ weak solutions this was open and in fact there were indications that constraints might exist [4, 11].”
Non–uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data (2018, methodology): “ even in the class of Lipschitz initial data there exists an example of wild data. The proof in [6] relied substantially on the convex integration method developed by De Lellis and Sz´ekelyhidi [12], [13] for the incompressible Euler equations combined with a clever analysis of the Riemann problem f”
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.”
On solvability and ill-posedness of the compressible Euler system subject to stochastic forces (2017, methodology): “ion 4, we rewrite the system as an abstract Euler problem in the spirit of [15]. Section 5 is the heart of the paper. Here, the apparatus of convex integration developed by De Lellis and Sz´ekelyhidi [9] is adapted to stochastic framework. The main result is a stochastic variant of the oscillatory l”
h-Principles for the Incompressible Euler Equations (2012, background): “In [7,8] these existence resultswere extended to solutionswith bounded velocity and pressure, and in any space dimensions.”
h -Principle and Rigidity for C 1, α Isometric Embeddings (2012, background): “This is still beyond reach, and in [10] a weak version of convex integration was applied instead, to produce solutions in L∞ (see also [11] for a slightly better space) and, moreover, to show that a weak version of the h-principle holds.”