Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations

By Sara Daneri. 2013

51 citations recorded by the source corpus.

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent $${\theta < \frac{1}{16}}$$θ<116 . Using the techniques introduced in De Lellis and Székelyhidi (Inventiones Mathematicae 9:377–407, 2013; Dissipative Euler flows and Onsager’s conjecture, 2012), we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b210348a0b1c25d4b6a34a46a9b0cca8b73416fa

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

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).”
Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, background): “A somewhat different notion of admissible subsolution was previously introduced in [5], but there the subsolution depends on a specific iteration scheme and cannot be used in our setting for Theorem 1.”
Anomalous dissipation for 1/5-Hölder Euler flows (2015, background): “A similar issue has been recently treated in the paper [6].”
Global Ill‐Posedness of the Isentropic System of Gas Dynamics (2013, methodology): “te that, in analogy to what has been shown recently for the incompressible Euler equations, even a \piecewise Holder regularity" might not be enough; see [13], [14], [16], [2] and in particular [8]. This paper draws also heavily from the work [22] where Szekelyhidi coupled the methods introduce”