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Computational Mathematics and Mathematical Physics

, Volume 59, Issue 10, pp 1742–1752 | Cite as

On Smooth Vortex Catastrophe of Uniqueness for Stationary Flows of an Ideal Fluid

  • O. V. TroshkinEmail author
Article
  • 14 Downloads

Abstract

It is well known that the steady-state plane-parallel or spatial axisymmetric flow of an ideal incompressible fluid in a finite-length plane channel or pipe that can be decomposed in powers of spatial coordinates (i.e., is an analytical and, hence, exactly computable flow) is uniquely determined by the inflow vorticity. Under the same boundary conditions, an infinite number of uncomputable phantoms, i.e., infinitely smooth, but nonanalytical flows exist if the domain of a unique analytical flow contains a sufficiently intense vortex cell where the maximum principle is violated for the stream function. A scheme for obtaining an uncomputable vortex phantom for the Euler fluid dynamics equations is described in detail below.

Keywords:

steady-state ideal incompressible fluid plane or axisymmetric finite channel inflow vorticity unique analytical flow violation of the maximum principle nonuniqueness of smooth nonanalytical flows 

Notes

FUNDING

This work was performed within the state assignment of the Federal Research Center Scientific Research Institute for System Analysis of the Russian Academy of Sciences (fundamental scientific research, GP 14) on subject no. 0065-2019-0005 “Mathematical Modeling of Dynamic Processes in Deformable and Reacting Media on Multiprocessor Computer Systems,” no. AAAA-A19-119011590092-6.

