Homogeneous solutions to the 3D Euler system

Shvydkoy, Roman

doi:10.1090/tran/7022

Homogeneous solutions to the 3D Euler system
HTML articles powered by AMS MathViewer

by Roman Shvydkoy PDF

Trans. Amer. Math. Soc. 370 (2018), 2517-2535 Request permission

Abstract:

We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated by recent exclusions of self-similar blowup for Euler and its relation to the Onsager conjecture and intermittency. We reveal several new classes of solutions and prove rigidity properties in specific categories of genuinely 3D solutions. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function, and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler flow on the sphere. We further show that in the case when homogeneity corresponds to the Onsager-critical state, the anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme $0$-dimensional intermittencies in dissipative flows.

References

D. J. Acheson, Elementary fluid dynamics, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1990. MR 1069557
G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
Anne Bronzi and Roman Shvydkoy, On the energy behavior of locally self-similar blowup for the Euler equation, Indiana Univ. Math. J. 64 (2015), no. 5, 1291–1302. MR 3418442, DOI 10.1512/iumj.2015.64.5657
Dongho Chae and Roman Shvydkoy, On formation of a locally self-similar collapse in the incompressible Euler equations, Arch. Ration. Mech. Anal. 209 (2013), no. 3, 999–1017. MR 3067830, DOI 10.1007/s00205-013-0630-z
A. Cheskidov and R. Shvydkoy, Euler equations and turbulence: analytical approach to intermittency, SIAM J. Math. Anal. 46 (2014), no. 1, 353–374. MR 3152734, DOI 10.1137/120876447
Camillo De Lellis and László Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math. 193 (2013), no. 2, 377–407. MR 3090182, DOI 10.1007/s00222-012-0429-9
Philip Isett, Hölder continuous Euler flows with compact support in time, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Princeton University. MR 3153420
Mikhail Korobkov, Konstantin Pileckas, and Remigio Russo, The Liouville theorem for the steady-state Navier-Stokes problem for axially symmetric 3D solutions in absence of swirl, J. Math. Fluid Mech. 17 (2015), no. 2, 287–293. MR 3345358, DOI 10.1007/s00021-015-0202-0
L. Landau, A new exact solution of Navier-Stokes equations, C. R. (Doklady) Acad. Sci. URSS (N.S.) 43 (1944), 286–288. MR 0011205
Xue Luo and Roman Shvydkoy, 2D homogeneous solutions to the Euler equation, Comm. Partial Differential Equations 40 (2015), no. 9, 1666–1687. MR 3359160, DOI 10.1080/03605302.2015.1045073
R. Monneau, L. Paszkowski, and L. Xue, Notes of self-similar singular solutions for the Euler equations, preprint, personal communication.
L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. Supplemento, 2 (Convegno Internazionale di Meccanica Statistica), 279–287. MR 36116, DOI 10.1007/BF02780991
Steven Rosenberg, The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, vol. 31, Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. MR 1462892, DOI 10.1017/CBO9780511623783
Roman Shvydkoy, Lectures on the Onsager conjecture, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 3, 473–496. MR 2660721, DOI 10.3934/dcdss.2010.3.473
V. Šverák, On Landau’s solutions of the Navier-Stokes equations, J. Math. Sci. (N.Y.) 179 (2011), no. 1, 208–228. Problems in mathematical analysis. No. 61. MR 3014106, DOI 10.1007/s10958-011-0590-5