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

This article is cited in 15 scientific papers (total in 15 papers)

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.


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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
Received: 27.10.2013

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
\by C.~Bardos, L.~Sz\'ekelyhidi, Jr., E.~Wiedemann
\paper Non-uniqueness for the Euler equations: the~effect of the boundary
\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
\issue 2(416)
\pages 3--22
\jour Russian Math. Surveys
\yr 2014
\vol 69
\issue 2
\pages 189--207

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    2. 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  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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  crossref  mathscinet  zmath  isi
    4. 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  crossref  mathscinet  zmath  isi  scopus
    5. P. Constantin, V. Vicol, “Remarks on high Reynolds numbers hydrodynamics and the inviscid limit”, J. Nonlinear Sci., 28:2 (2018), 711–724  crossref  mathscinet  zmath  isi  scopus
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    7. C. Foerster, L. Szekelyhidi, “Piecewise constant subsolutions for the Muskat problem”, Commun. Math. Phys., 363:3 (2018), 1051–1080  crossref  mathscinet  zmath  isi  scopus
    8. 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  crossref  mathscinet  zmath  isi
    9. T. D. Drivas, H. Q. Nguyen, “Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit”, J. Nonlinear Sci., 29:2 (2019), 709–721  crossref  mathscinet  isi  scopus
    10. C. Bardos, E. S. Titi, E. Wiedemann, “Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit”, Commun. Math. Phys., 370:1 (2019), 291–310  crossref  isi
    11. P. Constantin, M. C. Lopes Filho, H. J. Nussenzveig Lopes, V. Vicol, “Vorticity measures and the inviscid limit”, Arch. Ration. Mech. Anal., 234:2 (2019), 575–593  crossref  isi
    12. G.-M. Gie, J. P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, “The vanishing viscosity limit for some symmetric flows”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 36:5 (2019), 1237–1280  crossref  isi
    13. Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J., “Weak Vorticity Formulation of the Incompressible 2D Euler Equations in Bounded Domains”, Commun. Partial Differ. Equ., 45:2 (2020), 109–145  crossref  isi
    14. Akramov I., Debiec T., Skipper J., Wiedemann E., “Energy Conservation For the Compressible Euler and Navier-Stokes Equations With Vacuum”, Anal. PDE, 13:3 (2020), 789–811  crossref  mathscinet  zmath  isi
    15. Buckmaster T., Vicol V., “Convex Integration Constructions in Hydrodynamics”, Bull. Amer. Math. Soc., 58:1 (2021), 1–44  crossref  mathscinet  zmath  isi
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