2D Homogeneous Solutions to the Euler Equation

Xue Luo Beihang University Roman Shvydkoy University of Illinois at Chicago

Analysis of PDEs mathscidoc:1701.03037

Communications in Partial Differential Equations, 40, 1666-1687, 2015
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.
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  title={ 2D Homogeneous Solutions to the Euler Equation},
  author={Xue Luo, and Roman Shvydkoy},
  booktitle={Communications in Partial Differential Equations},
Xue Luo, and Roman Shvydkoy. 2D Homogeneous Solutions to the Euler Equation. 2015. Vol. 40. In Communications in Partial Differential Equations. pp.1666-1687. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170129103349711110130.
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