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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 2(416), Pages 3–22 (Mi umn9578)

Non-uniqueness for the Euler equations: the effect of the boundary

C. Bardosa, L. Székelyhidi, Jr.b, E. Wiedemanncd

a Université Paris VII – Denis Diderot, Paris, France
b Universität Leipzig, Mathematisches Institut, Leipzig, Germany
c University of British Columbia, Vancouver, Canada
d Pacific Institute for the Mathematical Science, Vancouver, Canada

Abstract: Rotational initial data is considered for the two-dimensional incompressible Euler equations on an annulus. With use of the convex integration framework it is shown that there exist infinitely many admissible weak solutions (that is, with non-increasing energy) for such initial data. As a consequence, on bounded domains there exist admissible weak solutions which are not dissipative in the sense of Lions, as opposed to the case without physical boundaries. Moreover, it is shown that admissible solutions are dissipative if they are Hölder continuous near the boundary of the domain.
Bibliography: 34 titles.

Keywords: Euler equations, non-uniqueness, wild solutions, dissipative solutions, boundary effects, convex integration, inviscid limit, rotational flows.

 Funding Agency Grant Number European Research Council 277993 Fondation Sciences Mathématiques de Paris The research of the second author was supported by ERC Grant Agreement No. 277993. Part of this work was done while the third author was a~visitor to the project "Instabilities in Hydrodynamics" of the Fondation Sciences Mathématiques de Paris. He gratefully acknowledges the Fondation's support.

DOI: https://doi.org/10.4213/rm9578

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English version:
Russian Mathematical Surveys, 2014, 69:2, 189–207

Bibliographic databases:

UDC: 517.958+517.951
MSC: 35D30, 35Q35, 76B03

Citation: C. Bardos, L. Székelyhidi, Jr., E. Wiedemann, “Non-uniqueness for the Euler equations: the effect of the boundary”, Uspekhi Mat. Nauk, 69:2(416) (2014), 3–22; Russian Math. Surveys, 69:2 (2014), 189–207

Citation in format AMSBIB
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• https://doi.org/10.4213/rm9578
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This publication is cited in the following articles:
1. P. Gwiazda, A. Świerczewska-Gwiazda, E. Wiedemann, “Weak-strong uniqueness for measure-valued solutions of some compressible fluid models”, Nonlinearity, 28:11 (2015), 3873–3890
2. C. Bardos, T. T. Nguyen, “Remarks on the inviscid limit for the compressible flows”, Recent Advances in Partial Differential Equations and Applications, Contemporary Mathematics, 666, eds. Radulescu V., Sequeira A., Solonnikov V., Amer. Math. Soc., Providence, RI, 2016, 55–67
3. P. Constantin, T. Elgind, M. Ignatova, V. Vicol, “Remarks on the inviscid limit for the Navier–Stokes equations for uniformly bounded velocity fields”, SIAM J. Math. Anal., 49:3 (2017), 1932–1946
4. P. Constantin, V. Vicol, “Remarks on high Reynolds numbers hydrodynamics and the inviscid limit”, J. Nonlinear Sci., 28:2 (2018), 711–724
5. E. Wiedemann, “Localised relative energy and finite speed of propagation for compressible flows”, J. Differential Equations, 265:4 (2018), 1467–1487
6. C. Foerster, L. Szekelyhidi, “Piecewise constant subsolutions for the Muskat problem”, Commun. Math. Phys., 363:3 (2018), 1051–1080
7. T. D. Drivas, H. Q. Nguyen, “Onsager's conjecture and anomalous dissipation on domains with boundary”, SIAM J. Math. Anal., 50:5 (2018), 4785–4811
8. Drivas T.D., Nguyen H.Q., “Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit”, J. Nonlinear Sci., 29:2 (2019), 709–721
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