REFERENCES

  1. 1.
    L. Euler, “Principes généraux de l’etat d’equilibre des fluids,” Mém. Acad. Sci. Berlin 11, 217–273 (1757); “Principes généraux du mouvement des fluids,” Mém. Acad. Sci. Berlin 11, 274–315 (1757); “Continuation des recherrches sur la theorie du mouvement des fluids,” Mém. Acad. Sci. Berlin 11, 316–361 (1757).Google Scholar
  2. 2.
    G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge, 1967).zbMATHGoogle Scholar
  3. 3.
    L. G. Loitsyanskii, Mechanics of Liquids and Gases (Nauka, Moscow, 1987; Begell House, New York, 1996).Google Scholar
  4. 4.
    I. S. Gromeka, “Some cases of incompressible fluid flows,” Collected Works (Akad. Nauk SSSR, Moscow, 1952) [in Russian].Google Scholar
  5. 5.
    H. Lamb, Hydrodynamics (Dover, New York, 1945).Google Scholar
  6. 6.
    L. M. Milne-Thomson, Theoretical Hydrodynamics (Macmillan, London, 1960).zbMATHGoogle Scholar
  7. 7.
    O. V. Troshkin, “On some properties of Eulerian fields,” Differ. Uravn. 18 (1), 138–144 (1982).Google Scholar
  8. 8.
    O. V. Troshkin, “Topological analysis of the structure of hydrodynamic flows,” Russ. Math. Surv. 43 (4), 153–190 (1988).MathSciNetCrossRefGoogle Scholar
  9. 9.
    E. M. Landis, Second-Order Elliptic and Parabolic Equations (Nauka, Moscow, 1971) [in Russian].zbMATHGoogle Scholar
  10. 10.
    F. Rellich, “Ein Satz über mittlere Konvergenz,” Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl., 30–35 (1930).zbMATHGoogle Scholar
  11. 11.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Am. Math. Soc., Providence, R.I., 1963; Nauka, Moscow, 1988).Google Scholar
  12. 12.
    S. L. Sobolev, “On a theorem of functional analysis,” Mat. Sb. 4 (3), 471–497 (1938).Google Scholar
  13. 13.
    K. O. Friedrichs, “The identity of weak and strong extensions of differential operators,” Trans. Am. Math. Soc. 55, 132–151 (1944).MathSciNetCrossRefGoogle Scholar
  14. 14.
    O. V. Troshkin, “A two-dimensional flow problem for steady-state Euler equations,” Math. USSR Sb. 66, 363–382 (1990).MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces (Fizmatgiz, Moscow, 1959) [in Russian].zbMATHGoogle Scholar
  16. 16.
    I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).zbMATHGoogle Scholar
  17. 17.
    O. V. Troshkin, “Admissibility of the set of boundary values in a steady-state hydrodynamic problem,” Dokl. Akad. Nauk SSSR 272 (5), 1086–1090 (1983).MathSciNetGoogle Scholar
  18. 18.
    S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I,” Commun. Pure Appl. Math. 12 (4), 623–727 (1959).MathSciNetCrossRefGoogle Scholar
  19. 19a.
    L. E. Fraenkel and M. S. Berger, “A global theory of steady vortex rings in an ideal fluid,” Bull. Am. Math. Soc. 79 (3), 806–810 (1973);MathSciNetCrossRefGoogle Scholar
  20. 19b.
    Acta Math. 132 (1), 13–51 (1974).Google Scholar
  21. 20.
    H. K. Moffatt, “Generalized vortex rings with and without swirl,” Fluid Dyn. Res. 3, 22–30 (1988).CrossRefGoogle Scholar
  22. 21.
    V. I. Yudovich, “Two-dimensional unsteady problem of inviscid incompressible flow through a given region,” Mat. Sb. 64, 562–588 (1964).MathSciNetGoogle Scholar
  23. 22.
    A. V. Kazhikhov and V. V. Ragulin, “Flow problem for the equations of an ideal fluid,” J. Sov. Math. 21 (5), 700–710 (1983).CrossRefGoogle Scholar
  24. 23.
    W. Wolibner, “On theoreme sur l’existence du movement plan d’un fluid parfait, homogene, incompressible, pendant um temps infiniment longue,” Math. Z. 37, 698–726 (1933).MathSciNetCrossRefGoogle Scholar
  25. 24.
    T. Kato, “On classical solutions of the two dimensional nonstationary Euler equations,” Arch. Ration Mech. Anal. 25 (3), 188–200 (1967).MathSciNetCrossRefGoogle Scholar
  26. 25.
    O. Glass, “Existence of solutions for the two-dimensional stationary Euler system for ideal fluids with arbitrary force,” Ann. I.H. Poincaré 20, 921–946 (2003).MathSciNetzbMATHGoogle Scholar
  27. 26.
    V. I. Arnold, “Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique de fluids parfaits,” Ann. Inst. Fourier (Grenoble) 16, 319–361 (1966).MathSciNetCrossRefGoogle Scholar
  28. 27.
    G. V. Alekseev, “On the uniqueness and smoothness of plane vortex flows of an ideal fluid,” in Dynamics of Continuum Media (Novosibirsk, 1973), Vol. 15, pp. 7–17 [in Russian].Google Scholar
  29. 28.
    A. B. Morgulis and V. I. Yudovich, “Asymptotic stability of a stationary flowing regime of an ideal incompressible fluid,” Sib. Math. J. 43 (4), 674–688 (2002).CrossRefGoogle Scholar
  30. 29.
    C. L. M. H. Navier, “Memoire sur les lois du mouvement des fluides,” Mem. Acad. R. Sci. Inst. France 6, 389–440 (1823).Google Scholar
  31. 30.
    O. V. Troshkin, “Algebraic structure of the two-dimensional stationary Navier–Stokes equations and nonlocal uniqueness theorems,” Dokl. Akad. Nauk SSSR 298 (6), 1372–1376 (1988).MathSciNetGoogle Scholar
  32. 31.
    A. A. Dezin, On a Class of Vector Fields: Collected Papers on Complex Analysis and Its Applications (Nauka, Moscow, 1978), pp. 203–208 [in Russian].Google Scholar
  33. 32.
    O. V. Troshkin, “On the stability of reverse flow vortices,” Comput. Math. Math. Phys. 56 (12), 2062–2067 (2016).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Scientific Research Institute for System Analysis, Federal Research Center, Russian Academy of SciencesMoscowRussia

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