Abstract
We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.
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@article{BressanFlores, MRKEY = {1320474},
AUTHOR = {Bressan, Alberto and Flores, Fabi{á}n},
TITLE = {On total differential inclusions},
JOURNAL = {Rend. Sem. Mat. Univ. Padova},
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MRCLASS = {35R70 (49J24)},
MRNUMBER = {96b:35244},
URL = {http://www.numdam.org/item?id=RSMUP_1994__92__9_0},
ZBLNUMBER = {0821.35158},
MRREVIEWER = {Corneliu Ursescu},
} -
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[Shnirelman1]
A. Shnirelman, "On the nonuniqueness of weak solution of the Euler equation," Comm. Pure Appl. Math., vol. 50, iss. 12, pp. 1261-1286, 1997.
@article{Shnirelman1, MRKEY = {1476315},
AUTHOR = {Shnirelman, A.},
TITLE = {On the nonuniqueness of weak solution of the {E}uler equation},
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author = {De Lellis, Camillo and Széklyhidi Jr., Lászlo},
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}