Global Existence and Uniqueness of Weak Solutions of Three-Dimensional Euler Equations with Helical Symmetry in the Absence of Vorticity Stretching

Abstract

We prove uniqueness and existence of the weak solutions of Euler equations with helical symmetry, with initial vorticity in $L^{\infty}$ under “no vorticity stretching” geometric constraint. Our article follows the argument of the seminal work of Yudovich. We adjust the argument to resolve the difficulties which are specific to the helical symmetry.

MSC codes

  1. 76B03
  2. 35Q35
  3. 35D05
  4. 76B47

Keywords

  1. inviscid helical flows
  2. three-dimensional Euler equations

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I and II. Comm. Pure Appl. Math., 12 (1959), pp. 623–727 and 17 (1964), pp. 35–92.
2.
C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., 40 (1972), pp. 769–790.
3.
C. Bardos and E. S. Titi, Euler equations of incompressible ideal fluids, Russian Math. Surveys, 62 (2007), pp. 409–451.
4.
J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94 (1984), pp. 61–66.
5.
D. Chae and O. Yu. Imanuvilov, Generic solvability of the axisymmetric 3-D Euler equations and the 2-D Boussinesq equations, J. Differential Equations, 156 (1999), pp. 1–17.
6.
P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. (N.S.), 44 (2007), pp. 603–621.
7.
P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1988.
8.
J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), pp. 553–586.
9.
R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), pp. 667–689.
10.
A. Dutrifoy, Existence globale en temps de solutions hélicoïdales des équations d'Euler, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), pp. 653–656.
11.
A. Ferrari, On the blow-up of solutions of the 3-D Euler equations in a bounded domain, Comm. Math. Phys., 155 (1993), pp. 277–294.
12.
G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Linearized Steady Problems, Springer Tracts in Natural Philosophy. 38, Springer-Verlag, New York, 1994.
13.
T. Y. Hou and C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), pp. 661–697.
14.
T. Kato, A remark on a theorem of C. Bardos on the 2-D Euler equation, UC Berkeley, Berkeley, CA, 1992, Preprint.
15.
O. A. Ladyženskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), pp. 155–177.
16.
A. Mahalov, E. S. Titi, and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations, Arch. Rational Mech. Anal., 112 (1990), pp. 193–222.
17.
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, L. Cambridge Texts Appl. Math., 27, Cambridge University Press, Cambridge, UK, 2002.
18.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ, 1970.
19.
M. R. Ukhovskii and V. I. Yudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech., 32 (1968), pp. 52–61.
20.
M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. École Norm. Sup. (4), 32 (1999), pp. 769–812.
21.
V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), pp. 805–808.
22.
V. I. Yudovich, Some bounds for solutions of elliptic equations, Mat. Sb. (N.S.), 59 (1962), pp. 229–244.
23.
V. I. Yudovich, Nonstationary flows of an ideal incompressible fluid, Z. Vyčisl. Mat. i Mat. Fiz., 3 (1963), pp. 1032–1066.
24.
V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett., 2 (1995), pp. 27–38.

Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 269 - 296
ISSN (online): 1095-7154

History

Submitted: 14 February 2008
Accepted: 5 January 2009
Published online: 13 May 2009

MSC codes

  1. 76B03
  2. 35Q35
  3. 35D05
  4. 76B47

Keywords

  1. inviscid helical flows
  2. three-dimensional Euler equations

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media