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Onsager’s conjecture for the incompressible Euler equations in bounded domains. (English) Zbl 1390.35241

Summary: The goal of this note is to show that, in a bounded domain \({\Omega \subset \mathbb{R}^n}\), with \({\partial \Omega\in C^2}\), any weak solution \({(u(x,t),p(x,t))}\), of the Euler equations of ideal incompressible fluid in \({\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t}\), with the impermeability boundary condition \({u\cdot \vec n =0}\) on \({\partial\Omega\times(0,T)}\), is of constant energy on the interval \((0,T)\), provided the velocity field \({u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))}\), with \(\alpha > \frac13\).

MSC:

35Q31 Euler equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Bardos C., Titi E.S.: Mathematics and turbulence: where do we stand?. J. Turbul., 14(3), 42-76 (2013) · doi:10.1080/14685248.2013.771838
[2] Buckmaster, T., De Lellis, C., Székelyhidi, L. Jr., Vicol, V.: Onsager’s conjecture for admissible weak solutions, arXiv:1701.08678. · Zbl 1480.35317
[3] Cheskidov A., Constantin P., Friedlander S., Shvydkoy R.: Energy con- servation and Onsagers conjecture for the Euler equations. Nonlinearity 21(6), 1233-1252 (2008) · Zbl 1138.76020 · doi:10.1088/0951-7715/21/6/005
[4] Constantin, P., W. E., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Commun. Math. Phys., 165, 207-209 1994 · Zbl 0818.35085
[5] Drivas, T.D., Eyink, G.L.: An Onsager singularity theorem for turbulent solutions of compressible Euler equations, 2017. arXiv:1704.03409. · Zbl 1397.35191
[6] Duchon J., Robert R.: Inertial energy dissipation for weak solutions of incompressible Euler and NavierStokes equations. Nonlinearity 13, 249-255 (2000) · Zbl 1009.35062 · doi:10.1088/0951-7715/13/1/312
[7] Eyink G. L.: Energy dissipation without viscosity in ideal hydrodynamics, I Fourier analysis and local energy transfer. Phys. D, 78(3-4), 222-240 (1994) · Zbl 0817.76011 · doi:10.1016/0167-2789(94)90117-1
[8] Feireisl E., Gwiazda P., Świerczewska-Gwiazda A., Wiedemann E.: Regularity and energy conservation for the compressible Euler equations. Arch. Rational Mech. & Anal., 223(3), 1375-1395 (2017) · Zbl 1365.35113 · doi:10.1007/s00205-016-1060-5
[9] Guillemin, V., Sternberg, S.: Geometric Asymptotic, American Mathematical Society, Providence, 1977 · Zbl 0364.53011
[10] Gwiazda, P., Michálek, M., Świerczewska-Gwiazda, A.: A note on weak solutions of coversation laws and energy/entropy conservation, arXiv:1706.10154. · Zbl 1172.35352
[11] Isett, P.: A Proof of Onsager’s conjecture, 2016, arXiv:1608.08301. · Zbl 1335.58018
[12] Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Holder Spaces. Graduate Studies in Mathematics, Vol. 12. American Mathematical Society , Providence, 1996. · Zbl 0865.35001
[13] Lions, J.-L., Magenes, E.: Problémes aux Limites Non Homógenes et Applications, Vol. 1, Dunod, Paris 1968 · Zbl 0165.10801
[14] Onsager, L.: Statistical hydrodynamics, Nuovo Cimento, (9) 6, Supplemento, 2(Convegno Internazionale di Meccanica Statistica), 279-287 1949
[15] Schwartz, L.: Théorie des distributions á valeurs vectorielles. II. Ann. Inst. Fourier. Grenoble, 8, 1-209 1958 (French) · Zbl 0089.09801
[16] Temam R.: On the Euler equations of incompressible perfect fluids. J. Funct Anal 20, 32-43 (1975) · Zbl 0309.35061 · doi:10.1016/0022-1236(75)90052-X
[17] Yu C.: Energy conservation for the weak solutions of the compressible Navier-Stokes equations. Arch. Rational Mech. & Anal., 225(3), 1073-1087 (2017) · Zbl 1375.35336 · doi:10.1007/s00205-017-1121-4
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