Abstract
We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Bertozzi and P. Constantin. Global regularity for vortex patches. Commun. Math. Phys., 152 (1993), 19–28.
G. Ben-Dor. Shock wave reflection phenomena. Springer (1992).
J.-Y. Chemin. Persistance de structures geometriques dans les fluides incompressibles bidimensionnels. Ann. Ec. Norm. Supér., 26(4) (1993), 1–16.
R. Danchin. Evolution temporelle d’une poche de tourbillon singulière. Commun. Partial. Diff. Eqns., 22 (1997), 685–721.
J.-M. Delort. Existence de nappes de tourbillon en dimension deux. J.Amer.Math. Soc., 4(3) (1991), 553–586.
R. DiPerna and A. Majda. Oscillations and concentrations in weak solutions of the incompressible euler equations. Commun. Math. Phys., 108 (1987), 667–689.
V. Elling. A possible counterexample to well-posedness of entropy solutions and to Godunov scheme convergence. Math.Comp., 75 (2006), 1721–1733.
V. Elling. The carbuncle phenomenon is incurable. Acta Math. Sci. (ser. B), 29(6) (2009), 1647–1656.
V. Elling. Existence of algebraic vortex spirals. Hyperbolic problems. Theory, Numerics and Applications., Ser. Contemp. Appl. Math. CAM, 17, vol. 1, World Sci. Publishing, Singapore, (2012), 203–214.
V. Elling. Algebraic spiral solutions of 2d incompressible euler. J. Diff. Eqns., 255(11) (2013), 3749–3787.
V. Elling. Self-similar 2d euler solutions with mixed-sign vorticity, (submitted), (2014).
H. Kaden. Aufwicklung einer unstabilen Unstetigkeitsfläche. Ingenieur-Archiv, 2 (1931), 140–168.
R. Krasny. Computing vortex sheet motion. Proceedings of the International Congress of Mathematicians, I,II (1991), 1573–1583.
M.C. Lopes-Filho, J. Lowengrub, H.J. Nussenzveig Lopes and Yuxi Zheng. Numerical evidence of nonuniqueness in the evolution of vortex sheets. ESAIM:M2AN, 40 (2006), 225–237.
P.-L. Lions. Mathematical topics in fluid mechanics. Oxford University Press, (1996).
C. De Lellis and L. Székelyhidi. The Euler equations as a differential inclusion. Ann. Math., 170(3) (2009), 1417–1436.
C. De Lellis and L. Székelyhidi. On admissibility criteria for weak solutions of the Euler equations. Arch. Rat. Mech. Anal., 195(1) (2010), 225–260.
A. Majda and A. Bertozzi. Vorticity and incompressible flow. Cambridge University Press (2002).
D.W. Moore. The rolling-up of a semi-infinite vortex sheet. Proc. Roy. Soc. London A, 345 (1975), 417–430.
M. Nitsche, M.A. Taylor and R. Krasny. Comparison of regularizations of vortex sheet motion. Comp. Fluid Solid Mech. (2003).
D. Pullin. The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. FluidMech., 88(3) (1978), 401–430.
D. Pullin. On similarity flows containing two-branched vortex sheets. Mathematical aspect of vortex dynamics (R. Caflisch, ed.), SIAM, (1989), 97–106.
N. Rott. Diffraction of a weak shock with vortex generation. J. Fluid Mech., 1 (1956), 111–128.
P.G. Saffman. Vortex dynamics, Cambridge University Press (1992).
V. Scheffer. An inviscid flow with compact support in space-time. J. Geom. Anal., 3 (1993), 343–401.
A. Shnirelman. On the nonuniqueness of weak solutions of the Euler equation. Comm. Pure Appl. Math., 50 (1997), 1261–1286.
A. Shnirelman. Weak solutions with decreasing energy of the incompressible Euler equations. Comm. Math. Phys., 210 (2000), 541–603.
M. van Dyke. An album of fluid motion. The Parabolic Press, Stanford, California (1982).
M. Vishik. Incompressible flows of an ideal fluidwith vorticity in borderline spaces of Besov type. Ann. Sci École Norm. Sup. (4), 32 (1999), 769–812.
V. Yudovich. Non-stationary flow of an ideal incompressible liquid. Comp. Math. Math. Phys., 3 (1963), 1407–1457.
V.I. Yudovich. Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett., 2 (1995), 27–38.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Elling, V. Algebraic spiral solutions of the 2d incompressible Euler equations. Bull Braz Math Soc, New Series 47, 323–334 (2016). https://doi.org/10.1007/s00574-016-0141-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-016-0141-2