Abstract
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 \({{\mathbb T}^2}\).
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Communicated by V. Šverák
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Choffrut, A. h-Principles for the Incompressible Euler Equations. Arch Rational Mech Anal 210, 133–163 (2013). https://doi.org/10.1007/s00205-013-0639-3
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DOI: https://doi.org/10.1007/s00205-013-0639-3