h-Principles for the Incompressible Euler Equations

By Antoine Choffrut. 2012

53 citations recorded by the source corpus.

In Dissipative Euler Flows and Onsager’s Conjecture. arxiv.1205.3626, preprint, 2012, De Lellis and Székelyhidi construct Hölder continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $${{\mathbb T}^3}$$. The construction consists of adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the H−1-norm by exact solutions. Furthermore, we prove that the flows thus constructed on $${{\mathbb T}^3}$$ are genuinely three-dimensional and are not trivially obtained from solutions on $${{\mathbb T}^2}$$.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/fe30b71e0032099311cd5024e25f333148bbbef6

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

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].”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, background): “12 Indeed, the construction in [10], combined with ideas from [18, 20] or from this paper shows that h-principles are available for 2D Euler with velocity fields in C ̨ t C ̨ x for any ̨ < 1=5.”
On Nonperiodic Euler Flows with Hölder Regularity (2014, methodology): “It is a basic principle of convex integration that approximate solutions used in the construction turn out to be weak limits of solutions to the partial differential equation (or inclusion) that is being solved (see, for example, Appendix A below, or [8,13,23]).”
Convex integration and phenomenologies in turbulence (2019, background): “Currently, the best known result in 2D is that there exist non-conservative weak solutions to the Euler equations lying in any Hölder space with Hölder exponent less that 1/5 [29, 28, 19, 155].”
Dissipative Euler Flows with Onsager‐Critical Spatial Regularity (2014, methodology): “t/ is not constant was presented in [12]; moreover, the method of [12] was further adapted to show also the nonuniqueness of continuous and Hölder-continuous admissible weak solutions in [8] (see also [5] for associated h-principle statements).”